Tailored Light - Creation and Characterization of...

Tailored Light - Creation and Characterization of Ultrashort Laser Pulses

Advanced Lab Course Physics, Bachelor

Fraunhofer ILT Aachen

Date of creation: 17.02.2017

Version 1.4 b en

Chair for Laser Technology

Prof. Dr. rer. nat. R. Poprawe M. A.

Prerequisite knowledge:

Absorption and emission of light in matter, creation of population inversion by optical pumping, 4-level laser system, laser resonators, stability conditions, continuous wave (cw) and pulsed operation, laser optics, creation of ultrashort pulses by mode locking, diffrac-tion, polarisation and laser safety.

Suggested literature for preparation:

Otter,G: Atome – Moleküle – Kerne: Band I Atomphysik, 2nd edition 1998 / Honecker,R: (this book offers a short introduction into laser physics)

Poprawe, R.: Lasertechnik I, Skript zur Vorlesung, 3rd edition, May 2008 (very elaborate lecture notes)

Lange, W.: Einführung in die Laserphysik, 2nd edition 1994

Demtröder, W.: Experimentalphysik 2: Elektrizität und Optik, 5th edition 2009

Siegmann, A. E., Lasers, University Science Books, 1986

Inhalt

Advanced Lab Course Physics, Bachelor I

Table of contents

1 Introduction 1

2 Laser safety 2

3 Theoretical basics 4 3.1 Interaction of light with matter 5 3.1.1 Einstein’s rate equations and the 2 level system,

process of pumping 5 3.1.2 The four-level laser system 6 3.1.3 Stationary operation 9 3.2 The Gaussian beam, transverse laser modes 13 3.2.1 Properties of the Gaussian beam 13 3.3 Optical resonators, longitudinal modes 17 3.4 Pulsed laser radiation 18 3.4.1 Creation of ultra-short pulses by mode locking 19 3.4.2 Active and passive modelocking, absorbers 24 3.4.3 Measuring the pulse durations 26 3.5 Etalon 28

4 Experimental Realisation 30

5 Conduction 34 5.1 Characterising the laser diode (pump source) 35 5.1.1 Implementing the beam path for the pump

radiation 35 5.1.2 Measuring the power curve of the diode laser 36 5.1.3 Preparing the beam profile measurement 37 5.1.4 Measuring the diode caustic 37 5.1.5 Preparations for Measuring the USP laser 38 5.2 Beam characterisation of the ultra short pulse laser

(USP laser) 38 5.2.1 Basic calibration of the measuring track and

powercurve of the mode-locked laser 38 5.2.2 Measurement of the PER (polarisation extinction

ratio) 39 5.2.3 Measurement of the (pulse-) repetition rate 39 5.2.4 Meassurement of the beam quality 39 5.2.5 Measurement of the spectrum 41 5.2.6 Measurement of the pulse duration 42

6 Analysis 43 6.1 Power curve and laser threshold for laser diode and

USP laser 43

Inhalt

Advanced Lab Course Physics, Bachelor II

6.2 Analysis of the repetition rate 43 6.3 Analysis of the beam quality 43 6.4 Analysis of pulse duration and spectrum 44 6.5 Analysis of the polarisation extinction ratio (PER) 45

7 Questions for self examination 46

8 Referenced documents 47

9 Appendix 48 9.1 Deriving the Gaussian beam from the wave

equation 48 9.2 Deriving the longitudinal conditions in the resonator 49 9.3 q-eigenparameter of a resonator 49 9.4 Pulse width of the interference of N equiphase

waves 52 9.5 Derivation of the time-bandwidth product 53 9.6 Locking of Modes 54

Introduction

Advanced Lab Course Physics, Bachelor 1

1 Introduction

Laser beam sources have been established to be important tools in various appli-cations since their first demonstration by Theodore Maiman in 1960. While in the first decades research focused on power scaling of various laser concepts, interest in ultra-short pulsed laser sources for industrial and scientific applications is on the rise. Many applications depend on precise control over relevant tem-poral, spatial, spectral and energetic beam properties (hence the importance of precise laser beam diagnostics). In this experiment, you will gain experience with both the creation of ultrashort laser pulses and the corresponding laser beam diagnostics. The main goals are:

Understanding the fundamental principles:

o Creation of ultrashort pulses by mode locking

o Time-bandwidth-limitation

o Diffraction limit of laser beams

Handling laser sources:

o Functionality and handling of laser sources and laser beam diagnos-tic devices

o Complete characterisation of laser radiation

o Laser safety

Dangers of working with laser radiation will be discussed first in this instructions manual and the resulting mandatory behavioural rules will be derived. These are to be applied under any circumstance!

In the consecutive chapter you will get acquainted with the theoretical basics. Be advised, that they contain a detailed set of formulas for fundamental laser pro-cesses, which has no immediate practical use for this experiment. The fundamen-tals, however, should be understood. For further clues refer to the chapter “Questions for self-examination”(ch. 7).

