Thesis - MBMason

A Sub-60 fs Titanium-Sapphire Chirped Pulse Amplification Laser

System

Michael Brett Mason

Thesis submitted in partial fulfilment of the requirements for the degree of Doctor of

Philosophy of the University of London

April 2000

Laser Consortium

Blackett Laboratory

Imperial College of Science, Technology and Medicine

Prince Consort Road

London SW7 2BW

United Kingdom

[email protected]

www.lsr.ph.ic.ac.uk

(www.mikemason.co.uk)

http://www.mikemason.co.uk/

2

Abstract

This thesis describes the construction of a sub-60 femtosecond (fs) titanium-sapphire

based Chirped Pulse Amplification (CPA) laser system. It outlines the procedure involved in

designing and building high intensity, ultra-short pulse systems in general, and presents a

specific solution to the problem. It then documents the construction of the laser and its

characterisation.

The thesis starts with an introduction to the field of chirped pulse amplification

including sources of femtosecond pulses, methods of measurement and methods of

amplification. In addition, the original Blackett Laboratory titanium-doped sapphire system is

described and the projected upgrade outlined. A summary of the laser-matter interaction

experiments that the author has been involved in is also included. The problem of stretching,

amplifying and recompressing ultrashort laser pulses is then discussed. A description of a

purpose-written, three-dimensional ray-tracing model for chirped pulse amplification is

presented; followed by results for specific systems from the literature and the authors own

solution. This code forms the basis of the theoretical part of the thesis. The implementation of

the system is split into four sections: the new short-pulse oscillator, the new pulse stretcher,

adapted and upgraded amplification stages and new compressor. The stretcher and compressor

construction is briefly described, followed by full characterisation showing how they compare

with modelling results. Each stage of amplification is described, but not in as much detail - the

main thrust of this thesis is to describe the details of ultra-short pulse stretching and

compression. Finally, the system characteristics at the time of writing are summarised and

directions for further improvements discussed.

3

DEFINITIONS

DEFIN. I. Ray By the Rays of Light I understand its least Parts, and those as well Successive in the same Lines, as Contemporary in several Lines. For it is manifest that Light consists of Parts, both Successive and Contemporary; because in the same place you may stop that which comes one moment, and let pass that which comes presently after; and in the same time you may stop it in any one place, and let it pass in any other. For that part of Light which is stoppd cannot be the same with that which is let pass. The least Light or part of Light, which may be stoppd alone without the rest of the Light, or propagated alone, or do or suffer anything alone, which the rest of the Light doth not or suffers not, I call a Ray of Light. Sir Isaac Newton

The First Book of Opticks, Part I, 1730

4

Acknowledgements My first acknowledgment must go to my parents. They have allowed me to travel through

twenty three years of education providing financial support and enthusiasm every step of the way. Thankyou to Dad for instilling in me enthusiasm for finding out how things work, mainly by not being too hard on me for taking everything I owned to bits. Thankyou to Mum for providing me with a career thus far by always telling me I would be perfect in research. This thesis is for them.

Secondly, I would like to thank my two supervisors: Henry Hutchinson and Roland Smith. Henry always managed to steer me onto the right track after wandering between our weekly meetings in the first year and a half of my PhD. Roland took over as my acting supervisor when Henry left and has always provided enthusiasm and even incentives. Thankyou also Roland for doing most of my proof reading.

I would also like to thank all the people I have worked with in the lab over the last four and a half years: Thanks Nick for answering my dumb computer questions, for sneaky beers down the pub and shenanigans afterwards (nobody will see the photos!). Thanks Emma for correcting my spelling. Thanks John for the windsurfing holiday in Italy and steering my writing style by rewriting my abstracts. Thanks Jon for the in-pub entertainment and continuing Monday morning meetings in the beardy-one style. Thanks Todd for banging your right fist into your left palm. Thanks Andy for singing Maiden. Thanks Kirstie for helping me to talk tidy. Thanks Thomas for introductions to German culture. And thanks Dan for my introduction to laser questing.

A mention must go to Peter Ruthven and Andy Gregory for excellent technical support throughout my work in the lab. Peter never really got much from me in terms of technical drawings, more like technical ramblings, but still managed to come up with something pretty special. I must also apologise for when he proudly brought me nice shiny pieces if kit only to find that I needed more than just the one sorry Peter. Thankyou also to Andy who produced the best damn grating and mirror mounts a man designing a new stretcher and compressor could hope for. Thanks also to Shahid Hanif for electronic support, Martin Horton for optical support and Archie Wallace for electrical support.

Many friends have supported me through my PhD, all of whom I cannot mention. I would though like to mention two in particular: Alaric Marsden and Jan Troska. Thankyou both for relieving the monotony that comes from working all day at home on my thesis by telephoning me and making me laugh. Thanks to Koka Noodles too for sustaining me through my thesis writing. Thankyou especially to Jan for being such a good friend since we first stepped through the Imperial College doors eight and a half years ago. Particularly when the going got tough a year ago as many people must have thought I was the offender. By the way, when do my snowboarding lessons in Geneva start?

If I can write acknowledgements, then I must be able to write unacknowledgements. I wouldnt like to thank a certain producer of battered fishy products for my introduction to the English small claims court system, especially when I was supposed to be writing my thesis.

Finally and definitely not lastly, I would like to say thankyou to the forbidden daffodil in my life. She has done her best to relieve my fedupedness over the last year and has always helped and encouraged me to get the book out the door.

5

Contents

Title Page.....................................................................................................1

Abstract .......................................................................................................2

Quote ...........................................................................................................3

Acknowledgements ....................................................................................4

Contents ......................................................................................................5

List of figures ..............................................................................................9

List of tables..............................................................................................19

1. Introduction...............................................................................................21

1.1. Thesis introduction .....................................................................................21

1.2. Ultra-short pulses........................................................................................23 1.2.1. Ultrashort pulse sources.......................................................................24

1.2.1.1. Introduction .................................................................................24 1.2.1.2. Kerr-lens modelocking.................................................................25 1.2.1.3. Cavity dispersion compensation..................................................28

1.2.2. Ultrashort pulse measurement..............................................................29 1.2.2.1. Far-field monitor ..........................................................................29 1.2.2.2. Single-shot autocorrelator ...........................................................30 1.2.2.3. FROG .........................................................................................33 1.2.2.4. SPIDER.......................................................................................34

1.3. Chirped pulse amplification........................................................................36

1.4. Short pulse amplifiers.................................................................................40 1.4.1. Multi-pass configuration........................................................................42 1.4.2. Regenerative amplification ...................................................................43

1.5. Blackett Laboratory Laser Consortium titanium-doped sapphire chirped

pulse amplification laser system................................................................44

6

1.6. PhD chronology...........................................................................................46

1.7. Laser-matter interaction experiments........................................................48 1.7.1. Experimental summary.........................................................................48 1.7.2. Measurement of the spatiotemporal evolution of high-order harmonic

radiation using chirped laser pulse spectroscopy .................................50 1.7.3. Explosion of C60 irradiated with a high-intensity femtosecond laser pulse

.............................................................................................................54

2. Design considerations for high power, short pulse laser systems......59

2.1. Considerations for high intensity laser-matter interaction experiments.59 2.1.1. Pulse contrast ......................................................................................59 2.1.2. Pedestal ...............................................................................................60 2.1.3. Pulse energy fluctuations .....................................................................61 2.1.4. Beam focusability .................................................................................62 2.1.5. Beam pointing stability .........................................................................63 2.1.6. Pulse duration fluctuations ...................................................................64

2.2. Optical component considerations............................................................65 2.2.1. Optical bandwidth.................................................................................65

2.2.1.1. Thin film coatings ........................................................................65 2.2.1.2. Polarisers and waveplates...........................................................66 2.2.1.3. Bandwidth clipping ......................................................................67 2.2.1.4. Gain narrowing and gain shifting .................................................69

2.2.2. Dispersion ............................................................................................70 2.2.2.1. Group-velocity dispersion............................................................70 2.2.2.2. High order dispersion ..................................................................71

2.2.3. Nonlinearities .......................................................................................73 2.2.3.1. Self-focusing ...............................................................................73 2.2.3.2. Self-phase modulation.................................................................74 2.2.3.3. B-integral.....................................................................................75

2.3. The need for modelling ...............................................................................76

3. The Chirped Pulse Amplification model .................................................77

3.1. Model suite ..................................................................................................78 3.1.1. CPQuick...............................................................................................78 3.1.2. CPTrace...............................................................................................82

7

3.1.3. CPTrace3D ..........................................................................................83

3.2. Detailed model description.........................................................................84 3.2.1. Pulse definition.....................................................................................84 3.2.2. Fourier transforms................................................................................86 3.2.3. Three-dimensional ray-tracing..............................................................90

3.2.3.1. Ray transfer equations ................................................................90 3.2.3.2. Snell’s law ...................................................................................93 3.2.3.3. Diffractive optics and the grating equation...................................94

3.2.4. Surface imperfections...........................................................................96 3.2.5. Pulse combination and output methodology .........................................97 3.2.6. Code limitations....................................................................................98

4. Modelling results ....................................................................................100

4.1. Code verification results...........................................................................100 4.1.1. Dispersionless runs ............................................................................100 4.1.2. Dispersion only calculations ...............................................................102 4.1.3. Ray transfer tests ...............................................................................103 4.1.4. Surface imperfections.........................................................................105

4.2. Systems modelled in detail.......................................................................106 4.2.1. Livermore system...............................................................................107

4.2.1.1. Livermore system specification .................................................107 4.2.1.2. Livermore system performance .................................................109

4.2.2. Öffner triplet system ...........................................................................110 4.2.2.1. Öffner triplet system specification..............................................110 4.2.2.2. Öffner triplet system performance .............................................111

4.3. Reflective doublet system ........................................................................113 4.3.1. Reflective doublet specification ..........................................................115 4.3.2. Reflective doublet performance..........................................................116

4.4. Additional modelling.................................................................................118 4.4.1. Introduction ........................................................................................118 4.4.2. Extra modelling results .......................................................................119

4.4.2.1. Adapted reflective doublet design..............................................120 4.4.2.2. Adapted Livermore design.........................................................121 4.4.2.3. Adapted Ross design ................................................................122

4.4.3. Extra modelling conclusions ...............................................................124

8

4.5. Summary....................................................................................................125

5. System implementation and characterisation......................................126

5.1. Short pulse oscillator upgrade.................................................................126

5.2. Stretcher implementation .........................................................................129

5.3. Amplification upgrade...............................................................................132 5.3.1. Regenerative amplifier .......................................................................132

5.3.1.1. Cavity design.............................................................................133 5.3.1.2. Pumping....................................................................................134 5.3.1.3. Pulse switching .........................................................................135 5.3.1.4. Unwanted pulse suppression ....................................................137 5.3.1.5. Spectral characteristics .............................................................139 5.3.1.6. B-integral...................................................................................140

5.3.2. Pre-amplifier.......................................................................................141 5.3.3. Power amplifier ..................................................................................141

5.4. Compressor implementation ....................................................................143

5.5. Laser characteristics.................................................................................147 5.5.1. Temporal............................................................................................147 5.5.2. Spatial ................................................................................................149 5.5.3. Energy................................................................................................150

5.6. Full system ................................................................................................152

6. Summary, conclusions and further work .............................................156

6.1. Summary....................................................................................................156

6.2. Conclusions...............................................................................................157

6.3. Further work ..............................................................................................159

References ..............................................................................................161

9

List of figures

Figure 1.1 Ponderomotive energy as a function of laser intensity for a laser of centre

wavelength 800 nm. Boxes show examples of accessible phenomena at the

given intensity (QED = quantum electrodynamics, e+e- = electron positron

pair). Adapted from figure 4 of [Perry 94].......................................................... 22

Figure 1.2 Logarithmic time line showing time scales of some common events.

Geometric mean between a 100 attosecond light pulse [Papadogiannis 99]

and estimated age of the Universe (~10 billion years) is about one second.

Adapted from figure 1 [Kapteyn 99].................................................................... 23

Figure 1.3 Schematic showing the configuration of mirrors around the gain medium in

a Kerr-lens modelocked cavity. Mirrors arranged such that, with the help

of the Kerr effect, a high intensity beam is collimated either side of mirrors

M1 and M2, whereas a low intensity beam diverges and suffers higher loss. ..... 27

Figure 1.4 Schematic of a sequence of prisms that give adjustable group velocity

dispersion without net angular dispersion............................................................ 28

Figure 1.5 Schematic of a far field monitor. CCD = charge-couple device. Image

viewed on a video monitor connected to the camera. .......................................... 30

Figure 1.6 Schematic of a single-shot autocorrelator. BS = beamsplitter and CCD =

charge-couple device. Image viewed on a video monitor connected to the

camera. ................................................................................................................. 31

Figure 1.7 Schematic of a second harmonic frequency resolved optical gate device for

retrieving the full complex field of the signal pulse. Prism acts as a

rudimentary spectrometer to illustrate how the FROG trace is produced.

Adapted from figure 5 of [Kapteyn 99]. BS = beamsplitter................................ 34

Figure 1.8 Schematic of a typical spectral phase interferometer for direct electric-field

reconstruction set up. Dispersive delay often comprised of a grating pair.

BS = beamsplitter................................................................................................. 35

Figure 1.9 Schematic of a parallel diffraction grating pair compressor. Green arrow

labelled λ shows manner in which rays of increasing wavelength are

diffracted On exit from grating pair, beam is spatially dispersed. Figure

adapted from [Treacy 69], figure 1. ..................................................................... 37

Figure 1.10 Antiparallel diffraction grating pair forming a Martinez [Martinez 87] style

stretcher that adds opposite chirp to the parallel grating pair of Figure 1,9.

10

f is the focal length of les L1 and f′ is the focal length of lens L2. G1′ is

the image of grating G1........................................................................................ 38

Figure 1.11 Schematic of the technique of chirped pulse amplification. ................................ 39

Figure 1.12 Schematic showing the principle of regenerative amplification. M#

represents cavity mirrors, P# represents polarisers and PC# represents

Pockels cells. ........................................................................................................ 43

Figure 1.13 Schematic of the Blackett Laboratory Laser Consortium titanium-doped

sapphire chirped pulse amplification laser system before being upgraded.

Proposed upgrades shown by dotted lines and half-tone colours......................... 45

Figure 1.14 Approximate chronology of work conducted during PhD.

S-T E of HHG = Spatio-Temporal Evolution of High Harmonic

Generation. ........................................................................................................... 46

Figure 1.15 The 13th harmonic studied was dispersed in a high-resolution imaging

vacuum-ultraviolet spectrometer and recorded with a microchannel plate

detector (MCP)..................................................................................................... 51

Figure 1.16 The space and time resolved images show that the harmonic beam becomes

annular on a sub-picosecond time-scale. The peak intensity in the Xe jet

was 1.3×1014Wcm-2. ............................................................................................. 52

Figure 1.17 Comparison of microchannel plate images at different peak laser intensities

with numerical simulations showing that ionisation depletion is the cause

of the annular harmonic beam.............................................................................. 53

Figure 1.18 Schematic of the time-of-flight spectrometer showing the C60 source, laser

focus, extraction and drift regions and microchannel plate (MCP) detector.

The laser propagates out of the page and is focused into the C60 beam in the

centre of the extraction region.............................................................................. 55

Figure 1.19 Spectra recorded at three intensities from 5×1014 Wcm-2 to 5×1015 Wcm-2

with horizontal polarisation (polarisation vector parallel to detection axis). ....... 56

Figure 1.20 The difference between horizontal polarisation and vertical polarisation

(polarisation vector perpendicular to detection axis) showing the

anisotropic distribution of the C2+, C3+ and C4+ ions............................................ 57

Figure 1.21 Comparison of experimental and simulated time of flight spectra using 150

fs, 780 nm, vertically polarised pulses. ................................................................ 58

Figure 2.1 Pulse profiles. (a) Short timescale profile showing pulse shoulder and pre-

/post-pulses due to incomplete recompression. (b) Longer timescale profile

showing pulse pedestal and pre-/post-pulses from sources within the

amplifier chain ..................................................................................................... 60

11

Figure 2.2 Distribution of beams of different wavelength in the compressor, (a) for

small diameter input beams and (b) for large diameter input beams ................... 67

Figure 2.3 Soft and hard spectral clipping for the case of large and small beam

diameters, respectively. Spectral bandpass profiles represent wavelengths

transmittable by the stretcher and compressor. FT represents Fourier

transform and ⊗⊗⊗⊗ represents mathematical convolution........................................ 68

Figure 2.4 Sech2 pulse bandwidth (blue curve) and temporal broadening (red curve)

due to 1 centimetre of BK7 glass, as a function of input pulse duration.

Calculated using chirped pulse amplification model described in chapter 3 ....... 71

Figure 2.5 Comparison between the variation of refractive index with wavelength for

BK7 and TGG ...................................................................................................... 72

Figure 2.6 Calculation of self-phase modulation of a Gaussian pulse (red) in the

absence of group-velocity dispersion. Blue line shows the instantaneous

frequency shift of a 30 fs Gaussian pulse after propagation through 1 cm of

fused silica............................................................................................................ 75

Figure 3.1 Calculated contributions of third and fourth order phase as a function of

input pulse duration for a reflective stretcher. High order phase

contributions become increasingly important as the pulse duration is

reduced. Third order phase is shown in red and fourth order phase is

shown in blue. ...................................................................................................... 77

Figure 3.2 Configuration of the pulse compressor showing the definition of grating

separation (b, metres), grating angle (θ0), diffracted angular spread (∆θ)

and input angle (γ)................................................................................................ 79

Figure 3.3 Compressor configurations. (a) Configuration with smallest second

diffraction grating size. (b) Littrow configuration giving largest diffraction

efficiency. (c) Compromise configuration that allows input and output of

finite sized beams................................................................................................. 79

Figure 3.4 Flow diagram for the CPTrace model. Blue lines represent

spectra/temporal phase and red lines represent spectral/temporal amplitude.

FT represents the Fourier transform and FT-1 represents the inverse Fourier

transform. ............................................................................................................. 82

Figure 3.5 Pulse definition. I represents intensity, x and y are the beam coordinates

and t is the time parameter. The time window is split into

NumTimeSteps time steps with the full-width-half-maximum pulse

duration described by FWHMTimeSteps steps. The spatial array is

12

defined by a grid of NumBeams × NumBeams, with the full-width-half-

maximum beam diameter described by FWHMSpaceBeams beams. ............... 84

Figure 3.6 Profiles of 30 femtosecond Gaussian (red) and sech2 (black) pulses. Inset

graph shows the same profiles on a log intensity scale........................................ 85

Figure 3.7 The discrete Fourier transform sample points, represented by blue dots,

are illustrative only. (a) Shows the sampled input pulse. (b) Shows how

this is transformed into frequency space. (c) Shows the non-aliased result

where values of A(ωn) are zero at the extremes of the sample set (where

|ωn| = π/∆). (d) Shows the aliased result where values of A(ωn) outside the

sample range (red) are flipped back into range creating a spectrum that is

elevated near the edges (blue). ............................................................................. 87

Figure 3.8 Rearrangement of the pulse in time and frequency in order that phase is

zero across the centre of the pulse. This reformatting also places zero time

and wavelength in the correct positions. .............................................................. 88

Figure 3.9 Definition of transformed pulse. Each beam now contains NumRays

frequency components. The amplitude of each spectrum is modified by the

amplitude of the beam at each position in the beam profile array. ...................... 89

Figure 3.10 Ray and surface parameter conventions used by the ray transfer equations.

The ray and surface are only shown in the z-y plane. The origin of this

plane represents the centre of the surface the ray is emerging from. Inset

shows axis convention and plane of dispersion.................................................... 91

Figure 3.11 Phase addition due to grating surface structure. Left hand diagram

represents grating surface, right hand diagram is a graph of phase addition.

Ray (a) starts with phase φ. Ray (b) is one groove away and so the phase at

this point is the original phase plus 2π. Ray (c) is 5 grooves away and so

the phase addition at this point is 5 times 2π. ...................................................... 95

Figure 3.12 Calculation of ray phase upon diffraction through an ideal grating pair

compressor. AA and CC are both wavefronts, but path ABC is longer

than ABC. Adapted from figure 3 of [Treacy 69]. Phase correction of

equation 3.28 must be added to compensate for this............................................ 95

Figure 3.13 Slow (left diagram) and fast (right diagram) variations added to surfaces to

simulate surface imperfections. Slow and fast variations added in

amplitude ratio ~6:1. ............................................................................................ 96

Figure 3.14 Surface pistoning. Left hand diagram represents ideal spherical concave

surface (black) with high spatial frequency irregularities (red). Right hand

diagram shows how these irregularities are simulated by pistoning. ................... 97

13

Figure 3.15 Each beam traced through the stretcher and compressor is brought to the

focus of a numerical achromatic, aspheric lens. This has the effect of

bringing each beam to the same point in space and time to produce a single

output pulse temporal profile. .............................................................................. 98

Figure 4.1 Spectral and temporal outputs of the code in the absence of material

dispersion on a logarithmic scale. The input pulse in this case was a 30 fs

Gaussian pulse. The red curve shows the pulse spectrum, the blue curve

shows the input temporal profile and the green curve shows the, laterally

displaced, output temporal profile. Dotted line represents the detection

limit of laboratory pulse profile and spectrum measurement devices................ 101

Figure 4.2 Comparison of calculations using the code (solid lines) and performed

manually (black circles) testing material (red) and diffractive (blue)

dispersion. .......................................................................................................... 102

Figure 4.3 Schematic showing layout in the z-x plane for the ray transfer test.

M = 1 m radius of curvature mirror, G = diffraction grating and S = screen.

Grey lines represent input rays........................................................................... 104

Figure 4.4 Image of a nine-by-nine grid of beams diffracted from a diffraction grating

via a 1 m radius of curvature concave mirror placed 1 m away from the

grating onto a screen placed at the centre of the grating. ................................... 104

Figure 4.5 Comparison of calculated recompressed pulse profiles of a 30 fs pulse

stretcher and those measured by Cheriaux et al [Cheriaux 96]. The left

hand graph shows the results presented in the paper by Antonetti et al and

the right hand graph shows the results of the chirped pulse amplification

model for three different surface figures of the large stretcher optic. ................ 106

Figure 4.6 Stretcher schematics designed by: (a) Lemoff et al; (b) Zhou et al and (c)

Itatani et al. Orange lines represent diffraction gratings, cyan lines

represent mirrors and grey lines represent input/output paths. .......................... 107

Figure 4.7 Schematics of the Livermore stretcher as it is folded a→b→c→d from the

standard Martinez configuration (a). Orange lines represent diffraction

gratings, cyan lines represent mirrors and grey lines represent input/output

paths. .................................................................................................................. 108

Figure 4.8 Calculated profile of a pulse stretched using a Livermore-style stretcher.

Blue curve shows 30 fs input pulse and red curve shows stretched and

recompressed pulse profile. Calculated using the code..................................... 109

Figure 4.9 Schematic of the Öffner triplet based stretcher design of Antonetti et al

[Cheriaux 96]. Input/output through System double pass mirror...................... 111

14

Figure 4.10 Comparison of 30 fs input pulse profile (black curve), output pulse profile

from an Öffner triplet based stretcher with perfectly flat optics (blue curve)

and the output pulse profile from an Öffner triplet based stretcher with

realistic surface figures (red curve). Calculated using the code. Inset

shows figure 2 from [Antonetti 95] showing independent calculations for

blue curve. .......................................................................................................... 112

Figure 4.11 Calculated residual phase of a 30 fs pulse after passing through an Öffner

triplet based stretcher and corresponding compressor, with perfect optics

(blue curve) and with optics having non-zero surface figure (red curve).

Calculated using the code................................................................................... 113

Figure 4.12 Schematics of two stretchers: (a) the Öffner triplet based and (b) doublet

based. Definitions: M = Mirror, G = Grating, GI = Grating Image,

C = Centre of curvature and N = the null stretch position. Grey lines

represent input/output paths. .............................................................................. 114

Figure 4.13 Three-dimensional schematic of the reflective doublet based stretcher

design. Input and output is via the single ray at the top of the diffraction

grating. ............................................................................................................... 115

Figure 4.14 Detailed schematic of final doublet stretcher specifications. roc = radius of

curvature, egs = effective grating separation, d = grating constant,

w = width of optic and φ = diameter of optic..................................................... 116

Figure 4.15 Comparison of 30 fs input pulse profile (black curve), output pulse profile

from a reflective doublet based stretcher with perfectly flat optics (blue

curve) and the output pulse profile from a reflective doublet based stretcher

with realistic surface figures (red curve). Calculated using the code................ 117

Figure 4.16 Calculated residual phase of a 30 fs pulse after passing through a reflective

doublet based stretcher and corresponding compressor, with perfect optics

(blue curve) and with optics having non-zero surface figure (red curve).


Figure 4.17 1 Input 30 fs pulse profile (black curve) and recompressed pulse profile

(red curve) for the doublet stretcher. The input pulse was stretched to

600 ps and the recompressed full width half maximum pulse duration of

55.7 fs................................................................................................................. 121

Figure 4.18 Input 30 fs pulse profile (black curve) and recompressed pulse profiles

(green, red and blue curves) for the Livermore stretcher. The input pulse

was stretched to 600 ps. The recompressed pulse duration and wing level

changes as the focal length of the curved mirror is changed.............................. 122

15

Figure 4.19 Schematic of the Ross stretcher design. Orange lines represent the two

diffraction gratings and the grey line represents the input beam path. The

first grating need not be as large as shown it needs only to be large

enough to accommodate the input beam. ........................................................... 123

Figure 4.20 2 Input 30 fs pulse profiles (black curves) and output profiles (red, green

and blue) for (a) double-double pass and (b) double-size Ross stretcher

designs. Different coloured curves show how the output pulse profile

changes as a function of optic surface flatness................................................... 123

Figure 4.21 Input 30 fs pulse profiles (black curves) and output profiles (red, green,

blue and magenta) for (a) double-double pass and (b) double-size Ross

stretcher designs. Different coloured curves show how the output pulse

profile changes as a function of input beam full width half maximum

diameter.............................................................................................................. 124

Figure 5.1 Schematic of the Murnane and Kapteyn short pulse oscillator. RM

represents the rear cavity mirror; B represents a beam block; M represents

concave mirrors with a radius of curvature of 10 cm; and L is a lens of

focal length 10.5 cm........................................................................................... 127

Figure 5.2 Single-shot autocorrelation of pulses from the oscillator showing a duration

of 37.3 fs. Inset shows the same profile on a logarithmic scale. Dynamic

range limited by the CCD camera used in the autocorrelation measurement .... 128

Figure 5.3 Spectrum of pulses from the oscillator showing a bandwidth of 31.1 nm

(red curve). Blue curve shows the spectrum of lines from an argon

discharge lamp. The pulse spectrum is centred and the spectrometer is

calibrated using a strong argon line at 801.5 nm................................................ 129

Figure 5.4 Picture of the completed stretcher with the oscillator in the foreground.

M1 = concave mirror; M2 = small flat mirror; M3 = convex mirror;

M4 = double pass mirror; G = diffraction grating; I = alignment irises;

C = oscillator crystal; and P = oscillator prism. ................................................. 130

Figure 5.5 Measurement of the spectrum across the beam (in the plane of dispersion)

for a misaligned stretcher (top image) and a corrected stretcher (bottom

image). Red represents no light and increasing light levels are represented

by orange-yellow-green-blue. ............................................................................ 131

Figure 5.6 An image taken from the oscilloscope showing the profile of the stretched

pulse. Measured using a fast diode [New focus] and sampling scope

[Tektronix]. The diode has a rise-time of approximately 50 ps that has the

effect of stretching the measurement. ................................................................ 132

16

Figure 5.7 Schematic of the regenerative amplifier. TFP1, TFP2 = thin film polariser;

M1, M2 = cavity mirrors. Titanium-doped sapphire crystal has length

20 mm and a doping level of 0.10%................................................................... 133

Figure 5.8 Traces from a digital oscilloscope showing the pump pulse (A, at 0 ns) and

the seeded build-up (around 200 ns) of the regenerative amplifier. Switch-

out is just after 200 ns. Inset shows the position of the seed (red trace)

compared to the pump pulse (green trace). The ringing after each short

pulse is the response of the photodiode to a 300 ps pulse.................................. 136

Figure 5.9 Schematics of the input (a) and output (b) regenerative amplifier slicers.

The removable λ/2 waveplate of the input slicer allows an alignment beam

to pass through the second polariser. The output energy of the output slicer

is controlled by allowing energy to be rejected (via route G) by rotating the

first λ/2 waveplate. Rays travel from left to right. ............................................ 137

Figure 5.10 Comparison of pulse spectra before (red curve) and after (blue curve) the

regenerative amplifier and its free-run spectrum (blue curve). Curves show

the effect of gain shifting. .................................................................................. 139

Figure 5.11 Schematic of the pre-amplifier. B = beam block blocks excess pump

energy that passes through the Brewster-cut titanium-doped sapphire

crystal. ................................................................................................................ 141

Figure 5.12 Schematic of the bow-tie power amplifier...................................................... 142

Figure 5.13 Detailed schematic of the final compressor design. RR = gold plated

retroreflector. Grating constant is 1200 lines/mm............................................. 144

Figure 5.14 Duration addition to a 35 fs pulse as the separation between the gratings in

the compressor is changed. Black curve shows duration change due to

second order phase, red curve from third order phase and green from fourth

order phase. Maximum stretch due to third and fourth order phase is 20 fs

and 4 fs respectively across the 1 cm grating separation displacement.

Calculated using the code described in chapter 3............................................... 144

Figure 5.15 Duration addition to a 35 fs pulse as the two diffraction gratings in the

compressor are rotated. Positive displacement angle corresponds to

rotating the grating anti-clockwise in Figure 5.13. Red curve shows

contribution from third order phase and green curve shows contribution

from fourth order phase. Blue curve shows how much the compressor

grating separation was changed in order to keep the contribution from

second order phase zero as the gratings are rotated. Calculated using the

code described in chapter 3. ............................................................................... 145

17

Figure 5.16 Change in contrast of a 35 fs pulse as the two diffraction gratings in the

compressor are rotated. Positive displacement angle corresponds to

rotating the grating anti-clockwise in Figure 5.13. Red curve shows

contribution from third order phase and green curve shows contribution

from fourth order phase. Contrast level assumes a pulse with originally

zero third and fourth order residual phase. Calculated using the code

described in chapter 3......................................................................................... 146

Figure 5.17 Temporal profile of the output pulse of the laser. Full-width-half-

maximum duration is 59 fs. Inset shows the profile of the pulse on a

logarithmic intensity scale.................................................................................. 148

Figure 5.18 Spectral profile of the output pulse of the laser (red curve). Full-width-

half-maximum bandwidth of the pulse is 20 nm. Spectrum centred at

797 nm The green curve shows the emission lines of rubidium used to

calibrate the spectrum. ....................................................................................... 149

Figure 5.19 Beam shape at the focus of a 50 cm focal length lens. Arrow has length

50 µm. Right image is the same as the left grey-scale image but shows full

dynamic range (two orders of magnitude) of measurement on a linear

colour scale (ROYGBIV). 1/e2 radius is 23 µm which is 1.9 times

diffraction limited............................................................................................... 150

Figure 5.20 Accumulation of 4199 laser shots showing the variation of energy on a

shot-to-shot basis for the laser before it was upgraded. Average energy

11.4 mJ with a standard deviation of 1.5 mJ...................................................... 151

Figure 5.21 Picture of the completed laser system. A = oscillator, B = stretcher,

C = oscilloscope traces showing spectrum (top) and pulse train (bottom)

from the oscillator, D = Millennia, E = Surelite, F = regenerative amplifier,

G = input and output slicers, H = pre-amplifier, I = BMI, J = power

amplifier (bow-tie) and K = Quanta ray. Photograph courtesy of Nick

Jackson - Physics department photographer at Imperial College....................... 152

Figure 5.22 Schematic of the laser system. A represents autocorrelators, S is a

spectrometer, T is a Newtonian telescope and FFM is a far-field monitor.

The grey outlines represent the three optical tables that accommodate the

laser system. ....................................................................................................... 153

Figure 5.23 Laser timing diagram. Time scales are nonlinear but pulse order is

accurate. OSC represents 89.4 MHz pulse train from oscillator; 10 Hz is

the oscillator pulse train divided by factor of ~89.4×105; InS = input slicer;

SFL = Surelite flashlamp trigger; SQS = Surelite Q-switch; Rin =

regenerative amplifier switch in; Rout = regenerative amplifier switch out;

18

OutS = output slicer; QRFL = Quanta Ray flashlamps; QRQS = Quanta

Ray Q-switch; BMIFL = BMI flashlamps; BMIQS = BMI Q-switch. .............. 154

19

List of tables

Table 1.1 Summary of characteristics for a range of common, short pulse

amplification media ([Diels 96], [Nees 98] and [Nikogosyan 97]).

λ0 = gain centre wavelength, ∆λ = gain bandwidth, σ = emission cross-

section, τ = upper-state lifetime, Usat = saturation fluence and Λ = thermal

conductivity. ٭ the thermal conductivity of organic dyes depends on the

solvent used; it is also of less importance as the dye is circulated through a

jet, therefore the thermal restrictions on repetition rate are relaxed..................... 41

Table 3.1 Surface definition parameters for the three different types of surface:

Spherical, Plane and Reflecting diffraction Grating. P8 represents the

value of parameter number 8................................................................................ 90

Table 4.1 Differences between second, third, fourth and fifth order phase

contributions of the Livermore stretcher and corresponding compressor.


Table 4.2 Summary of surface specifications of Öffner triplet based stretcher optics.

(* = flatness estimated due to lack of information in [Antonetti 97]). ............... 112

Table 5.1 Summary of optical components in the regenerative amplifier, their lengths

and n2 values [Nikogosyan 97]........................................................................... 140

Table 5.2 Summary of optical components in the output slicer, their lengths and n2

values [Nikogosyan 97]..................................................................................... 141

Table 5.3 Summary of allowable beam diameters as a function of mirror diameter.

D = mirror diameter, DU = usable mirror diameter, 1/e2 = 1/e2 beam

diameter, FWHM = full-width-half-maximum beam diameter, FG = peak

fluence at the grating surface assuming 200 mJ pulse energy, PB = peak

beam power after compression assuming a 50% efficient compressor and

DB = propagation distance required to accumulate a B-integral of one

radian.................................................................................................................. 147

Table 5.4 Summary of B-integral contributions from each laser stage, the total

accumulated B-integral and output energy from each stage. ............................. 153

Table 6.1 Summary of focused intensities mentioned at various points through this

thesis. E = pulse energy; ∆t = full width half maximum pulse duration;

20

λ = laser wavelength; ω = input beam 1/e2 radius; ω0 = focal spot 1/e2

radius; f = focal length of focusing optic; N = number of times diffraction

limited focal spot; and I0 = peak focused intensity. ........................................... 159

1. Introduction

1.1. Thesis introduction The last fifteen years has seen a revolution in the shortest duration pulses and highest

intensities available from laboratory scale lasers. Three main technological developments are

responsible for this: the technique of passive mode-locking a laser to produce pulses on the sub-

picosecond timescale; the development of titanium-doped sapphire as a gain medium for

femtosecond duration pulses; and the technique of chirped pulse amplification.

High intensity laser radiation may be produced in two ways: by having a large amount

of energy (many Joules) in a reasonably short pulse (nanoseconds); or by having a modest

amount of energy (millijoules) in an ultrashort pulse (pico - femtoseconds). The first method

requires large aperture beams in order to avoid damage to optical components through the

nonlinear response of optical materials. This method is therefore reserved for national facility

scale lasers such as those at the Rutherford Appleton Laboratory in the United Kingdom and at

the Lawrence Livermore National Laboratory in the United States. The latter method of

producing high intensity laser radiation uses the technique of chirped pulse amplification to

temporally stretch an ultrashort pulse thereby reducing the peak intensity for safe, damage-free

amplification to modest energies in the millijoule to few Joule range. Laser powers in the

terawatt regime have been produced by lasers of this type and a combination of chirped pulse

amplification and large aperture optics has given rise to lasers with petawatt powers [Perry 94].

The development of laboratory-scale high intensity, ultrashort lasers has opened up

many opportunities in the field of high intensity laser-matter interactions. Above intensities of

about 1012 Wcm-2, electric field of the laser is no longer a perturbation to the electric field inside

an atom, and so the perturbation theory of light-matter interactions breaks down. Figure 1.1

shows the increasingly rich range of physical phenomena available as the focused intensity of a

laser increases. The ponderomotive potential is the energy given to an electron in an atom by

the oscillating electric field of the laser and is given by:

20

20

2

4 ωeP m

EeU = (1.1)

where e is the electron charge (C), E0 is the maximum laser electric field (Vm-1), me is the

electron mass (kg) and ω0 is the laser centre angular frequency (radians s-1). The energy of the

electron, in the presence of the electric field of the laser, increases and then decreases as the

Introduction

22

laser pulse passes. Energy is transferred to the electron only if it interacts with a third body, for

example by colliding with an ion.

The fact that this high intensity laser radiation is in the form of an ultrashort pulse is

also important experimentally. In many experiments, we need to interact with the most intense

part of the pulse; if the laser pulse is temporally long, the target can fragment, or a pre-plasma

can be formed, changing the intended target before the main peak of the pulse arrives. Very

short probe pulses are also required to freeze extremely rapid processes such as energy

transport by keV electrons in a cluster plasma in pump/probe experiments. In addition, high-

order harmonic generation has been shown to be more efficient for femtosecond duration pulses

[Zhou 96] as the level of ionisation (which reduces harmonic efficiency) is reduced for shorter

pulses.

1013 1015 1017 1019 1021

100

102

104

106

108 QED,nuclear reactions,pion production

Fusion ignition,e+e- production

Relativistic plasmas,hard x-rays

High-brightnessx-ray sources,

x-ray lasers

Molecular andcluster coulomb/

hydrodynamic explosion

Perturbativeatomic physics and

nonlinear optics

Pon

dero

mot

ive

pote

ntia

l (eV

)

Laser intensity (Wcm-2) Figure 1.1 Ponderomotive potential as a function of laser intensity for a laser of centre

wavelength 800 nm. Boxes show examples of accessible phenomena at the given intensity (QED = quantum electrodynamics, e+e- = electron positron pair). Adapted from figure 4 of

[Perry 94].

In order to remain competitive in the field and to investigate the physics of ultrashort,

high intensity laser-matter interactions, the Blackett Laboratory Laser Consortium, titanium-

doped sapphire, chirped pulse amplification laser system was to be upgraded in terms of both

pulse duration and pulse energy. The subject of this thesis is to describe the process of this

upgrade. This includes: a discussion of the key issues involved when designing an ultrashort,

high power laser system; a description of a detailed three-dimensional ray-tracing model written

Introduction

23

to aid the design of an ultrashort pulse stretcher; the design and specification of a reflective

doublet based stretcher; and the implementation of this design into the laser system.

Primarily this thesis is concerned with the design and implementation of a high power,

chirped pulse amplification laser system. There is however, a section in this chapter that

outlines the authors involvement in experimental work using the original form of the laser

before it was upgraded.

1.2. Ultra-short pulses It is difficult to imagine the scale of a femtosecond light pulse. A 10 fs pulse travelling

through air or vacuum has a width in the direction of propagation of three microns. These

pulses are difficult to characterise as the fastest photodiodes available can only measure

durations of the order of fifty picoseconds (see section 5.2). Figure 1.2 gives an idea of the

scale of such short pulses by comparing common events on a logarithmic time scale. One

femtosecond is as small compared to one minute as the age of the Universe is large.

100 103 106 109 1012 1015 101810-310-610-910-1210-1510-18

100 as High Harmonic Generated Pulse100 fs Oscillator Pulse

5 ns Doubled Nd:YAG pulseCamera flash

Time (seconds)

One MinuteOne Month

Age of UniverseOne Millennium

Figure 1.2 Logarithmic time line showing time scales of some common events. Geometric mean

between a 100 attosecond light pulse [Papadogiannis 99] and estimated age of the Universe (~10 billion years) is about one second. Adapted from figure 1 of [Kapteyn 99].

In this section, sources of femtosecond light pulses and different ways of measuring

them are discussed. The method used to produce ultrashort pulses for the laser system

described in this thesis - Kerr-lens modelocking - is covered in detail. Pulse measurement

devices including autocorrelation, FROG and SPIDER are discussed as well as a simple device

for detecting beam aberrations.

Introduction

24

1.2.1. Ultrashort pulse sources

1.2.1.1. Introduction In recent years, the duration of available femtosecond pulses has dropped dramatically

[Steinmeyer 00]. The first laser to be modelocked was a helium-neon laser in 1964 by Hargrove

et al [Hargrove 64], which produced nanosecond duration pulses. In 1974, Shank and Ippen

produced the first modelocked pulses with a duration of <500fs in a free-flowing dye jet [Shank

74]. It is now possible to produce pulses with a duration of only 5 fs at the nanojoule level from

a modelocked titanium-doped sapphire oscillator [Baltuska 97]. Higher power pulses

(microjoules per pulse) are available from other techniques such as hollow fibre compression

[Nisoli 96] and optical parametric amplification [Ross 97]. These techniques rely on longer

pulses (tens of femtoseconds) from chirped pulse amplification lasers to seed and/or pump a

process that generates shorter pulses. Sources for chirped pulse amplification systems need

only be at the nanojoule per pulse level due to the large factors of gain available in a well

designed laser.

There are many different types of oscillator, based on different gain media, which

produce pulses at the nanojoule level. A review paper by [French 95] discusses many

modelocking techniques that produce pulses in the picosecond to femtosecond regime. These

include active modelocking (for example [Spence 91b]), passive modelocking with a saturable

absorber (for example [Mellish 93]) and passive modelocking with the Kerr effect including

additive pulse modelocking (for example [Mark 89]) and Kerr-lens modelocking (for example

[Proctor 92], [Lemoff 92], [Curley 93] and [Asaki 93]).

The bandwidth of the gain medium dictates the minimum pulse duration that an

oscillator can produce. Dye and rare-gas excimer systems typically have bandwidths of the

order 20-30 nm, which can support pulses as short as 30 fs [Hutchinson 89]. Pulses as short as

6 fs have been produced in organic dyes [Fork 87], but they pose significant disadvantages.

Dye lasers often suffer from problems including photo-induced degredation, jet-flow

instabilities and thermal effects, which are difficult to control and can lead to unstable operation.