Laser safety


The sections below concern the experimental realisation, functional principles of the measuring devices as well as execution and analysis of the experiment.

We hope you will enjoy preparing and conducting the experiment. If questions arise during the preparation, feel free to contact one of the tutors. Contact in-formation can be extracted from the Hompage of the Advanced Lab Course Physics

Please bring a USB flash drive for data storage. The finished test report should be turned in (as a .pdf file) as an email to your tutor.

2 Laser safety

Figure 1: Laser warn-ing signs: Laser radiation. Avoid exposure of eyes or skin to direct or scattered laser ra-diation. Laser class 4.

You will handle a class 4 laser in this experiment. Laser class 4 means that the accessible laser radiation is very dangerous to eyes and dangerous to the skin. Even scattered radiation can be dangerous, hence it is mandatory to take off watches and other reflective jewellery. The laser radiation can cause fire and explosive hazard. Wear cotton clothing (since it is hard to ignite) in order to prevent the burning in of synthetical material in case of a hazardous situation. Furthermore the laser radiation is part of the infrared spectrum and thus invisible to the naked eye.

Before carrying out the experiment your tutor will conduct a workplace related health and safety briefing. In general the following hazards are to be taken into account:

Primary hazards: Laser beam (also if reflected or scattered)

http://institut2a.physik.rwth-aachen.de/de/teaching/praktikum-physikbachelor_versuche

http://institut2a.physik.rwth-aachen.de/de/teaching/praktikum-physikbachelor_versuche

Laser safety


Secondary hazards:

direct (laser systems) Electricity (power supply of the diode) Scattered and pumping radiation Beam dumps (can get hot)

indirect (exposed material) Ignition of explosive material Fire hazard

Also do consider the safety instructions of the laser manufacturer (see [6]).

Turning on the power supply of the laser will also turn on the laser warning light outside the room. In order to minimize hazards, the following safety instructions are to be heeded by all personnel in the lab:

Before turning on the laser:

1. Check the magnetic plate with information of the laser and fix it to the outer side of the door.

2. Wait until all personnel in the lab has put on protective equipment.

While the laser is on:

1. Don’t cross the optical path with tools or your body

2. Don’t lower your head to the same height as the optical path

3. NEVER look into the beam

Turning off the laser:

1. Push the stop button on the power supply first. NEVER directly turn off the laser with the key.

2. Confirm the shutter on the laser housing being closed.

Always turn off the laser source electronically with the stop button and physically by closing the shutter. These redundant safeguards serve the security of all in-volved.

Theoretical basics


You will be equipped with laser protection goggles, which are also suited for wearer of glasses. Handle these goggles carefully and don’t put them down on their glasses. Use the appropriate boxes for storage. Don’t try to clean the glasses without instructions of the tutor. In case your sight is being diminished by visual impairments, inform your tutor, he will clean them properly.

Always work together in the lab and never on your own. Always follow the in-structions of the tutor. Disregarding the safety instructions leads to immediate expulsion from the experiment. The experiment will then be considered failed.

3 Theoretical basics

The required components for a working laser source are an active laser me-dium (which emits and amplifies light), an energy input (called pump light source) and an optical resonator (which reflects the photons back into the laser medium for further amplification). To create ultra-short pulses, there are an absorber with a time-dependent absorption and a spectral filter built into the resonator. Basic knowledge of the physical processes of absorption and emission of light in matter, as well as knowledge of the general laser principle are a premise to this instructions manual. The instructions manual contains a brief summary of essential relations. More detailed information can be obtained from the literature suggested earlier in this instructions manual.

Figure 2: Mandatory components of a la-ser source

Theoretical basics


3.1 Interaction of light with matter

3.1.1 Einstein’s rate equations and the 2 level system, process of pumping

In this section we will discuss the fundamental processes of absorption, sponta-neous emission and stimulated emission in the context of a 2 level system and then expand the discussion to a 4 level system.

In a system of 2 energy levels 𝐸1 and 𝐸2 we assume a constant density of atoms N, which is comprised of population densities Ni of both energy levels i: N = N1 + N2. Since the total population density is supposed to be constant, it is sufficient to describe the population densities of one of the energy levels. The photon den-sity 𝜌(𝜈21) is the density of photons with the frequency 𝜈21 = ℎ(𝐸2 − 𝐸1).

Three fundamental processes changethe population densities over time.

1) Absorption of photons by atoms increase the population density N2 of the upper energy level E2 and are proportional to the photon density 𝜌(𝜈21) and the population density N1 of the initial energy level E1. This is expressed through the transition rate for absorption processes:

3.1

𝑑

𝑑𝑡𝑁2,𝐴𝑏𝑠 = −

𝑑

𝑑𝑡𝑁1,𝐴𝑏𝑠 = 𝐵12 𝑁1 𝜌(𝜈21) (absorption)

The proportionality constant is called Einstein coefficient for absorption 𝐵12.