Solid state lasers have now almost completely superseded dye-based lasers as they offer large

tunability ranges and are more convenient to operate. Chromium-doped LiSAF (Cr3+:LiSrAlF6)

for example, offers ~250 nm of tunability centred at ~830 nm and may, conveniently, be

flashlamp pumped [Diels 96]. Titanium-doped sapphire (Ti3+:Al2O3) offers the largest

tunability of any laser gain medium: ~450 nm centred at ~780 nm [French 95]. Due to its short

upper-state lifetime, it is usual to pump titanium-doped sapphire using another laser, typically a

neodymium-doped YAG laser or CW argon-ion or solid-state equivalent. Lasing in titanium-

Introduction

25

doped sapphire was first achieved in 1986 by Moulton [Moulton 86], who also characterised the

gain medium.

For the purposes of this thesis, Kerr-lens modelocking in titanium-doped sapphire will

be described. This is the method and medium used in both the original and upgraded oscillators

of the Blackett Laboratory Laser Consortium ultrashort pulse, chirped pulse amplification laser

system.

1.2.1.2. Kerr-lens modelocking

In a laser cavity, there exists an infinite series of resonant frequencies defined by the

cavity length. The separation between these modes, in frequency space, is given by: δν = c/2L,

where c is the speed of light (ms-1) and L is the optical length of the cavity (m). The laser

transition in the gain medium has a finite width (∆ν) as a result of line broadening mechanisms

(e.g. homogeneous vibronic broadening as in titanium-doped sapphire, or inhomogeneous

broadening as in glass-based media). This gain bandwidth limits the number of longitudinal

modes that can oscillate in the laser cavity. The spectrum of the cavity is then given by the

superposition of the gain profile over the continuum of discrete cavity modes. Normally these

modes lase independently giving a noisy, incoherent output with irregular coherent spikes

(formed when the modes oscillate in phase). The length of these spikes can be as short as the

coherence time of the laser, tC = 1/∆ν. Modelocking is the process of forcing these modes to

oscillate coherently, i.e. in phase or with a constant phase difference. When modelocking is

achieved, the output of the laser is a regular train of pulses separated by the cavity round-trip

time, TRT = 1/δν given by the Fourier transform of the cavity spectrum described above. The

temporal profile of each pulse is given by the Fourier transform of the spectral gain profile (in

the case of transform limited pulses). When all the modes of the laser oscillate in phase or with

some constant phase difference, the pulses are said to be transform limited.

The methods used to mode lock a laser can be split into two main techniques: active and

passive modelocking. Active modelocking requires the radiation in the cavity to be modulated

at a rate matched to the cavity round trip time (by a device such as an acousto-optic or electro-

optic modulator). This modulation generates sidebands on the laser radiation modes, coupling

adjacent modes allowing them to oscillate in phase. With passive- (or self-) modelocking, the

intracavity radiation modulates itself through a nonlinear device (such as a saturable absorber or

an optical Kerr medium) placed inside the cavity. A fast saturable absorber (where the cavity

pulses are longer than the absorber recovery time) introduces a high loss in the cavity for low

intensity pulses, but low loss (and even pulse compression) for intense, pulsed radiation. The

shortest pulses produced have durations of the order of the absorber recovery time (typically as

short as a few picoseconds). A slow saturable absorber (where the cavity pulses are shorter than

Introduction

26

the absorber recovery time), in conjunction with a saturable amplifier, can produce the same

pulse compression as a fast saturable absorber and generate pulses shorter than the absorber

recovery time. This occurs when the parameters of the cavity are contrived such that the

saturable absorber steepens the leading edge of the pulse and the gain is saturated after the peak

of the pulse has passed, steepening the trailing edge. A medium that exhibits the optical Kerr

effect can act as a very fast (~1 fs) saturable absorber in the cavity. This provides an intensity

dependent loss in the cavity (through an induced Kerr-lens) and pulse compression is achieved

through temporal self-phase modulation. As the Kerr effect is essentially instantaneous, the

leading and trailing edges of the pulse see a weaker induced Kerr-lens than the central, more

intense part of the pulse. The following is a description of self-modelocking using the optical

Kerr effect.

In the regime of low intensity optics, the polarisation of a dielectric as a (linear)

function of electric field can be written as EP χεο= (Cm-2), where εο is the permittivity of free

space (Fm-1), χ is the electric susceptibility (dimensionless) and E is the electric field (Vm-1).

As the electric field increases, the polarisation is no longer a linear function of electric field and

can be written generally as an expansion in E as:

( )...3)3(2)2()1( +++= EEEP χχχεο . (1.2)

These higher order terms in electric field arise from the anharmonic motion of bound

electrons in a medium under an intense electric field (~GW/cm2). In a centro-symmetric

system, the χ(2)E2 term (the Pockels effect) is zero. In the case of a high electric field, the χ(3)E3

term becomes significant and can be written as n2(E)|E2|, where n2(E) is the nonlinear index of

refraction (m2V-2). This term produces two effects, third harmonic generation (which is

typically weak in non phase-matched systems and will be ignored here) and a time-averaged,

signal-induced refractive index change given by:

Innn 20 += (1.3)

where I is intensity (I ≡ 2/ Eµε , Wm-2), n0 is the linear refractive index and n2 is the

nonlinear refractive index (also written as γ, m2W-1) (see also section 2.2.3). This is known as

the Kerr effect [Siegman 86]. The two definitions of the nonlinear refractive index are related

by n2 = 3.77×10-3n2(E)/n0 (m2W-1). In nearly all materials, the Kerr effect has a positive sign

(n2 > 0).

For a short pulse in a laser cavity the peak intensity can be high, and the Kerr term can

become significant. If the pulse has a Gaussian-type spatial profile, then the intensity dependent

refractive index change creates a positive lens, focusing the pulse (self-focusing, see section

2.2.3.1). The process of self-focusing within a Kerr medium in an oscillator cavity can give rise

Introduction

27

to Kerr-lens modelocking (KLM, also known as magic modelocking). The laser cavity is set

up so that the intense pulses focused in the gain medium through the Kerr effect suffer less loss

than lower intensity (including cw) radiation (Figure 1.3). In this way, the modelocked situation

is favoured over cw operation and intensity dependent pulse shaping occurs (compression of

pulses or noise spikes). Kerr-lens modelocking was first observed by Spence et al [Spence 91]

in a titanium-doped sapphire oscillator and explained as being due to self-focusing by Piche

[Piche 91].

To DispersionCompensation &2nd Cavity Mirror

CavityMirror

Low Intensity Beam

Gain Crystal

Pump

Mode-Locked Beam

Z-CavityMirror

M1

M2

Figure 1.3 Schematic showing the configuration of mirrors around the gain medium in a Kerr-

lens modelocked cavity. Mirrors arranged such that, with the help of the Kerr effect, a high intensity beam is collimated either side of mirrors M1 and M2, whereas a low intensity beam

diverges and suffers higher loss.

A schematic of the technique of Kerr-lens modelocking is shown in Figure 1.3. Aided

by the optical Kerr effect, mirror M1 produces a beam waist smaller for intense pulses than for

less intense cavity radiation. The smaller beam is then recollimated by mirror M2, whereas the

lower intensity radiation diverges on exit from M2. By placing an aperture after M2, the loss

for low intensity radiation is higher compared to the high intensity pulses. The waist of the

pump beam can be set to be slightly smaller than that of the high intensity beam. In this case,

the high intensity beam sees a better volume overlap with the pump beam than the larger waist,

low intensity beam, increasing gain extraction and so aiding mode locking this is known as

gain-guiding. If gain-guiding is optimised sufficiently, this soft-aperture can replace the

hard-aperture that is often placed after M2. A hard aperture was used in the original Laser

Consortium oscillator and gain-guiding in the upgraded, ultrashort pulse oscillator.

In order for a Kerr-lens modelocked oscillator to start mode locking, there needs to be

an intensity spike in the cavity large enough to experience strong Kerr-lens self-focusing. This

can be introduced into the cavity in a number of ways: acousto-optic modulators, regenerative

mode-locking, synchronous pumping, saturable absorption using a dye, semiconductor absorber,

moving mirror mode-locking in an external cavity, or by moving/tapping any intracavity mirror

Introduction

28

or optical component. These methods generate an intensity spike that can be used as a seed for

modelocking. The last method was used in both the original and upgraded Laser Consortium

short pulse oscillators.

1.2.1.3. Cavity dispersion compensation

The technique described in the preceding section is sufficient to construct a modelocked

cavity producing a train of short pulses. In order to produce the shortest possible pulses, there

must also be some form of compensation for the group-velocity dispersion that the short pulse

accumulates in the cavity [Brabec 92]. This group-velocity dispersion is introduced mainly

through temporal self-phase modulation in the gain medium. Compensation is usually in the

form of a prism sequence ([Proctor 92], [Zhou 94] and [Christov 94]) or chirped mirrors [Stingl

94] and [Stingl 95] in ultrashort pulse oscillators based on titanium-doped sapphire. Figure 1.4

shows a schematic of the double-passed prism pair used in both the original and upgraded Laser

Consortium short pulse oscillators.

∆GVD

Prism 1Prism 2

Mirror

Figure 1.4 Schematic of a sequence of prisms that give adjustable group velocity dispersion

without net angular dispersion.

The first prism introduces angular dispersion (negative group-velocity dispersion) and

the second prism recollimates the beam. The mirror after the second prism is a cavity mirror

that also acts to double pass the dispersive system, removing the spatial chirp (this cavity mirror

clearly cannot be used as the output coupler). Frequency masking may be performed at this

mirror to tune the laser centre wavelength and/or control the bandwidth. The amount of group-

velocity dispersion introduced is proportional to the prism separation (measured between the

two prism apices) and to (dn/dω)2 (where ω is the laser angular frequency, radians s-1). There is

a positive group-velocity dispersion contribution from the prism material itself. This

contribution is proportional to d2n/dω2 and the amount of glass the beam passes through. The

net group-velocity dispersion can be controlled by translating the prisms along their axis of

symmetry. This is given, for Brewster angled prisms, by the second derivative of phase with

respect to angular frequency:

−

−

+−= β

λβ

λλπλ

ωφ cos2sin124

2

2

3

2

2

2

2

3

2

2

ddn

nn

ddn

dndl

cdd

(1.4)

Introduction

29

where l is the separation of the prisms between the apices (metres), β is the angular deviation of

the dispersed light from a reference drawn between the two prism apices and λ is the laser

wavelength (metres) (see also section 2.2.2) [Fork 84].

Further limitations to the duration of the oscillator output pulse arise from residual third

order dispersion introduced by the prism pair [Proctor 92]. For durations below 10 fs, the

length of the titanium-doped sapphire crystal itself is also a limiting factor [Christov 96].

1.2.2. Ultrashort pulse measurement

In this section, a selection of measurement techniques suitable for ultrashort pulses is

discussed. The first device, a far-field monitor, is not necessarily specific to short pulse

systems, but provides a means for setting up the compressor gratings and identifying beam

aberrations at each stage of the laser system. This is important as geometric aberrations

translate into temporal errors when the pulse is compressed. The second device, a single-shot

autocorrelator, is an instrument that has been built and used on both the original and upgraded

laser systems in order to optimise performance. The last two devices, FROG and SPIDER, have

not yet been used on the laser system, but represent the state-of-the-art in measurement of the

electric field of pulses from ultrafast laser systems. They are briefly included here as it is

intended that at least one of them will be used to optimise the output of the upgraded laser and

for reasons of completeness.

1.2.2.1. Far-field monitor

The far-field monitor is a simple device that images the focal spot of a laser beam. A

schematic of the monitor is shown in Figure 1.5. The input beam first encounters a lens similar

to one used to focus the laser in an experiment. Theoretically, a lens focuses rays from infinity

to a single point, therefore the focal plane is an image of the beam in the far field. The focal

spot is imaged, using a microscope objective lens (typically with a magnification between 10X

and 40X), onto the active array of a charge-coupled device (CCD) camera. The camera is a

sensitive device and the laser is very bright, so the input beam must be attenuated either by

taking reflections from (optical quality) glass plates or by using (optical quality) neutral density

filters. The picture, displayed on a video monitor connected to the camera, is an image of the

laser focal spot and of the beam in the far field. The dimensions of the image on the monitor

screen can be calibrated using an object of known dimensions (typically a 50 µm wire).

The focal spot of the laser beam can be examined by translating the microscope

objective and camera backwards and forwards. Assuming the focusing lens and microscope

objective do not impose any aberrations, the aberrations of the input beam will be evident on the

monitor screen. Astigmatism, for example, would show up as a line focus that turns into a spot

Introduction

30

of least confusion and then back to a line focus (90º to the first line), as the focus is scanned in

the z direction.

InputBeam

CCDCamera

x

y

MicroscopeObjective

Lens

z

Figure 1.5 Schematic of a far field monitor. CCD = charge-coupled device. Image viewed on a video monitor connected to the camera.

Aberrations throughout the laser system can be detected and eliminated using this

device. The performance of the compressor is very sensitive to aberrations on the input beam

and imposes aberrations itself when misaligned (such as spatial chirp and astigmatism). If the

compressor diffraction gratings are not parallel to each other, the output beam will be

astigmatic. This will reduce the available focused intensity by increasing both the size of the

focused beam and the recompressed pulse duration. The far-field monitor may also be used to

measure the size of the focused beam for use in calculations of focused intensity. These are

typically cross-checked with independent measurements of the focused intensity (such as ion

appearance intensities).

1.2.2.2. Single-shot autocorrelator

The single-shot autocorrelator is a popular short pulse measurement device as it is

relatively simple to construct and a rapid measurement of the pulse duration may be made.

Many groups have used variations of this device, including [Mindl 83] and [Miyamoto 93], a

high dynamic range version has been used by [Braun 95] and [Curley 95] and even a version

where the correlation medium is a gas [Kobayashi 00].

Correlations are generally performed by combining a signal and reference pulse in a

nonlinear medium and using a detection device of some form to measure the resulting

correlation function. In general, an intensity correlation of order (n+1) may be written as:

( ) ( ) ( ) ( )∫∞

∞−+ −= dttItIC n

rsn ττ1 (1.5)

where τ is the temporal delay between the two pulses, ( )tI s is the signal pulse intensity profile

to be measured and ( )tI nr is the reference pulse intensity profile. For an intensity

autocorrelation, ( )tI s and ( )tI nr represent the same pulse, split using a Michelson

interferometer. This gives a correlation function that is symmetric and that can only provide a

measurement of the signal pulse duration when a specific temporal profile is assumed. If the

signal pulse profile is assumed to be sech2, the width of the autocorrelation must be divided by

1.543. If the signal pulse profile is Gaussian, the deconvolution factor is 1.414.

Introduction

31

In the limit where the reference pulse tends towards a delta function,

( ) ( )ttI nrn δ∝∞→lim , the correlation function ( ) ( )tC n 1+ tends towards the signal pulse, ( )tI s . In

this manner, a third order autocorrelation, which uses a third order nonlinearity (third harmonic

generation), produces a correlation that gives a better reproduction of the shape of the signal

pulse. The correlation performed is between the signal pulse at centre laser frequency ω0, and

the frequency doubled signal pulse at 2ω0 in the nonlinear medium producing a correlation at

3ω0. It is also asymmetric so features that occur before and after the main pulse can be

distinguished.

InputPulse

CCDCamera

Second HarmonicCrystal

t

Lens

LensBS

Figure 1.6 Schematic of a single-shot autocorrelator. BS = beamsplitter and CCD = charge-

coupled device. Image viewed on a video monitor connected to the camera.

A second-order single-shot autocorrelator is constructed by splitting the signal pulse in

a Michelson interferometer and combining the resulting pulses in a second harmonic crystal

(such as KDP, KTP or BBO). Figure 1.6 shows a schematic of such a device. In order that the

autocorrelation is single-shot, the two replica pulses are brought together at an angle θ in the

harmonic crystal. By doing this, the phase fronts of the two pulses are tilted with respect to

each other, at an angle related to θ and the refractive index of the harmonic crystal. The two

arms of the interferometer must be arranged such that each pulse travels exactly the same

optical distance between the beamsplitter and the harmonic crystal. Additionally, the crystal

must be angled such that the wave vectors of the replica signal pulses satisfy the phase-

matching condition: ks1 + ks2 = kC, where ks1 and ks2 are the wave vectors of the two signal pulse

replicas and kC is the wave vector of the second harmonic correlation beam. The spatial width

of the overlap of the two beams in the crystal, d (full-width-half-maximum diameter), is related

to the full-width-half-maximum duration of the signal pulse by:

( )

cdt 2/sin θγ=∆ (1.6)

Introduction

32

where c is the speed of light (ms-1) and γ is the form factor that depends on the shape of the

pulse (γ = 1.414 for a Gaussian temporal profile and 1.543 for a sech2 pulse) [Kolmeder 79].

It is usual to image the beam crossover onto the active array of a charge-coupled device

(CCD) camera. The image on the screen can either be calibrated using the magnification given

by the lens, or by translating one arm of the interferometer and equating the change in path

length to a temporal delay. If an arm of the interferometer is translated by x metres and the

autocorrelation on the screen moves by n pixels, then:

cn

xpixel 21 = . (1.7)

Because the signal and reference beams are crossed in the harmonic crystal, the single-shot

autocorrelation is background-free (except for scatter in the crystal). There is no contribution to

the correlation signal from a single beam; the detected correlation is zero in the absence of

either the signal or reference beam.

As an aside, an interferometric autocorrelation may be obtained by a small modification

to the single-shot device. The Michelson interferometer is arranged such that the two emerging

pulses are collinear. These are then passed through a harmonic crystal and the resulting

radiation detected in a photomultiplier. An interferogram is produced when one arm of the

interferometer is translated (slowly) so the reference pulse passes through the signal pulse [Diels

85]. The form of the interferogram for a second order nonlinear process is as follows:

( ) ( ) ( )[ ] ∫∞

∞−

+−= dttEtEC srI22

2 ττ (1.8)

where Es(t) and Er(t) are the signal and reference pulse electric field functions. This device

however is not single-shot; multiple pulses are required to build up the interferogram.

Additionally the interferogram is not background-free as the signal and reference beams are

collinear. There is a factor of eight difference between the background and peak signal of an

interferometric autocorrelation. If the interferometer arm is translated quickly, only the

envelope of the interferogram is recorded. There is a factor of only three difference between the

background and peak signal in this case. It has been demonstrated that the harmonic crystal and

photomultiplier of this type of autocorrelator may be replaced by an ordinary light emitting

diode (LED) reducing cost and complexity [Reid 97]. The diode is used as a voltage source

with a second order response to the incident light intensity (Vdiode ∝ Iin2).

For ultrashort pulses great care must be taken not to pre-stretch the pulse before it is

measured. Pre-stretching will occur in the autocorrelator if the beamsplitter is excessively thick.

To work around this problem an autocorrelator was developed that did not use a conventional

beamsplitter. Instead, it used the apex of a 90º gold-plated prism to separate the input beam into

two halves. As long as the beam is large (>~10 mm diameter) and there is no difference in

Introduction

33

pulse duration across the beam, then this is a viable technique. Problems arise when the beam is

small, as diffraction from the hard edge of the prism (that can be apertured close to the harmonic

crystal when large beams are used) interferes with the correlation measurement. Cross-

correlations between different parts of the beam may be performed by aperturing after the beam

has been split. This would be a convenient way of measuring how the pulse duration changes

across a beam after propagating through a chirped pulse amplification laser system, where a

combination of aberrations in the stretcher and self-phase modulation may cause this to occur.

1.2.2.3. FROG FROG is an acronym for frequency-resolved optical gating [Kane 93a]. This technique

consists of two parts: experimental generation of a FROG trace and a phase-retrieval algorithm

that converts this trace into the temporal phase and intensity of the signal pulse. The generation

of the FROG trace may be achieved in several different ways, but all have in common a

correlation of the signal pulse followed by a measurement of the pulse spectrum. The FROG

trace is a two-dimensional plot with one axis representing the time delay of the correlator and

the other axis representing the wavelength components of the pulse from a spectrometer. The

intensity distribution in the trace may be described by the following expression:

( ) ( ) ( )2

exp,~, ∫∞

∞−

−= dttitEI sFROG ωττω (1.9)

where ( )τ,~ tEs is the signal field, which depends on the method used to correlate the pulse.

Reference [DeLong 96] gives a good description of frequency resolved optical gating and

describes four different correlation methods.

For the purposes of this discussion the second harmonic FROG will be discussed

[DeLong 94]. Figure 1.7 shows an example of a schematic for producing a FROG trace. The

signal electric field produced by the second harmonic correlation is given by:

( ) ( ) ( )ττ −= tEtEtEs~~,~

(1.10)

where τ is the delay between the two replica pulses, ( )tE~ is the complex envelope of the pulse

with the carrier frequency removed and ( )τ−tE~ is the gate pulse. In the same way as the

second harmonic correlation is restrictive, as it always returns a symmetric trace, second

harmonic FROG is also restrictive. The second harmonic FROG is always symmetric about τ

and leads to an ambiguity in the retrieved electric field with respect to time.

Introduction

34

InputPulse

CCDArray

Second HarmonicCrystal

t

λ

LensBS

Prism

Figure 1.7 Schematic of a second harmonic frequency resolved optical gate device for retrieving the full complex field of the signal pulse. The prism acts as a rudimentary

spectrometer. Adapted from figure 5 of [Kapteyn 99]. BS = beamsplitter.

This leads into the computational aspect of FROG, the conversion of the FROG trace

into the temporal phase and intensity of the signal pulse. The reconstruction of the complex

electric field of the signal pulse reduces to extracting ( )τ,~ tEs from the FROG trace (as

( )τ,~ tEs is proportional to ( )tE~ when the gate function is infinitely long). The FROG trace

intensity function can be rewritten in terms of the Fourier transform of ( )τ,~ tEs ( )( )τω,~ tEs

with respect to τ as:

( ) ( ) ( )2

exp,~, dtdtititEI sFROG τττ ωωωωτω ∫ ∫∞

∞−

∞

∞−

+−= . (1.11)

To get ( )τω,~ tEs (which leads to ( )tE~ via the inverse Fourier transform and the

relation ( )tE~ = ∫ ( )τ,~ tEs dτ, for all time) from equation 1.11 is a two-dimensional (t and τ)

phase-retrieval problem. Details of the iterative reconstruction algorithm can be found in

[Trebino 93] and [Kane 93b]. One drawback of the FROG is that the algorithm can return

ambiguous results for the pulse properties [Fittinghoff 95].

1.2.2.4. SPIDER SPIDER is an acronym for spectral phase interferometry for direct electric-field

reconstruction ([Iaconis 99], [Gallmann 99]). Like FROG, the SPIDER technique is split into

two halves: generation of a SPIDER trace and an algorithm to retrieve the temporal phase and

intensity of the signal pulse. SPIDER is based on a concept known as spectral interferometry

(SI) [Froehly 73], where the signal pulse is combined with a delayed reference pulse (whose

electric field profile is known) in an interferometer. The resulting spectral interferogram is

given by

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]ωτωφωφωωωωτω +−++= srsrsrSI EEEEI cos~~~~,22

(1.12)

Introduction

35

where ( )ωrE~ is the electric field of the reference pulse, ( )ωsE~ is the electric field of the signal

pulse and τ is the temporal delay between the two pulses.

The retrieval of spectral phase and intensity from this interferogram is simple and

noniterative. The first step is to take the Fourier transform of the spectrogram. As long as the

temporal delay of the two pulses is longer than the pulse duration, one of the sidebands of the

transform can be filtered out. The inverse transform of this sideband yields the complex

spectral phase of the difference between the signal and reference pulses ( ) ( )( )ωφωφ sr − .

Experiments by Fittinghoff et al [Fittinghoff 96] characterised ultrashort pulses by this method,

using FROG to determine the electric field of the reference pulse. A combination of spectral

interferometry and FROG is known as: temporal analysis by dispersing a pair of light electric

fields (TADPOLE).

In order to make a measurement of a signal pulse, without requiring a known reference

pulse, a technique known as spectral shearing interferometry (SSI) can be used. This method is

similar to spectral interferometry, but in this case, the signal and reference pulses are frequency-

shifted versions of the input pulse. The two frequency shifted replicas are combined in an

interferometer, and the resulting interferogram is given by:

( ) ( ) ( )

( ) ( ) ( ) ( )[ ]ωτωφωφωωωωτω

+Ω+−Ω+Ω+Ω+

+Ω++Ω+=ΩΩ

2121

2

2

2

121

cos~~

~~,,,

ssss

ssSSI

EE

EEI (1.13)

where the two input pulse replicas have been frequency shifted by Ω1 and Ω2 ([Zubov 91],

[Wong 94]).

The technique known as SPIDER is a version of spectral shearing interferometry where

the frequency shifted replicas are produced by mixing two temporally delayed pulses with a

stretched pulse in a nonlinear medium [Dorrer 98]. A typical schematic of the technique is

shown in Figure 1.8.

The input pulse is initially split into two halves. One is sent through a dispersive delay

line (typically a grating pair) to stretch and chirp the pulse. The other is sent through a

Michelson interferometer to produce a pair of pulses with temporal delay τ. For the case that

the nonlinear crystal is a χ(2) medium, when the pulse pair and stretched pulse are combined two

pulses emerge frequency upconverted but separated by a frequency shift proportional to τ and

the magnitude of the long pulses chirp. The two upconverted pulses then enter a spectrometer

and an interferogram described by equation 1.13 is produced. In this case Ω1 = ω0 and Ω2 = ω0+

Ω and the spectral interferometry inversion routine described above can be used to find the

phase difference between the sheared spectra. A review of the history of the SPIDER is given

in [Walmsley 99] and its use is reported on in [Shuman 99], [Dorrer 99a] and [Dorrer 99b].

Introduction

36

InputPulse

CCDArray

ω

I

DispersiveDelay

SpectrometerBS BS

Lens

Nonlinear Crystal

τBS

Figure 1.8 Schematic of a typical spectral phase interferometer for direct electric-field

reconstruction. Dispersive delay section is often comprised of a grating pair. BS = beamsplitter.

1.3. Chirped pulse amplification In order to produce high intensities in a laser system, a technique called chirped pulse

amplification is employed [Strickland 85] and [Maine 88]. Direct amplification of femtosecond

pulses to high intensities leads to nonlinear pulse degradation and optical damage through self-

phase modulation and self-focusing. In order to avoid these effects, advantage is taken of the

fact that a temporally short pulse must be accompanied by a large associated bandwidth through

the following relation:

=∆∆(Gaussian)0.441

)(sech0.315 2

20λ

λtc (1.14)

where ∆t is the full-width-half-maximum pulse duration (seconds), ∆λ is the full-width-half-

maximum pulse bandwidth (metres), c is the speed of light (metres s-1) and λ0 is the laser centre

wavelength. This is a specific result for two common pulse shapes which follows from the

more general uncertainty relationship 2/h≥∆∆ Et . When a short pulse propagates through a

dispersive delay line, it emerges temporally stretched with a time dependent frequency or chirp.

The consequent increase in pulse duration decreases the peak intensity, so avoiding damage due

to the nonlinear response of optical media. The equation for the electric field of a Gaussian

pulse can be written as:

( ) ( )( ) ..2

022

0 ccEtE ee ttita +

= +− φω (1.15)

where E0 is the peak electric field (Vm-1), the first exponent is the Gaussian pulse envelope

where a is related to the pulse duration by ∆t = (2ln2)½/a, the second exponent is the oscillating

Introduction

37

light field with central frequency ω0 (radians s-1) and temporal phase φ(t); c.c. represents the

complex conjugate. If d2φ(t)/dt2 = β is non-zero, the light field frequency changes through the

pulse; β is called the chirp parameter (or frequency sweep rate, radians s-2). If d2φ(t)/dt2 > 0, the

carrier frequency increases with time, this is known as up-chirp. If d2φ(t)/dt2 < 0, the carrier

frequency decreases with time, this is known as down-chirp.

γθ

P A

BB´

Q´

Q

FG

λ

Figure 1.9 Schematic of a parallel diffraction grating pair compressor. The green arrow

labelled λ shows manner in which rays of increasing wavelength are diffracted. On exiting the grating pair, the beam is spatially dispersed. Figure adapted from [Treacy 69], figure 1.

A parallel diffraction grating pair, as shown in Figure 1.9, will give a negative chirp

(higher frequencies lead lower frequencies). The separation between, and the angles of the

gratings determines the magnitude of the chirp. In Figure 1.9, the ray path length PABQ in

metres is equal to:

( ) ( )( )( ) ( )ωωφτωθγωθ

ddccGp ==−+= cos1sec (1.16)

where G is the perpendicular distance between the gratings (metres), γ and θ(ω) are the grating

input and diffracted angles respectively, τ is the associated group delay (seconds), φ(ω) is the

frequency dependent phase and ω is angular frequency (radians s-1). The relationship between γ

and θ(ω) for first order diffraction is written as sinθ(ω) + sinγ = dλ, the grating equation, where

d is the grating constant (lines m-1). The frequency dependent phase for the grating pair can be

written as:

( ) ( )ωθπωωφ tan2 dGcp −= . (1.17)

The origin of the second term on the right is in the nature of the diffraction grating (see also

section 3.2.3.3). An addition of -2π radians phase delay must be added for every groove on the

diffraction grating from a point perpendicularly opposite point A in the figure, to the

intersection of the diffracted ray at point B, F = Gtanθ(ω). G is related to the separation

Introduction

38

between the two gratings along the path of the central wavelength ray at λ0, b, by b = Gsecθ(ω0).

b is referred to as the grating separation throughout this thesis.

The expression for phase as a function of angular frequency may be Taylor expanded

about the centre frequency, ω0, as follows:

( ) ( ) ( ) ( ) K+−+−+−+= 30

320

2010 !3!2

ωωβωωβωωβφωφ (1.18)

where βi is defined as:

( )

0ωωωφβ i

i

i dd= si. (1.19)

The first two terms on the right-hand side of equation 1.18 add constant phase for all

frequencies; β1 is simply p/c, the group delay. The third term corresponds to the linear chirp; β2

is given by:

( )

θπγθβ 32

2

2 cos2sinsin

dcG +−= . (1.20)

where θ0 = θ(ω0). Higher order terms add high order phase, or high order chirp it is these

terms that cause problems in chirp pulse amplification with ultrashort pulses.

For chirped pulse amplification, a system that adds opposite chirp to the grating pair is

required. Such a system, conceived by Martinez in 1987 [Martinez 87a] is shown in Figure

1.10. A unit magnification telescope is placed between two antiparallel diffraction gratings

giving a negative effective grating separation.

γθ(ω0)

´ ´

z

G1G2 L2 L1

z´

G1´

z

P

Q

AB

Figure 1.10 Antiparallel diffraction grating pair forming a Martinez [Martinez 87a] style

stretcher that adds opposite chirp to the parallel grating pair of Figure 1.9. f is the focal length of lens L1 and f′ is the focal length of lens L2. G1′ is the image of grating G1.

For the general case of a telescope of angular magnification M (= ff ′/ ) between two

diffraction gratings, the angular dispersion from grating G1 is magnified by a factor M giving

Mdθ(ω)/dω. In order that the output from grating G2 is collimated, the grating constant of G2

must be Md1, where d1 is the grating constant of G1 (lines m-1). To allow folding about the

focal plane between the two lenses and so as not to have mismatched grating constants, the

magnification of the telescope is set to M = 1, f = f′. Mismatched grating constants in the

Introduction

39

stretcher and compressor have been used in chirped pulse amplification by some groups to

compensate for material dispersion in the amplifier chain ([Squier 98], for example).

The unit magnification telescope produces an image of grating G1 behind grating G2

giving a negative effective grating separation, b = -(z + z′). This distance is the separation

between the gratings along the ray with centre wavelength λ0. b is related to the perpendicular

grating separation, G, by b = -Gsecθ(ω0). Now the path length of a ray along PABQ in Figure

1.10 can be written as:

( ) ( )( )( )ωθγωθ −+−= cos1secGp . (1.21)

This exactly matches the expression for path length in equation 1.16 but with opposite sign,

meaning the two systems are complementary. Consequently, the expressions for group delay

and linear chirp are also complementary. The only difference between them is the two lenses in

the telescope system. As long as the lenses introduce no aberrations, the group delay and chirp

added by the telescope system will be exactly removed by the parallel grating pair. Aberrations

add high order phase in the telescope system that is uncompensable by the simple parallel

grating pair.

We now have two systems that add opposite chirp, which may be used at the start and

end of a chirped pulse amplification laser system. It is usual to place the telescope-based

system at the beginning of the laser to act as a stretcher and to use the parallel grating pair at the

end of the system as a compressor. This is because high fluences at the end of the system would

cause nonlinear pulse degradation in the lenses of the telescope of the antiparallel grating

system. Stretch factors as high as 10,000 have been produced by antiparallel grating pairs of

this type [Salin 92]. A schematic of the technique of chirped pulse amplification is shown in

Figure 1.11. Nonlinear effects and material dispersion in the amplification system can cause

mismatches between the stretcher and compressor. For this reason, material path length and

pulse intensities are kept to a minimum throughout the amplification stages of the laser system.

Amplification

InputPulse

StretcherStretched Pulse Amplified,

Stretched Pulse

OutputPulse

Compressor

Figure 1.11 Schematic of the technique of chirped pulse amplification.

Introduction

40

1.4. Short pulse amplifiers An amplification medium ideally suited for chirped pulse amplification should have

(amongst other things) the following properties:

• High saturation fluence. The saturation fluence gives a measure of the size of amplifier

aperture required to extract a given amount of energy (related to emission cross-section

of the gain medium).

• Broad gain bandwidth. In order to amplify femtosecond pulses, the gain of the

amplification medium must be larger than the bandwidth of the pulse to avoid the

effects of gain narrowing (see section 2.2.1.4).

• Long upper-state lifetime. The upper-state lifetime determines the pumping method

employed to provide gain in the amplifier medium. A long upper-state lifetime allows

flashlamps or laser diodes to be used as a pump source. A short upper-state lifetime

requires a laser-pumping source. The upper-state lifetime also has a bearing on the

contribution from amplified spontaneous emission from the amplifier.

• High gain coefficient. Related to the emission cross-section, the gain of the amplifier

medium must be high to reduce the material the short pulses pass through. This is

important for ultrashort pulse systems, as excess group velocity dispersion leads to

reduction in fidelity of recompressed pulses.

• Good thermal conductivity. The thermal conductivity has a large influence on the

maximum repetition rate of the laser system. Distortions on the seed beam can occur as

it passes through the gain medium due to the large amounts of thermal energy deposited

by pump sources. Neodymium-doped glass (Nd:glass) and chromium-doped LiSrAlF6

(Cr:LiSAF) with their low thermal conductivities tend to be used at a low repetition

rate, whereas titanium-doped sapphire (Ti:Al2O3/Ti:Saph) can be used at rates up to a

few kilohertz.

In practice, no one material gives the best performance for all these requirements and some

compromise must be made. Table 1.1 gives a summary of these characteristics for some,

common short pulse gain media.

Introduction

41

Medium Usat (Jcm-2) λ0 (nm) ∆λ (nm) τ (µs) σe (cm2) Λ (WmK-1)

Organic dyes ~10-3 300-1000 ≥50 10-2-10-6 ≥10-16 ٭

Nd:glass** ~7 ~1053 ~21 ~350 3×10-20 ~1

Cr3+:LiSrAlF6 5 ~830 ~250 ~60 3×10-20 ~1

Ti3+:Al2O3 0.8 ~780 ~450 ~3 3×10-19 34

Table 1.1 Summary of characteristics for a range of common, short pulse amplification media ([Diels 96], [Nees 98] and [Nikogosyan 97]). Usat = saturation fluence, λ0 = gain centre

wavelength, ∆λ = gain bandwidth, τ = upper-state lifetime, σe = emission cross-section and Λ = thermal conductivity. ٭ the thermal conductivity of organic dyes depends on the solvent used; it is also of less importance as the dye is circulated through a jet, therefore the thermal

restrictions on repetition rate are relaxed. ** glass here refers to phosphate glas.

The saturation fluence (Jcm-2) for a four-level laser gain medium is given by the

following relation:

e

esat

hUσν= (1.22)

where h is Plancks constant and σe is the emission cross-section (cm2) at frequency νe (Hz)

[Svelto 93]. This quantity represents the maximum energy per unit cross-sectional area that can

be extracted from the gain medium and is defined as the fluence required to deplete the

available energy to e-1 of its original value. For example, a typical organic dye would require an

aperture of around 500 cm2 to generate 1 J of optical energy. In contrast, titanium-doped

sapphire would require an aperture of just 1.3 cm2 and neodymium-doped glass an aperture of

0.14 cm2 to generate the same amount of energy. Saturation of the gain in chirped pulse

amplification laser systems is typically avoided as this can lead to a spectrally weighted gain,

which broadens the recompressed pulse duration. Although, working close to saturation can

benefit stability for example in regenerative amplifiers (section 1.4.2).

The minimum duration attainable by a gain medium is related to 1/∆ν, where ∆ν is the

gain bandwidth of the medium. Due to the effects of gain narrowing (see section 2.2.1.4) in

short pulse amplifiers, the bandwidth of the gain medium must be many times greater than the

bandwidth of the required amplified pulse. After n passes of a medium with gain bandwidth ∆ν,

a pulse with an initially flat spectral profile will have a bandwidth of (∆ν)n. Titanium-doped

sapphire has the largest percentage bandwidth of any current laser medium [French 95].

The upper-state lifetime (τ) of a medium is a measure of how long gain is available (in

the limit where the pump pulse duration is shorter than τ), or how quickly stationary gain is

reached (if the pump pulse duration is longer than τ) after it has been pumped. A medium that

has a long upper-state lifetime may be pumped efficiently with flashlamps or laser diodes (e.g.

neodymium:glass and chromium:LiSAF). A medium with a short upper-state lifetime must be

Introduction

42

pumped by another laser (e.g. organic dyes and titanium:sapphire). The magnitude of the

upper-state lifetime also determines the contributions from amplified spontaneous emission.

The power of the radiation from amplified spontaneous emission is inversely proportional to the

upper-state lifetime [Svelto 93].

Both saturation fluence and gain coefficient are related to the emission cross section of a

medium. A low emission cross-section means the seed pulse will interact weakly with the gain

medium, requiring long interaction lengths. High emission cross-sections lead to low saturation

fluences and therefore, rapid depletion of the available gain. Titanium-doped sapphire has a

small emission cross-section and so requires many passes to extract maximum gain

(regenerative amplification, section 1.4.2). Conversely, organic dyes have a high emission

cross-section requiring only single or multi stage amplification.

For ultrashort pulses (<50 fs), the best choice of medium has to be titanium-doped

sapphire, mainly due to its broad gain bandwidth. It has a short upper-state lifetime and must

therefore be pumped by another laser (typically, an argon-ion or solid-state equivalent for

continuous wave operation, and a frequency-doubled YAG for pulsed operation). It has a

moderately large saturation fluence, so can only be used to amplify to the hundreds of

millijoules to a few Joules level. Many passes of the inverted medium must be made in order to

extract all the available energy due to its small emission cross-section. This does not pose a

significant problem as its upper-state lifetime is long compared to the transit time of a seed

pulse on a standard optical table. Its relatively high thermal conductivity also allows it to be

used at up to kilohertz repetition rates.

For the purposes of this thesis, two amplifier configurations are described: multi-pass

configuration and regenerative amplification. Each method is outlined with respect to

amplifying ultrashort pulses in titanium-doped sapphire.

1.4.1. Multi-pass configuration

In order to efficiently extract the energy in a gain medium, the seed pulse must have

close to (order of magnitude) the saturation fluence. For this reason multiple stages of

amplification are required, with increasing aperture, in order to meet this condition. The multi-

pass configuration is a popular form of amplifier, used by many groups as the sole method of

amplification of ultrashort pulses ([Antonetti 97], [Backus 97]). The gain medium has to be

passed several times in systems with small emission cross sections (e.g. titanium-doped

sapphire) in order to extract all the available gain. The method uses a single gain region passed

once by one or more pump beams, and passed several times by the seed pulse ([Downer 84],

[Curley 96]). This is distinct from multi-stage single pass configurations, where several gain

media (with low saturation fluences) are individually pumped and consecutively single-passed

Introduction

43

by the seed pulse (a method often used on low repetition rate, high pulse energy systems, [Fork

82], [Danson 98]).

In order to make the amplifier suitable for short pulses, the total material path length

should be minimised and the gain should not be heavily saturated. In this type of amplifier, the

only way to do this is by decreasing the length of the gain medium. Consequently, the active

ion doping level must be increased in order to maintain the same total gain factor. In chirped

pulse amplification, the wavelength changes with time through the pulse. Saturation of the

amplifier causes uneven spectral gain and therefore a reduction in pulse bandwidth and an

increase in recompressed pulse duration (see section 2.2.1.4). Saturation should thus be avoided

in chirped pulse amplification laser systems.

1.4.2. Regenerative amplification As mentioned in the previous section, gain media such as titanium-doped sapphire that

have a low interaction cross-section need to be multi-passed in order to extract all the available

energy. An efficient method of doing this is to use the method of regenerative amplification

[Sizer 81]. A schematic of the general principle of regenerative amplification is shown in

Figure 1.12 although various configurations are possible (see for example references below).

gainPC1 PC2P1 P2

M1 M2

In Out Figure 1.12 Schematic showing the principle of regenerative amplification. M1/M2 represent

cavity mirrors, P1/P2 represent polarisers and PC1/PC2 represent Pockels cells.

In regenerative amplification, a pulse is switched into a cavity containing a gain

medium, makes multiple passes, and is then switched out using a series of Pockels cells. The

regenerative amplifier cavity is defined by mirrors M1 and M2. A seed pulse enters via

polariser P1 and is locked into the cavity when Pockels cell PC1 is switched quarter wave. Just

before the seed pulse arrives at the gain medium, the pump pulse arrives, providing population

inversion. The seed pulse grows in energy as it passes up and down the cavity until it has

gained the required amount of energy. After the final pass of the gain medium, Pockels cell

PC2 is switched quarter wave so the pulse is rejected by polariser P2. Regenerative amplifiers

can provide total gain factors as high as ~107. Many groups use variations of this design to

amplify picojoule to nanojoule pulses up to the millijoule level as part of a larger chirped pulse

amplification laser system ([Rudd 93], [Takeuchi 94] and [Faure 99], for example).

In order to make regenerative amplifiers support short pulses, the total material path

length must be kept to a minimum and the optical components must support large bandwidths.

Introduction

44

The total path length can be minimised by reducing the thickness of each cavity component, for

example, by exchanging the cube polarisers for thin film polarisers. The total path can also be

reduced by increasing the gain per pass (or reducing single pass losses or both), by increasing

the pump energy and/or increasing the doping level in the crystal so that the number of round-

trips can be reduced. Bandwidth sensitive components include polarisers and all components

that have thin film coated surfaces (Pockels cells and cavity mirrors). The gain medium -

titanium-doped sapphire - is often Brewster cut for minimal losses. See section 2.2.1.3 and

section 2.2.1.4 for discussions of how the bandwidth of the seed pulse can be reduced and

shifted after many passes of the gain medium.

Regenerative amplifiers may be run slightly into saturation in order to reduce shot-to-

shot fluctuations, which can arise from variations in the input pulse energy and pump pulse

energy. They also provide isolation between the oscillator and the rest of the amplifier chain as

the profile of the switched-out beam and its pointing stability are set by the mode and stability

of the amplifier cavity, rather than that of the oscillator.

1.5. Blackett Laboratory Laser Consortium titanium-doped sapphire chirped pulse amplification laser system

The Blackett Laboratory Laser Consortium has two complementary chirped pulse

amplification systems. The first is based on neodymium:glass as the amplification medium and

produces 2 ps pulses with up to a Joule of energy at 1053 nm. The other (the subject of this

thesis) is a short pulse system based on titanium-doped sapphire as the amplification medium.

Before upgrade, the system delivered 150 fs pulses with up to 40 mJ of energy at 780 nm

[Fraser 96]. Figure 1.13 shows a schematic of this laser.

The front-end of the system was a home-built short pulse oscillator pumped by a

mainframe argon-ion laser. The oscillator required 5.5 W all lines visible (blue and green) from

the argon-ion to produce 100 fs pulses with a sech2 temporal profile and an average power of

160 mW. The oscillator forms the basis for the whole system as it provides the short pulses to

be amplified and a trigger to time subsequent pump lasers and Pockels cells. For this reason it

is the most critical part of the laser system. Unfortunately, the argon-ion laser had no means of

beam stabilisation and this meant the beam moved as the laboratory temperature changed and as

the laser warmed up. This in turn made the oscillator and consequently the rest of the system

unstable. The first part of the upgrade is to replace the argon-ion with a solid-state device that

reliably pumps the oscillator. This then forms the basis for a stable laser system as a whole.

Introduction

45

Solid State Replacement

Nd:YAG Laser 1

Argon Ion Laser

Nd:YAG Laser 3

Femtosecond Oscillator

KML Oscillator

Regenerative Amplifier

5(4) Pass Amplifier

2 Pass Amplifier

Pulse Stretcher

Pulse Compressor

Short-Pulse Stretcher

Short-Pulse Compressor

Nd:YAG Laser 2

5.5W All Lines Visible

100fs, 2nJPulse Train@ 780nm

4.5W @ 532nm

260ps, 2nJChirped Pulses

@ 780nm

260ps, 2mJChirped Pulses

@ 780nm

90mJ @ 532nm

280ps, 80mJChirped Pulses

@ 780nm

400mJ @ 532nm

800mJ@ 532nm

~100mJ@ 532nm

35fs, 2nJPulse Train@ 800nm

300ps, 2nJ ChirpedPulses @ 800nm

300ps, 2mJ ChirpedPulses @ 800nm



40fs, 100mJ PulseTrain @ 800nm & 10Hz

150fs, 40mJ PulseTrain @ 780nm & 10Hz

Figure 1.13 Schematic of the Blackett Laboratory Laser Consortium titanium-doped sapphire

chirped pulse amplification laser system before being upgraded. Proposed upgrades shown by dotted lines and half-tone colours. KML = Kapteyn and Murnane Laboratories.

The next stage in the laser system forms the first component in chirped pulse

amplification. The 100 fs pulses are stretched by a factor of 2600 to 260 ps by a double-passed

lens-based stretcher. The stretcher is based on the design by Martinez [Martinez 87a] described

in section 1.3. It is folded at the plane between the two lenses so that only one lens and one

diffraction grating are required. The holographic diffraction grating has a ruling density of

2000 lines/mm and dimensions 57 mm × 37 mm. The input and output paths are separated by a

small vertical displacement of the beams. The system is double-passed to remove the spatial

chirp added by a single pass. This stretcher is suitable for long-pulse (100 fs) operation but

limits the final recompressed pulse duration to an increase of 50% over the input duration. This

is due to chromatic and spherical aberration imposed by the lens.

The next stage of the laser system is the first stage of amplification a regenerative

amplifier. This efficiently increases the pulse energy by a factor of 106 and selects a train of

pulses at 10 Hz. The 20 mm titanium-doped sapphire crystal is pumped with ~90 mJ from a

Continuum Surelite II neodymium:YAG laser [Continuum] in a 5 ns pulse. The seed pulse

passes the amplifier cavity 36 times before being switched out with an energy of 2 mJ. The

Introduction

46

oscillator is isolated from the regenerative amplifier by a Faraday isolator this stops leakage

pulses from the amplifier interrupting the modelocking of the oscillator. A significant problem

with this amplifier is that it can generate too much intensity in the cavity before switch-out. The

stretched pulse duration is a little too short and results in the Pockels cell being surface burned

and drilled on a regular basis. The upgrade will include an increase in stretched pulse duration.

The second and final stage of amplification is provided by a five-pass power amplifier.

A 15 mm cube titanium-doped sapphire crystal is pumped with ~400 mJ from a BMI

neodymium:YAG [BMI] in a 5 ns pulse. The beam from this pump laser is split into two halves

of equal energy that pump the crystal from both sides to produce a more uniform pump volume.

After five passes, the seed pulse has an energy of 80 mJ, an amplification factor of 40. Due to

thermal lensing and the odd number of passes of the gain, the output beam has slight residual

astigmatism.

Finally, the pulse is recompressed using a parallel diffraction grating pair. The

holographic diffraction gratings have a ruling density of 2000 lines/mm and dimensions

110 mm × 40 mm. The diffraction efficiency after four encounters of the gratings is 50%,

therefore the output pulse energy is 40 mJ. The recompressed pulse duration is 150 fs, 50%

longer than the input duration due to a mismatch between the stretcher and compressor arising

from the use of lenses in the stretcher. At the peak of performance (full 40 mJ), the laser

provided a focused intensity of 6.7×1016 Wcm-2, assuming a sech2 temporal profile and a lens of

focal length 30 cm producing a two-times diffraction limited spot.

1.6. PhD chronology In order to give an idea of the order in which the work for this PhD was conducted, this

section includes a PhD chronology. Figure 1.14 is a schematic of the approximate chronology

split into laser (upper half) and experimental (lower half) work.

Oct/95 Oct/96 Oct/97 Oct/98

Cluster IonsCluster HHG

S-T E of HHG C60Two Colour

LaserApprenticeship

CPQuick

CPTrace CPTrace3D

Modelling Upgrade

CLEO

Varenna

Figure 1.14 Approximate chronology of work conducted during PhD.

S-T E of HHG = Spatio-Temporal Evolution of High Harmonic Generation.

Introduction

47

At the beginning of the PhD work (October 1995) there existed the titanium doped

sapphire laser system described in the previous section. My first task was to learn how to

operate and more importantly optimise this system. This facilitated the rapid learning of the

physics involved in the technique of chirped pulse amplification and of each associated

component (oscillator, amplifiers, etc.). During this laser apprenticeship, the initial stages of

the modelling were conducted.

In order to design the components necessary for an upgrade of the laser, it was felt that

a model was needed in order to compare existing systems (from the literature) as well as to test

our own ideas. At this time, there was no commercial package (to which we had access) that

could model this type of laser in detail. The first general model written (CPQuick, see chapter

3), was designed in order to gain an understanding of how the parameters of chirped pulse

amplification lasers were related (such as finite optic sizes, stretching factor, grating separation

etc.). The first ray-tracing model written (CPTrace, see chapter 3) was derived from an

achromatic doublet design code written during my MSc work. In September 1997, the results of

this two-dimensional model were presented as a poster at a conference in Varenna, Italy [Mason

98a]. From discussions with Professor Christopher Barty at this conference, it was realised that

a full three-dimensional model was needed. Work on this (CPTrace3D, see chapter 3) was

started in September 1997, immediately after the conference. In March of 1998, the first results

of this model were realised and just before the CLEO conference in May, the reflective doublet

based stretcher was devised and then presented as a talk [Mason 98b]. During the period of the

writing and results of the three-dimensional model, the components necessary for construction

of the stretcher and compressor were ordered. Some of these components had very long lead-

times. The lab work involved in the upgrade of the laser was conducted between November

1998 and September 1999.

In addition, apart from conducting the work directly involved in this thesis, I was

involved in a number of experiments using the original laser system. These experiments are

described briefly in the next section, along with a more detailed description of two of them. The

chronology of these experiments is as follows. In July 1996, a four-month experiment started

which investigated the production of high charge state, high energy (MeV) ions from the

explosion of atomic clusters in an intense laser field (Cluster Ions, [Ditmire 97a/b]). In

October 1996, there was a two-month experiment to investigate the efficiency of high order

harmonic generation in atomic clusters (Cluster HHG, [Tisch 97]). In December 1996, there

was a short collaboration with Dr D Meyerhofer from the University of Rochester that

investigated the spatio-temporal evolution of high harmonic generation using chirped pulse

spectroscopy (S-T E of HHG, [Tisch 98]). My involvement in these experiments gave a good

Introduction

48

understanding of the experimental requirements of a chirped pulse amplification laser system.

This helped later in the design of many of the components of the upgraded laser system.

During the year gap between involvement in experimental work on the laser, I made

some substantial changes to the laser system. The most major of these was the rebuilding of the

stretcher and compressor to accommodate larger beam sizes, using three large diffraction

gratings on loan from the Laurence Livermore National Laboratory. This was necessary as the

beam was found to be breaking up as it propagated long distances through the air, causing

considerable damage to optics and problems for experiments. The exact specification of the

stretcher and compressor was devised using the two-dimensional ray-tracing model.

The next experiment, which started in January 1998, investigated the interaction of an

intense laser field with carbon-60 molecules (C60, [Hay 99]). My final involvement in

experimental work performed during the four years of this PhD, was an investigation of the

interaction of two intense laser fields of different wavelength and temporal delay with atomic

clusters (Two Colour, [Springate 00a]).

1.7. Laser-matter interaction experiments Although the primary subject of this thesis is the modelling that leads to the design of a

suitable stretcher for the upgraded Blackett Laboratory Laser Consortium chirped pulse

amplification laser system and its implementation, the author has also been involved in a

number of laser-matter interaction experiments. Involvement in these experiments has led to a

better understanding of the laser characteristics required by a laser user (for example, pre-

pulse level and beam quality), as well as producing a number of publications in peer review

journals. A brief summary of the authors experimental involvement is given below. These

experiments were all performed using the Blackett Laboratory Laser Consortium, titanium-

doped sapphire, chirped pulse amplification laser system at intensities between 1014 and

1016 Wcm-2, a wavelength of 780 nm and pulse durations between 150 and 250 fs. Two of these

experiments are discussed in detail in sections 1.7.2 and 1.7.3.

1.7.1. Experimental summary The first experiment compared the efficiency of high harmonic generation [Gavrila 92]

when the laser interacted with xenon atoms and xenon atom clusters. The clusters consisted of

several thousand xenon atoms bound by Van-der-Waal forces. The experiment comprised a

harmonic generating pump beam, a cluster dissociation beam, a xenon cluster source and a

vacuum ultraviolet spectrometer to measure the harmonic yield. The pump beam was the

Introduction

49

fundamental of the original titanium-doped sapphire chirped pulse amplification laser system

with a pulse duration of ~150 fs. The cluster dissociation beam was the third harmonic of a

neodymium:YAG laser with a pulse duration of a few nanoseconds. The pump beam was

focused under the cluster source, a pulsed gas jet [Smith 98], and the harmonic intensity was

recorded with and without the cluster dissociation beam present. A mild enhancement of high

harmonic generation efficiency was observed when clusters were present compared to the

efficiency of monatomic xenon [Tisch 97].

The next experiment aimed to investigate the spatio-temporal evolution of high

harmonic generation. This was done by generating harmonics in a similar way to the previous

experiment but using a linearly chirped pulse with a duration of ~1 ps as the pump. The chirped

pulse was generated by changing the separation of the diffraction gratings in the compressor.

The generated harmonics were measured using the same vacuum ultraviolet spectrometer as

before. The chirping of the pulse meant that the instantaneous frequency of the harmonic

generation pulse changed with time, therefore temporal evolution was mapped to harmonic

frequency. The change in harmonic frequency was then mapped to a measurable spatial

displacement by the spectrometer. It was found that early in time, high harmonic generation

occurred on the laser axis. As the intensity of the laser pulse increased the centre of the

interaction region was ionised, quenching the harmonic generation. Harmonic generation

moved into the wings of the beam as the intensity there reached the threshold for high harmonic

generation [Tisch 98]. More details in section 1.7.2.

Next was an experiment to investigate the ion energy spectrum of C60 in the presence of

a high intensity laser pulse. Previous work had seen that small molecules in an intense laser

field are rapidly ionised and undergo Coulomb explosion. Atomic clusters form a nanoplasma

in the intense field and explode due to a combination of Coulombic and hydrodynamic

pressures. C60 is intermediate to these two extreme cases and can be regarded as both a large

molecule and a small (sixty atom) cluster. By looking at the ion spectra the experiment aimed

to determine whether this intermediate target behaves like a small molecule or a cluster in an

intense laser field. We found that the observed ion spectra could be well matched using a model

of a simple Coulomb explosion [Hay 99]. Others found that a plasma model was required to

match their results [Constantinescu 98]. We conclude that C60 is truly intermediate, being both

a large molecule and a small cluster. More details in section 1.7.3.

It had been found previously that when an intense laser pulse interacts with an atomic

cluster larger than a few hundred atoms, there is a resonance in the electron-heating rate when

the electron density reaches a value of three times the critical density (ncrit is the electron density

at which the plasma becomes optically opaque). The enhanced heating of the electrons in turn

leads to enhanced heating of the ions. The idea of the next experiment carried out was to

Introduction

50

irradiate a cluster with a sequence of two pulses of differing frequency to hit the resonance

twice and further enhance the electron heating. One pulse was the fundamental laser radiation

(780 nm) and the other was the laser second harmonic (390 nm). The critical density is

wavelength dependent and is given by:

( )2

213

223

][101.1][

10][

mcmn

emcmn

crit

ecrit

µλ

λπ

×=

=

−

−

; (1.23)

therefore, the lower frequency pulse should be resonant at a lower density. We observed a

factor of two enhancement in the electron temperatures and the order and temporal separation of

the pulses was predicted by our nanoplasma model [Springate 00a].

Many of the experiments performed using the titanium-sapphire system involved the

interaction of short, intense laser pulses with clusters. Our first experiments found that we were

producing very hot (keV) electrons in the clusters [Shao 96]. These high electron energies

should lead to high ion temperatures, and indeed this is what we found. In a paper in Nature in

1997, we reported the production of MeV ion energies from high intensity interactions with

xenon clusters [Ditmire 97a]. Further studies are reported in [Ditmire 97b] and [Ditmire 97c].

A summary of the interactions of short, intense laser pulses with atomic clusters is presented in

[Springate 00b] and [Smith 99]. These experiments include scaling laws for cluster size, cluster

species, laser intensity and laser wavelength.

1.7.2. Measurement of the spatiotemporal evolution of high-order harmonic radiation using chirped laser pulse spectroscopy

High-order harmonic generation [LHuillier 92] has been studied for years, as a source

of high brightness, coherent radiation in the extreme ultraviolet. The temporal evolution of the

process is the most difficult aspect to measure as it occurs on the timescale of the laser pulse. In

the experiment presented, the evolution of the production of a high order harmonic radiation

was measured both temporally and spatially for the first time [Tisch 98]. The radiation studied

was the 13th harmonic of the 780 nm laser, generated at intensities between 1013 and 1014 Wcm-

2. In order to perform this measurement, a technique called chirped laser pulse spectroscopy

was used. In this technique, output pulses from a high intensity laser are temporally chirped.

This chirp is imposed onto the harmonic radiation, so that the instantaneous harmonic

wavelength changes with time. High resolution, time-integrated spectral measurements of the

harmonic thus yield the temporal resolution. Pulses were stretched to 1 ps with both positive

and negative linear chirp.

Introduction

51

Production of chirped pulses from a chirped pulse amplification laser system is

straightforward. Pulses can be given positive or negative chirp simply by reducing or increasing

the diffraction grating separation (respectively) in the compressor. Equation 1.20 shows how

the β2 (s2), related to the chirp parameter, changes with respect to the grating separation, G. The

two-dimensional ray-tracing model (CPTrace), described in chapter 3, was used to check that

the partially recompressed pulses would not change shape as the compressor grating was

translated. Fluctuations in the regenerative amplifier caused the centre wavelength of the laser

to jitter by approximately ±1 nm. The same model was used to check that the shape of the

partially recompressed pulses would not change as the spectrum centre wavelength changed.

The high-order harmonics were generated by focusing the output of the laser with an f-

17 lens through a jet of xenon atoms. The entrance slit of a one metre imaging vacuum-

ultraviolet spectrometer was placed 20 cm behind the jet. The spectrometer was tuned to the

13th harmonic of the incident laser radiation (60 nm) and a dual microchannel plate placed at the

spectrometer output. The image on the microchannel plate showed harmonic wavelength in the

plane of dispersion of the spectrometer and harmonic divergence in the orthogonal direction.

Figure 1.15 shows a schematic of the experimental set-up.

f/17

Dual Plate MCP

Xe Gas Jet

LaserFocus

Imaging SpectrometerResolution 0.05nm

Entrance Slit

5mm

PulsedValve

800 nmUp to 60 mJ160 fs 1 ps

Figure 1.15 The 13th harmonic studied was dispersed in a high-resolution imaging vacuum-

ultraviolet spectrometer and recorded with a microchannel plate detector (MCP).

The instantaneous frequency as a function of time of a linearly chirped laser pulse can

be written as tt βωω ′+= 0)( , where ω0 is the laser centre angular frequency and β΄ is the

chirp parameter ( 222 /1// βωφβ ≈==′ dtddtd , s-2). The value of β΄ for a stretched pulse

duration of ~1 ps is ±1.25×1025 s-2 depending on the sign of the chirp. In the limit where time-

dependent frequency shifts of the harmonic radiation ([Wood 91] and [Lewenstein 95]) are

small compared to the harmonic bandwidth, the harmonic frequency can be written as qω(t),

where q is the harmonic order. Therefore, the harmonic frequency follows the temporal

evolution of the laser pulse. If the unchirped laser pulse has duration τ, which has an associated

transform limited bandwidth ∆ω and the chirped pulse duration (τc) is much longer than τ, then

the chirp parameter is given by cc τωττβ /)/(2 ∆=≈′ . Therefore, the temporal dispersion at

Introduction

52

the harmonic wavelength is given by dt/dλ|λ/q = qτc/∆λ, where λ and ∆λ are the laser wavelength

and bandwidth respectively. The temporal resolution is given by τres = qτc∆λspec/∆λ, where ∆λspec

is the resolution of the spectrometer (valid provided τres is greater than the transform limit

associated with the spectral resolution). Figure 1.16 shows the image on the microchannel plate

for the case of positive and negative chirped laser pulses. Both images show harmonic

wavelength, or time, along the horizontal axis and harmonic divergence along the vertical axis.

1 ps 1 ps

1 nm λ 1 nm λ

Negative Chirp Positive Chirp

20m

rad

Div

erge

nce,

θ

time time Figure 1.16 The space and time resolved images show that the harmonic beam becomes annular

on a sub-picosecond time-scale. The peak intensity in the Xe jet was 1.3×1014Wcm-2.

In both cases, for positively and negatively chirped laser pulses, the image is a crescent.

It can be deduced from these two images that at early time, harmonic generation is along the

laser axis and at later times moves into the wings of the pulse (spatially) creating radiation with

an annular spatial profile. The images of Figure 1.16 show a slice of this annular profile, as

defined by the entrance slit of the spectrometer. The radius of this annular profile increases

with time.

The evolution of harmonic generation depicted here is consistent with modelling of

harmonic emission in the presence of ionisation [Muffett 94, Ditmire 96]. The peak laser

intensity used to produce these two images was 1.3×1014Wcm-2, an intensity at which significant

ionisation occurs. Harmonic generation in the gas jet on the leading edge of the laser pulse is

along the laser axis. As the intensity increases, ionisation occurs which quenches harmonic

generation on axis. The intensity around the laser axis is high enough to produce harmonic

radiation and so the spatial profile becomes annular. As the laser intensity increases further,

ionisation quenches emission in this annular region and the harmonic generation zone is pushed

away from the laser axis. Harmonic generation stops when the intensity in the spatial wings of

the laser falls below the threshold for emission this point defines the edge of the crescent.

Introduction

53

Figure 1.17 shows how the spatiotemporal image at the exit of the spectrometer changes

as a function of peak laser intensity (upper four images) for positively chirped laser pulses.

Each image is averaged over ~100 laser shots. The lower four images are numerically

simulated spectrometer outputs using a simple model of harmonic generation that includes the

effects of ionisation of the nonlinear medium (low density gas jet). The first image on the left in

each row represents a peak intensity where ionisation levels are quite low (~10%). In this case,

the harmonic distribution follows the laser distribution as ionisation has little effect. For higher

intensities, ionisation of the nonlinear medium plays an increasing role in the shaping of the

harmonic emission. The experimental results are well matched by numerical simulations, as

shown.

5.3x1013 W/cm2 7.3x1013 W/cm2 9.6x1013 W/cm2 1.3x1014 W/cm2

Expe

rimen

tM

odel

500 fs time

Divergence

Figure 1.17 Comparison of microchannel plate images at different peak laser intensities with numerical simulations showing that ionisation depletion is the cause of the annular harmonic

beam.

Another important feature of the images shown in Figure 1.17 is the relative arrival of

the harmonic emission compared to the peak of the laser pulse. The peak of the laser pulse

occurs on the far right of each image in Figure 1.17. As the peak intensity of the laser increases,

the radially integrated harmonic signal shows a peak moving earlier in time. Correspondingly,

ionisation occurs increasingly early during the rising edge of the laser pulse. At the lowest

intensity the harmonic duration is 250 fs±25%. At the maximum laser intensity, the harmonic

duration is reduced by ~10%, due to ionisation terminating the emission. An interesting feature

emerges when examining the harmonic duration taken from slices of the images of Figure 1.17.

In this case, as the laser intensity increases, the duration of the harmonic emission decreases

substantially. The harmonic emission on axis for the highest laser intensity shows a duration of

~120 fs. This is a factor of two shorter than the radially integrated duration and ~8 times shorter

than the incident laser pulse.

Introduction

54

In conclusion, the time evolution of the harmonic source distribution has been measured

for the first time. This has been enabled by the technique of chirped laser pulse spectroscopy,

which creates spectral images that include the time history of the high harmonic generation with

a resolution of ~100 fs. The results show that as the laser intensity increases, the harmonic

source becomes annular with an increasing radius. The changing source distribution can be

explained by ionisation of the nonlinear medium. The results also show a harmonic emission

duration much shorter than the incident pulse. This could be extended to shorter laser pulses

where, for example, a 20 fs pulse may produce a harmonic duration of ~3 fs, which could be

isolated with appropriate spatial filtering.

1.7.3. Explosion of C60 irradiated with a high-intensity femtosecond laser pulse

There have been extensive experimental and theoretical investigations of the ionisation

and dissociation of small molecules in strong laser fields [Codling 93, Sheehy 96]. In recent

years, work on atomic clusters having near solid density, but sub-wavelength dimensions, has

produced a range of unexpected and exciting results [Ditmire 97a]. These include very efficient

coupling of the laser field, multi keV electron production and MeV ion generation. Molecules

and clusters lie between the two extreme cases of single atoms and bulk, solid density targets

that initially dominated research into high intensity laser-matter interactions. They share some

of the characteristics of these extreme cases, but also demonstrate much behaviour that is unique

to them alone.

Coulomb explosion has been shown to be an effective model for describing the

dissociation of small molecules in intense (> 1014 Wcm-2) laser fields [Posrhumus 96]. In this

mechanism, the atoms in the molecule are photoionised and the resulting ions then repel each

other through Coulomb forces. The effect of the photoionised electrons on the dynamics of the

explosion is found to be negligible. Ions with rather low mean kinetic energy typically ~10-100

eV are produced in this way. This contrasts with the highly energetic interaction between large

(~1000 atom) atomic clusters and intense laser fields. The key difference between this and the

Coulomb explosion of small molecules is that here, the photoelectrons are retained in close

proximity to the ions. This interaction can be well described by a plasma model in which

hydrodynamic forces dominate the expansion and electron-ion collisional heating is resonantly

enhanced in the laser field [Ditmire 97b] as the result of a dielectric resonance.

The regime between small molecules and clusters has not been well explored. C60 was

chosen as a suitable target that might give information on the transition from exclusively

Coulomb explosion behaviour to a regime where plasma effects become dominant. It has a

well-characterised structure and size (unlike the atomic cluster sources typically used for high

Introduction

55

intensity interaction experiments, where a distribution of cluster sizes is present). Modelling of

cluster explosions [Ditmire 98] shows that electron collisional heating can be important even in

small clusters of 55 atoms or less, suggesting that this mechanism may also be present in C60. In

the experiment presented, ion kinetic energy spectra produced by fragmentation of C60 into

single Cn+ ions in high intensity femtosecond laser pulses at 780 nm are studied [Hay 99].

C60 powder (99.9%, Aldrich) was resistively heated to ~900 K in a small stainless steel

oven contained within a vacuum chamber with a background pressure of ~10-8 mbar. A 1 mm

diameter nozzle in the oven created an effusive beam of C60 vapour that was collimated by a

0.5 mm diameter aperture placed 10 mm from the nozzle. The laser was focused into the C60

beam, 40 mm from the aperture using an f-20 lens. The C60 density in the interaction volume

was ~1×1011 molecules cm-3, corresponding to ~1000 molecules within the 1/e2 iso-intensity

shell of the laser focus.

The ions were analysed using a time of flight spectrometer consisting of extraction and

drift regions and a dual microchannel plate detector (Figure 1.18). An extraction field of

~200 Vcm-1 orthogonal to both the incident laser and the C60 beam was used to extract the ions

created in the interaction volume. The ions are accelerated in proportion to their charge to mass

ratio, enabling different ion species to be resolved. After acceleration, the ions pass through a

35 cm long field-free drift region, before striking the microchannel plate.

Drift Region

MCPExtraction Region

LaserFocus

C60 Beam

C60 Powder in Oven

Figure 1.18 Schematic of the time-of-flight spectrometer showing the C60 source, laser focus,

extraction and drift regions and microchannel plate (MCP) detector. The laser propagates out of the page and is focused into the C60 beam in the centre of the extraction region.

It was important for this experiment to minimise laser pre-pulse to avoid pre-ionising or

dissociating the target, which would change the nature of the interaction between the target and

the main pulse. Pre-pulses arising from leakage from the regenerative amplifier were reduced

from 10-4 to < 10-7 of the main pulse energy by the addition of a Pockels cell/polariser pair

operating as a pulse-slicer. Removal of any pre-pulse is also important when working with

large clusters of atoms. The pre-pulse destroys the cluster before the main laser pulse interacts

Introduction

56

with it (as was intentionally done in an experiment described in section 1.7.1, where a second

laser was used to destroy the cluster medium).

The ionisation and fragmentation of C60 by 150 fs, linearly polarised, 780 nm laser

pulses over a range of peak focused intensities from 1×1013 Wcm-2 to 5×1015 Wcm-2 was

investigated. It was found that above 5×1014 Wcm-2 the C60 molecules underwent complete

dissociation into Cn+ ions. In this regime, the C60n+ signal was reduced to zero, but some smaller

fragments still survived in detectable quantities in the low intensity wings of the laser focus.

The behaviour of the Cn+ ions will only be discussed here, as it is from this data that information

about the interactions occurring in the highest intensity part of the laser focus can be extracted.

As expected, peaks in the time of flight spectrum corresponding to carbon ions from C+

to C4+ appear at successively higher intensities. Figure 1.19 shows time of flight spectra

recorded as the intensity of the laser is increased from 5×1014 Wcm-2 to 5×1015 Wcm-2 using an

extraction field of 185 V.cm-1. The Cn+ peaks are broad compared to the sharp O2+ peaks arising

from background contamination in the chamber. This broadening is due to the ions being

produced with a distribution of kinetic energies. For the peaks to be broadened to the extent

observed, the ions must have initial kinetic energies comparable with the kinetic energy

imparted to them by the extraction field (~400 eV for a 1+ ion).

3 4 50

10

20

30

40

50

5x1015Wcm-2

1x1015Wcm-2

5x1014Wcm-2

Flight Time (µs)

Ion

Sign

al (a

rb. u

nits

)

C4+

C3+

C2+

O2+

C+

Figure 1.19 Spectra recorded at three intensities from 5×1014 Wcm-2 to 5×1015 Wcm-2 with

horizontal polarisation (polarisation vector parallel to detection axis).

By rotating the laser polarisation vector, anisotropies in the ion distribution would

manifest themselves as changes in the ion spectra. At lower intensities, no polarisation

dependence was found. However, at higher intensities a strong anisotropy became apparent.

The spectra in Figure 1.19 were recorded with the laser polarisation vector oriented parallel to

the detection axis (horizontal polarisation). As the intensity was increased, additional structure

appeared in the broadened peaks. This additional structure was never apparent when the laser

polarisation vector was perpendicular to the detection axis (vertical polarisation).

Introduction

57

The anisotropy is illustrated clearly by Figure 1.20 which shows two time of flight

traces taken with the same peak intensity (5×1015 Wcm-2), but with vertical and horizontal

polarisation. The Cn+ peaks in the vertical polarisation case are smooth. Only their height

increased with peak intensity, their form was unchanged. This contrasts with the additional

structure apparent in the horizontal case. The C3+ and C4+ signals are much stronger for

horizontal polarisation. The differences between the two polarisation cases show that some

ions were preferentially ejected along the laser polarisation direction. The C+ peaks were

identical because these ions were produced in the low intensity regions of the focus

(<5×1014 Wcm-2).

3 4 50

10

20

30

40

50

C4+

C3+

C2+

O2+

C+

horizontal vertical

Ion

sign

al (a

rb. u

nits

)

Flight time (µs) Figure 1.20 The difference between horizontal polarisation and vertical polarisation

(polarisation vector perpendicular to detection axis) showing the anisotropic distribution of the C2+, C3+ and C4+ ions.

Small molecules exhibit Coulomb explosion behaviour when irradiated by an intense

ultrashort laser pulse. Their behaviour can be well reproduced by models that tunnel ionise the

constituent atoms, remove the resulting electrons from the interaction and then propagate the

ions under the influence of the Coulomb potential [Posthumus 96]. A simple numerical model

was developed to investigate this type of behaviour in C60.

The initial condition for the model is 60 neutral carbon atoms placed at rest in the

equilibrium positions of the carbon atoms in the C60 molecule. A 780 nm, 150 fs, linearly

polarised temporally and spatially Gaussian laser pulse is then propagated through the target.

At each time step the atoms are allowed to tunnel ionise in the electric field of the laser. The

ionised electrons are removed from the system and the ions are allowed to move under the

influence of their combined Coulomb potentials and the electric field of the laser.

To compare the modelling with the experiment the code was run over a range of

intensities from the lowest intensity at which ionisation occurred (3×1013 Wcm-2) to a peak

intensity in the laser focus of 1×1016 Wcm-2. At each intensity, the model was run many times

with random C60 orientations and the final ion charge states and energies were accumulated. In

Introduction

58

this way, ion kinetic energy distributions were built up for each ion stage at each intensity.

Even at the highest intensities, the explosion was isotropic, as expected from Coulomb

explosion of a spherically symmetric molecule. A numerical model of ion trajectories through

the time of flight spectrometer was used to recreate experimental spectra. Figure 1.21 shows the

results of the numerical modelling compared with experimental data. The general form of the

experimental spectrum is reproduced very well. The correct charge states are present and the

ion energies and abundances are in reasonable agreement.

Our model demonstrates that Coulomb explosion provides a good description of the C60

dissociation dynamics and kinetic energy distribution over an intensity range of

1014-1015 Wcm-2. This is in contrast to the model invoked by Wülker et al [Wülker 94]. The

possibility of electron-ion collisional heating has not been ruled out, but there is no strong

resonant enhancement as in the case of clusters. The observed heating is adequately described

by the Coulomb explosion model and C5+was not observed.

3 4 50

1

2

3

4

5

6

7

8

9

10

C3+

C2+

C+ experiment simulation

Ion

sign

al (a

rb. u

nits

)

Flight time (µs) Figure 1.21 Comparison of experimental and simulated time of flight spectra using 150 fs, 780

nm, vertically polarised pulses.

In conclusion, the dissociation of C60 in a linearly polarised, 780 nm, 150 fs laser pulse

is consistent with Coulomb explosion for laser intensities between 1×1014 Wcm-2 and

1×1015 Wcm-2. This contrasts with the case of large atomic clusters where ion heating is

dramatically increased by resonantly enhanced electron-ion collisions. The observed kinetic

energy distribution of Cn+ ions (n ≤ 4) is isotropic about the laser propagation direction. Above

1×1015 Wcm-2 the distribution becomes anisotropic. A simple Coulomb explosion model

including tunnelling ionisation and Coulomb repulsion between ions, but neglecting the effect

of the photoionised electrons gives good qualitative agreement with the experimental results up

to 1×1015 Wcm-2. There is no evidence of significant extra heating of the dissociating molecule

by resonantly enhanced electron-ion collisions.

2. Design considerations for high power, short pulse laser systems

In this chapter, the design considerations for a titanium-doped sapphire, chirped pulse

amplification laser system are described. The points mentioned are specific to the Blackett

Laboratory Laser Consortium system before and after it was upgraded the primary subject of

this thesis. They are also relevant to similar chirped pulse amplification systems based on

titanium-doped sapphire. Firstly, the desired pulse characteristics from the point of view of an

experimenter wishing to use this kind of laser are considered. The main problems are

highlighted here along with some examples of possible solutions. The implementation of

solutions specific to the upgraded system at the Blackett Laboratory are presented in chapter 4.

Secondly, problems specific to ultra-short pulses, high powers and high intensities are

considered. These include the effects of optical bandwidth, group-velocity dispersion and the

nonlinear response of optical materials. This chapter will highlight the difficulties and

compromises involved in chirped pulse amplification laser design.

2.1. Considerations for high intensity laser-matter interaction experiments

2.1.1. Pulse contrast As the peak intensity available from chirped pulse amplification lasers increases, so

does the pulse contrast requirements for many high intensity experiments. For example, a

focused intensity of 1018 Wcm-2 demands a contrast ratio of about eight orders of magnitude to

prevent the preformation of plasma on a solid target, before the arrival of the main pulse. The

formation of a pre-plasma changes the way energy is deposited in the target and so can

adversely affect solid target experiments. Here contrast is defined as the ratio of the peak

intensity, at the centre of the pulse in time, to that of the most intense pre- or post-pulse. These

extra pulses can be separated into two main groups. The first arise from multiple reflections in

parallel-faced optical components or from cavity leakage, particularly from regenerative

amplifiers. The timescale for arrival of these pulses relative to the main laser pulse is of the

Design considerations for high power, short pulse laser systems

60

order of twice the transit time of the optical component or multiples of the cavity round-trip

time (tens of picoseconds to tens of nanoseconds). Many of the picosecond pulses may be

avoided by careful choice of optics, for example having a small wedge angle between the

surfaces of thin optics or frosting the back surfaces of mirrors. The nanosecond pulses may be

quite easily removed with the use of optical switches (e.g. Pockels cells) which can operate with

rise-times on the order of a few nanoseconds. The second source of extra pulses is from

incomplete recompression of the amplified, stretched pulse, which are separated from the main

pulse by tens to hundreds of femtoseconds. These arise from mismatches between the stretcher

and compressor. The compressor is comprised of just two parallel diffraction gratings the

stretcher has two anti-parallel diffraction gratings separated by a unit magnification telescope.

Aberrations in this telescope contribute towards mismatches between the stretcher and

compressor; they therefore need to be carefully designed to keep aberrations to a minimum.

Section 3 describes a model that has been developed as an aid to this design process which is a

particularly important (and complex) problem for sub-100 fs pulses. Figure 2.1 shows the

relation of these extra pulses with respect to the main laser pulse (vertical scales greatly

exaggerated).

Time (femtosecond scale)

Inte

nsity

(arb

itrar

y un

its)

Pre-pulses Post-pulses

Pulse shoulder

(b)

Time (nanosecond scale)

Inte

nsity

(arb

itrar

y un

its)Main pulse Main pulse

Pre-pulse Post-pulses

Pedestal

(a)

Figure 2.1 Pulse profiles. (a) Short timescale profile showing pulse shoulder and pre-/post-

pulses due to incomplete recompression. (b) Longer timescale profile showing pulse pedestal and pre-/post-pulses from sources within the amplifier chain.

2.1.2. Pedestal Another form of pre/post-pulse that can be detrimental experimentally is pedestal (see

Figure 2.1(b)). This is a background level of light that can last for many times the pulse

duration, and can occur before and after the main pulse. It arises from amplified spontaneous

emission in the amplifiers and/or from incomplete recompression in the form of a pulse

shoulder (see Figure 2.1(a)). Spontaneous emission generally does not have the same

properties as the main laser beam; it is not very well collimated and propagates forwards and

backwards through the gain medium. It can however be a problem when it does travel in the


61

same direction as the main laser beam and is amplified by subsequent gain stages. It can be

controlled, to a certain extent, by correct timing of the pump and seed pulses arriving in the

amplifier gain media and correct timing of Pockels cells.

Pedestal may also be suppressed on the leading side of the pulse by the use of a

saturable absorber [Siegman 86]. This absorbs low intensity light (below about 1010 Wcm-2) but

the absorption is bleached at high intensities allowing the main pulse to propagate. If the

saturable absorber were used before the compressor (where peak intensities are low), the pulse

would be chirped, therefore its spectrum would be narrowed and the centre wavelength shifted.

This occurs because the leading edge of the chirped pulse has a low intensity, and is therefore

preferentially depleted by the saturable absorber [Diels 96]. Placing it after the compressor

where the pulse is no longer chirped would avoid these effects, but the intensities are higher so

nonlinear propagation effects can degrade the pulse. High intensities in the absorber may also

lead to continuum generation and frequency doubling of the recompressed beam, both of which

can cause problems experimentally. Saturable absorbers are generally used as a last resort as,

besides the problems associated with high intensity and/or chirped pulses, they are inconvenient

to handle and typically toxic.

2.1.3. Pulse energy fluctuations Broadly speaking, there are two classes of interaction experiment when concerned with

pulse energy. The first requires constant pulse energy, in order to build up a data set at a fixed

intensity. The second requires a range of energies, to show experimentally how a process

changes as a function of intensity (an intensity scan). Chirped pulse amplification lasers are

inherently unstable, producing pulses that vary in energy on a shot-to-shot basis. In all of the

amplification stages, the gain medium is pumped by one or more frequency doubled YAG

lasers. The pulses from these lasers suffer shot-to-shot energy fluctuations. These fluctuations

affect the amount of gain available per pass in the gain medium and so the final output energy of

an amplifier is unstable. One way around this is to make sure the gain is saturated, particularly

in the highest gain stages. Most titanium-doped sapphire amplifiers are used in a multi-pass

regime and so saturation is achieved by increasing the number of passes of the gain medium

until the available gain is fully depleted. Forcing the gain to saturate leads to increased output

energy stability by making sure all the available energy from the pump laser is utilised even

though the gain available per pass changes. Gain saturation can, however, have an adverse

effect on the pulse spectrum and consequently recompressed pulse duration as mentioned in

section 2.1.6.

In some situations energy fluctuations can be useful for intensity scans, so that a smooth

data set is produced rather than having a series of observations at discrete intensities. On the


62

other hand, experiments that need a fixed intensity require energy binning, where only laser

pulses with an energy in a narrow range are accepted. This increases the time required to take a

good data set.

Another issue associated with pulse energy is dynamic range. Often intensity scan

experiments require a dynamic range of up to ten orders of magnitude. The energy of the laser

is usually controlled using a λ/2 waveplate - polariser combination. These optics are expensive

and so need to be placed near the beginning of the laser system where beam sizes are small. On

the other hand, they do not provide exceptionally large dynamic range (102:1-103:1 at best), and

placing them before six or more orders of magnitude of amplification reduces this range. They

must, therefore be placed far enough through the amplifier chain to provide a reasonable

dynamic range without being too large. One way of increasing the available dynamic range

would be to use waveplate-polariser pairs in series.

Pulse energy cannot easily be controlled by changing the pump laser energy. This is

especially the case when the pump energy is high (above ~300 mJ) where thermal effects

become important. Large pump fluences cause the crystal temperature to rise in the centre of

the pump beam leading to a refractive index increase (dn/dT ≈ 1.3×10-5 K-1 for titanium-doped

sapphire). Cooling by conduction at the edge of the crystal creates a temperature gradient and

hence a positive thermal lens whose strength is proportional to the pump fluence. This can be

compensated for by making the seed beam slightly diverging so that the thermal lens

recollimates it. If the pulse energy was changed by varying the pump energy, the strength of the

thermal lens would change, therefore the divergence of the seed would have to be altered

accordingly - this would be impractical if the system is to remain well optimised for best beam

profile and recompressed temporal profile.

2.1.4. Beam focusability In order to achieve as high an intensity as possible, the beam must be able to be focused

to as close to the diffraction limit as possible. The final beam from the system described in this

thesis is designed to have a Gaussian spatial profile and a flat phase front. To achieve efficient

energy extraction in the amplifiers, a flat-top spatial profile would be better because the gain

media are pumped with flat-top YAG beams. These do not propagate well over long distances

unaided however, due to the high spatial frequency components associated with the step in

intensity at the edge of the beam. As a beam with a Gaussian spatial profile propagates over

long distances (several metres), the shape remains unchanged even if the waist size has

increased due to diffraction. For the same reasons, a Gaussian beam will focus to a Gaussian

spot, whilst a flat-top beam will focus to a spot with a sinc function profile. The profile of both

the focal spot and the beam in the far field is given by the Fourier transform of the initial beam


63

profile. For these reasons Gaussian spatial profile beams are convenient to use in small to

medium scale chirped pulse amplification lasers based on titanium doped sapphire.