2) Similarly the transition rate for stimulated emission is defined as:

3.2

𝑑

𝑑𝑡𝑁1,𝑆𝑡𝑖𝑚 = −

𝑑

𝑑𝑡𝑁2,𝑆𝑡𝑖𝑚 = 𝐵21 𝑁2 𝜌(𝜈21) (stimulated emission)

with the Einstein coefficient for stimulated emission 𝐵21.

3) The spontaneous emission on the other hand is independent of the photon density (the presence of a photon is not required for this process):

3.3

𝑑

𝑑𝑡𝑁1,𝑆𝑝𝑜𝑛𝑡 = −

𝑑

𝑑𝑡𝑁2,𝑆𝑝𝑜𝑛𝑡 = 𝐴21 𝑁2 (spontaneous emission)

The Einstein coefficient for spontaneous emission is 𝐴21.

The total transition rate is obtained by summing over the three processes. It can be simplified by using the constant total density of the atoms 𝑁 = 𝑁1 + 𝑁2 and the fundamental relation of Einstein coefficients 𝐵21 = 𝐵12:

Theoretical basics


3.4

𝑑

𝑑𝑡𝑁1 = −

𝑑

𝑑𝑡𝑁2 =

𝑑

𝑑𝑡𝜌(𝜈21) = 𝐴21 𝑁2 + 𝐵21 𝜌(𝜈21) 𝑁2 − 𝐵21 𝜌(𝜈21) 𝑁1

(Einstein’s rate equation

for population densities of the laser levels and the photon density)

These equations are applicable to equilibrium as well as non-equilibrium system states.

Population inversion is required for laser operation. This means, that more atoms are in an excited state then there are in the ground state. This is not the case for thermal equilibrium, which means, that energy must be externally fed to the sys-tem (pumping, e.g. optical by photons).

The two level system cannot be driven into population inversion by optical pump-ing. This is caused by pump photons simultaneously contributing to stimulated emission and absorption (since they need to have the same energy). This can be seen in the rate equations: In equation 3.4, the time derivative of N1 (the tem-poral change of the population of the lower laser level) is always positive. There-fore there are always more atoms releasing energy then there are excited. The best possible case is achieved, if the spontaneous emission vanishes (i.e. A21 = 0). Then both energy levels will in a stationary state be populated with 50% of the atoms in1, because the rate of change only depends on the population density. Photons will be absorbed until N1 = N2. If N2 > N1 the system will have more stimulated emission then absorption. If the spontaneous emission doesn’t vanish, equation 3.4 is always positive.

Conclusion: In a two level system the laser levels can both be filled half at best. There is no population inversion possible.

3.1.2 The four-level laser system

Population inversion is a prerequisite for laser operation. As shown in the previ-ous section, this requires more than two energy levels. Efficient laser materials usually have four energy levels (see Figure 3: scheme of a four level laser system). In addition to the two laser levels (in our case E1 and E2) those have an additional2 lower and upper pump level (in our case E0 and E3). This is convenient, because the pumping wavelength, which is required to induce population inversion (by absorption), does not contribute to stimulated emission at the same time, which would depopulate the upper laser level.

1 Assume equation 3.4 to equal 0 (stationary case) and solve for (e.g.) N. What happens for A21 = 0? What if A21 > 0? 2 As in mathematics: „a“ means „at least one“

Theoretical basics


For the discussion of the 4 level system we have to modify Einstein’s rate equa-tions of the previously discussed two level system.

Figure 3: scheme of

a four level laser sys-

tem

The atoms are excited from their ground state E0 to the upper pump state E3. These excited atoms release energy to lower their energy level to the upper laser level 𝐸2 by usually a non-radiative decay with the transition rate (pumping rate) 𝑃. The processes of spontaneous and stimulated emission as well as absorption take place between the laser levels 𝐸2 and 𝐸1. These transition rates are already known from Einstein’s rate equations for the two level system. The atoms from 𝐸1 decay to the ground state with an Einstein coefficient 𝐴10.3

The rate equations for levels 𝐸0 and 𝐸3 will be neglected because of the following approximations: Assuming the ground state will at all times have a much higher population then the laser levels, its population is approximately constant: 𝑁0 =𝑐𝑜𝑛𝑠𝑡. We therefore can assume the pump rate 𝑃 to be independant of the laser process and being constant. Assuming the upper pump level has a very short lifetime leads to its population being negligible: 𝑁3 = 0, since the atoms decay to the upper laser level 𝐸2 rapidly.

The rate equations of the population densities of the laser states of a 4-level-system can therefor be written as:

3.5

𝑑

𝑑𝑡𝑁2 = 𝑃 + 𝐵12𝜌𝑁1 − 𝐵21𝜌𝑁2 − 𝐴21𝑁2

𝑑

𝑑𝑡𝑁1 = 𝐵21𝜌𝑁2 − 𝐵12𝜌𝑁1 + 𝐴21𝑁2 − 𝐴10𝑁1

3 All other possible transitions (which actually do exist) e.g. from level E3 to E0 or the possibility of additional pump levels have been

disregarded here.