Modifications to the Gaussian spatial profile can arise from several components in the

laser system. When the beam passes through or reflects off an optic with an aperture smaller

than its diameter, it will be clipped. This clipping can be hard, from undersized optics, or soft,

for example from the limited size of gain volumes in gain media. Both cause the formation of

diffraction fringes across the beam that change as it propagates through the laser system.

Modifications may also arise from damaged optics affecting the beam at any point across its

profile, usually causing diffraction rings. Diffraction fringes across the beam lead to a reduction

in focused intensity as energy is transferred away from the centre of the beam focus and can

also seed nonlinear self-focusing.

Phase front modifications can arise from many sources including misaligned or

aberrated optics, thermal lensing in gain media, nonlinear response of optical materials and air

currents. The effect of the nonlinear response of optical components is dealt with fully in

section 2.2.3 and the effect of air currents is described in section 2.1.5. Individual aberrated

optics may occur in any part of the laser system, for example a mirror with a low surface figure

or one clamped too hard so that the surface bends. This can be avoided by proper specification

of optics and careful mounting arrangements. At many points through the laser system, the size

of the beam is changed with the use of a Galilean telescope. If one of the telescope lenses is

slightly tilted, the beam will be rendered astigmatic, reducing the ability to focus it tightly. This

can be avoided by careful alignment of the lenses, control of the beam pointing through the

telescope and correct orientation of the lenses. As mentioned in section 2.1.3 high pump beam

fluences lead to thermal lensing in the gain media. A beam passing through the crystal parallel

the to the pump beam will just be slightly focused, but a beam passing through at an angle will

be both focused and become astigmatic. There is some room for compensation here as the

astigmatism added by gain media can be partially compensated for by tilting the lenses of

Galilean telescopes. This would make the beam anastigmatic. However, this is not an ideal

solution as the tilted lens adds small amounts of other aberrations such as coma. A better

solution is to make sure that the number of passes through the crystal is even and the passes are

arranged symmetrically. In this case, the aberrations will cancel.

2.1.5. Beam pointing stability

The beam, as it passes through the chirped pulse amplification laser system, has a total

path length of several tens of metres. If an optic is tilted by only a small amount near the front

end of the laser, the beam will move by a large amount at the output. If a beam pointing

instability manifests itself as a change in angle, the beam focus will move from side to side.


64

This is detrimental to experiments as it can mean the beam moves by multiples of the beam

waist diameter, missing its intended target. If it occurs as a translation from side to side, with

the beam still perpendicular to the face of the focusing optic, then it will still be focused to the

same point. Whilst this may mean the beam is slightly aberrated, the focused intensity will not

change appreciably (as long as the f-number is large and the target is short compared with the

focal volume).

Air currents in the spaces between optics can cause the beam to move as it propagates.

Pockets of air with different temperatures, and therefore slightly different refractive indices,

move through the path of the beam due to air conditioners and the movement of people in the

laboratory. The effect of this is to steer the beam as the air flows and as the beam propagates

over long distances through the laser system. The easiest way of reducing these effects is to

enclose the beams in both pipes and boxes to limit the flow of air across the beam.

2.1.6. Pulse duration fluctuations

Shot-to-shot pulse duration fluctuations can arise from a number of different sources.

As mentioned in section 2.1.3, saturating the gain in the amplifiers can be useful for stabilising

output energy fluctuations, but can also be detrimental to the pulse duration. Gain saturation

leads to a reduction of the pulse bandwidth, which translates, into an increase in the

recompressed pulse duration (see section 2.2.1.4). Fluctuations in the amplifier pump pulse

energy causes instability in the degree of saturation of the gain. The exact dynamics depend on

the mean pump pulse energy. If the gain is just saturated when the pump pulse energy has its

mean value, then the seed pulse bandwidth will be reduced by a certain amount. If the pump

pulse energy is higher than the mean, the gain will saturate earlier, reducing the seed pulse

bandwidth further. If the pump pulse energy is lower than the mean, the gain is unsaturated so

the seed pulse bandwidth will not be affected (except by the ever present gain narrowing and

shifting see section 2.2.1.4). This will lead to shot-to-shot variations of the pulse bandwidth

and therefore the recompressed pulse duration.

Another source of pulse duration fluctuation arises from beam pointing instability in the

compressor. The compressor is very sensitive to alignment, as the phase it adds must exactly

oppose that added by the stretcher. Changing the angle of the input beam in the compressor is

equivalent to misaligning it. This can produce a pulse that is abnormally long, is chirped, has

short timescale pre- or post-pulses, or a combination of all three. A pure transverse translation

of the beam will not affect the pulse integrity as long as the compressor optics are large enough

so as not to clip the edges of the beam. Clipping would reduce the bandwidth of the pulse as

well as creating spatial structure in the beam due to diffraction.


65

On a day-to-day basis, the laser must be sufficiently easy to align so that the stretcher

and compressor are set up optimally every time they are used. This is achieved with the use of

alignment irises, crosshairs and a variety of pulse and beam diagnostics. Irises and crosshairs

are used in order that the beam travels along the same optical path through alignment sensitive

optical components. Diagnostics include an autocorrelator that measures the pulse duration and

a far-field monitor that monitors the beam aberrations (section 1.2.2).

2.2. Optical component considerations

2.2.1. Optical bandwidth Ultra-short optical pulses have a large associated bandwidth, for example, a 30 fs sech2

pulse has a transform limited bandwidth of ~23 nm. To put this in perspective, a 5 ns pulse

from a frequency doubled YAG laser will have an associated transform limited bandwidth of

~8×10-5 nm. The transform limited, full width half maximum bandwidth is calculated using an

equation derived from the time-bandwidth product, for a sech2 pulse ∆t∆ν = 0.315, and the

derivative of the relation between optical wavelength and optical frequency, ν = c/λ, and is

given by

tc∆

=∆2315.0 λλ (2.1)

where ν is the laser central frequency in Hertz, λ is the laser central wavelength in nanometres,

c is the speed of light in metres per second and ∆t is the full width half maximum pulse duration

in seconds. The constant may be replaced in expressions for bandwidth and time-bandwidth

product by 0.441 for pulses with Gaussian temporal profile. In order for the recompressed pulse

duration from a chirped pulse amplification laser system to be the same as the input duration,

this large bandwidth must be preserved throughout the system. Many components of the laser

system have some wavelength dependent characteristic associated with them. Each of these

components has the ability to modify the pulse spectrum in some generally detrimental way.

2.2.1.1. Thin film coatings

Reflective optics generally have a dielectric thin film coating to produce a high

reflection coefficient. Coatings are designed to work at a specific central wavelength and the

bandwidth changes depending on the polarisation used. For example, a 45û S-polarisation

mirror with central wavelength of 800 nm has a bandwidth of ~150 nm, whereas a

45û P-polarisation mirror has a bandwidth of ~110 nm [CVI]. If the profile of this bandwidth


66

had a top-hat shape, the P-polarisation mirror would be able to support sech2 pulses with a

duration of 12 fs. In reality the mirror bandwidth has more of a super-Gaussian profile so when

several tens of mirrors are used, bandwidth narrowing occurs reducing the bandwidth to, for

example ~44 nm (equivalent to 31 fs pulses) calculated assuming forty reflections.

One solution to the limited bandwidth of dielectric mirrors is to use metal mirrors.

Aluminium coated mirrors offer a modest reflectivity at a large range of wavelengths -

reflectivity greater than 85 % between 400 nm and 10 µm [CVI]. Silver mirrors offer a higher

reflectivity than aluminium, but only for wavelengths greater than about 450 nm reflectivity

greater than 90 % between 450 nm and 10 µm. Gold mirrors offer the best performance for

wavelengths longer than the near infra-red reflectivity greater than 96 % between 750 nm and

10 µm. An advantage of using metal mirrors rather than dielectric mirrors is they are less

sensitive to incidence angle and polarisation. A major disadvantage, especially in a high power

laser system, is their low damage threshold. Dielectric mirrors can have damage thresholds in

excess of 40 Jcm-2 (measured for an 8 ns pulse at 1064 nm [CVI]), whereas a metal coating will

only tolerate a fluence of between 10 and 100 times less than this before damage occurs.

Another type of dielectric coating used throughout the laser system is the antireflection

coating. These can be found in places such as normal-incidence crystal faces, windows, Pockels

cell environment protection windows and the rear surfaces of thin film polarisers. For the latter

three cases, the coatings are standard (off-the-shelf) and offer almost constant, low reflectivity

over a range from 670-1064 nm (reflectivity less than 0.5 % average [CVI]). These impose

little or no bandwidth degradation to the pulse. Amplifier crystals however, have to be dual-

wavelength antireflection coated as they are pumped at one wavelength and amplify at another.

The compromise of trying to have low reflectivity at two different wavelengths sometimes

means the bandwidth at the seed wavelength suffers (depending on the coating manufacturer).

2.2.1.2. Polarisers and waveplates

One example of the use of polarisers and λ/2 waveplates is to control laser energy, see

section 2.1.3. As the waveplate is rotated, the polarisation of a linearly polarised input beam is

also rotated. This is followed by a polariser with its axis orientation fixed with respect to the

input beam polarisation. Maximum energy is transmitted when the beam polarisation is in line

with the polariser axis and minimum when they are crossed (perpendicular).

Zero order waveplates are used whenever possible for this application because of the

increased bandwidth they offer over multiple order waveplates. Multiple order waveplates work

by having a thickness of birefringent material (for example crystal quartz) that is an integer

multiple of 2π phase shifts plus a fractional shift (½ or ¼). This makes them less expensive but

they only have a very limited bandwidth, as a given thickness of birefringent material is a


67

waveplate at many different wavelengths. Zero order waveplates, on the other hand, work by

having two plates of birefringent material whose thickness differ by an amount equivalent to the

required phase shift. The two plates are then aligned with the slow axis of one plate against the

fast axis of the other. A zero order waveplate used between parallel polarisers offers extinction

better than 100:1 over a bandwidth of 95 nm centred at 800 nm [CVI].

Two types of polariser are used in the laser system: cube polarisers and thin film

polarisers. Cube polarisers offer large extinction ratios of the order of 104:1, but the bandwidth

is limited and the optic is thick (optic depth is of the order of its clear aperture). They are useful

in places where a large extinction ratio is required and where the optic is passed only a few

times, so as not to stretch the pulse (due to group-velocity dispersion, see section 2.2.2). In the

regenerative amplifier (see section 1.4.2) however, two polarisers are used and are passed many

times and so the bandwidth and thickness is of critical importance. For this application, thin

film polarisers are used as they offer bandwidths that can support sub-50 fs pulses, even after

many tens of passes. These optics are also thin, so the temporal stretching due to group-velocity

dispersion is considerably less than for cube polarisers. The drawback of these large bandwidth

optics is that the extinction ratio can be as low as 5:1. In the regenerative amplifier, there is a

lot of gain available, so the losses from many passes of low efficiency polarisers can be

compensated for [Newport 00]. A problem does arise however, when trying to remove the

leakage pulses from the polarisers. These pulses can be at the millijoule level in the final few

passes of the regenerative amplifier and propagate in the same direction as the output beam and

in the opposite direction to the input beam, back towards the oscillator. This is where cube

polarisers in combination with either a Pockels cell or waveplate become most useful (see

section 5.3.1.4).

2.2.1.3. Bandwidth clipping The stretcher and compressor are the only components in the laser system where the

limited size of the optics can cause spectral clipping. This occurs because the beam is spectrally

dispersed on more than one of the component surfaces.

(a) (b) Figure 2.2 Distribution of beams of different wavelength in the compressor, (a) for small

diameter input beams and (b) for large diameter input beams

Taking the compressor as an example, when an input beam with a small diameter is

incident on the first grating, each ray of different wavelength hits the second grating in a


68

discrete position (Figure 2.2a). If clipping occurs due to the finite size of, for example, the

second grating, the pulse spectrum will have a sharp cut-off between wavelengths that hit the

optic and wavelengths that miss (Figure 2.3, frame 5). This is known as hard clipping. When

an input beam with a large diameter is incident on the first grating, each point across the beam is

dispersed in the same way as the case for small beams (Figure 2.2b). Rays diffracted from the

bottom of the beam will lose longer wavelengths due to clipping. Rays from the centre of the

beam will not be clipped and rays from the top of the beam will lose shorter wavelengths. This

is known as soft clipping as there is a gentle cut-off of wavelengths. The function of

wavelengths passed by the stretcher and compressor can be calculated from the convolution of

beam profile and the window of wavelengths that would be passed if the beam were infinitely

small (Figure 2.3, frames 1, 2 and 3).

The difference between hard and soft clipping is of critical importance in the

recompressed pulse. Hard spectral clipping produces sharp features in frequency space, which

translate into ringing in the recompressed pulse in the same way that a hard aperture will

produce diffraction fringes across a beam in the far field (Figure 2.3, frames 5 and 6, small

beam). Soft spectral clipping is similar to imposing an apodising aperture on a beam with a flat-

top profile. The aperture acts to remove the high spatial frequencies around the edge of the

beam, smoothing the profile in the far field. Soft spectral clipping removes any high-frequency

structure in the pulse spectrum profile, smoothing the recompressed pulse profile (Figure 2.3,

frames 5 and 6, large beam). See [Trentelman 97] for a discussion of bandwidth clipping

applied to pulse compressors.

SpatialBeam Profile

SpectralBandpass

ModifiedBandpass

Larg

e B

eam

S

mal

l Bea

m

x λ λModified

Pulse Spectrum

λ

FT

FT

RecompressedPulse Profile

tPulse

Spectrum

λ

Figure 2.3 Soft and hard spectral clipping for the case of large and small beam diameters,

respectively. Spectral bandpass profiles represent wavelengths transmittable by the stretcher and compressor. FT represents Fourier transform and ⊗⊗⊗⊗ represents mathematical convolution.

Care must be taken with the choice of diffraction grating in terms of the specification of

ruling density and incidence angle. Diffraction efficiency is a difficult to predict function of

wavelength and, in certain circumstances, discontinuities can occur in the efficiency curve due,

amongst other things, to Woods anomalies [Mansuripur 99].


69

2.2.1.4. Gain narrowing and gain shifting

Titanium doped sapphire is a popular choice of amplifier due to, amongst other things,

its large gain bandwidth. It has been reported to support pulses as short as 5 fs [Baltuska 97].

The profile of the gain of this material is approximately Gaussian and centred around 780 nm

[Silfvast 96]. When a pulse with a Gaussian spectral profile is amplified many times, gain

narrowing occurs due to a lack of gain in the wings of the pulse spectrum [Le Blanc 96]. When

gain media are used in a multi-pass regime, the effective supportable bandwidth reduces with

increasing numbers of round trips, according to the following relation:

( ) ( )( ) )( tripsroundgaineffective gg −= λλ (2.2)

where, g(λ)effective is the effective supportable gain profile, g(λ)gain is the gain profile of the gain

medium and (round-trips) is the number of passes of the gain medium. The single pass gain

bandwidth of titanium sapphire is ~140 nm, however after 36 passes for example (as in a

regenerative amplifier), this is reduced to ~23 nm (equivalent to sech2 pulse of 29 fs). If the

centre wavelength of a seed pulse spectral profile coincides with the centre of the gain spectral

profile, the centre wavelength of the amplified pulse will be unchanged. If however, the centre

of the gain spectral profile is at a shorter wavelength than that of the seed pulse, shorter

wavelengths will be preferentially amplified. The result is a shift of the centre of the seed pulse

spectral profile towards shorter wavelengths. This is known as gain shifting and can produce an

effect on the order of several nanometres over a gain of around 106 [Le Blanc 96].

Another form of spectral shifting occurs when an amplifiers gain is saturated (see

section 2.1.6). In chirped pulse amplification the seed pulse is chirped, so if the gain is

saturated before it has propagated fully through the crystal, the leading edge will be

preferentially amplified. Pulse stretchers normally chirp the pulse such that the longer

wavelengths arrive first, so the peak of the pulse spectrum is shifted toward the shorter

wavelengths. This can be used to balance the effect of gain shifting. Due to the reduction in

amplitude of the trailing edge of the chirped pulse, the pulse bandwidth is also reduced,

increasing the recompressed pulse duration that can be obtained.


70

2.2.2. Dispersion

2.2.2.1. Group-velocity dispersion Group-velocity dispersion is defined as the rate of change of group-velocity with

wavelength:

2

2

2

222

2 ωφ

ωπυω

λυ

dd

dkd

cdd

GVD gg === (2.3)

where υg = dω/dk is the group velocity (ms-1), λ is the laser wavelength (m), ω is the laser

frequency (radians s-1), c is the speed of light (ms-1) and k = ωn(ω)/c is the propagation constant

(m-1). It describes how, because an optical material has a different refractive index at different

wavelengths, a pulse comprising many wavelengths will be temporally stretched. In a material

exhibiting normal dispersion, higher frequencies travel more slowly (refractive index, n ~ λ-1),

giving a positive chirp. This is particularly important when considering ultra-short pulses, i.e.

sub 50 fs, because of the very large associated bandwidths. For example, a 30 fs sech2 pulse has

an associated transform limited bandwidth of 22.4 nm. All optical components in the laser

system that the pulse travels through will stretch the pulse by a certain amount, some more than

others. For example, 1 cm of BK7 optical glass will stretch a 30 fs pulse by 26.1 fs to 56.1 fs

(calculated for a sech2 pulse at 800 nm). Comparatively, 1 cm of SF8 (a high dispersion glass

often used in prisms) would stretch the same pulse to 111.9 fs.

Figure 2.4 shows, as a function of input pulse duration, how much a pulse will be

stretched as it propagates through 1 cm of BK7 glass (calculated using the chirped pulse

amplification model described in chapter 3). As the pulse duration is reduced, the associated

bandwidth rapidly increases (blue curve). The amount a particular optical medium stretches the

pulse by is proportional to the pulse bandwidth; therefore, the fractional increase in duration

grows as the pulse gets shorter. It can be seen from this graph that the shorter the pulse required

from the system, the more care needs to be taken to control the amount of dispersion picked up

in the amplifier chain and on the way to the target. It also shows that as the input pulse duration

increases, the output duration asymptotically approaches the τout = τin line. Therefore, longer

pulses are not stretched so much and less care needs to be taken to control material dispersion

through the laser system.

Other optical materials used in the laser system described in this thesis include titanium

doped sapphire and KD*P (Potassium Di-Hydrogen (Deuterated) Phosphate) used in Pockels

cells. Both of these materials behave in much the same way as most standard glasses, as far as

dispersion is concerned. Group-velocity dispersion contributions from KD*P are especially low


71

which is advantageous given the large number of passes of the Pockels cell inside a regenerative

amplifier cavity. For example, a 30 fs sech2 pulse, after propagating 1 cm, will be stretched to

46.9 fs in KD*P (ordinary ray) and 64.4 fs in sapphire (ordinary ray) compared to 56.1 fs in

BK7. Calculations for KD*P and sapphire shown for ordinary ray as both crystals are

birefringent stretching for extraordinary ray would be 52.7 fs and 63.5 fs respectively (all

values calculated using the model described in chapter 3).

30 40 50 60 70 80 90 100

30

40

50

60

70

80

90

100

τout = τin

Out

put P

ulse

Dur

atio

n (fs

)

Input Pulse Duration (fs)

6

8

10

12

14

16

18

20

22

Pulse Bandwidth (nm

)

Figure 2.4 Sech2 pulse bandwidth (blue curve) and temporal broadening (red curve) due to 1

centimetre of BK7 glass, as a function of input pulse duration. Calculated using chirped pulse amplification model described in chapter 3.

This linear dispersion can be compensated, to a certain extent, by controlling the

separation between the diffraction gratings of the compressor. See section 5.4 for a discussion

of phase contributions from the compressor as the configuration is changed.

2.2.2.2. High order dispersion In a linear system, the phase added by an optical material is given by

φ(ω) = -n(ω)ω0L/c, where n(ω) is the wavelength dependent refractive index of the material, ω0

is the laser centre frequency (radians s-1) and L is the propagation length (m). The accumulated

phase through a laser system may be Taylor expanded about the centre laser frequency as

follows:

( ) ( ) ( ) ( ) ( ) K+−+−+−+= 303

32

02

2

00 61

21 ωωφ

ωωωφ

ωωωφ

ωωφωφ

dd

dd

dd

(2.4)


72

The first term in the expansion, φ(ω0), represents the static accumulated phase at the centre

wavelength. The next term represents the linear phase component or the total group delay

through the system. The double differential of phase with respect to angular frequency

represents the quadratic phase or linear chirp. The next term represents second order chirp or

third order phase. Higher order components become increasingly more important the shorter the

pulse and hence the larger the pulse bandwidth.

As described in section 2.2.2.1, most optical materials in the laser system add linear

chirp to the pulse as it propagates through them. The blue curve of Figure 2.5, refractive index

as a function of wavelength, is typical of most optical media. Calculating the phase expansion

of this material using equation 2.4 would give strong zero, first and second order components

and little higher order contributions. This corresponds to the addition of linear chirp and a small

amount of higher order terms. Linear chirp can be compensated for in the compressor as

changing the separation of the gratings changes the amount of linear chirp added (see sections

1.3 and 5.4). Other optical materials can add higher order phase terms that cannot be

compensated for in the compressor in such a simple way. These should be avoided wherever

possible - however, if they cannot, they may be compensated for by stretcher alignment

([Sullivan 95], [White 93]).

500 600 700 800 900 1000 1100

1.950

1.955

1.960

1.965

1.970

1.975

1.980

1.985

1.990

Ref

ract

ive

Inde

x of

TG

G

Wavelength (nanometres)

1.508

1.510

1.513

1.515

1.517

1.520

1.522

Refractive Index of BK7

Figure 2.5 Comparison between the variation of refractive index with wavelength for BK7 and

TGG.

An example of a medium that adds high order phase is the crystal used inside a Faraday

isolator, known as TGG (Terbium Gallium Garnet). TGG is a Faraday rotator material that,


73

when a static magnetic field is applied, rotates the polarisation of an incident beam in the same

sense, independent of the direction of propagation (forwards and backwards). Figure 2.5 shows

the comparison between the variation of refractive index with wavelength for BK7 and TGG

[Halbo Optics]. As can be seen, the high order phase contributions from TGG will be

significantly larger than those from BK7 (calculation of phase using equation 2.4 would give a

strong third order component, i.e. second order chirp).

2.2.3. Nonlinearities

2.2.3.1. Self-focusing In the technique of chirped pulse amplification, short pulses are temporally stretched to

avoid damage to the laser system through self-focusing during amplification. Nevertheless,

high intensities are unavoidable in lasers of this type and lead to nonlinear responses from

practically all optical media. The optical Kerr effect describes how the refractive index of an

optical material changes with the intensity of incident radiation, as follows:

Innn 20 += (2.5)

where n0 is the standard (linear) refractive index, n2 is the nonlinear refractive index (m2W-1)

and I is intensity (Wm-2). Titanium-doped sapphire for example, has a value of n2 =

3.1 × 10-16 cm2W-1 [Nikogosyan 97] to increase the refractive index by 0.01% requires an

intensity of 5.5 × 1011 Wcm-2. In a beam with a uniform Gaussian spatial intensity profile, the

refractive index change is strongest in the centre, which can produce a lens through the effect of

nonlinear refractive index. Similarly, in an otherwise uniform beam, regions of high intensity or

hot spots lead to lenslets. In the oscillator, this effect is used to our advantage to discriminate

between high and low intensity beams by careful arrangement of the optical cavity, resulting in

a strong mode-locking mechanism (see section 1.2.1.2). In the amplifier chain, the same effect

can lead to self-focusing, self-phase modulation and beam break-up.

In most materials, the value of n2 is positive; therefore, a nonlinear lens created due to a

beam being more intense in the centre than around the edges will be positive. This, initially

weak, lens focuses the beam making it more intense and therefore making the induced lens

stronger. This process can run away catastrophically until the damage threshold of the optical

medium is reached and damage occurs. A critical peak power at which self-focusing becomes a

problem can be derived from the point at which beam focusing due to nonlinear effects

outweigh the effect of divergence due to diffraction. The critical power is given by [Diels 96]

20

2

32)22.1(nn

Pcrπλ= (2.6)


74

where λ is the laser wavelength (m) and Pcr is on the order of a few megawatts. Self-focusing

can also occur at lower average intensities due to small-scale intensity variations across the

beam profile. Modulations in the intensity profile become more pronounced exponentially with

distance due to induced lenslets caused by the Kerr effect. This leads to a single or several

beam filaments forming which can cause optical damage or optical breakdown. This effect may

be counteracted by the use of spatial filtering before hot spots reach the critical power for self-

focusing. A spatial filter is a pair of lenses arranged with their foci at a single point where a

pinhole is placed. As the pinhole is in the focal plane of the lenses and therefore the Fourier

plane, it will remove the high spatial frequency components of the beam.

2.2.3.2. Self-phase modulation

Self-phase modulation is analogous to self-focusing but acts in the time domain rather

than the spatial domain. Over the duration of an optical pulse the time-dependent intensity I(t)

causes a time dependent change in refractive index, n(t) = n0 + n2I(t) (from equation 2.5). This

in turn produces a time-dependent phase change, ∆φ(t) = -n2I(t)ω0L/c. At each point in time the

pulse experiences an instantaneous frequency shift (∆ω) given by the derivative of the phase

change at that point [Siegman 86]:

dt

tdIc

Lntdtd )()( 02ωφω −=∆=∆ (2.7)

This leads to the production of new frequency components and a broadening of the pulse

spectrum. In a pulse with a Gaussian temporal profile, this adds chirp that is only linear over

the central part of the pulse (Figure 2.6). The chirp in the wings of the pulse is highly nonlinear

and difficult to compensate for as a result. In the presence of group-velocity dispersion, self-

phase modulation becomes more complicated. Group-velocity dispersion will change the

duration of the pulse as it propagates, changing the temporal intensity profile and therefore the

action of self-phase modulation. For this reason, modelling requires either a segmented

approach where the effects of self-phase modulation and group-velocity dispersion are added

sequentially in small steps, or by solution of the wave equation. In the same way that spatial

variations of intensity across a beam become more pronounced due to self- focusing, temporal

variations of phase through the pulse can be amplified causing the pulse to break up spectrally

and temporally.


75

-60 -40 -20 0 20 40 60

0.0

5.0x1010

1.0x1011

1.5x1011

Puls

e in

tens

ity (W

/cm

2 )

Time (femtoseconds)

-2x1014

-1x1014

0

1x1014

2x1014

Frequency shift, ∆ω (rads/s)

Figure 2.6 Calculation of self-phase modulation of a Gaussian pulse (red) in the absence of

group-velocity dispersion. Blue line shows the instantaneous frequency shift of a 30 fs Gaussian pulse after propagation through 1 cm of fused silica.

2.2.3.3. B-integral

The effect of these spatial and temporal optical Kerr effects may be generally described

as beam break-up. In a linear system, phase is given by φ = -nω0L/c, therefore (from equation

2.5), the phase contribution due to nonlinear effects is be given by φNL = -n2Iω0L/c. The

accumulation of this nonlinear phase through the system is known as the break-up integral or B-

integral and is given by:

∫≡L

dzzIznB0

2 )()(2λπ

(2.8)

where I(z) is the intensity as a function of distance along the optical axis (Wcm-2) and n2(z) is

the nonlinear refractive index of the different materials the beam encounters along the optical

axis (cm2W-1). In a high power laser system, it is generally required that the total B-integral

should not exceed a value of between three and five to avoid the effects of self-focusing and

self-phase modulation [Siegman 86]. This criterion may be relaxed by making sure the

accumulated B-integral between spatial filters (section 2.2.3.1) does not exceed two [Tisch 95].

This allows the total B-integral accumulated through the system to be as large as ten without

incurring physical damage due to nonlinear spatial effects. This does not include however

temporal nonlinear effects that are particularly critical in the type of chirped pulse amplification

laser system described here. In the best case, B-integral should be kept as small as possible.


76

2.3. The need for modelling The most critical aspects in the design of a short-pulse, chirped pulse amplification laser

system are the design of the stretcher and compressor. In these two systems, the size of the

optics determines the supportable bandwidth and therefore the duration of the final output pulse.

The angle of the diffraction gratings not only determines their size, but also their efficiency, this

is especially important in the compressor where energy throughput is important. A diffraction

grating is most efficient when used in Littrow configuration (equal input and output angles), and

departure from this reduces the efficiency by a factor of roughly the cosine of half the difference

between the input and output angles [Learner]. In order for there to be zero residual high order

phase in the recompressed pulse, the stretcher and compressor must be well matched. The

compressor contains only diffraction gratings and mirrors, whereas the stretcher has a unit

magnification telescope. Aberrations in this telescope cause mismatches between the stretcher

and compressor and therefore result in residual phase and incomplete compression. Aberrations

can arise from the design of the telescope (e.g. spherical aberration) or from surface

irregularities in the optical components. The latter is important in ultra-short pulse systems, as

the beam is large and spectrally dispersed on many components which adds spectral phase

distortions directly to the pulse. Any stretcher design should also be realisable that is the

alignment tolerances should not be so tight that it is impossible to construct and use on a routine

basis.

It is imperative to be able to gauge the importance of the effects mentioned above and

also to compare the many different designs of stretcher described in the literature. A full, three-

dimensional ray-tracing model specifically designed for chirped pulse amplification is required.

It must include all aspects relevant to short pulse systems such as taking account of high-order

phase contributions and be able to cope with large pulse bandwidths. Most importantly, it must

allow a compromise to be found between the size of optics required (large curved mirrors and

diffraction gratings) and the performance of the required system (in terms of pulse contrast and

recompressed duration) such that as good a system can be constructed for a given investment in

equipment. Such a model is described in the next chapter along with other models that facilitate

the design process at all levels [Mason 98a] [Mason 98b].

3. The Chirped Pulse Amplification model

A central part of the work described in this thesis has been the development of a

detailed model to aid the design of pulse stretchers for chirped pulse amplification. Specifically

the model has been used to design a stretcher (and corresponding compressor) for use in the

ultrashort pulse upgrade of the Blackett Laboratory Laser Consortium, titanium-doped sapphire

system.

As the temporal pulse duration required from chirped pulse amplification laser systems

decreases, the contribution of high order phase becomes more important. Figure 3.1 shows how

third and fourth order phase contributions increase exponentially as the input pulse duration

decreases (calculated using the code described in this chapter). For this reason and because

there are many different stretcher designs to choose from, a model is needed to compare

different systems and optimise the overall design chosen.

50 100 150 200 250

-2.80x109

-2.82x109

-2.84x109

-2.86x109

-2.88x109

-2.90x109

Thi

rd o

rder

pha

se (s

3 )

Pulse duration (femtoseconds)

3.36x1010

3.33x1010

3.30x1010

3.27x1010

3.24x1010 Fourth order phase (s4)

Figure 3.1 Calculated contributions of third and fourth order phase as a function of input pulse

duration for a reflective stretcher. High order phase contributions become increasingly important as the pulse duration is reduced. Third order phase is shown in red and fourth order

phase is shown in blue.

In this chapter, three models for chirped pulse amplification are described. The three

codes were designed sequentially as the need arose for more complex modelling. The first,

looks at macroscopic aspects of the stretcher and compressor and provides a starting point

design. The second traces rays through a two-dimensional stretcher and compressor to calculate

The Chirped Pulse Amplification model

78

projected output pulse profiles [Mason 98a]. The third is a full three-dimensional ray-tracing

code that takes into account displacements of the beam in the plane perpendicular to the plane of

dispersion [Mason 98b]. The codes have also been used to aid the design of stretcher

configurations for the ASTRA ([Langley 99a] and [Langley 99b]) and Vulcan [Danson 99]

lasers at the Rutherford Appleton Laboratory.

3.1. Model suite The primary aim of the chirped pulse amplification model is to compare the many

existing stretcher designs and to design a specific system that performs as required with the

limited resources available for this project. Traditionally refractive optics have been used to

form the unit magnification telescope required for the stretcher of chirped pulse amplification

laser systems ([Martinez 87b], [Fraser 96], [Sullivan 96], [Kalashnikov 97]). This is no longer

appropriate for pulse durations below about fifty femtoseconds due to the large associated

bandwidths and consequent intolerable chromatic aberration. The natural step away from

refractive optics is the use of reflective optics - there have been many examples of how these

can be used in the stretcher telescope ([Rudd 93], [Cheriaux 96]). Most methods aim to produce

an aberration-free design so that the stretcher exactly matches the compressor. The models

developed and described here are capable of comparing these (and any other new) designs to

different degrees of detail.

3.1.1. CPQuick

The first level model looks at macroscopic aspects of the laser system such as input

pulse duration, centre wavelength and the stretcher/compressor characteristics, such as physical

grating separation. Using these it calculates the stretched pulse duration and the approximate

sizes of the optics required to accommodate the associated pulse bandwidth. This is useful in

the initial stages of design so that a basic system that produces the required stretched pulse

duration using reasonably sized optics can be devised.

The specification of the compressor is calculated first as it is the configuration of the

compressor that determines the losses in amplified pulse energy at the end of the system due to

diffraction efficiency, and a large fraction of the cost associated with stretching and

compression of femtosecond pulses (the size of the diffraction gratings). The separation of the

gratings (b, metres) and the input angle (θ0) determine the stretched pulse duration (see Figure

3.2). A combination of the grating input angle, the input beam size and required final pulse

duration determines the size of the gratings needed.


79

Grating Separation = b

Input Angle = γ

GratingAngle = θ0

OpticalAxis ∆θ

Figure 3.2 Configuration of the pulse compressor showing the definition of grating separation

(b, metres), grating angle (θ0), diffracted angular spread (∆θ) and input angle (γ).

Figure 3.3 shows three different compressor configurations. When the diffraction

gratings are perpendicular to the optical axis, this gives the smallest size of the second grating

(Figure 3.3(a)). For an input pulse with a given bandwidth, the rays from the first grating will

have a certain angular spread. The further away from normal incidence the second grating is,

the larger it has to be to accommodate all the rays. Reducing the size of the grating reduces the

cost dramatically, but at the expense of diffraction efficiency. Diffraction efficiency is

maximum when the input and output angles of the grating are equal (Figure 3.3(b)). This is

known as the Littrow configuration. As the angle between the input and output rays increases,

the diffraction efficiency decreases as roughly the cosine of half the difference between the

input and output angles [Learner]. This configuration will give maximum output energy, but

can be costly given the size of diffraction gratings required, and difficult to build as the input

and output beams are inside the compressor. The ideal configuration is a compromise

between efficiency and grating size. Another factor that needs to be taken into account when

finding this compromise is the finite size of the input and output beams (Figure 3.3(c)). The

two sets of beams must be large enough so as not to damage the gratings due to large fluences

and small enough so as not to be clipped by the opposite grating (or grating mount).

(c)(a) (b) Figure 3.3 Compressor configurations. (a) Configuration with smallest second diffraction

grating size. (b) Littrow configuration giving largest diffraction efficiency. Input and output in this case is along the optical axis defined by the dotted line. (c) Compromise configuration that

allows input and output of finite sized beams.


80

The grating constant (d), defined here as the number of grooves per unit distance along

the grating (lines/metre), must also be chosen. If the grating constant is high, the beam is highly

dispersed, so the two compressor gratings do not need to be so far apart to compress a given

stretched pulse. If d is small, the two gratings have to be far apart in order to compress the same

stretched pulse. Diffraction gratings have preferred grating constant values as set by the grating

manufacturers and the techniques used to create them. For this reason the choice of d is quite

limited, or a premium must be paid for non-standard ruling densities. It is more a case of

choosing a ruling density from the set easily available and designing a compressor around it

rather than decide the density from the design. For a given stretched pulse duration, a

compromise must be reached between high dispersion gratings giving a compact system, and

low dispersion gratings giving a physically long system. A long system allows a configuration

closer to Littrow, but is difficult to fit on an optical table. A short system will be compact but

far from Littrow configuration making it inefficient.

All these factors have to be taken into consideration if we are to create a system that

performs well, is practical to build and economical in terms of optic size and cost. CPQuick

allows this to be done by automating the macroscopic calculations associated with chirped pulse

amplification. The stretched pulse duration (T, seconds) is calculated from the input pulse

bandwidth (∆λ, metres) and compressor parameters using the analytical equation [Faldon 92]:

0

2

20

cos2

θλλ

cbdT ∆= (3.1)

where λ0 is the laser centre wavelength (m) and c is the speed of light (ms-1), which assumes

that β2 (section 1.3) is approximately given by T/∆ω where ∆ω is the angular frequency

bandwidth of the input pulse. For example, a pulse with 30 nm of bandwidth centred at 800 nm

injected into a compressor with a grating separation of 1 m, a grating angle of 30° and a ruling

density of 1200 lines/mm, will emerge stretched to a duration of ~300 ps. For a given grating

input angle, γ, measured from the grating normal (Figure 3.2), the first order diffracted angle is

calculated from the grating equation:

λγθ d=+ sinsin (3.2)

where θ is the diffracted angle, measured from the grating normal, for a ray of wavelength λ

(m). The grating angle is often defined as the centre laser wavelength diffraction angle for pulse

compressors, so that sinθ0 = dλ0 - sinγ (see Figure 3.2). CPQuick also calculates the projected

maximum focused intensity (I, Watts/m2) of the laser after compression. It is assumed that the

total energy after compression (E, Joules) is given by the integral of intensity over space and

time of the pulse,


81

( ) dtdydxtyxIE ∫ ∫ ∫∞

∞−

∞

∞−

∞

∞−

= ,, . (3.3)

A pulse with a sech2 temporal profile with a full-width-half-maximum duration of ∆t (seconds)

and a Gaussian spatial profile with 1/e2 beam radius ω (metres) is written as:

( ) ( ) eeyx

tt

ItyxI 2

2

2

2 22

76.120

cosh,, ωω

−−

∆

= (3.4)

where I0 is the peak intensity (Wm-2). The 1/e2 radius of the beam is related to the full-width-

half-maximum (FWHM) diameter by FWHMωω ×= 85.0 . Equation 3.3 evaluated for all time

and space using equation 3.4 gives the following expression for the peak intensity:

( ) tNEI

∆= 2

00

76.1ωπ

η (3.5)

where the projected size of the focal spot is N-times the diffraction limit and η is the compressor

diffraction efficiency. The constant in equation 3.5 may be replaced by 1.88 for a pulse with

Gaussian temporal profile. The diffraction limited 1/e2 focal spot radius for a lens (or reflective

focusing optic) with focal length f (metres), an incident beam with 1/e2 radius of ω (metres) and

centre wavelength λ0 (metres) is given by:

πωλω f0

0 = . (3.6)

For example, an amplified pulse with 200 mJ of energy, centred at 800 nm, recompressed to a

duration of 40 fs, with a 1/e2 beam radius of 20 mm, in a compressor that is 50% efficient and

focused by a 30 cm focal length lens to a 2-times diffraction limited spot would produce a

focused intensity of 2.4×1018 Wcm-2.

The size of optical components is simply calculated geometrically by using the full-

width-half-maximum bandwidth angular spread (∆θ, radians) of the beam from the grating,

given by:

λθ

θ ∆≅∆0cos

d. (3.7)

where θ0 is the grating angle (degrees) and ∆λ is the bandwidth of the input pulse (nm). For

example, a beam with 30 nm of bandwidth incident at a diffraction grating at an angle of 30°,

with a grating constant of 1200 lines/mm would give an angular spread of 2.38°. The definition

of angular spread is given in Figure 3.2. This definition assumes θ0 is related to γ by λ0 and the

grating equation (equation 3.2)


82

3.1.2. CPTrace

Armed with the characteristics of the compressor (grating ruling density, grating

separation and input angle) it is possible to design the stretcher. In order for the stretcher to

exactly match the compressor (in terms of high order phase components), the stretcher imaging

system must be as aberration free as possible. CPTrace models this by tracing rays through a

two-dimensional stretcher and compressor.

t

Output Phase& Amplitude

FT-1

ω

SpectralClipping

ω

2-D Ray-Traceof Compressor

t ω ω

1 Beam 10s 1000s

of rays2-D Ray-Trace

of Stretcher

FT

Figure 3.4 Flow diagram for the CPTrace model. Blue lines represent spectral/temporal phase and red lines represent spectral/temporal amplitude. FT represents the Fourier transform and

FT-1 represents the inverse Fourier transform.

A single beam that has been sampled with between a few tens to a few thousand time

components is injected into the numerical system (Figure 3.4 frame 1). When this beam

encounters a diffraction grating, it splits into many rays, each one travelling in a different

direction according to its frequency, but always in the plane defined by the input ray and the

grating normal (limited to two-dimensions for simplicity). The frequency components to be

traced are calculated from the Fourier transform of an input pulse with a specific temporal

profile, duration and central wavelength (Figure 3.4 frame 2). Phase contributions from the

stretcher and compressor are added to each frequency component by calculating the

accumulated optical path length of each ray as it travels from optic to optic (Figure 3.4 frames 3

and 4). Rays are traced by applying ray transfer equations and/or the grating equation at each

surface. Spectral clipping due to finite sized optics is simulated by aperturing the pulse

spectrum (Figure 3.4 frame 5). The output temporal profile of the pulse is calculated through

the inverse Fourier transform of the input pulse modified by the stretcher and compressor


83

(Figure 3.4 frame 6). Residual phase components are calculated by fitting a polynomial to the

calculated phase profile as a function of frequency.

Each different stretcher/compressor system has to be manually entered into the code,

but the properties of each optical component (such as position, radius of curvature, grating

constant, etc.), is input from a parameter file. This allows specific stretcher designs to be

optimised to match the compressor and different stretcher designs to be compared.

This code is limited to rays traced in a plane (two-dimensions) this is a drawback of

CPTrace. It is usually impossible to design a stretcher or compressor that only works in two-

dimensions. This is because they are normally double-passed in order to double the stretch

factor and remove spatial chirp. Therefore, the beam needs to be vertically displaced in order

that the output does not travel back down the input beam. This vertical displacement introduces

off-axis aberrations as optical surfaces can be curved in the plane perpendicular to the plane of

dispersion. To model this requires a three-dimensional ray-tracing code such a code has been

developed and is outlined in section 3.1.3. CPTrace makes it possible to design the basic

stretcher layout and compare the effects of limiting apertures on the recompressed pulse

duration.

3.1.3. CPTrace3D This program is the natural progression of CPTrace into three dimensions. Instead of

tracing a single beam that splits into many thousand rays of different frequency, it traces a

bundle of these beams. The bundle comprises a two-dimensional grid that represents a finite

size input beam. Each beam in this grid splits into many thousand rays of different frequency,

traced through the system in the same way as in CPTrace, but now taking into account the three-

dimensional nature of each surface (details in section 3.2.3). As well as the stretcher telescope

introducing geometrical aberrations, aberrations can also arise from imperfect optic surfaces.