Theoretical basics


The rate equation of the photon density is also inherited from the two level sys-tem and modified by the resonator selectivity 𝐹 and the resonator loss factor 𝛽:

3.6

𝑑

𝑑𝑡𝜌 = 𝐵21𝑁2𝜌 − 𝐵12𝑁1𝜌 + 𝐹𝐴21𝑁2 − 𝛽𝜌

The resonator selectivity 𝐹 is the ratio of the photons, which are confined inside the resonator within the solid angle given by the resonator mirrors, and the total amount of photons, which stem from the process of spontaneous emis-sion.

3.7

𝐹 =

1

4𝜋∫ ∫ 𝑑𝜃𝑑𝜑 sin 𝜃 ≈

𝜃𝐵

0

1

2∫ 𝑑𝜃 𝜃 =

𝜃𝐵

0

1

4𝜃𝐵

22𝜋

0

Typical values for resonator selectivities are F = 10-4 to 10-6.

The resonator loss factor 𝛽 represents the outcoupling of photons through the resonator mirrors. Every time the beam is reflected at a resonator mirror, only the percentage given by the reflectivity 𝑅1 and 𝑅2 respectively of the beam re-mains within the resonator. Dividing by the round-trip time of the resonator yields the loss factor:

3.8

𝛽 = (1 − 𝑅1𝑅2)𝑐

2𝐿

There is no exact solution for the rate equations 3.5 and 3.6 in this general form. A simplification arises from an approximation regarding the equation of 𝑁1: Pop-ulation inversion can only be achieved, if the lower laser level 𝐸1 depopulates much faster than the upper laser level 𝐸2. This means the transition coefficient 𝐴10 must be much higher then 𝐴21 and 𝐵21𝜌. Then the term proportional to 𝐴10 dominates the rate equation of 𝑁1 and the other terms can be neglected:

3.9

𝑑

𝑑𝑡𝑁1 = −𝐴10𝑁1 ⇒ 𝑁1 ~ 𝑒−𝐴10𝑡 → 0.

The lower laser level depopulates rapidly. Now we can replace the population inversion 𝐷 = 𝑁2 − 𝑁1 with the population density of the upper laser level. The problem is now reduced to two equations4:

3.10

𝑑

𝑑𝑡𝐷 = 𝑃 − (𝐵21𝜌 + 𝐴21)𝐷

𝑑

𝑑𝑡𝜌 = 𝐵21𝐷𝜌 + 𝐹𝐴21𝐷 − 𝛽𝜌

4 The fundamental relation B21 = B12 does still apply

Theoretical basics


The rate equations are a system of two coupled differential equations. Their exact solution becomes more complicated due to the non-linear coupling by the prod-uct of 𝐷 and 𝜌.

3.1.3 Stationary operation

If the laser medium operates in a dynamic equilibrium5, the operation is called stationary. In that case one can deduce exact expressions for the photon density and population inversion from the coupled differential equations 3.10.

Pulse creation by mode locking can be assumed to be stationary, due to averag-ing over many pulses, since the dynamic of singular pulses is much faster than the lifetime of the involved energy levels.

3.11

𝑑

𝑑𝜏𝐷 = 0 ,

𝑑

𝑑𝜏𝜌 = 0

The population inversion now reads:

3.12

𝐷 =

𝑃

𝐵21𝜌 + 𝐴21

This can be plugged into the rate equation of the photon density and leads to the solution of a quadratic equation:

3.13

𝜌 =1

2[𝑃

𝛽−

𝐴21

𝐵21+ √(

𝑃

𝛽−

𝐴21

𝐵21)

2

+ 4 𝐹

𝛽

𝐴21

𝐵21𝑃]

The second solution of the quadratic equation (the one with the minus sign in front of the square root) can be discarded, since the photon density has to be positive. The solution of equations 3.12 and 3.13 is discussed in two regimes depending on the parameter 𝑃.

5 Refer to the experiment F08 LAS for non-stationary methods for pulse creation like Q-switsching

Theoretical basics


Weak pumping: Spontaneous emission

If the laser medium is only pumped weakly, spontaneous emission and outcou-pling from the resonator mirrors dominate:

3.14

𝑃 ≪

𝛽

𝐵21∙ 𝐴21

Next the square root in equation 3.13 has to be expanded according to relation equation 3.14. The photon density then is:

3.15

𝜌 ≈

𝐹

𝛽𝑃 ≪

𝑃

𝛽

Plugging this into equation 3.12 yields:

3.16

𝐷 =

𝑃

𝐴21(1 + 𝜌)≈

𝑃

𝐴21

for the population inversion.

The photon density and the population inversion are both linear in the pump rate 𝑃. The inversion results from the equilibrium of pump rate and total spontaneous emission from the upper laser level. The photon density is fixed by the loss factor 𝛽 and the resonator selectivity 𝐹. The dependence on 𝐹 shows that spontaneous emission is the dominant process: Since spontaneous emission emits into all spa-tial directions equally, only the percentage 𝐹 of the photons is retained and con-tributes to the laser process. Good resonators have a very small 𝐹 (10−4 to 10−6), hence the photon density rises very slowly with the pump rate.

Strong pumping: Laser operation

In case of high pump rates

3.17

𝑃 ≫

𝛽

𝐵21∙ 𝐴21

one can expand the square root for the photon density and obtains:

3.18

𝜌 ≈

𝑃

𝛽+

𝐹

𝛽

𝑃

𝐵21𝛽𝐴21

⁄ − 1− 1 ≈

𝑃

𝛽

This leads to the population inversion

Theoretical basics


3.19

𝐷 =

𝑃

𝐴21 +𝐵21𝛽

𝑃≈

𝛽

𝐵21.

For high pump rates the saturation inversion is achieved and won’t rise further with stronger pumping (Figure 4: The normalized population inversion 𝐷′ plottet against the normalized pump rate 𝑃′. Here: 𝛼 =

𝛽𝐴21

⁄ ). The saturation inversion is implied by the equilibrium of resonator losses and the stimulated emission of the medium; spontaneous emission is negligible for high pump rates. The photon density does rise further, since it only depends on the resonator losses. The res-onator selectivity looses importance due to the spontaneous emission being neg-ligible, hence the photon density rises much faster with the pump rate (Figure 5: normalized photon density 𝜌′ on a logarithmic scale plotted against the normal-ized pump rate 𝑃′ for various resonator selectivities. One can observe the change of the dependence while raising the pump rate 𝑃′).

Hence in this pumping regime laser operation can take place.

Figure 4: The nor-malized population inversion 𝐷′ plottet against the normal-ized pump rate 𝑃′. Here: 𝛼 =

𝛽𝐴21

⁄

The lasing threshold

The two limiting cases of the last section have shown that beneath a certain threshold the laser behaves like a thermal light source (since radiation is pri-marily caused by spontaneous emission). This threshold is called lasing threshold. Above that threshold the stimulated emission dominates (see Figure 5: nor-malized photon density 𝜌′ on a logarithmic scale plotted against the normalized pump rate 𝑃′ for various resonator selectivities. One can observe the change of the dependence while raising the pump rate 𝑃′). The lasing threshold is defined as:

Theoretical basics


3.20

𝑃𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 = 𝛽

𝐴21

𝐵21

The lasing threshold is therefor defined as the ratio of total losses (resonator losses and spontaneous emission) to stimulated emission. Plugging the photon density (equation 3.18) into the condition for the laser threshold (equation 3.20) leads to the following condition:

3.21

𝜌𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 =

𝑃𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑

𝛽 ⇒ 𝐵21𝜌𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 = 𝐴21 ⇔ 𝑤𝑠𝑡𝑖𝑚 = 𝑤𝑠𝑝𝑜𝑛

with the transition probability for stimulated 𝑤𝑠𝑡𝑖𝑚 and spontaneous emission 𝑤𝑠𝑝𝑜𝑛𝑡. At the threshold the probabilities of a photon being emitted by spon-

taneous or stimulated emission are equal. Above the threshold the photon den-sity is so high, that the probability of stimulated emission dominates. Most of the atoms transition into the lower laser level by stimulated emission before they can do so by spontaneous emission.

The threshold, however, only exists, if the system is enclosed in an effective res-onator. This is expressed by the dependence of the curve on the resonator selec-tivity 𝐹 in Figure 5: normalized photon density 𝜌′ on a logarithmic scale plotted against the normalized pump rate 𝑃′ for various resonator selectivities. One can observe the change of the dependence while raising the pump rate 𝑃′. The closer 𝐹 equals one, the lower is the suppression of spontaneous emission by the reso-nator. In the limiting case 𝐹 = 1 there is no resonator and the system emits purely thermally like a black body. The transition to laser operation does not occur.

Determining the laser threshold is one of the tasks in this experiment.

Figure 5: normalized photon density 𝜌′ on a logarithmic scale plotted against the normalized pump rate 𝑃′ for various resonator selectivi-ties. One can ob-serve the change of the dependence while raising the pump rate 𝑃′

Theoretical basics


3.2 The Gaussian beam, transverse laser modes

The transverse properties of laser radiation specify how field amplitude and in-tensity of the laser beam change during propagation.

3.2.1 Properties of the Gaussian beam

A general solution for the paraxial wave equation in empty space (see appen-dix 9.1) is any linear combination of fundamental solutions. Fundamental solu-tions are often called modes in the context of laser beams6. A mode is a special way of propagation of the radiation. A real laser beam usually is a superposition of multiple modes.Similarly to all other wave solutions in free space the solution of the paraxial wave equation are transverse waves, i.e. electric and magnetic field vector are perpendicular to the axis of propagation (called TEM-modes).