This is modelled in CPTrace3D by pistoning the surfaces on a small scale, in other words small

sections of the surface are randomly shifted backwards and forwards along the direction of the

optical axis (details in section 3.2.4). This allows the calculation of acceptable surface figure,

which plays an important part in the cost of large optics. At the end of the system the beams are

recombined to produce an overall pulse profile at an imaginary beam focus (details in section

3.2.5).

Different systems to be traced are entirely encapsulated in an input file, so changing the

system configuration does not involve rewriting and recompiling the code. This makes the

model far more flexible than CPTrace allowing many more systems to be traced easily. Section

3.2 explains how each part of CPTrace3D works in detail.


84

3.2. Detailed model description This section describes the routines that are used in both CPTrace and CPTrace3D.

CPTrace uses all the routines that will be described except the surface imperfection and pulse

combining routines. In addition, the ray-tracing part of the code only works in two dimensions

in CPTrace. CPTrace3D follows the same flow diagram as that in Figure 3.4, but the spatial

(and therefore spectral) clipping is done on a surface-by-surface basis. As a result, when large

beam sizes are used, bandwidth clipping may be soft rather than hard. CPTrace simulates

clipping by aperturing the bandwidth with a hard edge causing unrealistic fast modulations in

the output pulse temporal profile. See section 2.2.1.3 for an explanation of the difference

between soft and hard clipping.

3.2.1. Pulse definition The input pulse is defined by seven parameters: two spatial, four temporal and one

spectral. Two definitions are used throughout this thesis to describe the pulse properties: the

word ray is defined as the path of light of a specific wavelength and the word beam is defined as

a bundle of rays all travelling in the same initial direction. In the pulse definition section of the

model, the input temporal pulse profile is sampled by a number of time steps. To calculate the

frequency components that should be traced, the pulse is Fourier transformed (see section

3.2.2). The centre of the calculated pulse spectrum is set by the input parameter CentWavl. The

number of frequency components traced is equal to the number of time steps in the pulse

definition and is the same as the number of rays.

NumBeams

FWH

MSp

aceB

eam

s

x

y

tI

I

FWHMTimeSteps

NumTimeSteps Figure 3.5 Pulse definition. I represents intensity, x and y are the beam coordinates and t is the time parameter. The time window is split into NumTimeSteps time steps with the full-width-

half-maximum pulse duration described by FWHMTimeSteps steps. The spatial array is defined by a grid of NumBeams × NumBeams, with the full-width-half-maximum beam

diameter described by FWHMSpaceBeams beams.


85

Finite sized input beams are described by defining a two-dimensional array of beams

(the overall beam profile is assumed to be Gaussian). The array is a square grid with side length

NumBeams, which means a total of NumBeams2 beams are traced (see Figure 3.5). The number

of beams contained within the full-width-half-maximum beam diameter is defined as

FWHMSpaceBeams. If FWHMSpaceBeams is set to equal NumBeams, then only beams

contained within the full-width-half-maximum beam diameter will be traced. The real world

full-width-half-maximum size of the beam is defined for each optical system that is traced (see

section 3.2.3).

Every beam in the spatial array contains NumTimeSteps elements (see Figure 3.5). The

number of steps contained within the full-width-half-maximum pulse duration is defined as

FWHMTimeSteps. The temporal duration of the pulse is defined as PulseDur. Consequently,

the time window (Twindow) in which the pulse is described is defined as:

psNumTimeSteepsFWHMTimeSt

PulseDurTwindow ×= . (3.8)

-40 -20 0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

Gaussian

Sech2

Duration

30 fs

Nor

mal

ised

Inte

nsity

Time (femtoseconds)

-90 -60 -30 0 30 60 90

1E-4

1E-3

0.01

0.1

1

Figure 3.6 Profiles of 30 femtosecond Gaussian (red) and sech2 (black) pulses. Inset graph

shows the same profiles on a log intensity scale.

The pulse temporal profile may take two shapes: Gaussian or sech2. Although the

model always uses the electric field (amplitude, A) of the pulse, the pulse duration, PulseDur, in

equations 3.9 and 3.10 is defined as the duration of the square of the electric field (intensity) of

the pulse. The equations for the pulse amplitude at time t are defined as:

( )PulseDur

aetA ta 2ln2,22

== − (3.9)


86

( ) ( )PulseDur

bee

tA btbt12ln2,2 +=

+= − (3.10)

for Gaussian and sech2 pulses respectively. The reason for defining PulseDur as the full-width-

half-maximum of the pulse intensity profile is that pulses in any real system are measured in

intensity rather than amplitude (for example with an autocorrelator see section 1.2.2.2). The

peak amplitude of each pulse at a given position in the input grid (NumBeams×NumBeams) is

modified by the intensity at that point, which follows the intensity profile of the input beam

(Figure 3.5). The pulse profile, Gaussian or sech2, is set in the model by the variable PulseType.

Figure 3.6 shows the profiles of Gaussian and sech2 pulses of 30 fs duration on a linear and log

intensity scale.

3.2.2. Fourier transforms

In order to calculate the frequency components that should be traced through the

system, the Fourier transform of the input pulse is taken. Each beam in the input array (Figure

3.5) contains an input pulse temporal profile whose amplitude is modified by the beam profile.

The Fourier transform of each of these pulse profiles gives the transform-limited spectral profile

of that pulse:

( ) ( ) ( )( )tAFTdtetAA ti == −∞

∞−∫ ωω (3.11)

where A(ω) is the amplitude spectrum, A(t) is the temporal pulse amplitude profile and ω is the

laser angular frequency (radians s-1). The inverse transform is given as follows:

( ) ( ) ( )( )∫∞

∞−

−== ωωωπ

ω AFTdeAtA ti 1

21

(3.12)

The functions A(ω) and A(t) are generally complex, with amplitude information

contained within the real, and phase information the imaginary part of each function. The input

pulse, defined as A(t), is discretely sampled and so the discrete Fourier transform must be used.

A(t) is described by a number of samples, defined in the model as NumRays samples and is

written as:

( ) ...,2,1,0,1,2...,, −−=∆= nnAAn (3.13)

where ∆ is the sampling interval defined in the model as ∆ = PulseDur/FWHMTimeSteps. In

frequency space, the frequency interval is then ∆ω = 2π/(NumRays×∆) and the frequency range

is given by -π/∆ ≤ ω ≤π/∆ (see Figure 3.7). The numerical Fourier transform is written as:

( ) ∑−

=

−≈1

0

/N

k

Niknkn eAA ω (3.14)


87

where N = NumRays, ωn = 2πn/(NumRays×∆) and n takes values from -NumRays/2 to

NumRays/2 [Press 92]. It would appear from this that there are NumRays+1 frequency

components output from the discrete transform. In fact, the two frequency components at the

extreme ends of the range, ±NumRays/2, are equal giving only NumRays frequency components

as expected. The inverse numerical transform is written as:

( ) ∑−

=

≈1

0

/

21 N

n

Niknnn eA

NtA

π (3.15)

If the input temporal pulse is to retain its integrity after being operated on by the Fourier

transform, the frequency window of the transformed pulse must be large enough to contain the

bandwidth of the pulse. That is, the value of the sampled Fourier transformed pulse, A(ωn),

must be zero at |ωn| = π/∆. If this condition is satisfied, the pulse is completely determined by

its samples, i.e. bandwidth limited [Press 92]. This condition is known as the sampling theorem

(see Figure 3.7c). If the sampling theorem condition is not met, then non-zero values of A(ωn)

that would normally lie outside the range |ωn| = π/∆ will be reflected back into this range. This

is known as aliasing and creates a calculated pulse spectrum that differs considerably from the

analytically transformed pulse case (see Figure 3.7d).

FT

A(ω)

ω

(d)

0-π/∆ π/∆

A(ω)

ω

(c)

0-π/∆ π/∆

A(t)

t

∆

(a)

0NumTimeSteps

A(ω)

ω

∆ ω(b)

0NumRays

Figure 3.7 The discrete Fourier transform sample points, represented by blue dots, are

illustrative only. (a) Shows the sampled input pulse. (b) Shows how this is transformed into frequency space. (c) Shows the non-aliased result where values of A(ωn) are zero at the

extremes of the sample set (where |ωn| = π/∆). (d) Shows the aliased result where values of A(ωn) outside the sample range (red) are flipped back into range creating a spectrum that is

elevated near the edges (blue). Note NumTimeSteps = NumRays.

To avoid aliasing, the time step size, ∆, and total number of time steps, NumTimeSteps,

must be chosen so that the frequency window is large enough to contain the whole pulse

spectrum. This often involves having a time window that is considerably larger than the


88

duration of the input pulse. The time around the pulse at the centre is padded out with zeros to

make up a continuous profile. The numerical Fourier transform assumes that the pulse to be

transformed repeats for all time, i.e. a train of pulses. In order that the phase profile of the

transformed pulse is flat across the centre frequency components, the pulse must be shifted in

time by half the number of time steps. Another way of think about this is that the pulse defined

in Figure 3.7a, has time equal to zero in the centre of the array. The transform requires that time

starts at zero at the first array element. The input for the transform needs to be a pulse train so

the first half of the pulse that is lost by shifting the time origin to the left, is wrapped around to

the end of the array (see Figure 3.8). The output of the transform also has zero frequency at the

beginning of the array, so in order to have a spectrum that is centred in the middle of the array, a

similar shift to that of Figure 3.8 must be performed. The spectrum is shifted to the correct

centre frequency by adding ω0 = 2πc/CentWavl (the centre frequency of the input pulse) to ω.

0 0max/min+ve +ve-ve -ve

Array LengthRepeat to ∞ Repeat to ∞

Figure 3.8 Rearrangement of the pulse in time and frequency in order that phase is zero across the centre of the pulse. This reformatting also places zero time and wavelength in the correct

positions.

The numerical transform used is similar to that described in Chapter 12 of [Press 92]

(but with a different sign convention in the exponent) and is called the Fast Fourier Transform

or FFT. This routine reduces the computational time by requiring only N(lnN) steps rather than

N2 steps as with conventional numerical techniques (N is the array length). The transform used

in the model is taken from an IMSL mathematical functions library [IMSL]. This routine is

most efficient when the array length is a product of small primes - this limits the choice of

suitable array sizes and always means that the array size is odd.

The output of this part of the code is a transform of Figure 3.5 into an array of beams

each containing a number of frequency components, or rays. Figure 3.9 shows how the input

pulses of Figure 3.5 have been transformed.


89

NumBeams

FWH

MSp

aceB

eam

s

x

y

ωI

A

NumRays

ω0

Figure 3.9 Definition of transformed pulse. Each beam now contains NumRays frequency

components. The amplitude of each spectrum is modified by the amplitude of the beam at each position in the beam profile array.


90

3.2.3. Three-dimensional ray-tracing

The ray-tracing part of the model can be split into three sections. The first is the

transfer of rays from surface to surface using geometry and trigonometry, extended to three

dimensions. The second is the calculation of the direction a ray will travel when it encounters a

discontinuity in refractive index, using Snells law in three dimensions. The final section is the

calculation of the direction of each ray as it diffracts from a diffraction grating. Each section is

discussed in detail below.

3.2.3.1. Ray transfer equations The optical system to be traced is split up into several surface sections. Each section is

described by up to twelve parameters, the distance from the current position to the next surface

(D, metres), the surface type and ten other parameters summarised in Table 3.1. The incident

ray is described by six parameters: the ray starting position (X,Y,Z), the ray direction angles in

the x- and y-planes (Θx and Θy) and the ray wavelength (λ, metres). The full-width-half-

maximum diameter of the input beam is additionally set by the parameter BeamSize. BeamSize

is divided into FWHMSpaceBeams samples.

Surface type Spherical Plane Reflection Grating

1 x-tilt, θx x-tilt, θx x-tilt, θx 2 y-tilt, θy y-tilt, θy y-tilt, θy 3 optical medium 1 optical medium 1 optical medium 1 4 optical medium 2 optical medium 2 - 5 x-shift, xs x-shift, xs ruling density, d 6 y-shift, ys y-shift, ys wavelength 7 Radius of Curvature, R - diffraction order, n 8 flatness, λ/P8 flatness, λ/P8 ruling direction 9 x-surface width, Wx x-surface width, Wx x-surface width, Wx

Para

met

er

10 y-surface width, Wy y-surface width, Wy y-surface width, Wy

Table 3.1 Surface definition parameters for the three different types of surface: Spherical, Plane and Reflecting diffraction Grating. P8 represents the value of parameter number 8.

The ray starting position parameters and surface parameters for each section are

summarised in Figure 3.10. The ray starting position is defined as a displacement from the

position of the first optical component due to the components shape or tilt. The units of this

displacement (X,Y,Z) is metres. The ray direction is defined by the two angles it subtends in the

x-z and y-z planes. The unit of these angles (Θx and Θy) is degrees. The equation of the

incident ray is described by the intersection of two planes:

( ) ( ) 0tan =−−Θ− XxZz x (3.16)


91

( ) ( ) 0tan =−−Θ− YyZz y (3.17)

The parameter D (metres) defines the distance between the centre of the surface the ray

emerges from, to the centre of the surface it is travelling to (see Figure 3.10). The centre of the

second surface can be shifted in the x- and y-directions using parameters 5 and 6 (xs and ys,

metres). This allows spherical surfaces to be used off centre and surface positions to be

toleranced (important for testing the usability of various designs). The surface can be tilted

about its centre by defining a tilt angle in the x- and y-directions using parameters 1 and 2 (θx

and θy, degrees). This is useful for setting the input angles for diffraction gratings and the

angles of mirrors for retroreflecting beams. Parameters 3 and 4 set the type of optical medium

before and after the second surface respectively. A lens may be defined by creating two curved

surfaces close together with a glass material between them. The optical medium setting is used

in the calculation of Snells law (see section 3.2.3.2)

Zz

y

Y

ys

Θy θy

D

Wy

SurfaceCentre

z

y

x

Plane ofDispersion Intercept

(zi,yi)

PreviousSurfaceCentre

Figure 3.10 Ray and surface parameter conventions used by the ray transfer equations. The ray and surface are only shown in the z-y plane. The origin of this plane represents the centre of the surface the ray is emerging from. Inset shows axis convention and plane of dispersion.

At this point, it is possible to define the equation for plane surfaces and gratings.

Gratings are just defined as plane surfaces with diffractive properties and a phase correction to

compensate for the uneven surface profile (see section 3.2.3.3). The equation of a plane surface

is given in the model as:

( ) ( ) ( ) 0tantan =−−−+− Dzyyxx ysxs θθ (3.18)


92

Equations 3.16, 3.17 and 3.18 form simultaneous equations, the solution of which gives the

intercept of an incident ray with a plane surface (xi,yi,zi, metres). The ray-surface intercept is

given by:

( ) xi ZzXx Θ−+= tan (3.19)

( ) yi ZzYy Θ−+= tan (3.20)

( ) ( ) ( )

1tantantantantantantantantantan

−Θ+Θ−−−−−Θ+Θ

=yyxx

ysxsyyxxi

DyYxXZz

θθθθθθ

(3.21)

The length of the ray can now be calculated using the start and end coordinates of the

ray and Pythagoras theorem in three dimensions. The optical length of the ray is calculated by

multiplying the length of the ray by the refractive index of the medium the ray is propagating in.

The phase the ray of wavelength λ has accumulated is then given by:

( ) ( ) ( ) ( )λ

πωφ

22212 iii zZyYxXn −+−+−

= (3.22)

where n1 is the refractive index of the medium the ray propagates in.

The flatness of spherical and plane surfaces can be set with parameter 8 (P8, see Table

3.1). The parameter becomes the denominator in the fraction that defines the surface figure, in

other words λ/P8 (for example a standard optical surface figure might be λ/10). Parameter 8

must be an integer, the definition of surface figure is discussed in more detail in section 3.2.4.

The surface size can be set by defining widths in the x- and y-directions using

parameters 9 and 10 (Wx and Wy, metres). When a ray intercepts a surface outside the

dimensions specified by Wx and Wy, the ray amplitude is set to zero and it is no longer traced.

Setting the size of the optic in only two directions means that the surface can only be

rectangular. Other shapes can be simulated by setting a different width for different encounters

of the surface. This involves having to know exactly where the rays fall each time the surface is

encountered. For this purpose, and to calculate the used area of a surface, a file can be saved

which logs every point that a ray hits a surface.

Spherical surfaces are described in exactly the same way as plane surfaces with the

addition of an extra parameter radius of curvature (R, metres). The sign of R is positive when

the centre of curvature is to the positive z side of the surface. The equation of a spherical

surface is given in the model as:

( ) ( ) ( ) ( )( ) 0tantan 2222 =−−−−−−−+−+− RyyxxDRzyyxx ysxsss θθ (3.23)


93

Equations 3.16, 3.17 and 3.23 form simultaneous equations, the solution of which gives

the intercept of the incident ray with the spherical surface (xi,yi,zi, metres). The ray length is

calculated in the same way as for the plane surface case using equations 3.19, 3.20 and 3.22.

The result for zi is not given here due to its size and complexity, but is used in the code every

time a ray intercepts a spherical surface.

3.2.3.2. Snell’s law Snells law is used to calculate refractive and reflective effects on rays in the code. It is

simply defined by an equation and the statement that the input and output rays and the surface

normal lie in a plane [Newton 79]. The equation is written as n1sinθ1 = n2sinθ2, where n is the

refractive index, θ is the angle between the ray and the surface normal and the subscripts denote

the incident (1) and refracted (2) rays. This can be extended into three dimensions using the

following equation:

( ) ( )21221121 SnnSnn ×=× (3.24)

where n12 is the surface normal unit vector and S is the ray unit vector for incident (1) and

refracted (2) rays. This cross-product relation can be separated into three equations for the x-,

y- and z-directions as follows:

( ) ( )yzzyyyzy SnSnniSnSnni 11211212122122 −=− (3.25)

( ) ( )zxxzzxxz SnSnnjSnSnnj 11211212122122 −=− (3.26)

( ) ( )xyyxxyyx SnSnnkSnSnnk 11211212122122 −=− (3.27)

where i, j and k are the unit vectors representing the x-, y- and z-directions and the superscripts

x, y and z represent the three orthogonal components of the vectors n12, S1 and S2.

Snells law, in this form, is used in the model to calculate refracted ray angles. It is also

used to calculate rays in reflection by setting n2 equal to n1. When imperfect surfaces are used,

the surface normal vector at every position on a surface is modified. This is explained in more

detail in section 3.2.4.

When an optical medium is specified in the model, the parameter medium is set to, for

example, bk7 for BK 7 glass. The refractive index as a function of wavelength for the

material is then calculated using, for example, the Sellmeier equation:

∑ −+=

j j

jAn 2

02

22 1

λλλ

(3.28)


94

where n is the refractive index of the medium, λ (microns) is the wavelength at which n is

required Aj are constants and λ0j2 are the vacuum wavelengths associated with a natural

frequency ν0j, such that λ0jν0j = c, the speed of light [Hecht 87]. For example, to calculate the

refractive index of sapphire, the following coefficients and wavelengths are used [Malitson 67]:

3616.3210122544.000377588.0280792.5058264.1023798.1

203

202

201

3

2

1

======

λλλAAA

(3.29)

accurate to ±3×10-6 between 400 nm and 750 nm, and ±5×10-6 between 750 nm and 1014 nm

[Melles Griot]

3.2.3.3. Diffractive optics and the grating equation The grating equation in two dimensions has already been described in section 3.1.1,

equation 3.2. In the plane orthogonal to the plane of dispersion, the rays obey Snells law. In

reality, this may not be strictly accurate, but in the systems considered here, diffraction gratings

are aligned perpendicular to the plane of dispersion. Any deviations from this case, for example

angling beams so that input and output paths do not overlap, are small and considered to be an

insignificant perturbation to the grating equation. Rigorous treatment of rays incident on

diffraction gratings at angles outside the plane of dispersion, requires the solution of Maxwells

equations at the grating surface [Jarem 99] this is beyond the scope of this thesis work.

The aim of the model is to calculate the frequency dependent phase of a pulse as it

propagates through a stretcher and compressor. When a ray encounters a mirror, the surface is

flat (except for imposed surface imperfections) and so measurements of ray lengths may be

made directly to the surface. Grating surfaces, however, are not flat and so compensation must

be made when calculating phase. This problem has been treated in a paper by Treacy [Treacy

69] and amounts to an addition of 2π phase for every additional groove along the grating from a

fixed reference point (see Figure 3.11).

In the model the distance that the ray is from an arbitrary reference point is calculated

and divided by the grating pitch (d-1, the distance between adjacent grooves) in order to

calculate the number of multiples of 2π to add. The phase addition due to the grating is

therefore given as:


95

Fdadd πφ 2= (3.30)

where F is the distance along the grating from the reference point measured positive according

to x in Figure 3.11 (metres) and d is the ruling density (lines/metre) see also figure 1.9.

φφ+2π

φ+5(2π)

φ/2π1 2 30 4 5 6

x

Phase Addition

ab

c

Reference Point

Figure 3.11 Phase addition due to grating surface structure. The left hand diagram represents the grating surface, the right hand diagram is a graph of phase addition. Ray (a) starts with phase φ. Ray (b) is one groove away and so the phase at this point is the original phase plus

2π. Ray (c) is 5 grooves away and so the phase addition at this point is 5 times 2π.

Another way of thinking about this phase addition is via Figure 3.12. AA and CC

represent surfaces of constant phase. The problem with using path length to calculate phase is

that the path ABC is longer than ABC. The phase addition of equation 3.30 must be added

along the grating surface BB in order to correct for this.

A

B

A

B

C

C

Figure 3.12 Calculation of ray phase upon diffraction through an ideal grating pair

compressor. AA and CC are both wavefronts, but path ABC is longer than ABC. Adapted from figure 3 of [Treacy 69]. Phase correction of the type given by equation 3.30 must be

added to compensate for this.


96

3.2.4. Surface imperfections

When trying to design a stretcher and compressor system that are matched up to fifth

order phase, the quality of optical surfaces becomes an important consideration. For example, if

an optical surface is considered to be flat to λ/10, then two sections of the surface that lie at the

extremes of this specification may differ in phase by ~0.5 radians (calculated for a laser

wavelength of 800 nm and a test wavelength of 633 nm). This is important when trying to

reduce residual phase at the end of the system to less than ±π as some optical surfaces are

passed many times. To quantify the surface quality allowable for a given system, a section of

the code was added which models imperfect surfaces.

Departures from a perfect flat optical surface fall into two groups: a low spatial

frequency undulation and a high frequency ripple (see Figure 3.13). The low frequency

imperfection is described by the peak-to-valley surface specification. The high frequency

imperfection is described by the RMS surface specification. The difference in magnitude

between these two types of surface irregularity is that the low frequency imperfections tend to

be roughly six times that of the high frequency imperfections [Optical Surfaces]. In other

words, of two surfaces with the same surface specification (e.g. λ/10), the RMS specified

surface would be flatter than the peak-to-valley specified surface.

In order to make a realistic estimate of how this should be implemented, a paper by

Antonetti et al at ENSTA [Antonetti 97] was used. In this paper, pulses from a 30 fs oscillator

are stretched and recompressed and an autocorrelation of the output pulse is made. On the third

page of the article, the autocorrelation profile is plotted while the flatness of the large concave

stretcher mirror is changed. The model was adjusted to match this data as closely as possible

(the results of this matching are presented in more detail in section 4.1.4).

-0.1

0

0.1

-0.1

0

0.1

-2

-1

0

1

2

-0.1

0

0.1

2

4

6

8

10

2

4

6

8

10

-1.0-0.50.00.51.0

Figure 3.13 Slow (left diagram) and fast (right diagram) variations added to surfaces to simulate surface imperfections. Slow and fast variations added in amplitude ratio ~6:1.

The surface of an optic in the model is assumed to vary from the ideal in a continuous

fashion. The fast and slow variations of the surface are represented in Figure 3.13. Surface


97

irregularities are modelled using a technique called pistoning [Maxwell]. In this technique,

the surface is split into many squares and each square is moved (pistoned) backwards or

forwards along the optical axis according to how it is perturbed from the ideal. Figure 3.14

illustrates this with a spherical concave surface that has a fast spatial frequency perturbation.

Each surface is only described by a discrete number of points so, in order to make the

simulation more realistic, points are interpolated between the pistoned points to give a more

continuous surface. This causes small portions of the surface to have an additional tilt that

steers rays away from their normally reflected or refracted path.

R R

SurfaceIrregularities

(RMS)

PistoningSurface

Simulation

RMSDisplacement

λ/a

λ/aIdeal

Sphere

Figure 3.14 Surface pistoning. Left hand diagram represents ideal spherical concave surface (black) with high spatial frequency irregularities (red). Right hand diagram shows how these

irregularities are simulated by pistoning.

3.2.5. Pulse combination and output methodology After light paths through the stretcher and compressor have been traced, each test beam

contains a number of rays of different wavelength. The amplitude of each of these rays will

either have the value it started with, or zero if it fell outside the physical bounds of any of the

optics in the system. The phase of each ray will have been modified by the path it took through

the stretcher and compressor. In order to transform this information back into the temporal

domain, the inverse Fourier transform is performed (see section 3.2.2). After this operation

each beam has a temporal pulse profile associated with it. In order to combine these pulses to

produce a single pulse profile (as would be seen on an experiment), the set of beams is focused

(numerically) using an achromatic, aspheric lens. This has the effect of bringing all the beams


98

to the same point in space and time. In the model this just means all the temporal profiles are

added with a weight according to their position in the beam (see Figure 3.15). The beam is

assumed to have a Gaussian spatial profile so individual beams in the centre are weighted more

than those at the edge.

Achromatic,Aspheric

Lens

OutputPulse

Profile

BeamAmplitude

Profile

Beam/PulseInfo

- 20

- 10

0

10

20

-20

-10

0

10

20

Figure 3.15 Each beam traced through the stretcher and compressor is brought to the focus of a

numerical achromatic, aspheric lens. This has the effect of bringing each beam to the same point in space and time to produce a single output pulse temporal profile.

The output of the code is a single pulse temporal profile and associated temporal phase.

As well as combining all the beams into one pulse profile, each individual beams pulse profile

can be saved to see how output pulse temporal profile changes across the beam. The code will

also calculate the pulse duration and contrast of each profile. The duration is found by

measuring the width of the largest pulse at the half maximum position (full-width-half-

maximum duration). The contrast is found by calculating the ratio of the centre peak intensity

(which is always normalised) to the intensity of the next highest peak in the whole time

window. These two values can be saved in a grid representing each of the beam positions. This

is useful when calculating how large beams can be before the overall pulse profile is degraded.

Generally, stretchers perform less well for beams away from the optical axis; and as a result off-

axis recompressed pulse durations can be intolerably long for large diameter beams.

As well as saving a temporal intensity profile for each beam, the temporal phase profile

can also be saved. This is useful when looking at how accumulated phase changes across the

beam, at a given wavelength. It also helps when deciding how large beam diameters should be,

or can be.

3.2.6. Code limitations

There are a few points to remember when using the code to model stretcher and

compressor designs. All variables within the code are defined to double precision, which

imposes no practical limit to the accuracy of the calculations. Errors associated with the

numerical Fourier transform can limit how many orders of magnitude are available to describe


99

pulse profile. These limitations, however, are at the ~10-20 level and so impose no real

problems. There is a problem though when defining the pulse temporal window, as mentioned

in section 3.2.2. If the time window is not large enough to encompass the bandwidth of the

pulse, calculated through the Fourier transform, the problem of aliasing can arise. This, as

described in section 3.2.2, is caused by the folding of frequency components that would

normally lie outside the frequency window, back into the frequency window, creating an

artificial Fourier transform. To overcome this problem, the temporal pulse profile, even though

short, needs to be padded-out with zeros to create a large time window.

Nowhere in the code are the effects of diffraction taken into account. The assumption is

adequate as long as large beam diameters are used. The criterion to follow is that the Rayleigh

range of a beam should be longer than the total path of the beam through each optical system

traced. The Rayleigh range is given by:

0

2

λπwzR = (3.31)

where w2 is the 1/e2 beam radius and λ0 is the laser wavelength [Silfvast 96]. For example, a

1 mm full-width-half-maximum diameter beam of wavelength 800 nm has a Rayleigh range of

2.8 m. The Rayleigh range is defined as the distance over which the beam increases in size by a

factor √2. As long as beams are kept relatively large, the fact that diffraction is not included in

the code should not be a problem.

4. Modelling results

This chapter presents results obtained from the codes used to investigate a range of

stretcher and compressor designs. There are many things the code can tell about how a specific

system might perform which are not mentioned here. They do however play an important role

in the fine-tuning of a particular system. These include: the calculation of spectral phase and

temporal profile of each beam across a finite beam profile to measure the effects of finite beam

size; tolerancing of surfaces to gauge whether a system is feasible to align; and calculation of

the temporal output pulse profile as a function of compressor grating separation to measure the

sensitivity to adjustment of the compressor.

The results shown fall into three categories: verification that the code models its major

components in the correct fashion; a summary of the systems considered prior to the final

design; and a specification and results of the final design. The design presented here will be

built and tested in the next chapter: System implementation and characterisation.

Some of the results of this work have been published in [Mason 98a] and [Mason 98b].

As well as being used to design a stretcher for use in the Blackett Laboratory Laser Consortium

titanium-doped sapphire chirped pulse amplification laser system, the codes have been used to

help design stretchers for the ASTRA ([Langley 99a] and [Langley 99b]) and Vulcan [Danson

99] lasers at the Rutherford Appleton Laboratory (see section 4.4).

4.1. Code verification results This section aims to verify the working of the code by setting simple tasks for which the

outcome can be calculated from first principles, using well-known analytical results. Four areas

of the code are tested: the temporal characteristics, the implementation of material and

diffractive dispersion, the transfer of rays from one surface to another using Snells law, and the

modelling of surface imperfections.

4.1.1. Dispersionless runs This section tests the capabilities of the code in terms of pulse definition and the Fourier

transform. Pulses of a relevant duration and either Gaussian or sech2 temporal profile are

defined. The pulses are Fourier transformed into frequency space to make sure they fit within

the frequency window. When the pulse spectra are inverse-transformed, in the absence of

material or added dispersion, they should closely resemble (within the limits of numerical

Modelling results

101

errors) the original pulse profiles. Deviations from this represent numerical limitations of the

code.

-2000 -1000 0 1000 2000

1E-43

1E-37

1E-31

1E-25

1E-19

1E-13

1E-7

0.1

OutputInput

Nor

mal

ised

inte

nsity

Time (femtoseconds)

600 800 1000 1200 1400

1E-21

1E-18

1E-15

1E-12

1E-9

1E-6

1E-3

1


Norm

alised amplitude

Figure 4.1 Spectral and temporal outputs of the code in the absence of material dispersion on a

logarithmic scale. The input pulse in this case was a 30 fs Gaussian pulse. The red curve shows the pulse spectrum, the blue curve shows the input temporal profile and the green curve

shows the, laterally displaced, output temporal profile. Dotted line represents the detection limit of laboratory pulse profile and spectrum measurement devices.

Figure 4.1 shows the result of such a dispersionless run with the code. The blue curve

shows the Gaussian intensity profile of the 30 fs input pulse. The red curve shows the profile of

the spectrum of the pulse calculated using the Fourier transform. The extent of the wavelength

scale (top axis) represents the wavelength window imposed by the code. The green curve

shows the (laterally shifted for clarity) output pulse profile. As can be seen from these plots

there is no practical dynamic range limit imposed by the code. The extent of the time scale

(bottom axis) represents the temporal window of the input pulse. In order to avoid internal

underflow errors the code limits amplitude dynamic range to twenty-five orders of magnitude.

This limits the intensity profile to fifty orders of magnitude (amplitude squared). It was thought

that given the best laboratory pulse profile measurement device is limited to only about eight

orders of magnitude, that this level of accuracy would be sufficient.

Modelling results

102

4.1.2. Dispersion only calculations

The code handles two types of dispersion to model chirped pulse amplification. The

first, material dispersion, enables calculations to be made of how pulses are stretched as they

propagate through different optical media. The second, diffractive dispersion, calculates how

rays are deviated according to their wavelength by diffraction gratings.

Material dispersion calculations have been checked by passing a short (30 fs) pulse

through a fixed amount of optical material (1 cm of BK7). The accumulated phase as a function

of wavelength calculated by the code is compared with the same calculation done manually for

selected wavelengths across the bandwidth of the pulse. The equation used for the manual

calculation of phase (φ) as a function of wavelength (λ, m) for a given length (L, m) of

dispersive material is as follows:

( ) ( )λλπλφ Ln2= (4.1)

where n(λ) is the refractive index of the dispersive medium at wavelength λ calculated using the

Sellemeier equation for BK7 (see section 3.2.3.2, equation 3.26).

Figure 4.2 shows the comparison of the manual calculation (black circles on red line)

for six discrete points in the frequency window and the calculation using the code (red line).

Both calculations agree across the whole frequency window to eleven decimal places (the limit

of accuracy of the calculator used for the manual calculations).

700 750 800 850 900 950

1.0x106

1.1x106

1.2x106

1.3x106

1.4x106

1.5x106

Accu

mul

ated

pha

se (r

adia

ns)


-0.15

-0.10

-0.05

0.00

0.05

0.10

Intercept position (metres)

Figure 4.2 Comparison of calculations using the code (solid lines) and performed manually

(black circles) testing material (red) and diffractive (blue) dispersion.

Modelling results

103

Diffractive dispersion calculations have been checked by diffracting a short (30 fs)

pulse from a 1200 lines/mm grating. The grating is arranged such that the centre wavelength

ray (800 nm) strikes the centre of a 20 cm screen placed 1 m away (grating input angle 33.8°

and centre wavelength diffracted angle 23.8°). The position that rays strike the screen in the

dispersion plane is compared with manual calculations as a function of wavelength. Figure 4.2

shows the result of this test. The manual calculations are shown by black circles on the blue

line and the blue line shows the result of calculations with the code. Again, both agree to within

the limitations of the manual calculation mentioned above.

4.1.3. Ray transfer tests As described in section 3.2.3.1, the ray transfer equations are used to calculate the

trajectories of rays from surface to surface. There are two important aspects of ray-tracing that

should be tested: ray-surface intercepts and Snells law (limited to the calculation of angles of

reflection for this test). To test these, a point source of rays is retro-reflected using a spherical

mirror. A convenient way of producing a point source in the code is by diffracting a short pulse

from a diffraction grating (there is no provision for defining a point source specifically). A

concave spherical mirror of radius of curvature 1 m is then placed 1 m away from the grating so

that the centre wavelength ray (800 nm) strikes its centre (Figure 4.3). A screen is placed at the

position of the grating to mark where the rays are imaged, perpendicular to the optical axis

(drawn from the centre of the diffraction grating to the centre of the concave mirror). This tests

the use of Snells law at the reflection off the spherical mirror. In order to test the off-axis

performance of the code a nine-by-nine array of beams is traced with an extent of 1 cm in the

dispersive and orthogonal planes.

Figure 4.4 shows the positions the rays strike the screen looking towards the concave

mirror. The colour index (displayed on the right hand side of the graph) represents rays of

different wavelength that originate from the grating. Longer wavelengths are diffracted in the

negative direction with respect to the horizontal axis of the graph. The centre circle at (0.0,0.0)

is the image of the point source located at the centre of the diffraction grating (point A in Figure

4.3). Due to the order that the ray images are drawn, this circle appears blue, but in fact, all

colour circles are superimposed. This shows that the ray transfer equations and calculations of

Snells law for reflection work.

Modelling results

104

A

B

SG

M

Figure 4.3 Schematic showing layout in the z-x plane for the ray transfer test. M = 1 m radius of curvature mirror, G = diffraction grating and S = screen. Grey lines represent input rays.

-0.4 -0.2 0.0 0.2 0.4

-0.4

-0.2

0.0

0.2

0.4 nm

Orth

ogon

al d

ispl

acem

ent (

cent

imet

res)

Displacement in dispersion direction (centimetres)

720.0

752.0

784.0

816.0

848.0

880.0

Figure 4.4 Image of a nine-by-nine grid of beams diffracted from a diffraction grating via a 1 m radius of curvature concave mirror placed 1 m away from the grating onto a screen placed at

the centre of the grating.

Rays traced in the plane of dispersion exhibit defocusing. The source of the ray bundles

is a diffraction grating tilted at an angle of 33º to the x-y plane in the x plane. The only ray

bundle that is imaged to itself is the one emitted from point A in Figure 4.3. All other ray

bundles are emitted from behind or in front of the screen S. Figure 4.3 shows how a ray bundle

emitted from a point behind the screen is focused in front of the screen and therefore the image

Modelling results

105

is spread laterally (middle right image in Figure 4.4). The ray bundle emitted from the opposite

side of the grating is imaged behind the screen and so the image is spread in the opposite

direction (middle left image in Figure 4.4).

Rays traced in the plane orthogonal to the dispersive plane suffer spherical aberration

(vertical columns of circles). The aberration manifests itself as a slight vertical smearing of

the image which gets worse the further the rays are from the plane of dispersion. The effect is

most noticeable on the rays traced at the extremes of the dispersive plane (top right, bottom

right, top left and bottom left ray bundles).

4.1.4. Surface imperfections This section details the performance of the surface imperfection part of the code

described in section 3.2.4. The two ways in which surfaces are perturbed control the

recompressed pulse in different ways. The first surface perturbation comprises a low spatial

frequency variation across the optic (with a period of order of 20 cm). In order that a surface is

perturbed in the same way every time a ray strikes the same point, a map is made of the detailed

surface shape. To save computer memory, the map has dimensions 20 cm by 20 cm and is

folded over to form a 40 cm square (optic size limit of the code). The low spatial frequency

surface variations must therefore be described by an even function so as not to produce

discontinuities along the fold lines. The effect of these perturbations is to broaden the

recompressed pulse and they have little effect on the pulse contrast.

The high spatial frequency (period of order of 100µm) perturbations are superimposed

on this map to simulate small-scale surface roughness. The perturbations are calculated by

adding a random shift to the surface (pistoning, section 3.2.4) proportional to the surface figure

description, λ/n (where n is an integer). The overall effect of these perturbations is to add a

pedestal in the wings of the stretched and recompressed pulse, and has little effect on the

recompressed pulse duration.

The exact choice of fast and slow spatial frequencies and the ratio of their relative

magnitudes have been chosen to attempt to represent the effect of realistically imperfect optics.

In a paper by Antonetti et al [Antonetti 97], figure 3 shows a graph of recompressed pulse

profile (obtained through second order autocorrelation) for different surface figures of the large

optic in the stretcher described in their paper. This graph is reproduced in Figure 4.5 (left hand

graph). Although this graph cannot possibly be reproduced exactly (as the exact small-scale

shape of the optic is unknown), it has been matched relatively closely given the limitations of

the imperfect surface description of the code. The right hand graph of Figure 4.5 shows the

result of the code for three surface figures of the large optic in the Antonetti et al stretcher

design.

Modelling results

106

The aim was to get an overall match between the code and the paper results in terms of

pulse duration and contrast. The paper result for the λ/2 optic shows considerable pulse

broadening and wings at the 4×10-2 level. The code result shows a little additional broadening

compared to measurement, but the wings are about the right level. The paper result for λ/10

optic shows also considerable broadening and wings at the 3×10-3 level. The code result shows

an underestimate of the broadening and level of the wings. Both the paper and code results

show good agreement for the λ/40 optic, the profile from the paper seems to be limited by the

autocorrelation measurement. Although not a perfect match, the surface imperfection section of

the code provides a reasonable measure by which the effect of imperfect optics can be

estimated.

-200 -100 0 100 200

1E-4

1E-3

0.01

0.1

1

λλλλ /40

λλλλ /10

λλλλ /2

Norm

alis

ed in

tens

ity (a

rbitr

ary

units

)

Time (femtoseconds) Figure 4.5 Comparison of calculated recompressed pulse profiles of a 30 fs pulse stretcher and

those measured by Cheriaux et al [Cheriaux 96]. The left hand graph shows the results presented in the paper by Antonetti et al and the right hand graph shows the results of the

chirped pulse amplification model for three different surface figures of the large stretcher optic.

4.2. Systems modelled in detail Many different systems were considered when deciding the design of the stretcher for

the upgrade of the Blackett laboratory laser consortium titanium doped sapphire chirped pulse

amplification laser system. Amongst these are a system designed by Lemoff et al [Lemoff 93]

that uses cylindrical mirrors to stretch 20 fs pulses to 300 ps (similar to a design by Du et al [Du

95], Figure 4.6a). The advantage of this design is that it has aberration compensation that

delivers 25 fs recompressed pulses after amplification to 125 mJ. Disadvantages include the use

of (expensive) cylindrical mirrors, the use of two diffraction gratings that increases cost and that

the whole system is reportedly very sensitive to alignment. Another system considered was

designed by Zhou et al [Zhou 95] consisting of a folded all-reflective version of the original

Modelling results

107

Martinez [Martinez 87a] design (Figure 4.6b). This system stretches 10 fs pulses to around

45 ps using a single spherical concave mirror and a low dispersion (600 lines/mm) grating. An

advantage of this design is its simplicity, the major disadvantage is the limited stretch factor it

offers. Finally (for the purposes of this discussion) a system designed by Itatani et al [Itatani

97] that uses a concave and convex mirror pair with two diffraction gratings, stretching 21 fs

pulses to 330 ps (Figure 4.6c). Advantages include the large stretch factor and aberration

compensation offered by the mirror pair. Disadvantages include difficulty of alignment

associated with having two diffraction gratings. There are many other designs not discussed

here that were also considered. The ones mentioned here represent a sample significant because

of their substantially different designs.

ac

b

Figure 4.6 Stretcher schematics designed by: (a) Lemoff et al; (b) Zhou et al and (c) Itatani et al. Orange lines represent diffraction gratings, cyan lines represent mirrors and grey lines

represent input/output paths.

Described in the following sections are two systems that show the extremes of

complexity of stretcher design. The Livermore design is a very well folded, reflective version

of the standard Martinez design. The Öffner triplet based design is aberration free in the zero

stretch position and consequently has low overall aberrations when used to stretch a pulse by a

factor of 10,000.

4.2.1. Livermore system

4.2.1.1. Livermore system specification

Figure 4.7d shows a schematic of the Livermore stretcher designed by P.S.Banks

[Banks 97]. The design is based around a well folded version of the standard two lens, unit

magnification telescope stretcher of Martinez [Martinez 87a]. The key to this compact stretcher

Modelling results

108

design is the diffraction grating which has a mirror strip across its centre. Figure 4.7 shows how

the stretcher is folded from the familiar Martinez lens design.