The lowest order solution for the wave equation in paraxial approximation is the Gaussian beam also known as TEM00- mode or fundamental transverse mode:

3.22

𝐸(𝑟, 𝑧) = 𝐸0

𝑤0

𝑤(𝑧)𝑒

−𝑟2

𝑤(𝑧)2𝑒−𝑖[𝜔𝑡−Ψ𝑇(𝑟,𝑧)−Ψ𝐿(𝑧)]

The field distribution is the combination of an amplitude factor and a phase fac-tor. The amplitude factor defines the transversal intensity distribution of the mode (here a Gaussian distribution), the phase factor is of concern for eigen-frequencies of spherical resonators primarily (see section 3.3).

Figure 6: Beam caus-tic: Evolution of the field amplitude of a Gaussian TEM00 mode along the axis of propagation.

6 The term „mode“ is a short form of „mode of propagation“

Theoretical basics


The transversal amplitude profile is Gaussian for any position z, hence the name Gaussian beam. 𝑤(𝑧) is the beam radius of the Gaussian beam at position z. For the infinitely expanded Gaussian beam this radius is defined as the distance from the beam axis at which the field amplitude is reduced by a factor of 1/e. The development of 𝑤(𝑧) is also called beam caustic.

The position 𝑧0 of minimal beam diameter is called beam waist 𝑤0.The point of origin (𝑧 = 0) of the propagation axis is usually put at the waist. The distance 𝑧𝑅,

at which the beam radius has increased by a factor of √2, is called Rayleigh length 𝑧𝑅.

The beam radius increases approximately linearly for large 𝑧 (𝑧 ≫ 𝑧𝑅). The far field divergence angle 𝜽 of a beam is defined as7:

3.23

𝜃 = lim

𝑧→∞

𝑤(𝑧)

𝑧

For a Gaussian beam this yields:

3.24

𝜃 ≔ 𝜃𝐵 =

𝑤0

𝑧𝑅=

𝜆

𝜋𝑤0

The Gaussian beam has the physically smallest possible angle of divergence. Lower divergence at the same beam diameter is impossible due to diffraction. Therefore 𝜃𝐵 is called the diffraction-limited angle of divergence or diffraction angle. Equation 3.24 is commonly written as:

3.25

𝜃 ∙ 𝑤0 =

𝜆

𝜋

This way the correlation of beam waist and angle of divergence is stressed, since 𝜆

𝜋⁄ is constant.

For real beams this equation is modified by the so called 𝑀2-factor, in order to quantify the deviation from an ideal beam. This yields the beam parameter product:

3.26

𝜃 ∙ 𝑤0 =

𝜆

𝜋⋅ 𝑀2

7 under consideration of the small-angle approximation

Theoretical basics


The smallest possible value is 𝑀2 = 1,0 for diffraction limited radiation. A laser beam with 𝑀2 < 1,4 is often called to operate in the “fundamental mode”.

The 𝑀2 factor (often called beam quality parameter) is an important parameter for the characterization of laser radiation. It dictates the focusability of the light. Considering an equal angle of divergence, a smaller 𝑀2 results in a smaller beam waist. Measuring the 𝑀2 factor is part of this experiment.

The phase factor

3.27

𝑃(𝑟, 𝑧) = 𝑒−𝑖(𝜔𝑡−Ψ𝑇−Ψ𝐿)

contains a transverse and a longitudinal contribution Ψ𝑇 and Ψ𝐿 respectively. The longitudinal term describes the oscillation of the phase along the axis of propa-gation and can be approximated as the phase factor of a plane wave8. The equi-phase surfaces are paraboloids of revolution. They can be approximated as spher-ical surfaces near the axis of propagation. They have the following radius of cur-vature:

3.28

(

𝜕2

𝜕𝑟2

Ψ𝑇(𝑟, 𝑧)

𝑘)

−1

= 𝑅(𝑧)

In the limiting cases

3.29

𝑧 → ∞: 𝑅(𝑧) → ∞ and 𝑧 → 0: 𝑅(𝑧) → ∞

the radius of curvature becomes infinite and the equiphase surfaces become planes after infinite propagation and at the beam waist. The minimum radius of curvature is achieved at the Rayleigh length: 𝑅(𝑧𝑅) = 2𝑧𝑅.

8 The actual deviation from this is called Guoy phase and has exciting consequences, which however exceed the scale of this experi-

ment.

Theoretical basics


Figure 7: transverse equi-phase surfaces of a TEM00 mode

The Gaussian beam is determined completely by two parameters (e.g. 𝑧0 and 𝑧𝑅). These parameters can be unified in the complex q-parameter9 𝑞 = (𝑧 − 𝑧0) +𝑖 𝑧𝑅.