The first fold is made by replacing the lenses with concave mirrors (a→b) this step is

essential to remove the chromatic aberration associated with refractive optics. The second fold

uses the mirror strip across the centre of the diffraction grating (b→c). Input rays strike the

grating above the central mirror strip and are diffracted towards the concave mirror. The rays

are then reflected down towards the centre of the grating hitting the mirror strip. The next step

folds the whole system in two (c→d). The rays from the grating mirror now strike the folding

mirror and back onto the grating mirror, back to the bottom of the concave mirror and onto the

diffraction grating again. At this point the beam is spatially dispersed and so is retroreflected

back through the system removing the dispersion and doubling the stretch factor. This design

stretches 30 fs pulses to approximately 600 ps, a stretch factor of ~20,000.

b

a

c

d

Figure 4.7 Schematics of the Livermore stretcher as it is folded a→b→c→d from the standard Martinez configuration (a). Orange lines represent diffraction gratings, cyan lines represent

mirrors and grey lines represent input/output paths.

The concave mirror has a radius of curvature of 226 cm (focal length = 113 cm). The

diffraction gratings are produced in-house and have a ruling density is 1480 lines/mm. This

Modelling results

109

system offers a compact design and large stretch factor at the expense of requiring a highly

customised diffraction grating.

4.2.1.2. Livermore system performance Although very well folded, the Livermore stretcher design represents only a first

iteration away from chromatically aberrated refractive optics to reflective optics. As such, there

is no attempt at aberration compensation in the stretcher design and consequently, there is a

mismatch between this and its corresponding compressor. Figure 4.8 shows the calculated

profile of a 30 fs pulse stretched to 600 ps in a Livermore-style stretcher and recompressed with

a grating pair.

-300 -200 -100 0 100 200 3001E-4

1E-3

0.01

0.1

1

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds) Figure 4.8 Calculated profile of a pulse stretched using a Livermore-style stretcher. Blue curve

shows 30 fs input pulse and red curve shows stretched and recompressed pulse profile.

Above the 3×10-2 level, the recompressed pulse represents well the input pulse being

only slightly longer at 36.8 fs. Below this, the profile has a rather unsatisfactory shape with

extensive structure. This is due to high order phase differences between the stretcher and

compressor. Table 4.1 details the differences between second, third, fourth and fifth order

spectral phase contributions of the stretcher and compressor.

Modelling results

110

2nd order (fs2) 3rd order (fs3) 4th order (fs4) 5th order (fs5)

Stretcher 2.616×107 -5.515×107 1.457×108 -4.067×108

Compressor -2.616×107 5.508×107 -1.489×108 4.346×108

Residual 6.000×103 -7.721×104 -3.190×106 3.677×107

Table 4.1 Differences between second, third, fourth and fifth order phase contributions of the Livermore stretcher and corresponding compressor. Calculated using the code.

The residual second order phase is relatively small and is the reason for the small

increase in overall pulse duration. In addition, the relatively small third order residual phase

results in a slight asymmetry in the profile and makes a small contribution to the ringing

nature of the wings of the output pulse. The large fourth order residual phase contribution

results in the large peaks either side of the main peak and the fifth order residual phase gives the

main contribution to the ringing in the wings.

This stretcher design is suitable for longer pulses (i.e. >~80 fs), but the high order

residual phase makes it unsuitable for very short pulses without the addition of external phase

compensation.

4.2.2. Öffner triplet system

4.2.2.1. Öffner triplet system specification

This stretcher is designed around an imaging system based on the Öffner triplet [Öffner

71]. The concave and convex mirrors are placed so that their common centre of curvature lies

250 mm to the left of the diffraction grating (see Figure 4.9). The radius of curvature of the

small convex mirror is half that of the large concave mirror (512 mm and 1024 mm

respectively). The Öffner triplet is aberration free for the position of common centre of

curvature (point C in Figure 4.12a). An image of this point is stigmatic but, deviations exhibit

spherical aberration and astigmatism. The grating is moved 250 mm away from this point in

order to give a non zero stretch factor (the image of the grating is 250 mm to the left of the

common centre of curvature giving an effective grating separation of 500 mm). A stretch

factor of 10,000 is achieved by double passing the whole system (system double pass mirror in

Figure 4.9) giving an effective grating separation of 1000 mm. The double pass mirror in the

figure removes the spatial dispersion added by a single pass of the stretcher (see section 1.3,

Figure 1.9 and figure 1.10 for illustration of spatial dispersion). The stretcher is described in

full in a paper by Cheriaux et al [Cheriaux 96] also in [Curley 96] and [Antonetti 97].

Modelling results

111

DiffractionGrating

System DoublePass Mirror

Double PassMirror

ConvexMirror

ConcaveMirror

Figure 4.9 Schematic of the Öffner triplet based stretcher design of Antonetti et al [Cheriaux

96]. Input and output is between two mirrors of System double pass mirror retroreflector.

The diffraction grating has a ruling density of 1200 lines/mm. This system offers a

large stretch factor and low aberrations, faithfully producing 33 fs pulses from 30 fs seed pulses

after compression and in the absence of the material dispersion of an amplifier chain.

4.2.2.2. Öffner triplet system performance The Öffner triplet based stretcher has a high level of aberration compensation and

consequently, matches its corresponding compressor very well. The Öffner triplet based

imaging system is only aberration free at the null stretch position (250 mm left of the diffraction

grating in Figure 4.9 and point C in Figure 4.12a). Small amounts of spherical aberration and

astigmatism are thus introduced when the grating is moved away from the null stretch position

to give the stretch factor of 10,000. The effect of these aberrations is apparent when comparing

the input (black curve) and output (blue curve) pulse profiles in Figure 4.10.

At the 1×10-2 level, the output profile starts to depart from the input pulse profile. At

the 4×10-5 level, wing structure starts to appear due to mismatches in fourth order dispersion.

Below the level shown in the graph, departures from the input pulse profile are shown by

ringing caused by mismatches in fifth (and higher) order dispersion.

Modelling results

112

Figure 4.10 Comparison of 30 fs input pulse profile (black curve), output pulse profile from an Öffner triplet based stretcher with perfectly flat optics (blue curve) and the output pulse profile

from an Öffner triplet based stretcher with realistic surface figures (red curve). Calculated using the code. Inset shows figure 2 from [Antonetti 97] showing independent calculations for

blue curve.

In order to achieve this performance, the quality of the optics used must be carefully

optimised. Figure 4.5 shows experimental output pulse profiles for the Öffner stretcher with

varying surface quality of the large concave mirror. As the quality of this optic increases, so

does the quality of the output pulse profile in terms of duration and contrast. The quality

(surface figure) of each of the surfaces in the Öffner stretcher are summarised in Table 4.2.

These surface figures were used to calculate the effect of surface figure on the performance of

the Öffner stretcher shown by the red curve in Figure 4.10.

Surface Flatness

Diffraction grating λ/10 Concave mirror λ/40 Convex mirror λ/20

Double pass mirror λ/20* System double pass mirror λ/20*

Table 4.2 Summary of surface specifications of Öffner triplet based stretcher optics. (* = flatness estimated due to lack of information in [Antonetti 97]).

Figure 4.11 shows the residual phase of a 30 fs pulse after passing through an Öffner

stretcher (stretching to 300 ps) and corresponding compressor. The blue curve represents the

phase calculated for perfectly flat optics and the red curve represents the phase for imperfect

optics with flatnesses summarised in Table 4.2. In the spectral window shown in Figure 4.11

(745 nm to 812 nm), the residual phase using perfect optics deviates from zero by ±0.26 radians.

Modelling results

113

The effect of the imperfect optics is superimposed on top of this and is characterised by a

maximum deviation of ±0.65 radians. Across the full spectral window of the stretcher (731 nm

to 825 nm, limited by the size of the optics in the plane of dispersion) the residual phase

deviates from zero by ±1.18 radians.

750 760 770 780 790 800 810-1.0

-0.5

0.0

0.5

1.0

Res

idua

l pha

se (r

adia

ns)

Wavelength (nanometres) Figure 4.11 Calculated residual phase of a 30 fs pulse after passing through an Öffner triplet

based stretcher and corresponding compressor, with perfect optics (blue curve) and with optics having non-zero surface figure (red curve). Calculated using the code.

This stretcher design works very well - it stretches 30 fs pulses by a factor 10,000 and is

well matched to its corresponding compressor, which produces pulses stretched by only 10%

from the original pulse duration. The limiting factor in the fidelity of the output pulses is the

flatness of the stretcher optics. In order to keep the aberrations low, the stretcher is double

passed to achieve the required stretch factor. This has the disadvantage that all surfaces are

struck twice as many times, which increases the minimum required surface figure by a factor of

two. Consequently, the large concave optic in the stretcher has a surface figure of λ/40.

Unfortunately, optic price does not increase linearly with surface figure; a λ/40 optic is roughly

six times more expensive than a λ/20 optic (price comparison from REOSC [REOSC]). The

performance of this stretcher comes at a significant price.

4.3. Reflective doublet system In an effort to design a stretcher that performs as well, if not better than the Öffner

based system but without the expensive high figure mirror, a stretcher based on a reflective

doublet was devised. In papers by Sullivan et al ([Sullivan 95] and [Sullivan 96]), a (refractive)

Modelling results

114

doublet replaces the lens of the telescope in a standard Martinez style stretcher. This doublet

can be adjusted to minimise the high order phase mismatches between the stretcher and

compressor. The idea behind using a reflective doublet to form the telescope is that chromatic

aberrations are removed immediately (the case for all sub 50 fs stretchers) and the

characteristics of the doublet can be arranged so the grating lies at the aberration free position

(point G in Figure 4.12b).

xC/N

M1

GGIM2

(a)

M1

GM2M3

GI

(b)

xN

Figure 4.12 Schematics of two stretchers: (a) the Öffner triplet based and (b) doublet based. Definitions: M = Mirror, G = Grating, GI = Grating Image, C = Centre of curvature and

N = the null stretch position. Grey lines represent input/output paths.

Figure 4.12(a) shows a schematic of the Öffner stretcher with point C representing the

centre of radius of curvature of the two curved mirrors. It is this point (the null stretch position)

that offers aberration free operation of the Öffner triplet imaging system. Figure 4.12(b) shows

a schematic of a doublet based stretcher that is designed to be virtually aberration free at point

G, the position of the grating. This is achieved by careful choice of the radius of curvature and

positions of the concave and convex mirrors (M1 and M2 in Figure 4.12(b)) and the position of

the flat mirror, M3. Figure 4.13 shows a three-dimensional schematic of the reflective doublet

based stretcher design. Comparing this design to the Martinez stretcher in Figure 4.7a, the flat

mirror represents the fold between the two lenses, the convex and concave mirrors form the

reflective doublet that replaces the two lenses and the double pass mirror retroreflects the

spatially dispersed beam after diffraction off the second grating.

The requirement for the stretcher is that it should stretch 35 fs pulses to at least 300 ps.

A suitable compressor that can compensate for this stretch factor using the guidelines in section

3.1.1 for a central wavelength of 780 nm can also be designed. This gives the required effective

grating separation, grating angle and input angle.

Modelling results

115

DiffractionGrating

Double PassMirror

ConvexMirror

FlatMirror

ConcaveMirror

Figure 4.13 Three-dimensional schematic of the reflective doublet based stretcher design. Input

and output is via the single ray at the top of the diffraction grating.

4.3.1. Reflective doublet specification The first stage of deciding the parameters of the doublet is to find a set of radii of

curvature for the two curved mirrors and their rough positions and the position of the flat mirror

so that parallel rays emerging from the left (Figure 4.12(b)) are focused to a point between the

two curved mirrors. This point is the position of the flat mirror and should be in front of the

concave mirror. If this was not the case, the large mirror would have to allow rays to pass

through its middle.

The second stage is to move the grating away from the null stretch position towards the

three mirrors to give the required stretch factor. The positions of the convex and flat mirror are

then adjusted to make sure all the rays of differing wavelength that leave the grating arrive back

at the grating with exactly the same angle. In other words, the grating has a perfect image

behind the null stretch position (at point GI in Figure 4.12(b)). The effective grating separation

is given by the negative of the distance between the position of the grating and its image (see

section 1.3, Figure 1.10).

The final step is to use the code to find out how the design performs. This process is

iterated until a satisfactory performance is achieved. Figure 4.14 summarises the final

parameters of the doublet stretcher after a total of 99 iterations.

The radius of curvature of the concave mirror (1175 mm) was chosen as a value that the

suppliers (Optical Surfaces [Optical Surfaces]) could provide easily (flatness λ/20). The small

convex mirror was also supplied by Optical Surfaces and has a flatness of λ/20; both have a

protected gold coating on the appropriate surfaces. The flat mirror and mirror used to

retroreflect the spatially dispersed beam back for a second pass of the stretcher both have

flatnesses of λ/10. The small flat mirror is made from ZERODUR® [HV Skan], polished by

Martin Horton [Horton] in the Blackett Laboratory optics section workshop and coated with

protected gold by Coherent Optics Europe [Coherent OE]. The (replicated) diffraction grating

Modelling results

116

was supplied by Milton Roy/Richardson [Richardson] and has a flatness of ~ λ/10. The sizes of

the optics were chosen to allow a bandpass of 66.6 nm a compromise between maximum

bandwidth and minimum cost.

d = 1200 l/mmw = 110 mm

γ = 33º θ0 = 23º

roc = -1000 mmw = 100 mm

egs = -1142 mm

w = 90 mm

roc = 1175 mmφ= 200 mm

693.82 mm978.37 mm

1100.00 mm

Figure 4.14 Detailed schematic of final doublet stretcher specifications. roc = radius of curvature, egs = effective grating separation, d = grating constant, w = width of optic and

φ = diameter of optic. In order to separate input and output beams, the input beam is vertically displaced by 25 mm away from the optical axis of the stretcher.

4.3.2. Reflective doublet performance The reflective doublet based stretcher has a high level of aberration compensation

designed around the position of the diffraction grating. This means there is no compromise

made by moving away from an aberration free null stretch position as there is with the Öffner

based stretcher. If the grating of the doublet stretcher were placed at the null stretch position

(point N in Figure 4.12b), the image of this point would exhibit spherical aberration and

astigmatism as the Öffner stretcher does at the grating position. Figure 4.15 shows a

comparison between the stretched and recompressed pulse profile (blue curve) and a 30 fs input

pulse (black curve).

The output pulse matches the input pulse very well, down to the 1×10-3 level. Below

this there is a little evidence for a third order phase mismatch between the stretcher and

compressor, but the ringing is mostly dominated by spectral clipping. The spectral clipping is

due to the finite size of the optics coupled with a small beam size (see section 2.2.1.3). A

comparison with the results of the Öffner stretcher in Figure 4.10 highlights a significant

difference between the performances of the stretchers in terms of pulse contrast. The two

stretchers use similarly sized optics (especially the grating and large concave mirror). The

reason for the difference is that a single pass of the Öffner stretcher gives a stretch of only 5,000

times because the diffraction grating is angled such that the bandwidth of the pulse is not

stretched so much across the optics. The doublet stretcher however, attains a stretch factor of

Modelling results

117

10,000 in a single pass and so the grating must disperse more across the optics. As a result, the

clipping is more severe and ringing appears in the wings of the pulse.

-300 -200 -100 0 100 200 300

1E-4

1E-3

0.01

0.1

1

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds) Figure 4.15 Comparison of 30 fs input pulse profile (black curve), output pulse profile from a reflective doublet based stretcher with perfectly flat optics (blue curve) and the output pulse profile from a reflective doublet based stretcher with realistic surface figures (red curve).

Calculated using the code.

The effect of this ringing is balanced by the fact that the contrast of the pulse is limited

by the effects of optic surface quality. The red curve of Figure 4.15 shows the output pulse

profile of a 30 fs pulse stretched by a doublet stretcher with imperfect optics to 300 ps and

recompressed with a corresponding compressor. The optical figure of each surface is described

in section 4.3.1. Even though the doublet stretcher uses at best λ/20 optics, it performs as well

as the Öffner stretcher with a λ/40 optic. The performance of the Öffner stretcher below the

level imposed by imperfect optics is superior to the doublet stretcher. If the size of the optics

were increased to accommodate more bandwidth, the performance of the doublet stretcher

would exceed that of the Öffner stretcher. This is born out in Figure 4.16 which shows the

residual phase calculated for a 30 fs pulse after passing through a doublet stretcher (stretched to

300 ps) and corresponding compressor.

The blue curve shows the residual phase using a stretcher with perfect optics and the red

curve shows the phase with non-perfect optics. The spectral window shown in the figure

represents the bandpass of the stretcher (745 nm to 812 nm) and the residual phase using perfect

optics deviates from zero by a maximum of ±0.03 radians across this range. The effect of non-

perfect optics has the same overall effect as for the Öffner stretcher (compare with Figure 4.11).

The residual phase after recompression for the doublet stretcher is a factor of 8.7 times smaller

than that from the Öffner stretcher. This illustrates the difference between the two stretcher

Modelling results

118

designs; the doublet stretcher is designed to be virtually aberration free for the position of the

grating not the centre of radius of curvature of the two curved mirrors, as is the case for the

Öffner stretcher.

750 760 770 780 790 800 810-1.0

-0.5

0.0

0.5

1.0

Res

idua

l pha

se (r

adia

ns)

Wavelength (nanometres) Figure 4.16 Calculated residual phase of a 30 fs pulse after passing through a reflective doublet based stretcher and corresponding compressor, with perfect optics (blue curve) and with optics

having non-zero surface figure (red curve). Calculated using the code.

4.4. Additional modelling In addition to the modelling done specifically for the design of a stretcher compressor

system for the Blackett Laboratory Laser Consortium, the code has been used to aid the choice

of stretcher for the upgrade of the ASTRA and VULCAN lasers at the Rutherford Appleton

Laboratory. This section describes how the code described in chapter 3 facilitated this choice

for the ASTRA laser (adapted from a report written for the Rutherford Appleton Laboratory in

February 1999). Reproduced by kind permission of the Rutherford Appleton Laboratory.

4.4.1. Introduction

The ASTRA laser at the Rutherford Appleton Laboratory is a high power, short pulse

titanium doped sapphire laser which services a broad base of United Kingdom users. The

stretcher must expand 30 fs pulses to at least 600 ps for safe and efficient amplification in the

ASTRA amplifier chain, due to the large gain factors these amplifiers provide. A corresponding

pulse compressor must be designed to restore the pulses as close as possible to their original

pulse duration with minimum wings. Here, wings are defined as those that arise from

Modelling results

119

incomplete recompression of the pulse, not from Amplified Spontaneous Emission in the

amplifier chain. Incomplete recompression leaves the pulse with residual high order phase

components due to aberrations in the stretcher. The stretcher needs to impart phase on the input

pulse that exactly opposes that imparted by the compressor to at least fourth order to obtain

clean laser pulses for use in interaction experiments.

There are three designs under consideration, all based around a reflective variation

of the standard Martinez style stretcher [Martinez 87a]. The three systems being considered are

as follows:

• A system devised by the author, used at Imperial College on the titanium doped sapphire

chirped pulse amplification laser system in the Blackett Laboratory Laser Consortium.

This design is based around the idea of using a positive and negative curved mirror to

make a reflective doublet. The radii of curvature of the two mirrors and their relative

positions can be chosen to make the imaging system aberration free for any given

diffraction grating position. The detailed specifications of the stretcher used at Imperial

College is given in chapter 4.3.

• A system devised by Lawrence Livermore National Laboratory in California, described in

the Ph.D. thesis of Paul Stuart Banks (see chapter 4.2.1). This design is an all-reflective

and well-folded version of the standard Martinez-Style stretcher. The clever part of the

design is the use of a reflective strip across the centre of the diffraction grating an optic

only available from the Livermore laboratory. This system is not aberration free, it is

simply a direct replacement of the lens in a standard stretcher with a spherical mirror.

• A system based on a design by Ian Ross at the Rutherford Appleton Laboratory. This

design uses a single spherical mirror and two diffraction gratings. Although the

introduction of a second diffraction grating means there is an extra degree of difficulty in

alignment, the system is aberration free for a point, on-axis, at the centre of the radius of

curvature of the spherical mirror.

Each system will be compared by inputting a 30 fs pulse into the stretcher to give a stretched

pulse of around 600 ps. The pulse is then recompressed with a corresponding compressor and

the relative performances compared by looking at recompressed pulse profiles. All comparisons

are done using the model described in chapter 3.

4.4.2. Extra modelling results

For a given input pulse duration, the central wavelength (800 nm) and bandwidth are

constant for the systems being considered and the grating ruling density is 1480

lines/millimetre. It can be seen from equation 3.1 that this leaves only the input angle and the

grating separation to adjust in order to give the required stretched pulse duration of 600 ps. The

Modelling results

120

input angle should be chosen such that the grating is used as close to Littrow configuration as

possible (so as to give efficient diffraction). This angle should also be large enough so that the

dispersed beam from the second grating misses the edge of the first grating (taking into account

the fact that the gratings must be held in a finite sized mount). The distance between the

gratings can then be chosen to give the required stretch duration.

4.4.2.1. Adapted reflective doublet design A schematic of the doublet-based design is shown in Figure 4.13. The important

parameters to adjust with this system are the radius of curvature of the convex and concave

mirrors, their separation, their distance from the diffraction grating and the position of the flat

mirror. These are adjusted to try to produce an aberration free system for rays originating from

the diffraction grating.

The doublet design is similar to a stretcher based on an Öffner triplet used at ENSTA,

Paris [Cheriaux 96]. In this stretcher, two curved mirrors are chosen to make a reflective triplet

imaging system (one mirror being used twice), which images the null-stretch position of the

stretcher without aberration. A stretch is achieved by moving the diffraction grating away from

the null-stretch position towards the triplet mirrors. Twice the distance from the grating to the

null-stretch position is the effective grating separation (negative). The problem with this

method is that as the grating is moved away from the null-stretch position, it is also moved

away from the aberration-free imaging regime. The idea behind the doublet design is that the

curved mirrors are chosen such that the grating is at the aberration-free position.

After a lot of perseverance with this design, a combination of parameters could not be

found that would give satisfactory recompression. The reason for this is probably the large

stretch factor required. In order to have a large stretch factor, the null stretch position needs to

be far behind the diffraction grating to give a large effective grating separation. A combination

of mirror radius of curvature and separations could not be found to give an aberration free

system. The designs that came closest to working involved putting the flat folding mirror

behind the large concave mirror, which is unphysical. An example of one of the best

recompressed pulse profiles is shown in Figure 4.17.

Modelling results

121

-400 -300 -200 -100 0 100 200 300 4001E-4

1E-3

0.01

0.1

1

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds) Figure 4.17 Input 30 fs pulse profile (black curve) and recompressed pulse profile (red curve) for the doublet stretcher. The input pulse was stretched to 600 ps and the recompressed full

width half maximum pulse duration of 55.7 fs.

4.4.2.2. Adapted Livermore design

A schematic of the Livermore design is shown in Figure 4.7. This design is a very

cleverly folded version of the original Martinez-Style stretcher. It uses a concave spherical

mirror to replace the lens and a horizontal mirror strip across the centre of the diffraction grating

to further fold the design. The construction of the stretcher is detailed in section 4.2.1.

Although cleverly folded, this design offers nothing in the way of aberration compensation.

The system does however perform fairly well for what it was designed for: a final pulse

duration of 30 femtoseconds and a stretched pulse duration of 300 picoseconds. Because of the

aberrations added by the stretcher, the input pulse duration must be shorter than the final output

duration.

The Livermore design was adjusted to give a stretched pulse duration of 600

picoseconds by changing the position and radius of curvature of the concave mirror. This

moves the null-stretch position further behind the diffraction grating, thus increasing the

effective grating separation. As with the doublet design, a system could not be found which

would give satisfactory recompression. Systems that came close to working had shorter radius

of curvature mirrors. Figure 3.2.2 shows how the radius of curvature of the concave mirror

affects the recompressed pulse profile.

Modelling results

122

-400 -300 -200 -100 0 100 200 300 4001E-4

1E-3

0.01

0.1

1

f=4.00 m f=3.00 m f=1.25 m input = 30 fs

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds) Figure 4.18 Input 30 fs pulse profile (black curve) and recompressed pulse profiles (green, red

and blue curves) for the Livermore stretcher. The input pulse was stretched to 600 ps. The recompressed pulse duration and wing level changes as the focal length of the curved mirror is

changed.

4.4.2.3. Adapted Ross design A schematic of the Ross stretcher design is shown in. This system has been designed so

that the centre of the first diffraction grating is at the centre of curvature of the concave mirror.

This means that, as long as small beams are used, the imaging system is aberration free. One

disadvantage of this design is the fact that, unlike the other designs, two diffraction gratings are

used making it more difficult to align (the two gratings have to be exactly parallel). This may

be balanced by the fact that there are fewer optical components overall to align. The beam

diffracted from the second grating is horizontally dispersed and so the stretcher is double passed

such that the output beam travels collinearly with, but in the opposite direction to the input

beam.

The original design for this stretcher stretched 30 femtosecond pulses to 300

picoseconds. A stretch to 600 picoseconds can be achieved in two ways: by double-double

passing the stretcher; or by doubling the radius of curvature of the concave mirror and moving it

twice as far away from the first grating.

Modelling results

123

FirstGrating

SecondGrating

ConcaveMirror

Input Output

Figure 4.19 Schematic of the Ross stretcher design. Orange lines represent the two diffraction gratings and the grey line represents the input beam path. The first grating need not be as large

as shown it needs only to be large enough to accommodate the input beam.

The first method has the advantage that the optics are kept small this is important as

usually the price of optics goes up disproportionately with size. The main disadvantage is that

the beam reflects or diffracts off each optical surface twice as many times as in single pass

configuration. This is important because it means the optics used have to be twice as flat as

they would normally be. Figure 4.20a shows how the recompressed pulse profile changes as a

function of surface flatness (excluding grating flatness).

The second method has the opposite advantages and disadvantages as the first method.

The optics are twice as big, making them more expensive, but the surfaces need only be half as

flat for the same performance. Figure 4.20b shows how the recompressed pulse profile changes

as a function of surface flatness (excluding grating flatness).

-400 -300 -200 -100 0 100 200 300 4001E-6

1E-5

1E-4

1E-3

0.01

0.1

1

(a)

Input 30 fs λ/20 λ/10 λ/5

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds)

-400 -300 -200 -100 0 100 200 300 4001E-6

1E-5

1E-4

1E-3

0.01

0.1

1

(b)

Input 30 fs λ/20 λ/10 λ/5

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds) Figure 4.20 Input 30 fs pulse profiles (black curves) and output profiles (red, green and blue)

for (a) double-double pass and (b) double-size Ross stretcher designs. Different coloured curves show how the output pulse profile changes as a function of optic surface flatness.

Whichever of these two options are chosen, the correct beam diameter must also be

chosen. The beam has to be large enough so that, as it propagates through the stretcher (whose

total path length can be several metres) it does not increase in size too much due to diffraction.

Conversely, as the beam size is increased, the edges of it move away from the ideal imaging

conditions that are offered at the centre of curvature of the concave mirror, increasing the

Modelling results

124

overall recompressed pulse duration. Figure 4.21 shows how the full width half maximum

diameter of the input beam affects the recompressed pulse profile for (a) the double-double pass

design and (b) the double size design.

-400 -300 -200 -100 0 100 200 300 4001E-6

1E-5

1E-4

1E-3

0.01

0.1

1

(a)

Input 30 fs 0.50 mm 1.00 mm 2.00 mm 3.00 mm

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds)

-400 -300 -200 -100 0 100 200 300 4001E-6

1E-5

1E-4

1E-3

0.01

0.1

1

(b) Original 30 fs 0.50 mm 1.00 mm 2.00 mm 3.00 mm

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Time (femtoseconds) Figure 4.21 Input 30 fs pulse profiles (black curves) and output profiles (red, green, blue and

magenta) for (a) double-double pass and (b) double-size Ross stretcher designs. Different coloured curves show how the output pulse profile changes as a function of input beam full

width half maximum diameter.

4.4.3. Extra modelling conclusions It is clear from the results of the three stretcher designs that to produce a stretched pulse

duration of 600 ps, the adapted Ross stretcher must be used. The other two designs work well

for producing a stretched pulse duration of ~300 ps, for which they were originally designed,

but they cannot easily be adapted for use on the ASTRA laser.

Having chosen to go with the Ross design, the method of adaptation of the original

design must be chosen. Considering the double-double pass design, there seems little penalty in

choosing this method over the double size design in terms of surface flatness requirements (see

Figure 4.20). The model does not include, however, the diffraction grating flatness. This is

generally poor (less than λ/10) and there are twice as many passes in the double-double pass

design than for the double size design, making the former perform less well. It is clear though

from Figure 4.21 that the double size design is more tolerant of large beams making it easier to

implement. This is because the radius of curvature of the concave mirror in the double-size

design is twice as large as in the double-double pass design. The double-size stretcher should

perform the same as the double-double pass stretcher with a beam approximately half the size.

This is analogous to the fact that the beam waist size approximately doubles when the focal

length of the focusing optic is doubled.

Overall, although it is more expensive, it seems logical to choose the double size design

in terms of system performance. This was the recommendation to the Rutherford Appleton

Laboratory as a result of this ray-tracing modelling.

Modelling results

125

4.5. Summary Several stretcher designs have been considered in an effort to find a system that

performs as required with the limited resources available for this project. The best of the

systems described in the literature is the reflective stretcher based on an Öffner triplet. This

system is reported to be aberration-free, but only when it does not stretch an input pulse (null-

stretch configuration. With inspiration from the air-spaced refractive doublet based stretcher

designed by Sullivan et al, a reflective version was designed that is virtually aberration-free at

the configuration that gives a stretch factor of ~10,000. The next chapter describes the

construction of this stretcher and its integration into the Blackett Laboratory Laser Consortium

titanium-doped sapphire chirped pulse amplification laser system.

5. System implementation and characterisation

In this chapter, the results of the modelling described in section 4.3 are put into practice.

In addition, the upgrade of the Blackett Laboratory Laser Consortium titanium-doped sapphire

chirped pulse amplification laser system described in section 1.5 is discussed. The only stages

of the old system to remain in place after the upgrade are the regenerative and power amplifiers.

The upgrade of these amplifiers is discussed in terms of accommodating the large bandwidth

associated with ultra-short pulses and an increase in the overall gain of the system. An

additional increase in total amplification is also achieved by adding an extra gain stage (pre-

amplifier) that uses a portion of the energy left over from pumping the regenerative amplifier.

The characteristics of each of these amplification stages are then presented. The replacement of

the oscillator and newly designed stretcher and compressor for ultrashort pulse operation is

discussed. The performance of each of these three components compared to their design

specification is presented. Finally, the integration of each stage into a complete system is

discussed including performance characteristics.

5.1. Short pulse oscillator upgrade The oscillator of the original Blackett Laboratory titanium-doped sapphire laser system

was built and developed in house (see section 1.5). Its replacement is a kit laser from the

Murnane and Kapteyn laboratories [Oscillator]. The new oscillator is capable of delivering

pulses as short as 20 fs centred in the region of 800 nm. It achieves this by having a thin but

highly doped crystal that gives high gain without excessive material dispersion. The material

dispersion is further controlled by the use of a pair of fused silica prisms (Figure 5.1). See

section 1.2.1 for a discussion of Kerr-lens mode-locked oscillators.

The pump source for the old oscillator was a mainframe argon ion laser [Coherent]. In

order to extract the most power from this laser, it was used on all visible lines (in the blue to

green part of the spectrum). Although this is an efficient mode of operation for the argon ion

laser, not all the argon lines pump titanium-doped sapphire very efficiently. Consequently, the

old oscillator required 6 W CW from the argon ion for stable operation. The pump source for

the new oscillator is a Spectra-Physics Millennia [Millennia]. This uses two diode bars, each

delivering 13 W at 809 nm to pump neodymium-doped yttrium vanadate (Nd:YV04). When

System implementation and characterisation

127

frequency doubled, this produces up to 5.5 W at 532 nm, CW. The centre of the absorption

band of titanium-doped sapphire is very close to this wavelength making the Millennia an

efficient pump source.

12% OutputCoupler

Fused SilicaPrism Pair

4.75mm, 0.15% DopedTi:Sapphire Crystal

4.5 W532 nm

LM

M

RMB

TuningSlit

Figure 5.1 Schematic of the Murnane and Kapteyn short pulse oscillator. RM represents the

rear cavity mirror; B represents a beam block; M represents concave mirrors with a radius of curvature of 10 cm; and L is a lens of focal length 10.5 cm.

The oscillator cavity is relatively long (~3.36 m) and so sensitivity to alignment of the

pump beam is critical. The old argon ion laser also has a long cavity (~2.0 m) and, as the

temperature of the laboratory changes, the beam pointing changes. This makes for a very

unstable combination, one that involves frequent oscillator and pump beam alignment. The

Millennia however, has a very stable, short cavity making it perfect for pumping short-pulse

oscillators of this kind. The need for constant oscillator re-alignment is eliminated and the start-

up time is reduced by a factor of approximately six, to just half an hour.

The output of the new oscillator is an 89.4 MHz train of pulses with an average power

of approximately 400 mW (the average power depends on day-to-day alignment). This gives a

pulse energy at the beginning of the chirped pulse amplification laser system of ~9.0 nJ. The

pulses have a temporal profile approximating a Gaussian with a full-width-half-maximum

duration set to ~35 fs (therefore the single pulse average power is 0.26 MW). Figure 5.2 shows

a second-order single-shot autocorrelation of these pulses (see section 1.2.2.2 for a description

of autocorrelation methods). The autocorrelation shows a duration of 37.3 fs, longer than stated

for the output of the oscillator. This measurement will be slightly longer than the duration of

the pulses emerging from the oscillator because the pulses pass through 1.4 mm of BK7 glass

(beamsplitter) before they reach a 200 µm KDP crystal in the autocorrelator. This serves to

broaden the pulse prior to and during the pulse length measurement. The inset of Figure 5.2

shows the pulse profile on a logarithmic scale illustrating the near Gaussian nature of the

profile. The dynamic range of this measurement is limited, in part, by the 2.5 orders of

magnitude that the CCD camera [Pulnix] provides.


128

-125 -100 -75 -50 -25 0 25 50 750.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

37.3 fs

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Delay (femtoseconds)

-100 -50 0 50 100

0.01

0.1

1

Figure 5.2 Single-shot autocorrelation of pulses from the oscillator showing a duration of

37.3 fs. Inset shows the same profile on a logarithmic scale. Dynamic range limited by the CCD camera used for the autocorrelation measurement.

Originally, the laser wavelength was centred at 780 nm (for historical reasons [Fraser

96]). The bandwidth of pulses from the old oscillator was ~8 nm compared to ~30 nm from the

new one. Most commercially available thin film anti-reflection and high reflectivity coatings

around this wavelength are centred at 800 nm (see section 2.2.1.1). When small bandwidth

pulses are used with these mirrors, only a small modification to the spectral profile occurs. The

spectral profile of large bandwidth pulses however, can be seriously compromised after many

coating encounters. For example, the spectrum of a 30 nm pulse centred at 780 nm is

preferentially reflected at longer wavelengths by a high reflectivity coating. This has the effect

of skewing the spectral profile and shifting the centre wavelength to longer wavelengths (see

section 2.2.1). For this reason the centre wavelength of the oscillator is set to ~800 nm. For

convenience, the actual centre wavelength of the oscillator is set to 801.5 nm, a prominent line

in the emission spectrum of argon used for spectrometer calibrations. Figure 5.3 shows the

spectral profile of the pulse (red curve) with the emission spectrum of argon (blue curve). The

time bandwidth product of the oscillator when the duration measurement shown in Figure 5.2

and bandwidth measurement shown in Figure 5.3 were made was 0.55 (characteristically large

for this type of short pulse oscillator).

The wavelength of the oscillator can be set by use of the tuning slit placed between the

two prisms (Figure 5.1). This only needs to be changed when the alignment of the oscillator is

changed after laser servicing or alignment creep over long periods of time. The bandwidth of

the pulses can be changed by a combination of the positions of the two prisms (into and out of

the cavity beam) and the width of the tuning slit).


129

760 770 780 790 800 810 820 830 8400.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

31.1 nm

Nor

mal

ised

inte

nsity

(arb

itrar

y un

its)

Wavelength (nanometres) Figure 5.3 Oscillator pulse spectrum showing a bandwidth of 31.1 nm (red curve). Blue curve shows the spectrum of lines from an argon discharge lamp. The pulse spectrum is centred and

the spectrometer is calibrated using a strong argon line at 801.5 nm.

The beam emerging from the oscillator diverges slightly and a Newtonian telescope is

used after the oscillator to collimate the beam. The positions of the telescope lenses are

adjustable to account for changes in divergence arising from oscillator realignment.

5.2. Stretcher implementation The design of the stretcher (described in section 4.3) is such that the convex mirror and

the small flat mirror (between the concave and convex mirrors) are required to have beams pass

above and below them. The two mirrors also need to be tilted about the mirror surface in order

to fine-tune their positions. To this end two mirror mounts were designed which project the

mirrors out into the middle of the stretcher and provide stable support. These two mounts can

be seen in Figure 5.4 marked as M2 and M3. Two brass-coloured thumb wheels are visible on

the rear of the mount for M3 these give control of the vertical and horizontal tilt of the mirror.

Alignment of the stretcher is aided by painting the optical table with engineers blue

(marking dye) and scribing the paths of crucial rays. The width of the ray path marks on the

table is ~0.2 mm. The mirrors are aligned in turn using a small beam from a laser diode and a

beam block with vertical lines scribed on one side. Beam alignment over a distance of 1 m

using this method gives an angular accuracy of ±54 seconds. The two irises in Figure 5.1 allow

the stretcher input beam alignment to be checked on a day-to-day basis. The input beam is

retro-reflected by mirror M4 back along its input path. In order to allow the beam to exit the


130

stretcher without causing the oscillator to drop-out (due to excess feedback), the input beam is

angled up slightly so that on exit it passes just above a cut mirror placed before the first

alignment iris. This tilt represents a change in beam height of ~5 mm over a distance of ~10 m,

an angle of 0.027º. It was thought that this would not impair the performance of the stretcher.

Figure 5.4 Picture of the completed stretcher with the oscillator in the foreground.

M1 = concave mirror; M2 = small flat mirror; M3 = convex mirror; M4 = double pass mirror; G = diffraction grating; I = alignment irises; C = oscillator crystal; and P = oscillator prism.

A very sensitive test of the alignment of the stretcher is to check the spatial chirp across

the beam (in the plane of dispersion) on its exit from the stretcher. Spatial chirp arises from

misalignment of the optics causing the image of the input beam on the grating to be aberrated.

The grating imparts a spectral component to this aberration causing the wavelength to change at

different positions across the beam. The spatial chirp can be measured quite easily by rotating

the beam through 90º so that when it strikes the entrance slit of a grating spectrometer, the plane

of dispersion of the beam is in line with the slit. This measures the spectrum at each position

across the beam exiting the stretcher. Figure 5.5 shows the results of such a measurement.

The top image of Figure 5.5 shows the spatial chirp measurement for a misaligned

stretcher. The y-axis represents the distance across the beam in the plane of dispersion and the

x-axis represents the spectrum at that point. It can be seen that the spectrum on one side of the

beam is skewed towards shorter wavelengths and the spectrum on the other side is skewed

towards longer wavelengths. The bottom image of Figure 5.5 shows the spatial chirp

measurement of the stretcher after correction. This correction involved moving the flat mirror

between the two curved mirrors by less than 0.5 mm.


131

775 780 785 790 795 800 805 810 815 820 825 8300

1

2

3

4

5

Beam

dim

ensi

on (m

illim

etre

s)

Wavelength (nanometres) Figure 5.5 Measurement of the spectrum across the beam (in the plane of dispersion) for a

misaligned stretcher (top image) and a corrected stretcher (bottom image). Red represents zero intensity and increasing intensity is represented by orange-yellow-green-blue.

The final measurement to check that the stretcher works correctly is to find the stretched

pulse duration. To do this a fast diode [New Focus] with a rise time of ~50 ps in conjunction

with a sampling scope [Tektronix] is used. The output of the stretcher is focused onto the small

active area of the diode to make the measurement. The result on the scope is a build-up of

samples of the diode signal. An image of the trace observed is shown in Figure 5.6. There are

two important things to notice in this image. Firstly, the slight bump on the right hand side of

the trace. Because the stretcher adds a frequency dependent delay producing a chirped pulse,

the stretched pulse takes on the form of the input pulse spectral profile. The bump corresponds

to the bump on the short wavelength side of Figure 5.3. Secondly, the rise time of the diode is

approximately 50 ps; therefore, the recorded duration should at least 50 ps longer than expected.

Figure 5.6 shows a duration of ~375 ps compared to the design stretched pulse duration of

300 ps. The exact duration of the stretched pulse depends on the bandwidth of the pulses from

the oscillator so can be expected to be longer than calculated (due to excess bandwidth,

mentioned in the previous section).


132

-800 -600 -400 -200 0 200 400 600 800 1000 1200

-20

0

20

40

60

80

100

120

140

Sign

al (m

illiVo

lts)

Scan time (picoseconds) Figure 5.6 An image taken from the oscilloscope showing the profile of the stretched pulse.

Measured using a fast diode [New Focus] and sampling scope [Tektronix]. The diode has a rise-time of approximately 50 ps, which has the effect of stretching the measurement.

5.3. Amplification upgrade The upgrade of the amplification stages of the laser system falls into two main

categories. First, the adaptation of existing amplifiers to cope with the increased bandwidth of

the short pulses from the new oscillator. Secondly, an increase in the total gain of the amplifier

chain. The regenerative amplifier is the main spectrally dependent component in the

amplification system due to many passes of polarisers, Pockels cell and the gain medium. The

increase in total gain is achieved by introducing an extra pre-amplifier and using a second pump

laser in the power amplifier. The following sections describe each of the amplifiers in terms of

spectral response (in the case of the regenerative amplifier) and the gain characteristics.

5.3.1. Regenerative amplifier The regenerative amplifier has been inherited from the old Blackett Laboratory Laser

Consortium titanium-doped sapphire system. Therefore, the design of this laser will not be

considered from scratch, only the adaptation of the design for ultra-short pulse operation.