Symbol Meaning

𝑤0 beam radius at 𝑧 = 0: the beam waist

𝑧𝑅 =𝜋𝑤0

2

𝜆 Rayleigh length

𝑤(𝑧) = 𝑤0√1 + (𝑧𝑧𝑅⁄ )

2

beam radius as a function of the distance z from the beam waist

𝑅(𝑧) = 𝑧 [1 + (𝑧𝑅

𝑧)

2

] radius of curvature of a equi-phase surface

Ψ𝑇(𝑟, 𝑧) =𝑘𝑟2

2𝑅(𝑧) transverse phase term

Ψ𝐿(𝑧) = 𝑘𝑧 − arctan (𝑧

𝑧𝑅) longitudinal phase term

9 Using the q-parameter helps dealing mathematicly describing beam-propagation and -mapping of more complex optical systems

(see chapter 9.3). This procedure is fundamental for calculation and analyzation of laser resonators (see chapter 3.3).

Theoretical basics


Table 1: Symbols for a Gaussian beam

3.3 Optical resonators, longitudinal modes

The purpose of the laser resonator is the feedback of the emitted radiation into the laser medium to achieve further amplification and the selection of one (or a few) modes from the many possible modes of the radiation field, which vary in frequency and orientation. Here the feedback is achieved by reflection and the selection of eigen-frequencies by superposition of the back fed fundamental waves. The eigenfrequencies are dictated by constructive interference of the respective fundamental waves, while all other fundamental waves undergo de-structive interference.

Figure 8: longitudi-

nal resonator modes

The longitudinal condition for eigen-modes in a cavity of length 𝐿, which is completely enclosed by ideal mirrors (𝑅 = 1), is discussed in appendix 9.2. The eigen-modes are standing waves (Figure 8: longitudinal resonator modes). The solution 𝐸 = 𝑐𝑜𝑛𝑠𝑡. is excluded due to the boundary conditions10; the lowest

possible eigen-frequency is 𝜈1 =𝑐0

2𝐿⁄ . The distance of neighbouring modes in

frequency space is:

3.30

Δ𝜈 =𝑐0

2𝐿

Hence the number of eigen-modes in a frequency interval is

3.31

𝑑𝑛 =

𝑑𝜈

Δ𝜈=

2𝐿

𝑐0𝑑𝜈 = 2

𝐿

𝜆

𝑑𝜈

𝜈

10 E = 0 for ideal mirrors on the surface of the mirror

Theoretical basics


Figure 9: Scheme for

a resonator round

trip

Now that the longitudinal modes are formulated as an eigenvalue problem, we can also understand the transversal properties (see chapter 3.2.1) as an eigen-value problem. An eigensolution is found if the transverse properties of a funda-mental mode reproduce within one resonator cycle. This is the case if the 𝑞-parameter 𝑞 = 𝑧 + 𝑖 𝑧𝑅 reproduces after every cycle (see Figure 9: Scheme for a resonator round trip ) 11. This means the curvature of the equiphase surface has to be equal to the curvature of the mirror, when reaching the mirror (see Figure 10: The Gaussian beam in a spherical resonator) 12. This corresponds to the caus-tic of a laser beam as in chapter 3.2.1 and hence a Gaussian beam.

Figure 10: The Gaussian beam in a spherical resonator

3.4 Pulsed laser radiation

For the description of pulsed laser radiation one distinguishes average power �̅� and pulse peak power 𝑃𝑃𝑒𝑎𝑘 (Figure 11: Time-resolved and average power of pulsed laser radiation.). Using pulsed laser radiation one can obtain pulse peak powers, which surpass the average power by several orders of magnitude. This is often beneficial for practical applications, because the desired effects often

11 The mathematical derivation can be found in chapter 9.3 12 Figure 10: The Gaussian beam in a spherical resonator is obtained by replacing some of the equiphase fronts in Figure 7: trans-

verse equi-phase surfaces of a TEM00 mode by mirrors.

Theoretical basics


depend on peak powers (e.g. material ablation for surface structuring), while undesired effects often scale with the average power (e.g. thermal effects).

Figure 11: Time-re-solved and average power of pulsed la-ser radiation.

If mode-locking is used to generate pulses, the pulses will have a fixed pulse spacing 𝑇 and thus a fixed repetition rate 𝜈𝑟𝑒𝑝. The time-resolved power 𝑃 fol-

lows a fixed temporal progression, which is characterized by pulse form (e.g. Gaussian), pulse duration 𝜏 (FWHM: full width half maximum) and pulse peak power 𝑃𝑃𝑒𝑎𝑘. The pulse energy is the temporal integral of the instantaneous power over one period. For Gaussian pulses the following formula applies:

3.32

𝐸𝑝𝑢𝑙𝑠𝑒 = ∫ 𝑑𝑡

period

𝑃(𝑡) ≈ 0,94 ⋅ 𝑃𝑃𝑒𝑎𝑘 ⋅ 𝜏 = �̅� ⋅1

𝜈𝑟𝑒𝑝

3.4.1 Creation of ultra-short pulses by mode locking

Pulsed laser radiation can be created using multiple methods. In non-stationary methods e.g. q-switching or cavity dumping, excitation energy is first deposited in the resonator and then released in a short burst of radiation energy. The in-ertness of the system limits the minimum pulse duration due to setting- and fading times.