133

5.3.1.1. Cavity design

As explained in section 1.4.2, regenerative amplifiers offer an efficient means of

increasing the energy of pulses for chirped pulse amplification. The amplifier is designed

around a stable laser cavity with elements for switching the seed pulse in and out (see Figure

5.7). The concave mirror (M1) has a radius of curvature of 10 m and is 1.67 m away from the

convex mirror (M2) that has a radius of curvature of 20 m. The radii of curvature of the two

mirrors are chosen to provide a large beam size in the cavity and a stable resonator design

[Kogelnik 66]. High fluences cause damage to intracavity optics, so a large beam diameter is

important. A stable resonator design is important with respect to beam pointing stability

(section 2.1.5) and energy stability (section 2.1.3).

PockelsCell

TFP2 IrisTi:Sapphire

Crystal 80mJ, 532nmPump

λ/2

InOut

M2M1 TFP1Iris

Figure 5.7 Schematic of the regenerative amplifier. TFP1, TFP2 = thin film polarisers;

M1, M2 = cavity mirrors. Titanium-doped sapphire crystal has length 20 mm and a doping level of 0.10%.

The size of the beam waists (radius) at the two mirrors (M1, M2) is given (respectively)

by the following equations:

−+−

−

=

−+−

−

=

LRRL

LRLRR

LRRL

LRLRR

212

12

2042

211

22

1041

πλω

πλω

5.1

where λ0 is the central laser wavelength (800 nm), R1 and R2 are the radii of curvature of mirrors

M1 and M2 respectively (metres) and L is the cavity length (metres) [Koechner 76]. The

stability condition is given by the modulus of the product of g1 = 1-L/R1 and g2 = 1-L/R2. |g| is

dimensionless and must lie in the region:

10 << g 5.2

in order for the cavity to be considered stable [Kogelnik 66]. The maximum length of the cavity

is fixed, by the amount of room on the optical table, at L=1.67 m. For the radii of curvature

stated above, the value of |g| is 0.902 giving a stable resonator. The beam diameters at mirrors

M1 and M2 are 2.49 mm and 2.18 mm respectively. If R1 is decreased, the cavity would

become more stable, but the difference between the beam waists would increase giving an


134

output that quickly diverges. The opposite happens if R1 is increased. If R2 is increased, the

cavity becomes unstable and the beam waists increase in size together. The opposite happens if

R2 is increased.

Another quantity associated with laser resonators is the Fresnel number. This describes

the losses in the cavity due to diffraction. The losses in the cavity due to the polarisers and

Pockels cell are far greater than this so the diffraction losses are not considered here.

The polarisers used in this amplifier have been specially designed for use in large

bandwidth regenerative amplifiers [Newport 00]. They have a large reflection and transmission

bandwidth (greater than 200 nm) to cope with the large bandwidth pulses that have to propagate

through them many times. A drawback of these polarisers is their relative inefficiency, offering

an extinction ration of only 5:1 (ratio of p-polarisation transmission to s-polarisation

transmission). The Pockels cell used is a dry type (i.e. no index matching fluid between the

crystal and the environmental isolation windows) consequently, there are many surfaces from

which stray reflections can occur. This makes the Pockels cell a rather inefficient component of

the amplifier also, but minimises material path length. Large beam waists allow other

transverse modes to oscillate in the cavity.

5.3.1.2. Pumping To overcome the losses from the components mentioned in the previous section, the

single pass gain has to be quite high. The titanium-doped sapphire crystal is pumped with a

portion of the energy from a Continuum Surelite II [Continuum]. Eighty millijoules of 532 nm

radiation is focussed through the crystal using a 50 cm focal length lens, giving a beam diameter

of ~2.5 mm at the crystal surface and a fluence of 1.63 Jcm-2 at 10 Hz. The single pass gain of

the regenerative amplifier, compensating for the losses of the polarisers and Pockels cell, is

1.468, calculated using the following equation:

n

in

out

EEg

1

= (5.3)

where Ein and Eout are the input and output energies respectively (Joules) and n is the number of

passes of the gain medium. This calculation assumes that the gain is not saturated, which is

normally true except for the last two or three passes of the gain when the pump energy

fluctuates high. With an input energy of ~2 nJ and an output energy of ~2 mJ, the total gain

after 36 passes is 1.0×106.

The timing of the pump pulse with respect to the seed pulse can help to suppress

amplified spontaneous emission (see section 2.1.2). If the seed pulse is timed to arrive just after

the pump pulse, then the seed pulse mode will most likely be the dominant one. If the seed

arrives much later than the pump pulse in the crystal, then spontaneous emission has more


135

chance to propagate from the crystal, along the amplifier cavity and back to the crystal seeing

gain. The effect is small, but everything that reduces amplified spontaneous emission helps to

reduce output pulse pedestal. The inset of Figure 5.8 shows the relative timing of the seed and

pump pulse at the crystal.

5.3.1.3. Pulse switching Pulse switching is controlled by the Pockels cell in the middle of the cavity (Figure 5.7)

(see section 1.4.1). The firing of the Pockels cell is performed by a thyratron switched, high

voltage power supply and charge line. The charge line (a length of high voltage coaxial cable)

is charged to ~12 kV by the high voltage power supply. The thyratron then fires, discharging

the charge line to the Pockels cell; the cell sees half the voltage of the charge line,

approximately 6 kV. The duration of the resulting high voltage pulse is set by the length of the

charge line. This switch-in pulse locks the seed pulse into the regenerative amplifier cavity.

It is important that the voltage pulse is sufficiently short so that the Pockels cell is fully

off when the switched-in seed pulse returns from reflection at the cavity mirror. If this is not the

case, a portion of the seed pulse will be rejected from the cavity the next time it encounters a

polariser. The voltage pulse should also be long enough so as not to clip the seed pulse

temporally. The seed pulse is chirped so any modifications to its profile will result in

modifications to the pulse spectrum. The voltage pulse has a full-width-half-maximum duration

of ~5 ns with an approximately Gaussian temporal profile.

A second voltage pulse is required to switch out the amplified seed pulse when it has

the required energy. This is achieved by having a relatively long delay line attached to the

Pockels cell. After the voltage pulse has switched the Pockels cell, it travels down the delay

line, reflects from the other (open) end, and returns to the Pockels cell. The delay between the

switch-in and switch-out pulses is controlled by the length of the delay line. The switch-out

voltage pulse must coincide with the amplified seed pulse as it propagates through the Pockels

cell after an even number of round trips of the amplifier cavity.

There are two main drawbacks to this method of producing two short voltage pulses

with a relatively long delay between them (~200 ns). Firstly, the delay is fixed by the length of

the cable, which restricts the flexibility of the amplifier design. Secondly, the second voltage

pulse does not have the same amplitude as the original. Even though the reflection efficiency at

the low/high impedance boundary (open end) of the delay line is very high, there are losses in

the delay line and Pockels cell. This has the effect that either the switch in voltage has to be set

high in order that the second voltage pulse drives the Pockels cell exactly half wave, or that the

second voltage pulse does not drive the Pockels cell half wave making it an inefficient optical

switch. The latter effect is usually the case given the difficulty of producing very short duration


136

high voltage pulses. The reflected pulse is also longer than the input pulse due to dispersion in

the delay line. The has the effect that on the second to last pass of the Pockels cell, the applied

voltage is non-zero and a small portion of the cavity radiation is switched out towards the

oscillator.

Figure 5.8 shows the trace from a digital oscilloscope of the pump pulse and build-up of

the seed pulse in the regenerative amplifier cavity. Point A is the pump pulse at zero time and B

is a reflection of this pulse down the probe cable, due to imperfect impedance matching at the

oscilloscope. Point C shows the amplified seed pulses as they begin to be strong enough to be

detected by the photodiode looking through the back of cavity mirror M2. Consecutive pulses

are separated by the cavity round trip time of 11.4 ns. After point C, pulses increase in intensity

until they start to saturate, just before point D. When the pulse shown at point D reflects off

cavity mirror M2, the Pockels cell begins to turn on. It switches half-wave by the time the pulse

reaches the Pockels cell (~2.5 ns, half the duration of the Pockels cell voltage pulse). The pulse

then reflects off polariser TFP1.

0 50 100 150 200 2500.0

0.5

1.0

1.5

2.0

F

E

D

CB

A

Dio

de s

igna

l (V

olts

)

Time (nanoseconds)

-4 -2 0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

0.10

Figure 5.8 Traces from a digital oscilloscope showing the pump pulse (A, at 0 ns) and the

seeded build-up (around 200 ns) of the regenerative amplifier. Switch-out is just after 200 ns. Inset shows the position of the seed (red trace) compared to the pump pulse (green trace). The

ringing after each short pulse is the response of the photodiode to a 300 ps pulse.

The pulse at point E is the pulse that remains in the cavity after switch-out. This pulse

is large due to the inefficiency of the polarisers and the fact that the Pockels cell doesnt quite

switch half wave. Consecutive pulses decrease in intensity due to cavity losses. The voltage

pulse that has switched the Pockels cell for the second time travels back down the transmission


137

line to the power supply, reflects back and switches the cell for a third time. This causes the

drop in pulse height at point F and a strong post-pulse as it coincides with the residual cavity

pulse passing though the Pockels cell towards the output polariser, TFP2. The contrast between

the main pulse and the post pulse is approximately 5:1. Although this level of post pulse is bad,

it will be suppressed by subsequent amplification stages and could be removed by changing the

length of the transmission line between the power supply and Pockels cell.

5.3.1.4. Unwanted pulse suppression

As mentioned in the previous section, the efficiency of the broad bandwidth polarisers

in the regenerative amplifier cavity is not the best. One consequence of this is that leakage

pulses from the polarisers can make its way to the oscillator. As an amplified pulse travels to

the right in the regenerative amplifier cavity (Figure 5.7), the pulse reflects off polariser TFP2 in

the opposite direction to the seed pulse. As the intensity of the cavity pulse increases, the

intensity of the leakage pulse increases until it reaches the stage where it upsets the mode

locking mechanism in the oscillator. This either results in mistimed pulses emerging from the

oscillator or it dropping out (mode locking stops). In order to avoid this, an input slicer is

employed to block the route of the leakage pulses back to the oscillator. Figure 5.9a shows a

schematic of the input slicer. This slicer replaces the Faraday isolator used in the original

system. Faraday isolators can add high order phase that is difficult to compensate for (see

section 2.2.2.2).

PockelsCell

Polariser Polariserλ/2Removable

Energy Control

A

B

C

D

E

Polariser PolariserPockels

Cell

λ/2 Polariserλ/2

F GH

I

(a)

(b)

Figure 5.9 Schematics of the input (a) and output (b) regenerative amplifier slicers. The

removable λ/2 waveplate of the input slicer allows an alignment beam to pass through the second polariser. The output energy of the output slicer is controlled by allowing energy to be

rejected (via route G) by rotating the first λ/2 waveplate. Rays travel from left to right.

Pulses from the oscillator arrive from the left into a cube polariser. Cube polarisers are

used due to the large extinction ratio they offer (up to 104:1). The first polariser rejects any light


138

from the oscillator that is not exactly horizontally polarised, via route A (Figure 5.9a). The

Pockels cell fires half-wave at 10 Hz, the same rate as the regenerative amplifier Pockels cell.

This serves, in the first instance, to select oscillator pulses at 10 Hz from the 89.4 MHz pulse

train. Under normal operation the half wave plate immediately after the Pockels cell is not in

place. This is only used to allow an alignment beam to pass through polariser TFP2 when the

Pockels cell is not fired. When the Pockels cell does fire, the polarisation of the beam is

switched to vertical allowing it to pass through the second Pockels cell. The cell remains in the

on state for ~5 ns, just long enough to allow one oscillator pulse through; subsequent pulses

are rejected via route C.

Leakage pulses from the regenerative amplifier arrive from the right of Figure 5.9a.

Unswitched leakage pulses are rejected at the first polariser via route D. Switched pulses,

switched out by the regenerative amplifier Pockels cell firing from reflected electrical pulses,

are rejected by the second polariser via route B. The Slicer Pockels cell is required to be off to

coincide with returning pulses.

Another consequence of the inefficiency of the regenerative amplifier intracavity

polarisers is pre-pulses in the output of the laser system as a whole. Amplified pulses travelling

to the left in the regenerative amplifier cavity (Figure 5.7) reflect off polariser TFP1 in the same

direction as the switched out, amplified pulse. As the intensity of the pulse in the cavity

increases, so does the intensity of the leakage pulse. These pulses, after propagating through the

rest of the amplification stages, can reach intensities as high as 10% of the main pulse. In order

to guard against this, an output slicer is used to block the output of the regenerative amplifier

until just before the main pulse arrives. The leakage pulses are separated by the round trip time

of the regenerative amplifier (11.4 ns), therefore a Pockels cell (with a rise time of ~2.5 ns) is

fast enough to remove these pulses.

Figure 5.9b shows a schematic of the output slicer that includes a waveplate-polariser

combination to control the laser energy. Output pulses from the regenerative amplifier arrive

from the left into a cube polariser. This rejects leakage pulses from the regenerative amplifier

polarisers that are horizontally polarised, via route F. The following waveplate and polariser

gives control over the output energy of the laser system as a whole by rotation of the waveplate.

Unwanted energy is rejected via route G. The Pockels cell-waveplate-polariser combination

that follows acts as the pulse selector. In normal operation the waveplate offers no rotation, this

is only used to bypass the slicer in cases of malfunction. When the Pockels cell is off, the

polariser rejects leakage pulses via route H. When the Pockels cell fires (half wave), just before

the arrival of the main pulse, the pulse polarisation is rotated by 90û allowing it to pass through

the polariser and to subsequent amplification stages. This combination of polarisers and

Pockels cell offers a contrast between pre pulse and the main pulse of ~105.


139

5.3.1.5. Spectral characteristics

As mentioned in section 2.2.1.4, the finite width of the gain profile of titanium-doped

sapphire can cause narrowing of the spectral bandwidth of the amplified pulse. In addition, if

the centre wavelength of the gain is different from that of the seed pulse, gain shifting can

occur. From a study of the literature on titanium-doped sapphire crystals, it is apparent that the

exact centre and width of the gain profile of this laser crystal depends on the doping level of

Ti3+ ions in the sapphire lattice. The doping level of the crystal used in the regenerative

amplifier is 0.10%.

The blue curve of Figure 5.10 shows the spectrum of the regenerative amplifier when it

free-runs or is unseeded. The centre of this spectrum is at 785 nm and it has a width of 21.6 nm.

The centre of this spectrum is not necessarily at the centre of the gain; it is a combination of this

and the other wavelength dependent components in the regenerative amplifier cavity. The width

of the spectrum does not represent the width of the gain profile; it is a combination of the

bandwidth of the wavelength dependent components and the bandwidth of the gain raised to the

power of the number of passes (in this case ~36, the approximate build-up time).

760 770 780 790 800 810 820 830

0.0

0.2

0.4

0.6

0.8

1.0 Oscillator Regen Output

Nor

mal

ised

inte

nsity

(ar

bitr

ary

units

)

Wavelength (nanometres) Figure 5.10 Comparison of pulse spectra before (red curve) and after (green curve) the

regenerative amplifier and its free-run spectrum (blue curve). Curves show the effect of gain shifting.

To illustrate the effects of gain shifting, the seed pulse was shifted to a slightly longer

wavelength than its normal centre value. The centre of the seed spectrum (red curve in Figure

5.10) is 805 nm and the bandwidth is 21.4 nm. The output of the regenerative amplifier after 36

passes of the gain, Pockels cell and two polarisers is shown as the green curve in Figure 5.10.

The width of the seed pulse spectrum has been reduced to 16.2 nm and the centre wavelength


140

has been shifted down to 799 nm. The input spectrum has been heavily compromised by the

regenerative amplifier due to the finite gain bandwidth. The input spectrum in this example was

a little on the small side, the normal operating bandwidth is 30 nm, which is reduced to 24 nm

by the amplifier. This bandwidth is enough to support a 28 fs pulse with a sech2 temporal

profile. The centre wavelength of the seed pulse is also normally set to 801 nm, which is shifted

to 795 nm by the regenerative amplifier. This shift to shorter wavelengths can be balanced in

the power amplifier by gain saturation where shorter wavelengths are clipped, bringing the

centre wavelength back up to ~800 nm.

5.3.1.6. B-integral The design parameters of the regenerative amplifier cavity have been chosen to keep the

accumulated B-integral to a minimum. It is considered that, in order to avoid the effects of self-

focusing and self-phase modulation in the amplifier chain, the accumulated B-integral should

not exceed a value of between three and five (see section 2.2.3.2). Table 5.1 shows a summary

of the optical components inside the regenerative amplifier cavity. The lengths of each material

after a single pass of the cavity are shown along with the associated value of the nonlinear

refractive index, n2.

Component Length (cm) n2 (cm2W-1)

Titanium:Sapphire 2.0 3.10×10-16

Polarisers (bk7) 1.0 (2 × 0.5) 3.98×10-16

Pockels Cell (KD*P) 2.0 2.65×10-16

Table 5.1 Summary of optical components in the regenerative amplifier, their lengths and n2 values [Nikogosyan 97].

Using equation 2.8 for B-integral and the data in the above table, the contribution from

the regenerative amplifier is 0.10 radians (assuming 36 cavity passes and input and output

energies of 2 nJ and 2 mJ respectively). Considering that the major contribution of B-integral

usually comes from the regenerative amplifier, given the large number of passes and high

intensities, this is a good basis for the rest of the laser system.

Further B-integral contributions arise from the output slicer. The optical components in

the slicer, their lengths and nonlinear refractive indices are summarised in Table 5.2. The B-

integral contribution from the output slicer is 0.0066 radians. In comparison, the input slicer

contributes 2.2×10-8 radians to the B-integral. The difference is due to the six orders of

magnitude difference in intensity between the input and output of the regenerative amplifier.


141

Component Length (cm) n2 (cm2W-1)

Waveplates (bk7) 0.1 (2 × 0.05) 3.98×10-16

Polarisers (bk7) 2.4 (3 × 0.8) 3.98×10-16

Pockels Cell (KD*P) 2.0 2.65×10-16

Lenses (bk7) 0.6 (2 × 0.3) 3.98×10-16

Table 5.2 Summary of optical components in the output slicer, their lengths and n2 values [Nikogosyan 97].

5.3.2. Pre-amplifier The pre-amplifier is a two-pass amplifier pumped with a portion of the energy from the

Surelite II [Continuum] (Figure 5.11). The output of the Surelite is split using a 30% (reflected)

beam-splitter to pump the 6 mm long, Brewster-cut titanium-doped sapphire crystal with

~65 mJ. The crystal has an effective aperture width of ~3 mm; the pump and seed beams are

matched to this. The pump beam is focused through the crystal with a 1 m focal length lens

placed 72 cm away. The seed beam size is set using a down collimating Newtonian telescope.

The pump fluence at the crystal surface is 0.92 Jcm-2. The pre-amplifier input energy is 2 mJ

and the output energy is 5 mJ giving a single pass gain of 1.58 (using equation 5.3) and total

gain factor of 2.5.

Ti:SapphireCrystal 65 mJ

532 nm

In

Out

B

Figure 5.11 Schematic of the pre-amplifier. B = beam block blocks excess pump energy that

passes through the Brewster-cut titanium-doped sapphire crystal.

The only optical material to be considered here in terms of calculating the B-integral

contribution is titanium-doped sapphire. The length of the crystal is 6 mm and the value for the

nonlinear refractive index is 3.10×10-16 cm2W-1 (from Table 5.1). After two passes of the

crystal, the B-integral contribution, given input and output energies of 2 mJ and 5 mJ

respectively, is 0.0099 radians.

5.3.3. Power amplifier The power amplifier is a scaled-up version of the pre-amplifier. The amplifying

medium in this case is a 15 mm cube of 0.10% titanium-doped sapphire. It has antireflection


142

coatings on two opposite surfaces for the pump wavelength at 532 nm and broad band centred at

800 nm. A schematic of the 4-pass amplifier is shown in Figure 5.12.

In

OutTi:SapphireCrystal

400 mJ532 nm

800 mJ532 nm

Figure 5.12 Schematic of the bow-tie power amplifier.

The crystal is pumped from both sides by two doubled YAG lasers, a BMI 500 Series

[BMI] delivering 400 mJ and a Spectra-Physics Quanta Ray [Spectra Physics] delivering

800 mJ. The distance between the pump laser and the crystal is on the order of 4 m, too far to

just let the beam propagate unaided. For this reason an imaging system is used that images the

doubling crystal in each pump laser to the amplifier crystal. This prevents the beams from

degrading as they propagate from the pump lasers and gives a uniform pump profile across the

amplifier crystal. The imaging system used is a Newtonian telescope, which means there are

intermediate foci. The intensity at the intermediate focal spot (~48µm 1/e2 radius) of the BMI is

2.1 GWcm-2; the region around the focal spot needs to be flushed with filtered air to stop the

beam breaking down on dust particles. The intensity at the intermediate focal spot (~25µm 1/e2

radius) of the Quanta Ray is 25.5 GWcm-2; the focal spot is flushed with filtered helium to stop

the beam breaking down in air due to multiphoton ionisation (ionisation potential of helium is

1.7 times higher than that of nitrogen).

The BMI is imaged so that the beam diameter at the crystal is 7 mm and the energy per

pulse is 400 mJ (5 ns full-width-half-maximum duration) giving a fluence of 1.0 Jcm-2. The

Quanta Ray is imaged so that the beam diameter at the crystal is 12 mm and the energy per

pulse is 800 mJ (in 3 ns) giving a fluence of 0.7 Jcm-2. A Galilean telescope is used to increase

the diameter of the seed beam into the amplifier crystal. Seed pulses have 5 mJ of energy from

the pre-amplifier and are amplified to 60 mJ by the power amplifier (before compression). The

total gain factor is 12 and the single pass gain is 1.86 (four passes). This is not the optimal

pump geometry; it is possible to pump the crystal much harder by decreasing the size of the

pump beams at the crystal surface. It has been possible to extract up to 200 mJ from the four-

pass amplifier. This corresponds to a total gain factor of 40 and a single pass gain of 2.52.

The large amount of energy deposited in the laser crystal by the two pump lasers causes

an effect called thermal lensing (see section 2.1.3). The induced lens is positive and so focuses

the beam each time it passes through the crystal. As long as the induced lens is not too strong,

the effect of thermal lensing can be eliminated by making the seed beam slightly diverging by

adjusting the separation of the lenses of the input telescope. By the time the beam has passed


143

through the crystal four times, it is recollimated. If the induced lens were too strong, the beam

would have to be so diverging that it would be clipped by the finite aperture of the crystal on the

first pass.

Another problem arising from the induced thermal lens is that of astigmatism, as the

beam passes through the crystal at an angle (see section 2.1.4). This is eliminated by passing

the seed beam though the crystal symmetrically, an even number of times and at small angles

(maximum 5.5º). The seed beam picks up astigmatism as it passes though the crystal on the

first pass, but as long as the beam is at the same angle on the second pass, the astigmatism will

be cancelled. The same occurs on the third and fourth passes.

The only optical material to be considered in the power amplifier in terms of calculating

the B-integral contribution is titanium-doped sapphire. The length of the crystal is 15 mm and

the value for the nonlinear refractive index is 3.10×10-16 cm2W-1 (Table 5.1). After four passes

of the crystal, the B-integral contribution, given input and output energies of 5 mJ and 60 mJ

respectively, is 0.032 radians. The B-integral contribution from the telescope before the power

amplifier (assuming two similar lenses to the telescope before the pre-amplifier) is

0.0059 radians.

The current titanium-doped sapphire crystal in the power amplifier has a defect, in that

it has inherent astigmatism. This is compensated for by tilting one of the lenses of the telescope

before the power amplifier. This is not an ideal situation as the astigmatism is not completely

removed and a small amount of coma is added. It also restricts the degree to which the crystal

can be pumped as the combination of the inherent astigmatism and the induced thermal lens is

awkward. A new crystal is on order at the time of writing this thesis; therefore, the full

potential of the power amplifier cannot be reported on here.

5.4. Compressor implementation The hardest part of the design of a chirped pulse amplification laser system has already

been achieved - the design of an aberration-free stretcher. The corresponding compressor

comprises two parallel diffraction gratings separated by the effective grating separation of the

stretcher. The grating input angles are the same as that for the stretcher. Figure 5.13 shows a

schematic of the compressor that corresponds to the stretcher described in section 4.3.1. The

diffraction gratings used are Jobin Yvon holographic, 1200 lines/mm, 140 mm × 120 mm [Jobin

Yvon]. The retroreflector shown in Figure 5.13 is a roof reflector made from two 130 mm ×

80 mm ZERODUR® [HV Skan] substrates, polished in house to λ/10 by Martin Horton

[Horton] and protected gold coated by Coherent Optics Europe [Coherent OE].


144

1142 mm

33 º

23º

OpticalAxis

RR

Figure 5.13 Detailed schematic of the final compressor design. RR = gold plated

retroreflector. Grating constant is 1200 lines/mm.

Second order dispersion, or chirp, from the compressor is controlled by changing the

separation of the two diffraction gratings along the optical axis. Optical material in the

amplifier chain stretches the pulse as it propagates and adds mostly linear chirp. This can be

compensated for by adjusting the compressor configuration. Figure 5.14 shows how a 35 fs

pulse is stretched (black curve) when the compressor grating separation is changed. The

contribution to the duration of the pulse imposed by higher order phase is shown by the red

(third order) and green (fourth order) curves.

-4 -2 0 2 4

-1000

-500

0

500

1000

Dur

atio

n in

crea

se (

fem

tose

cond

s)

Grating separation displacement (millimetres)

Phase order 2nd

3rd

4th

Figure 5.14 Duration addition to a 35 fs pulse as the separation between the gratings in the compressor is changed. Black curve shows duration change due to second order phase, red

curve from third order phase and green from fourth order phase. Maximum stretch due to third and fourth order phase is 20 fs and 4 fs respectively across the 1 cm grating separation

displacement shown. Calculated using the code described in chapter 3.

Small amounts of third and fourth order phase can be controlled by changing the angles

of the two diffraction gratings at the same time. This also adds a large amount of second order

phase that can be compensated for by changing the grating separation again. Figure 5.15 shows


145

how a 35 fs pulse is stretched when the two diffraction gratings are rotated about the centre of

their surfaces. The red and green curves show the stretch contributions from third and fourth

order phase respectively. The blue curve shows how much the grating separation has been

changed in order to keep the second order phase contribution zero as the gratings are rotated.

-0.4 -0.2 0.0 0.2 0.4

-30

-20

-10

0

10

20

D

urat

ion

incr

ease

(fe

mto

seco

nds)

Gratings angle displacement (degrees)

Phase order 3rd

4th

-25

-20

-15

-10

-5

0

5

10

15

20

25

Grating separation

compensation (m

illimetres)

Figure 5.15 Duration addition to a 35 fs pulse as the two diffraction gratings in the compressor are rotated. Positive displacement angle corresponds to rotating the grating anti-clockwise in

Figure 5.13. Red curve shows contribution from third order phase and green curve shows contribution from fourth order phase. Blue curve shows how much the compressor grating

separation was changed in order to keep the contribution from second order phase zero as the gratings are rotated. Calculated using the code described in chapter 3.

It can be seen from the previous two graphs that second order phase is the major

contributor to the duration of the recompressed pulse. Third and fourth order phase do

contribute to duration, but they have most effect on the contrast of the pulse. Figure 5.16 shows

how, given a pulse with initially zero residual third and fourth order phase, the contrast of the

pulse changes as the angle of the two compressor gratings are rotated. Pulse contrast is defined

here as the level of the second highest peak, either side of the main peak, when the main peak

height is normalised. It is obvious from the graph that third order phase dominates in its

contribution to pulse contrast. Pure fourth order phase at the extremes of the scale of gratings

angle displacement in Figure 5.16 would make the contrast 100:1.


146

-0.4 -0.2 0.0 0.2 0.4

0.00

0.05

0.10

0.15

0.20

0.25

Con

tras

t

Gratings angle displacement (degrees)

Phase order 3rd

4th

Figure 5.16 Change in contrast of a 35 fs pulse as the two diffraction gratings in the

compressor are rotated. Positive displacement angle corresponds to rotating the grating anti-clockwise in Figure 5.13. Red curve shows contribution from third order phase and green

curve shows contribution from fourth order phase. Contrast level assumes a pulse with originally zero third and fourth order residual phase. Calculated using the code described in

chapter 3.

The size of the diffraction gratings is an important consideration. If the gratings are too

small the input beam has to be small and there is a danger of damaging the gratings due to

excess fluence. If the gratings are too large, there is a limit to the geometry of the compressor;

the larger the gratings, the further away from Littrow configuration the compressor must be (see

section 3.1.1). There is also the issue of cost of diffraction gratings especially the holographic

variety, which are the most efficient for infrared wavelengths and consequently the most widely

used in chirped pulse amplification.

There is much debate over the damage thresholds of holographic diffraction gratings for

short pulse durations (femtoseconds). According to the Lawrence Livermore National

Laboratories in the US, who make such gratings, the damage threshold is believed to be

between 300 and 500 mJ/cm2 [Banks 97]. The maximum energy the Blackett Laboratory Laser

Consortium chirped pulse amplification laser system described could deliver before

compression is 200 mJ (see section 5.3.3), and the beam size depends on the choice of steering

mirror size used to transport the beam to experiments. Assuming that the maximum clear

aperture restriction on a mirror is when it is used at 45º, and that a 1.5 mm margin is unusable

around the circumference due to the limited coverage of the thin film coating (1 mm margin for

1 mirrors), Table 5.3 summarises the maximum beam sizes allowable for different off the

shelf mirror diameters (D).


147

Maximum beam size (mm)

D (inches) DU (mm) 1/e2 FWHM FG (mJcm-2) PB (Wcm-2) DB (m)

1 23.4 16.6 9.7 155.0 4.1×1012 0.03

2 47.8 33.8 19.9 37.4 9.8×1011 0.13

3 73.2 51.8 30.5 15.9 4.2×1011 0.31

4 98.6 69.7 41.0 8.8 2.3×1011 0.56

Table 5.3 Summary of allowable beam diameters as a function of mirror diameter. D = mirror diameter, DU = usable mirror diameter, 1/e2 = 1/e2 beam diameter, FWHM = full-width-half-

maximum beam diameter, FG = peak fluence at the grating surface assuming 200 mJ pulse energy, PB = peak beam power after compression assuming a 50% efficient compressor and

DB = propagation distance required to accumulate a B-integral of one radian.

The allowable beam diameters stated also assume that the beam is clipped at the 1/e2

intensity level by the sides of the mirror. The fraction of energy lost by clipping at this level is

small (~7%). Assuming a grating input angle of 33º and an input energy of 200 mJ, the peak

fluence on the grating is shown in the column labelled FG. The fluence on the grating does not

exceed the limit mentioned above for any beam size. This is not the limiting factor in the choice

of beam size; the size of the gratings is chosen to be large to accommodate the large pulse

spectrum. The column labelled PB shows the peak fluence of the recompressed beam, assuming

a compressor efficiency of 50% and a full-width-half-maximum pulse duration of 40 fs. The

column labelled DB shows how far this beam can propagate in air before a B-integral of one

radian is accumulated (n2(air) = 9.9×10-19 cm2W-1 [Nikogosyan 97]). The fluence of the

recompressed beam is obviously the limiting factor in beam size. A beam size of 30.5 mm

(FWHM) was chosen as a compromise and, at the highest energies, the beam is to be

propagated in argon whose nonlinear refractive index is a factor of 3.7 lower than that of air

(2.7×10-19 cm2W-1 [Nikogosyan 97]).

5.5. Laser characteristics This section summarises the output characteristics of the laser as a whole. Section 5.6

explains how each element of the laser system described in this chapter is joined together to

form the upgraded laser.

5.5.1. Temporal Using a second order, single-shot autocorrelator (see section 1.2.2.2) a measurement of

the duration of the output pulse has been made. Figure 5.17 shows the profile of this


148

measurement on a linear and logarithmic (inset) intensity scale. The dynamic range of the

measurement is limited to that of the CCD camera used (Pulnix). The duration of the pulse,

assuming a sech2 temporal profile, is 59±5 fs. This is higher than the predicted duration of 40 fs

and is most likely due to incomplete recompression.

-400 -300 -200 -100 0 100 200

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ised

Inte

nsity

Autocorrelation Time (femtoseconds)

-200 0 20010-2

10-1

100

Figure 5.17 Temporal profile of the output pulse of the laser. Full-width-half-maximum duration is 59±5 fs. Inset shows the profile of the pulse on a logarithmic intensity scale.

The spectrum associated with this pulse is shown in Figure 5.18. The time bandwidth

product of the measured temporal and spectral profiles is 0.55. The theoretical product for a

sech2 pulse is 0.315, which means there is excess bandwidth. Twenty nanometres bandwidth is

sufficient to support a 34 fs sech2 pulse.

It is impossible to infer exactly how to fix this problem without using a more

sophisticated pulse measurement device, such as a FROG [Kane 93a] or SPIDER [Iaconis 99]

(see section 1.2.2.3 and section 1.2.2.4). FROG or SPIDER should give an idea whether the

problem is due to lack of bandwidth or residual high order phase by measuring the electric field

and phase of the pulse both temporally and spectrally. With further optimisation, it should be

possible to reduce the pulse duration to the design duration of 40 fs, or even lower.


149

770 780 790 800 810 820

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Nor

mal

ised

inte

nsity

Wavelength (nanometres) Figure 5.18 Spectral profile of the output pulse of the laser (red curve). Full-width-half-maximum bandwidth of the pulse is 20 nm. Spectrum centred at 797 nm. The green curve

shows the emission lines of rubidium used to calibrate the spectrum.

5.5.2. Spatial An important aspect of the final laser intensity is the focal spot size. This is measured

using a far-field monitor, which comprises a lens and microscope objective (see section 1.2.2.1).

The microscope objective images the focal spot of the lens onto the array of a CCD camera.

The monitor is calibrated by imaging something of known size, such as a thin wire. Figure 5.19

shows the result of such a measurement on the beam at the output of the compressor. The spot

is formed using a lens of focal length 50 cm.

The left-hand image shows the focal spot measured directly from the CCD camera and

the right-hand picture shows the same image on a linear colour scale showing two orders of

magnitude of intensity. The spot has a 1/e2 radius of 23 µm, which is 1.9 times diffraction

limited (theoretical focal spot size calculated using ω0 = λf/ωπ). The near field spatial profile is

not as good as this as it suffers from aberration compensation in the power amplifier input

telescope. The beam has a more oval profile as it enters the compressor and has a nominal 1/e2

radius of 10.7 mm.

The far-field monitor is also useful for monitoring beam aberrations. When the gratings

in the compressor are not exactly parallel, the output beam is astigmatic. This manifests itself

as an astigmatic focus in the far-field monitor. If the beam has residual astigmatism before the

compressor, then it can be compensated for by misaligning the compressor gratings. This

causes the pulse to be incompletely recompressed. The far-field monitor can be used to measure


150

and eliminate this. The beam depicted in Figure 5.19 shows no residual astigmatism before or

after the compressor.

Figure 5.19 Beam shape at the focus of a 50 cm focal length lens. Arrow has length 50 µm.

Right image is the same as the left grey-scale image but shows full dynamic range (two orders of magnitude) of the measurement on a linear colour scale (ROYGBIV). 1/e2 radius is 23 µm

which is 1.9 times diffraction limited.

5.5.3. Energy As mentioned at the end of section 5.3.3, the laser system does not quite meet the design

specification in terms of pulse energy, due to aberrations in the power amplifier crystal. The

recompressed pulse energy is 30 mJ instead of the predicted 100 mJ (at the time of writing).

This is to be remedied as soon as a replacement amplifier crystal arrives. The shot-to-shot

energy stability is good with fluctuations characterised by a standard deviation of only 0.75 mJ.

Figure 5.20 is a histogram of the energy of consecutive shots before the laser was upgraded.

The graph is placed here to show how the energy fluctuates with a lognormal type distribution

around a central value. The upgraded laser fluctuates in the same fashion but with a higher

average energy and smaller standard deviation.

The maximum available focused intensity of the laser at the time of writing this thesis is

1.5×1017 Wcm-2, assuming an input 1/e2 beam radius of 10.7 mm, a lens of focal length 30 cm

(producing a 1.9 times diffraction limited focal spot) and a pulse duration of 60 fs. The

predicted best case focused intensity when the power amplifier is fully upgraded is

6.0×1019 Wcm-2, assuming an input beam 1/e2 radius of 25.9 mm, an f-3 off-axis parabolic

focusing optic and a pulse duration of 40 fs.


151

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

120

140

160

Num

ber i

n bi

n

Bin energy (millijoules) Figure 5.20 Accumulation of 4199 laser shots showing the variation of energy on a shot-to-shot basis for the laser before it was upgraded. Average energy 11.4 mJ with a standard deviation

of 1.5 mJ.


152

5.6. Full system Each section of the upgraded laser system has been independently described in this

chapter. In this section, the way in which these sections fit together to form the Blackett

Laboratory Laser Consortium, titanium-doped sapphire, chirped pulse amplification laser

system is described. Figure 5.21 shows a photograph of the completed laser (excluding the

compressor table). The whole laser fits onto three optical tables: two 3 m × 1.2 m tables for the

oscillator, stretcher and amplifiers and one 1.8 m × 0.9 m table for the compressor.

Figure 5.21 Picture of the completed laser system. A = oscillator, B = stretcher,

C = oscilloscope traces showing spectrum (top) and pulse train (bottom) from the oscillator, D = Millennia, E = Surelite, F = regenerative amplifier, G = input and output slicers, H = pre-amplifier, I = BMI, J = power amplifier (bow-tie) and K = Quanta ray. Photograph courtesy of

Nick Jackson - Physics department photographer at Imperial College.

The arrangement of each stage of the laser system on the optical tables is critical as

there is not a great deal of space. Individual sections of the laser are not necessarily isolated

from the next; a general schematic of the laser system is shown in Figure 5.22. In this

schematic, pink represents stages with amplification; green represents pump lasers; pale blue

represents passive optical stages; yellow represents measurement devices; red arrows show the


153

paths of infrared beams and green arrows show the paths of pump beams. The photograph in

Figure 5.21 was taken from the top left corner of the top table in Figure 5.22

Millennia

Surelite

A

Oscillator

S Slicers

Regenerative Amp

Pre-Amp

T

Power Amp (Bow-Tie)Quanta

RayBMI

Compressor

A FFM

Output

Stretcher

Figure 5.22 Schematic of the laser system. A represents autocorrelators, S is a spectrometer, T is a Newtonian telescope and FFM is a far-field monitor. The grey outlines represent the three

optical tables that accommodate the laser system.

For each amplification stage described in sections 5.3.1, 5.3.2 and 5.3.3, the

contribution to the B-integral is calculated. Table 5.4 shows a summary of the B-integral

contribution for each stage, the output energy from each stage and the total accumulated B-

integral of the whole laser. The accumulated B-integral is very low in the laser - due to the

careful choice of beam size in each amplification stage.

Laser Stage B-Integral (radians) Output Energy (mJ)

Regenerative Amplifier 0.10 2 mJ

Output Slicer 0.0066 2 mJ

Pre-Amplifier 0.0099 5 mJ

Power Amplifier Telescope 0.0059 5 mJ

Power Amplifier 0.032 60 mJ

Total = 0.15

Table 5.4 Summary of B-integral contributions from each laser stage, the total accumulated B-integral and output energy from each stage. All values of B-Integral stated to two significant

figures.


154

When adding all the individual stages of the laser together, it is important to make sure

they are timed together correctly. For example, when a Pockels cell fires, it is activated by a

voltage spike that is 5 ns wide and has an approximately Gaussian profile. The optical pulse

that it switches is 300 ps wide and is chirped. If the timing drifts or jitters by only 1 ns, the peak

intensity of the optical pulse is reduced and the spectrum is distorted. This affects both the

recompressed pulse duration and energy. The timing of the laser is controlled by two Stanford

Research Systems delay generators [SRS] that together give eight independently controllable

triggers. Figure 5.23 is a schematic of how each stage of the laser is timed with the rest.

11.2 ns

100.707 ms

OSC

10 Hz

InS

SFL

SQS

RIn

ROut

OutS

QRFL

QRQS

5.7 ns

4.5 ns

~3 µs

205.8 ns

~7 ns

BMIFL

BMIQS

~20 ns

118 ns

~190 µs

~173 µs

~227 µs

240 µs

172 µs

193 µs

Figure 5.23 Laser timing diagram. Time scales are nonlinear but pulse order is accurate. OSC represents 89.4 MHz pulse train from oscillator; 10 Hz is the oscillator pulse train divided by factor of ~89.4×105; InS = input slicer; SFL = Surelite flashlamp trigger; SQS = Surelite Q-

switch; Rin = regenerative amplifier switch in; Rout = regenerative amplifier switch out; OutS = output slicer; QRFL = Quanta Ray flashlamps; QRQS = Quanta Ray Q-switch; BMIFL =

BMI flashlamps; BMIQS = BMI Q-switch.

The first row, labelled OSC, represents the 89.4 MHz pulse train from the oscillator

monitored on a fast photodiode. This is divided down by a factor of ~89.4×105 by a divider

circuit built in-house to give a nominal 10 Hz train that forms the basis of the repetition rate of

the laser row labelled 10 Hz. The first trigger from the SRS delay generator is for the row

labelled InS, the Pockels cell of the input slicer. This trigger is timed ~3 µs after the 10 Hz

trigger to allow pre-triggering for experiments. The electrical pulse that fires the Pockels cell is

generated by the same driver as for the Pockels cell in the regenerative amplifier (see section

5.3.1.3). The delay between the input slicer and regenerative amplifier Pockels cell (row


155

labelled Rin) is controlled by the length of BNC cable connecting the two. The second firing of

the regenerative amplifier Pockels cell is controlled as described in section 5.3.1.3 (row labelled

Rout).

One problem with the SRS delay generators is a drift in the output delay as a function of

ambient temperature. The magnitude of the drift is also related to the length of the delay. For

example, a 1 µs delay suffers a drift of the order of 500 ps whereas a 100 µs delay suffers a drift

of the order of 5 ns (source: users manual for the delay generators [SRS]). Consequently, the

large (100s µs) delays required between the flashlamps and Q-switch of neodymium:YAG

lasers poses a problem. The exact position in time of the optical output of these lasers tends to

drift as the temperature in the laboratory changes (which it can do by as much as 10ºC through

the day). In addition, the infrared beams need to be timed in close relation to the Q-switch

trigger, so Pockels cell delays tend to be long and susceptible to drifts.

The solution to this problem is shown by the rows labelled SFL and SQS in Figure 5.23

the Surelite flashlamp and Q-switch triggers respectively. The Q-switch is triggered with a

short delay compared to the 10 Hz trigger so that the timing drift is small. The flashlamps are

triggered just before the next pulse in the 10 Hz train. The delay required to do this is the period

of the nominal 10 Hz trigger minus the 3 µs experimental delay and minus the 193 µs flashlamp

Q-switch delay required by the Surelite. The drift associated with this delay is large compared

to the timing of infrared triggers, but small compared to the flashlamp delay. It therefore has no

consequence to the output energy of the Surelite, or the arrival of the green optical pulse. The

row labelled SFL in the figure shows two triggers: the one on the right is the flashlamp trigger

for the next laser shot and the one on the left is the trigger timed from the previous pulse in the

10 Hz train that corresponds to the Q-switch trigger for the current laser shot. One problem

caused by this set-up is the Surelite does not fire correctly as soon as the SRS delay generators

are switched on. The first Q-switch trigger is not preceded by a flashlamp trigger so there is no

laser output it must wait for the next trigger in order to function correctly.

The next trigger is for the Pockels cell in the output slicer. The ~7 ns delay between the

output trigger for the regenerative amplifier and output slicer is set by the optical path between

the two Pockels cells (~2.1 m). This is true also of the delay between the output slicer and the

BMI Q-switch. The optical path between the output slicer Pockels cell and the power amplifier

crystal is ~6.1 m. The flashlamp and Q-switch triggers for the BMI and Quanta Ray are

controlled in the same way as for the Surelite. This ensures the green output pulses from the

two pump lasers arrive at the power amplifier crystal with the minimum of timing drift.

6. Summary, conclusions and further work

6.1. Summary This thesis has documented the modelling and upgrade of a state of the art high power,

chirped pulse amplification laser system. Motivations for the upgrade, a description of the

original system and methods of measuring ultrashort pulses are given in chapter one. Design

considerations for high power, short pulse laser systems were discussed in chapter two. These

include the effects of passing pulses with large bandwidths through many optical components

with limiting spectral characteristics, and the problem of the nonlinear response of optical

materials due to high intensity radiation.

To aid the design of the stretcher and compressor of the laser system, a suite of models

was written including a full three-dimensional ray-tracing model. These models form the basis

of the bulk of the theoretical part of the thesis and are described in detail in chapter three. A

description of how the three-dimensional model was rigorously tested (by modelling simple

optical configurations where the result is well described analytically) is given in chapter four.

The model was used to test many different stretchers systems, two of which are discussed in

detail. The second of these stretcher designs, based on the Öffner triplet, was shown to work

very well as it allows faithful recompression of 30 fs pulses stretched by a factor of 10,000. The

stretcher only performs well however, as long as optics with a high surface figure are used as

each optic is passed many times. This links high cost with high performance. To overcome this

problem, a stretcher based on a reflective doublet was designed. This system uses a positive

and negative focusing optic that is arranged such that the diffraction grating lies at an

aberration free position. This gives a large stretch factor with only a single pass (compared to

the double pass in the Öffner triplet based stretcher). The doublet-based design outperforms the

Öffner-based design as long as the doublet stretcher optic sizes are slightly increased. The two

stretchers perform comparably if the optic sizes are kept the same, but the surface figure

specification for the doublet stretcher is lower than that of the Öffner stretcher, much reducing

its cost.

In section five, the integration of the new stretcher design into the laser upgrade is

described. The upgrade includes the replacement of the original oscillator with an ultrashort

Summary, conclusions and further work

157

pulse version and upgrade of the amplifier chain. This includes increasing the overall gain

factor, and adapting the amplifiers so they are able to support short pulses with large associated

bandwidths. The implementation of the compressor that corresponds to the stretcher is then

discussed, followed by the final laser characteristics. In the final section of chapter five, the

system as a whole is described. This includes the timing of the pump lasers, a schematic of the

layout of the laser on three optical tables and a summary of the accumulated B-integral through

the laser.

6.2. Conclusions The primary aim of this PhD was to upgrade the Blackett Laboratory Laser Consortium,

titanium-doped sapphire, chirped pulse amplification laser system in terms of pulse duration and

energy. The first of these goals has been met by the implementation of a new short pulse

oscillator, stretcher and associated compressor. These components are completely new to the

upgraded system. The stretcher stretches the oscillator pulses from 30 fs to 300 ps, as designed,

avoiding damage in the amplifier chain due to self-focusing. The final pulse duration was

measured as 59 fs using a single-shot autocorrelator, reduced from 260 fs, just before the

upgrade commenced.

The laser system was designed to produce 40 fs pulses, however the final pulse duration

after the upgrade is currently 59 fs. There are several possible explanations for this discrepancy.

Firstly, a lack of bandwidth in the recompressed pulse. The oscillator produces pulses with a

time-bandwidth product of 0.55 compared to 0.441 for Gaussian pulses and 0.315 for sech2

pulses. If the cause of the extra, unused bandwidth is high order phase originating in the

oscillator, this will remain after recompression, as the effects of the stretcher and compressor

should cancel. These high order phase components may only be detected and removed with the

use of a FROG or SPIDER (see sections 1.2.2.3 and 1.2.2.4). The regenerative amplifier is the

major contributor to reductions in pulse bandwidth. This can be remedied by spectral filtering

in the amplifier cavity. If the increased pulse duration is caused by a lack of bandwidth, this

may alleviate the problem.

Secondly, high order phase will be added, to some degree, by the amplifier chain. The

compressor can compensate for this, to a certain extent, by careful arrangement of the

diffraction gratings (see section 5.4). If the added high order phase is too extreme, it cannot be

compensated in this way. Further compensation can be achieved by changes in the alignment of

the stretcher. Again, this can only be done with the help of a FROG or SPIDER. Thirdly, the

stretcher may well be aligned for aberration-free operation as described in section 5.2, but may


158

still be adding unwanted high order phase. This can only be detected and removed again by the

use of a FROG or SPIDER. Fourthly, after the beam has passed through the power amplifier, it

is slightly aberrated. The compressor is very intolerant to beam aberrations and consequently

causes the recompressed pulse to be longer in duration. This problem will be removed when the

new power amplifier gain crystal is incorporated into the system.

The final reason for the increase in recompressed pulse duration over that predicted is

probably a combination of these effects. Further optimisation will be achieved by replacement

of the power amplifier gain crystal, use of a measurement device that records the spectral and

temporal phase of the recompressed pulse and possibly by spectral filtering in the regenerative

amplifier.

The upgrade of the pulse energy has been partially implemented. The amplifiers have

been adapted to cope with the large bandwidth associated with these short pulses. This has

clearly been achieved, as the full-width-half-maximum bandwidth of the output pulse is 20 nm

compared with the 9 nm bandwidth of the original system. The oscillator pulses have been

successfully stretched and amplified up to an energy of 200 mJ. There is currently a problem

however with the gain crystal in the power amplifier which limits the available energy to 60 mJ

before compression. The laser consequently delivers an estimated 1.5×1017 Wcm-2 instead of

the best case predicted 6×1019 Wcm-2. This will be improved in the near future with a higher

quality, power amplifier crystal.

As a gauge of the accuracy of the focused intensity measurement, ion appearance

intensities in a time of flight spectrometer have been used. The appearance intensity of a

particular charge state is the intensity at which a corresponding number of electrons are ejected

from an atom via barrier suppression. The existence of this charge state is measured by looking

for a signal at a particular time (related to the mass and charge state of the ion) in an ion time of

flight spectrum. Using a 20 cm focal length lens (producing a 2.7 times diffraction limited focal

spot), argon with a charge state of 8+ (which has an appearance intensity for barrier suppression

ionisation of 2.65×1016 Wcm-2) was observed with a pulse energy of 7 mJ and duration of 70 fs.

However, neon 5+, which has an appearance intensity of 4.05×1016 Wcm-2 was not seen under

the same conditions. The amplitude of the Ar8+ peak in the ion spectrum was small compared to

those for lower charge states, suggesting the actual laser intensity was closer to the argon

appearance intensity. The estimated focused intensity for the same conditions, using equation

3.5, is 3.5×1016 Wcm-2.

The following table gives a summary of the focused intensities mentioned in sections

1.5, 5.5.3 and in this section. Equations 3.5 and 3.6 have been used in each case to calculate the

focused intensity, assuming a sech2 temporal profile. Original refers to the laser before the

upgrade; Upgraded refers to the laser at the time of writing this thesis; Predicted refers to the


159

best case focused intensity, using an f-3 off-axis parabolic focusing optic producing a

diffraction limited focal spot and using the maximum available energy from the amplifiers; and

Appearance I refers to the conditions of the laser when the appearance intensity measurement

was made.

E (mJ) ∆t (fs) λ (nm) ω (mm) ω0 (µm) f (cm) N I0 (Wcm-2)

Original 40 150 780 10.0 7.5 30.0 2.0 6.7×1016

Upgraded 30 60 800 10.7 7.1 30.0 1.9 1.5×1017

Predicted 100 40 800 25.9 1.5 15.5 (f-3) 1.0 6.0×1019

Appearance I 7 70 800 10.7 4.8 20.0 2.7 3.5×1016

Table 6.1 Summary of focused intensities mentioned at various points through this thesis. E = pulse energy; ∆t = full width half maximum pulse duration; λ = laser wavelength;

ω = input beam 1/e2 radius; ω0 = focal spot 1/e2 radius; f = focal length of focusing optic; N = number of times diffraction limited focal spot; and I0 = peak focused intensity.

6.3. Further work Future directions for work on the laser system are obvious. At the time of completion

of this thesis a replacement, titanium-doped sapphire crystal has arrived. The first step is to

exchange the old crystal, increase the fluence of the pump lasers in the crystal and remove the

aberration compensation prior to the power amplifier. This should bring the output pulse energy

up to the 100 mJ level as designed. The old power amplifier crystal will be used to replace the

crystal in the pre-amplifier, as this is undersized. Although aberrated, only a small volume of

the crystal will be used and it will only be passed twice, so the aberrations imparted should be

small.

Further reduction of the pulse duration should be possible given the available bandwidth

after compression. In order to be able to do this, either a FROG or SPIDER must be built to

measure the output pulses. By doing this, the reason for the increase in pulse duration over

predicted values can be discovered. With further optimisation it should be possible to reduce

the pulse duration to the design duration of 40 fs, or even lower.

The further work that is required to bring the laser up to specification has now to be

scheduled. The laser is a working system and as I finish this thesis, the third experimental run is

under way to investigate the explosion of atomic clusters in the presence of a 60 fs high

intensity laser field.

As well as the publications already mentioned in this thesis that have arisen from my

modelling work ([Mason98a] and [Mason98b]), the following publications have been enabled


160

by work that I have carried out on the titanium-doped sapphire laser system: [Springate 00a],

[Springate 00b], [Tisch 99], [Hay 99], [Smith 99], [Tisch 98], [Tisch 97], [Ditmire 97c],

[Ditmire 97b] and [Ditmire 97a]. Also, my work has featured in the highly successful

Engineering and Physical Science Research Council (EPSRC) programme grant number

GRL34334 and has formed a large part of the successful grant proposal for research over the

next four years (programme grant number GRN11292). As part of this proposal I shall be

working on, amongst other things, ray-tracing modelling for the implementation of a short-pulse

upgrade to the Laser Consortium glass-based laser, pulse tailoring on the titanium-doped

sapphire system, hollow fibre compression and/or optical parametric chirped pulse amplification

(OPCPA) and the construction of a SPIDER and/or FROG.

161

References

A Antonetti 97 A Antonetti, F Blasco, JP Chambaret, G Cheriaux, G Darpentigny, C

LeBlanc, P Rousseau, S Ranc, G Rey and F Salin, “A laser system

producing 5 × 1019 Wcm-2 at 10 Hz”, Applied Physics B 65 197 (1997)

Asaki 93 MT Asaki, C-P Huang, D Garvey, J Zhou HC Kapteyn and MM Murnane

“Generation of 11-fs pulses from a self-mode-locked Ti-Sapphire laser”,

Optics Letters 18 977 (1993)

B Backus 97 S Backus, CG Durfee III, G Mourou, HC Kapteyn and MM Murnane, “0.2-

TW laser system at 1 kHz”, Optics Letters 22 1256 (1997)

Baltuska 97 A Baltuska, “Optical pulse compression to 5 fs at a 1-MHz repetition rate”,


Banks 97 PS Banks, “Frequency Conversion of High-Intensity Femtosecond Laser

Pulses”, PhD Thesis, University of California, Davis (1997)

BMI Nd:YAG Laser, 500 Series, B.M. Industries, 7 Rue du Bois Chaland, Z.I. du

Bois Chaland – LISSES, 91029 EVRY Cedex, France

Brabec 92 T Brabec, C Spielmann and F Krausz, “Limits of pulse shortening in solitary

lasers”, Optics Letters 17 748 (1992)

Braun 95 A Braun, JV Rudd, H Cheng, G Mourou, D Kopf, JD Jung, KJ Weingarten

and U Keller, “Characterization of short-pulse oscillators by means of a

high-dynamic range autocorrelation measurement”, Optics Letters 20 1889

(1995)

162

C Cheriaux 96 G Cheriaux, P Rousseau, F Salin, JP Chambaret, B Walker and LF Dimauro,

“Aberration-free stretcher design for ultrashort-pulse amplification”, Optics

Letters 21 414 (1996)

Christov 94 IP Christov, MM Murnane, HC Kapteyn, JP Zhou and CP Huang, “4th-

Order dispersion-limited solitary pulses”, Optics Letters 19 1465 (1994)

Christov 96 IP Christov, VD Stoev, MM Murnane and HC Kapteyn, “Sub-10-fs

operation of Kerr-lens mode-locked lasers”, Optics Letters 21 1493 (1996)

Codling 93 K Codling and LJ Frasinski, “Dissociative ionisation of small molecules in

intense laser fields”, Journal of Physics B: Atomic and Molecular Optical

Physics 26 783 (1993)

Coherent Innova 100, Coherent –Ealing, Photonics Division, 15 Greystone Road,

Watford WD2 4PW, United Kingdom

Coherent OE Coherent Optics Europe Ltd, 27-35 Ashville Way, Whetstone, Leicester LE8

6NU, United Kingdom

Constantinescu 98 RC Constantinescu, S Hunsche, HB van Linden van den Heuvell, HG

Muller, C LeBlanc and F Salin, “Highly charged carbon ions formed by

femtosecond laser excitation of C-60: A step towards an x-ray laser”,

Physical Review A 58 4637 (1998)

Continuum Surelite II Continuum, Photonics Solutions Plc, Gracemount Business

Pavilions, 40 Captains Road, Unit A2/A3, Edinburgh EH17 8QF, United

Kingdom

Curley 93 PF Curley, Ch Speilmann, T Brabec, F Krausz, E Wintnet and AJ Schmidt,

“Operation of a femtosecond Ti-Sapphire solitary laser in the vicinity of zero

group-delay dispersion”, Optics Letters 18 54 (1993)

Curley 95 PF Curley, G Darpentigny, G Cheriaux, JP Chambaret and A Antonetti,

“High dynamic range autocorrelation studies of a femtosecond Ti:sapphire

oscillator and its relevance to the optimisation of chirped pulse

amplification systems”, Optics Communications 120 71 (1995)

Curley 96 PF Curley, C Le Blanc, G Chériaux, G Darpentigny, P Rousseau, F Salin, JP

Chambaret and A Antonetti, “Multi-pass amplification of sub-50 fs pulses up

to the 4 TW level”, Optics Communications 131 72 (1996)

CVI CVI Laser Corporation, Laser Optics and Coatings Catalogue

163

D Danson 98 CN Danson, J Collier, D Neely, LJ Barzanti, A Damerell, CB Edwards,

MHR Hutchinson, MH Key, PA Norreys, DA Pepler, IN Ross, PF Taday,

WT Toner, M Trentelman, FN Walsh, TB Winstone and RWW Wyatt, “Well

characterized 1019 Wcm2 operation of VULCAN – an ultra-high Nd:glass

laser”, Journal of Modern Optics 45 1653 (1998)

Danson 99 CN Danson, R Allott, G Booth, J Collier, CB Edwards, PS Flintoff, SJ

Hawkes, MHR Hutchinson, C HernandezGomez, J Leach, D Neely, P

Norreys, R Notley, DA Pepler, IN Ross, JA Walczak and TB Winstone,

“Generation of focused intensities of 5 X 1019 Wcm-2”, Laser And Particle

Beams 17 341 (1999)

DeLong 94 KW DeLong, R Trebino, J Hunter and WE White, “Frequency-resolved

optical gating with the use of second-harmonic generation”, Journal of the

Optical Society of America B 11 2206 (1994)

DeLong 96 KW DeLong, DN Fittinghoff and R Trebino, “Practical issues in ultrashort-

laser-pulse measurement using frequency-resolved optical gating”, IEEE

Journal of Quantum Electronics 32 1253 (1996)

Diels 85 J-CM Diels, JJ Fontaine, IC McMichael and F Simoni, “Control and

measurement of ultrashort pulse shapes (in amplitude and phase) with

femtosecond accuracy”, Applied Optics 24 1270 (1985)

Diels 96 J-C Diels and W Rudolph, “Ultrashort Laser Pulse Phenomena.

Fundamentals, Techniques and Applications on a Femtosecond Time Scale”,

(Academic Press, California 1996)

Ditmire 96 T Ditmire, K Kulander, JK Crane, H Nguyen and MD Perry, “Calculation

and measurement of high-order harmonic energy yields in helium”, Journal

of the Optical Society of America B – Optical Physics 13 406 (1996)

Ditmire 97a T Ditmire, JWG Tisch, E Springate, MB Mason, N Hay, JP Marangos, RA

Smith and MHR. Hutchinson, "High-Energy Ion Explosion of Superheated

Atomic Clusters", Nature 386 54 (1997)

Ditmire 97b T Ditmire, JWG Tisch, E Springate, MB Mason, N Hay, JP Marangos and

MHR Hutchinson, "High Energy Ion Explosion of Atomic Clusters:

Transition from Molecular to Plasma behaviour", Physical Review Letters

78 2732 (1997)

Ditmire 97c T Ditmire, E Springate, JWG Tisch, YL Shao, MB Mason, N Hay, JP

Marangos and MHR Hutchinson, "Explosion of Atomic Clusters Heated by

High Intensity, Femtosecond Laser Pulses", Physical Review A 57 369

(1997)

164

Ditmire 1998 T Ditmire, “Simulation of exploding clusters ionized by high-intensity

femtosecond laser pulses”, Physical Review A 57 R4094 (1998)

Dorrer 98 C Dorrer, C Le Blanc and F Salin, “Characterization of femtosecond kHz

amplifier chains by spectral shearing interferometry”, ” in Conference on

Lasers and Electro-Optics, Optical Society of America Technical Digest

Series 6, 517 (Optical Society of America, Washington DC 1998)

Dorrer 99a C Dorrer, “Implementation of spectral phase interferometry for direct

electric-field reconstruction using a simultaneously recorded reference

interferogram”, Optics Letters 24 1532 (1999)

Dorrer 99b C Dorrer, B de Beauvoir, C Le Blanc, S Ranc, J-P Rousseau, P Rousseau, J-

P Chambaret and F Salin, “Single-shot real-time characterization of chirped

pulse amplification systems by spectral phase interferometry for direct

electric-field reconstruction”, Optics Letters 24 1644 (1999)

Downer 84 MC Downer, RL Fork and M Islam, “3 MHz amplifier for femtosecond

optical pulses”, in Ultrafast Phenomena IV p27, (Springer-Verlag, Berlin,

Heidelberg 1984)

Du 95 DT Du, J Squier, S Kane, G Korn, G Mourou, C Bogusch and CT Cotton,

“Terawatt Ti-Sapphire laser with a spherical reflective-optic pulse

expander”, Optics Letters 20 2114 (1995)

F Faldon 92 ME Faldon, “A high intensity, short pulse neodynium laser”, PhD Thesis,

University of London (1992)

Faure 99 J Faure, J Itatani, S Biswal, G Chériaux, LR Bruner GC Templeton and G

Mourou, “A spatially dispersive regenerative amplifier for ultrabroadband

pulses”, Optics Communications 159 68 (1999)

Fittinghoff 95 DN Fittinghoff, KW DeLong, R Trebino and CL Laders, “Noise sensitivity

in frequency-resolved optical-gating measurements of ultrashort pulses”,

Journal of the Optical Society of America B 12 1955 (1995)

Fittinghoff 96 DN Fittinghoff, JL Bowie, JN Sweetser, RT Jennings, MA Krumbugel, KW

DeLong, R Trebino and IA Walmsley, “Measurement of the intensity and

phase of ultraweak, ultrashort laser pulses”, Optics Letters 21 884 (1996)

Fork 82 RL Fork, CV Shank and R Yen, “Amplification of 70 fs optical pulse to

gigawatt powers”, Applied Physics Letters 41 223 (1982)

165

Fork 84 RL Fork, OE Martinez and JP Gordon, “Negative dispersion using pairs of

prisms”, Optics Letters 9 150 (1984)

Fork 87 RL Fork, CH Brito-Cruz, PC Becker and CV Shank, “Compression of

optical pulses to six femtoseconds by using cubic phase compensation”,


Fraser 96 DJ Fraser and MHR Hutchinson, “A high-intensity titanium-doped sapphire

laser”, Journal of Modern Optics 43 1055 (1996)

French 95 PMW French, “The generation of ultrashort laser pulses”, Reports on

Progress in Physics 58 169 (1995)

Froehly 73 V Froehly et al, “Notions de réponse impulsionnelle et de fonction de

transfert temperelles des pupilles optiques, justifications experimentales et

applications”, Nouvelle Revue d’Optique 4 183 (1973)

G Gallmann 99 L Gallmann, DH Sutter, N Matuschek, G Steinmeyer, U Keller, C Iaconis

and IA Walmsley, “Characterization of sub-6-fs optical pulses with spectral

phase interferometry for direct electric-field reconstruction”, Optics Letters

24 1314 (1999)

Gavrila 92 M Gavrila, “Atoms in Intense Laser Fields”, (Academic Press 1992)

H Halbo Optics Private communication with Harry Koetser at Halbo Optics UK

Hargrove 64 LE Hargrove, RL Fork and MA Pollack, “Title unknown – Modelocked

Helium-Neon Laser”, Applied Physics Letters 5 4 (1964)

Hay 99 N Hay, E Springate, MB Mason, JWG Tisch, M Castillejo and JP Marangos,

"Explosion of C60 Irradiated with a High Intensity Femtosecond Laser

Pulse", Journal of Physics B 32 L17 (1999)

Hecht 87 E Hecht, “Optics”, Second Edition, (Addison-Wesley Publishing Company

1987)

Horton Martin Horton, Optics Section, Physics Department, Imperial College of

Science technology and Medicine

Hutchinson 89 MHR Hutchinson, “Terawatt lasers”, Contemporary Physics 30 355 (1989)

166

HV Skan H.V. Skan Ltd, 425/433 Stratford Road, Shirley, Solihull B90 4AE, United

Kingdom. ZERODUR is a registered trademark of SCHOTT

GLASWERKE, Mainz, Germany

I Iaconis 99 C Iaconis and AI Walmsley, “Self-referencing spectral interferometry for

measuring ultrashort optical pulses”, IEEE Journal of Quantum Electronics

35 501 (1999)

IMSL IMSL mathematical and statistical libraries, Visual Numerics Inc., URL:

http://www.vni.com/products/imsl/index.html

Itatani 97 J Itatani, Y Nabekawa, K Kondo and S Watanabe, “Generation of 13-TW,

26-fs pulses in a Ti:Sapphire laser”, Optics Communications 134 134 (1997)

J Jarem 99 JM Jarem, PP Banerjee, “Application of the complex Poynting theorem to

diffraction gratings”, Journal of the Optical Society of America A 16 1097

(1999)

Jobin Yvon Jobin Yvon – Spex, Groupe HORIBA, Instruments S.A. (UK) Ltd, 2-4

Wigton Gardens, Stanmore HA7 1BG, United Kingdom

K Kane 93a DJ Kane and R Trebino, “Characterization of arbitrary femtosecond pulses

using frequency-resolved optical gating”, IEEE Journal of Quantum

Electronics 29 571 (1993)

Kane 93b DJ Kane and R Trebino, “Single-shot measurement of the intensity and

phase of an arbitrary ultrashort pulse using frequency-resolved gating”,


Kapteyn 99 H Kapteyn and M Murnane, “Ultrashort light pulses: life in the fast lane”,

Physics World 12 No.1 31 (1999)

Kobayashi 00 Y Kobayashi, T Ohno, T Sekikawa, Y Nabekawa and S Watanabe, “Pulse

width measurement of high-order harmonics by autocorrelation”, Applied

Physics B 70 389 (2000)

167

Koechner 76 W Koechner, “Solid-State Laser Engineering”, (Springer-Verlag, New York

1976)

Kogelnik 66 H Kogelnik and T Li, “Laser Beams and Resonators”, Applied Optics 5

1550 (1966)

Kolmeder 79 C Kolmeder, W Zinth and W Kaiser, “Second harmonic beam analysis, a

sensitive technique to determine the duration of single ultrashort laser


L Langley 99a AJ Langley, EJ Divall, N Girard, CJ Hooker, MHR Hutchinson, A Lecot, D

Marshall, D Neely and PF Taday, “Development of a Multi-Terawatt,

Femtosecond Laser Facility – Astra”, Central Laser Facility, Rutherford

Appleton Laboratory Annual Report RAL-TR-1999-062 186 (1998-99),

URL: http://www.clf.rl.ac.uk

Langley 99b AJ Langley, N Girard, I Mohammed, IN Ross and PF Taday, “Astra

Development Phase 1 – a Femtosecond Terawatt Laser”, Central Laser

Facility, Rutherford Appleton Laboratory Annual Report RAL-TR-1999-

062 187 (1998-99), URL: http://www.clf.rl.ac.uk

Learner Private communication with Professor RCM Learner, Laser Optics and

Spectroscopy, Department of Physics, Imperial College of Science,

Technology and Medicine.

Le Blanc 96 C Le Blanc, P Curley and F Salin, “Gain-narrowing and gain-shifting of

ultra-short pulses in Ti:sapphire amplifiers”, Optics Communications 131

391 (1996)

Lemoff 92 BE Lemoff and CPJ Barty, “Generation of high-peak-power 20-fs pulses

from a regeneratively initiated, self-mode-locked Ti-Sapphire laser”, Optics

Letters 17 1367 (1992)

Lemoff 93 BE Lemoff and CPJ Barty, “Quintic-phase-limited, spatially uniform

expansion and recompression of ultrashort optical pulses”, Optics Letters 18

1651 (1993)

Lewenstein 95 M Lewenstein, P Salieres and A L’Huillier, “Phase of the atomic

polaraization in high-order harmonic-generation”, Physical Review A 52

4747 (1995)

L’Huillier 92 A L’Hiullier from “Atoms in Intense Laser Fields”, edited by M Gavrila

(Academic Press, Boston, 1992), pp 139-202

168

M Maine 88 P Maine, D Strickland, P Bado, M Pessot and G Mourou, “Generation of

ultrahigh peak power pulses by chirped pulse amplification”, IEEE Journal

of Quantum Electronics QE-24 398 (1988)

Malitson 67 IH Malitson, “Refraction and Dispersion of Synthetic Sapphire”, Journal of

the Optical Society of America 52 1377 (1967)

Mansuripur 99 M Mansuripur, L Lifeng and W-H Yeh, “Diffraction Gratings, Part 1”,

Optics and Photonic News 10 No.7 42 (1999) and “Diffraction Gratings,

Part 2”, Optics and Photonic News 10 No.8 44 (1999)

Mark 89 J Mark, LY Liu, KL Hall, HA Haus and EP Ippen, “Femtosecond pulse

generation in a laser with a nonlinear external resonator”, Optics Letters 14

48 (1989)

Martinez 87a OE Martinez, “3000 times grating compressor with positive group velocity

dispersion: application to fibre compensation in 1.3-1.6 µm region”, IEEE

Journal of Quantum Electronics QE-23 59 (1987)

Martinez 87b OE Martinez, “Design of high-power ultrashort pulse amplifiers by

expansion and recompression”, IEEE Journal of Quantum Electronics QE-

23 1385 (1987)

Mason 98a MB Mason and MHR Hutchinson, “Design considerations for a stretcher of

a 35-fs chirped pulse amplification laser system”, in Superstrong Fields in

Plasmas 426 Ch74 467 (AIP Press American Institute of Physics, AIP

Conference Proceedings 1998)

Mason 98b MB Mason and MHR Hutchinson, “Three-dimensional ray tracing model

for the design of femtosecond chirped pulse amplification laser systems” in

Conference on Lasers and Electro-Optics, Optical Society of America

Technical Digest Series 6, 281 (Optical Society of America, Washington DC

1998)

Maxwell Private communication with J Maxwell, Applied Optics, Department of

Physics, Imperial College of Science, Technology and Medicine

Melles Griot Melles Griot, The Practical Application of Light Catalogue (1999)

Mellish 93 R Mellish, PMW French, JR Taylor, PJ Delfyett and LT Florez, “Self-

starting femtosecond Ti-Sapphire laser with intracavity multiquantum-well

absorber”, Electronics Letters 29 894 (1993)

Millennia Millennia 5AC, Diode-pumped, cw Visible Laser, Spectra-Physics Ltd,

Boundary Way, Hemel Hempstead HP2 7SH, United Kingdom

Mindl 83 T Mindl, P Hefferle, S Schneider and F Dorr, “Characterization of a train of

subpicosecond laser-pulses by fringe resolved auto-correlation

169

measurements”, Applied Physics B - Photophysics and Laser Chemistry 31

201 (1983)

Miyamoto 93 Y Miyamoto, T Kuga, M Baba and M Matsuoka, “Measurement of ultrafast

optical pulses with two-photon interference”, Optics Letters 18 900 (1993)

Moulton 86 PF Moulton, “Spectroscopic and laser characteristics of Ti:Al2O3”,Journal

of the Optical Society of America B 3 125 (1986)

Muffett 94 JF Muffett, CG Wahlstrom and MHR Hutchinson, “Numerical modelling of

the spatial profiles of high-order harmonic”, Journal of Physics B – Atomic

and Molecular Optics 27 5693 (1994)

N Nees 98 J Nees, S Biswal, F Druon, J Faure, M Nantel, G Mourou, A Nishimura, H

Takuma, J Itatani, JC Chanteloup and Hönninger, “Ultrahigh intensity laser:

present and future”, in Superstrong Fields in Plasmas 426 Ch74 397 (AIP

Press American Institute of Physics, AIP Conference Proceedings 1998)

New Focus New Focus 6 GHz Photodiode 1437

Newport 00 Newport ltd., Optics and Mechanics Catalogue (1999/2000)

Newton 79 I Newton, “The First Book of Opticks”, (Dover Publications, New York

1979), first published in 1704

Nikogosyan 97 DN Nikogosyan, “Properties of optical and laser-related materials. A

handbook”, (John Wiley and Son Limited 1997)

Nisoli 96 M Nisoli, S De Silvestri and O Svelto, “Generation of high energy 10 fs

pulses by a new pulse compression technique”, Applied Physics Letters 68

2793 (1996)

O Öffner 71 A Öffner, US patent 3,748.015 (1971)

Optical Surfaces Optical Surfaces Ltd, Godstone Road, Kenley, CR8 5AA, United Kingdom

Oscillator Model TS Ti:sapphire laser kit, Kapteyn and Murnane Laboratories L.L.C.

212 Riverview Drive, Ann Arbour, MI 48104, United States. Also Asaki 93

170

P Papadogiannis 99 NA Papadogiannis, B Witzel, C Kalpouzos, and D Charalambidis,

“Observation of Attosecond Light Localization in Higher Order Harmonic

Generation”, Physical Review Letters 83 4289 (1999)

Perry 94 MD Perry and G Mourou, “Terawatt to Petawatt Subpicosecond Lasers",

Science 264 917 (1994)

Piche 91 M Piche, “Beam reshaping and self-mode-locking in nonlinear laser

resonators”, Optical Communications 86 156 (1991)

Posthumus 1996 JH Posthumus, AJ Giles, MR Thompson, W Shaikh, AJ Langley, LJ

Frasinski and K Codling, ”The dissociation dynamics of diatomic molecules

in intense laser fields”, Journal of Physics B: Atomic and Molecular Optical

Physics 29 L525 (1996)

Press 92 WH Press, SA Teukolsky, WT Vetterling and BP Flannery, “Numerical

recipes in FORTRAN”, Second Edition (Cambridge University Press 1992)

Proctor 92 B Proctor and F Wise, “Quartz prism sequence for reduction of cubic phase

in a mode-locked Ti:Al2O3 laser”, Optics Letters 17 1295 (1992)

Pulnix Interline transfer CCD camera, model TM-500-EX, Pulnix Europe Ltd,

Pulnix House, Aviary Court, Wade Road, Basingstoke RG24 8PE, United

Kingdom

R Reid 97 DT Reid, M Padgett, C McGowan, WE Sleat and W Sibbet, “Light-emitting

diodes as measurement devices for femtosecond laser pulses”, Optics Letters

22 233 (1997)

REOSC Division Optique haute performance de la Société REOSC, Avenue de la

Tour Maury, 91280 SAINT PIERRE DU PERRAY, France

Richardson Richardson Grating Laboratory (Formerly Milton Roy Company),

Spectronic Instruments Inc, 820 Linden Avenue, Rochester, NY 14625,

United States

Ross 97 IN Ross, P Matousek, M Towrie, AJ Langley and JL Collier, “The prospects

for ultrashort pulse duration and ultrahigh intensity using optical

parametric chirped pulse amplifiers”, Optics Communications 144 125

(1997)

171

Rudd 93 JV Rudd, G Korn, S Kane, J Squier and G Mourou, “Chirped pulse

amplification of 55-fs pulses at a 1-kHz repetition rate in Ti:Al2O3

regenerative amplifier”, Optics Letters 18 2044 (1993)

S Salin 92 F Salin, J Squire and G Mourou, “Large temporal stretching of ultrashort

pulses”, Applied Optics 31 1225 (1992)

Shank 74 CV Shank and EP Ippen, “Subpicosecond kilowatt pulses from a mode-

locked cw dye laser”, Applied Physics Letters 24 373 (1974)

Shao 96 YL Shao, T Ditmire, JWG Tisch, E Springate, JP Marangos and MHR

Hutchinson, "Multi-keV Electron Generation in the Interaction of Intense

Laser Pulses with Xe Clusters", Physical Review Letters 77 3343 (1996)

Sheehy 1996 B Sheehy and LF DiMauro, “Atomic and molecular dynamics in intense

optical fields”, Annual Review of Physical Chemistry 47 463 (1996)

Shuman 99 TM Shuman, ME Anderson, J Bromage, C Iaconis, L Waxer and IA

Walmsley, “Real-time SPIDER: ultrashort pulse characterization at 20 Hz”,

Optics Express 5 134 (1999)

Siegman 86 AE Siegman, “Lasers”, (University Science Books, Mill Valley, California

1986)

Silfvast 96 WT Silfvast “Laser Fundamentals”, (Cambridge University Press, 1996)

Sizer 81 T Sizer, JD Kafka, a Krisiloff and G Mourou, “Generation and amplification

of sub-ps pulses using a frequency doubled Nd:YAG pumping source”,

Optics Communications 39 259 (1981)

Smith 98 RA Smith, T Ditmire and JWG Tisch, "Characterization of a Cryogenically

Cooled High Pressure Gas Jet for Laser/Cluster Interaction Experiments",

Review of Scientific Instruments 69 3798 (1998)

Smith 99 RA Smith, JWG Tisch, T Ditmire, E Springate, N Hay, MB Mason, ET

Gumbrell, AJ Comley, LC Mountford, JP Marangos and MHR Hutchinson,

"The Generation of High Energy Ions by Photo-Induced Dissociation of

Atomic Clusters", Physica Scripta T80A 35 (1999)

Spectra Physics Quanta-Ray model GCR-270, Spectra-Physics Ltd, Boundary Way, Hemel

Hempstead, Hertfordshire HP2 7SH, United Kingdom

Spence 91a DE Spence, PN Kean and W Sibbett, “60-fsec pulse generation from a self-

mode-locked Ti-Sapphire laser”, Optics Letters 16 42 (1991)

172

Spence 91b DE Spence, JM Evans, WE Sleat and W Sibbett, “Regeneratively initiated

self-mode-locked Ti-Sapphire laser”, Optics Letters 16 1762 (1991)

Springate 00a E Springate, N Hay, JWG Tisch, MB Mason, T Ditmire, JP Marangos and

MHR Hutchinson, “Enhanced explosion of atomic clusters irradiated by a

sequence of two high-intensity laser pulses”, Physical Review A 61 044101

(2000)

Springate 00b E Springate, N Hay, JWG Tisch, MB Mason, T Ditmire, MRH Hutchinson

and JP Marangos, “Explosions of atomic clusters irradiated by high-intensity

laser pulses: scaling of ion energies with cluster and laser parameters”,

Physical Review A 61 to be published in June (2000)

Squier 98 J Squier, CPJ Barty, F Salin, C Le Blanc and S Kane, “Use of mismatched

grating pairs in chirped-pulse amplification systems”, Applied Optics 37

1638 (1998)

SRS Digital Delay/Pulse Generator model DG535, Stanford Research Systems

Inc, 1290-D Reamwood Avenue, Sunnyvale, California 94089, United States

Steinmeyer 00 G Steinmeyer, DH Sutter, L Gallman and U Keller, “Extreme Photonics:

Ultrafast Lasers”, Photonics Spectra 34 No.2 100 2000

Stingl 94 A Stingl, C Speilmann, F Krausz and R Szipocs, “Generation of 11 fs pulses

from a Ti:sapphire laser without the use of prisms”, Optics Letters 19 204

(1994)

Stingl 95 A Stingl, M Lenzner, C Speilmann, F Krausz and R Szipocs, “Sub-10-fs

mirror-dispersion-controlled Ti:sapphire laser”, Optics Letters 20 602

(1995)

Strickland 85 D Strickland and G Mourou, “Compression of amplified chirped optical


Sullivan 95 A Sullivan and WE White, “Phase-control for production of high-fidelity

optical pulses for chirped-pulse amplification”, Optics Letters 20 192 (1995)

Sullivan 96 A Sullivan, J Bondlie, DF Price and WE White, “1.1-J, 120-fs laser system

based on Nd:glass-pumped Ti:sapphire”, Optics Letters 21 603 (1996)

Svelto 93 O Svelto and DC Hanna, “Principles of Lasers”, Third Edition, (Plenum

Press, New York and London 1993)

T Takeuchi 94 S Takeuchi and T Kobayashi, “Highly efficient Ti:Sapphire regenerative

amplifier”, Optics Communications 109 518 (1994)

173

Tektronix Tektronix 7934 Storage Oscilloscope with following units: Tektronix

Sampling Unit 7S11 and Tektronix Sampling Sweep Unit 7T11

Tisch 95 JWG Tisch, “Studies of high-harmonic generation using high-power lasers”,

PhD Thesis, University of London (1995)

Tisch 97 JWG Tisch, T Ditmire, DJ Fraser, N Hay, MB Mason, E Springate, JP

Marangos and MHR Hutchinson, "Investigation of High Harmonic

Generation from Xenon Atom Clusters", Journal of Physics B 30 L709

(1997)

Tisch 98 JWG Tisch, DD Meyerhofer, T Ditmire, N Hay, MB Mason and MHR

Hutchinson, "Measurement of the Spatio-Temporal Evolution of High-Order

Harmonic Radiation using Chirped Pulse Spectroscopy", Physical Review

Letters 80 1204 (1998)

Tisch 99 JWG Tisch, N Hay, E Springate, ET Gumbrell, MHR Hutchinson and JP

Marangos, "Measurements of ion energies from the explosion of large

hydrogen iodide clusters irradiated by intense femtosecond laser pulses",

Physical Review A 60 3076 (1999)

Treacy 69 EB Treacy, “Optical pulse compression with diffraction gratings”, IEEE

Journal of Quantum Electronics QE-5 454 (1969)

Trebino 93 R Trebino and DJ Kane, “Using phase retrieval to measure the intensity and

phase of ultrashort pulses: frequency-resolved optical gating”, Journal of the

Optical Society of America A 10 1101 (1993)

Trentelman 97 M Trentelman, IN Ross and CN Danson, “Finite size compression gratings

in a large aperture chirped pulse amplification laser system”, Applied

Optics 36 8567 (1997)

W Walmsley 99 IA Walmsley, “Measuring ultrafast optical pulses using spectral shearing

interferometry”, Optics and Photonics News 10 No.4 28 (1999)

White 93 WE White, FG Patterson, RL Combs, DF Price and RL Shepherd,

“Compensation of higher-order frequency-dependent phase terms in chirped

pulse amplification systems”, Optics Letters 18 1343 (1993)

Wong 94 V Wong and IA Walmsley, “Analysis of ultrashort pulse-shape measurement

using linear interferometers”, Optics Letters 19 287 (1994)

174

Wood 91 WM Wood, CW Siders and DC Downer, “Measurement of femtosecond

ionisation dynamics of atmospheric density gases by spectral blueshifting”,

Physical Review Letters 67 3523 (1991)

Wülker 1994 C Wülker, W Theobald, D Ouw, FP Schäfer, and BN Chichkov, “Short-

Pulse Laser-Produced Plasma from C60 Molecules”, Optical

Communications 112 21 (1994)

Z Zhou 94 JP Zhou, G Taft, CP Huang, MM Murnane, HC Kapteyn and IP Christov,

“Pulse evolution in a broad-bandwidth Ti-Sapphire laser”, Optics Letters 19

1149 (1994)

Zhou 95 J Zhou, C-P Huang, MM Murnane and HC Kapteyn, “Amplification of 26-fs,

2-TW pulses near the gain-narrowing limit in Ti:Sapphire”, Optics Letters

20 64 (1995)

Zhou 96 J Zhou, J Peatross, MM Murnane, HC Kapteyn, “Enhanced high-harmonic

generation using 25 fs laser pulses”, Physical Review Letters 76 752 (1996)

Zubov 91 VA Zubov and TI Kuznetsova, “Solution of the phase problem for time-

dependent optical signals by an interference system”, Soviet Journal of

Quantum Electronics 21 1285 (1991)

Date post:	21-Apr-2015
Category:	Documents
Upload:	mike-mason
View:	43 times
Download:	1 times

Thesis - MBMason

Documents