In order to produce shorter pulses, stationary operation is necessary. This means, the temporal average of deposited energy inside the resonator is con-stant. By that, the inertness effects can be mostly avoided. One method of sta-tionary pulse creation is mode-locking. The mode-locking method is based on excitation and superposition of multiple longitudinal modes with a fixed phase relation. One can achieve pulse durations in the femtosecond regime and pulse peak powers in the order of Gigawatts with this method. It is known from section 3.3, that

3.33

𝑛𝜆 = 2𝐿

𝜔𝑛 = 𝑛𝜋𝑐

𝐿⇒ Ω = 𝜔𝑛+1 − 𝜔𝑛 =

𝜋𝑐

𝐿 (oder 𝑓𝑛+1 − 𝑓𝑛 =

𝑐

2𝐿)

applies for neighbouring resonator modes (see Figure 12: Scheme of a frequency spectrum of resonator modes.).

Theoretical basics


Figure 12: Scheme of a frequency spec-trum of resonator modes.

In theory there can be an unlimited number of modes within the resonator at the same time. These modes are initially independent of each other and oscillate with arbitrary statistically distributed relative phases and amplitudes. Now the total electric field in the resonator is the sum over the individual field strengths 𝐸𝑛 of all modes:

3.34

𝐸(𝑧, 𝑡) = ∑ 𝐸𝑛(𝑧, 𝑡) =

𝑛

∑ 𝐸0,𝑛𝑒𝑖(𝑘𝑛𝑧−𝑤𝑛𝑡) mit 𝐸0,𝑛 = |𝐸0,𝑛|𝑒𝑖𝜑𝑛

𝑛

with the complex amplitude 𝐸0,𝑛 and phase 𝜑𝑛 of the 𝑛-th mode. Due to the statistically distributed phases

3.35

∑ 𝐸𝑚(𝑧, 𝑡)𝐸𝑛∗(𝑧, 𝑡) = 0

𝑛≠𝑚

since single summands of the independent modes cancel each other on average. The total intensity in a resonator with 𝑁 ⋙ 1 modes is then:

3.36

𝐼(𝑧, 𝑡)~𝐸(𝑧, 𝑡)𝐸∗(𝑧, 𝑡) = ∑ ∑ 𝐸𝑚(𝑧, 𝑡)𝐸𝑛

∗(𝑧, 𝑡) = ∑ |𝐸0,𝑚|2𝑁

𝑚=1

𝑁

𝑛=1

𝑁

𝑚=1

If all amplitudes are equal, we get the total intensity:

3.37

𝐼(𝑧, 𝑡) = 𝑁|𝐸0|2 ≔ 𝑁𝐼0

Hence the superposition of modes leads to spatially constant total intensities, which scale linearly with the number of involved modes 𝑁, in the case of ran-domly distributed phases.

Theoretical basics


Figure 13: Equiphase superposition of four waves with a fre-quency discrepancy of Ω = 2𝜋𝜈

The pulse peak intensity rises, if all modes have the same phase (see Figure 13: Equiphase superposition of four waves with a frequency discrepancy of Ω =2𝜋𝜈). Assuming the simplification of the amplitudes of all modes being equal again, the total intensity yields:

Theoretical basics


3.38

𝐼(𝑧, 𝑡)~|𝐸0|2 ∑ ∑ 𝑒𝑖(𝑘𝑚−𝑘𝑛)𝑧−𝑖(𝜔𝑚−𝜔𝑛)𝑡 = |𝐸0|2 ∑ ∑ 𝑒𝑖(𝑚−𝑛)

Ω𝑐

(𝑧−𝑐𝑡)

𝑁

𝑛=1

𝑁

𝑚=1

𝑁

𝑛=1

𝑁

𝑚=1

Whenever the condition

3.39

Ω

𝑐(𝑧 − 𝑐𝑡) = 2𝜋 ⋅ 𝑗 ⇔ 𝑧 − 𝑐𝑡 = 2𝐿 ⋅ 𝑗 , 𝑗 = 0,1,2, …

is fulfilled, the exponential function in equation 3.38 becomes one for any sum-mand. The total intensity then has a maximum with the value

3.40

𝐼𝑚𝑎𝑥 = 𝑁2|𝐸0|2 = 𝑁2𝐼0

and scales with the square of the number of involved modes. Spatial and tem-poral distance of these maxima can be read from equation 3.39:

3.41

Δ𝑧 = 2𝐿 , Δ𝑡 =

2𝐿

𝑐≔ 𝑇

The maxima succeed each other at intervals of the time it takes for a photon to traverse the resonator once (𝑇), i.e. there is always one such maximum in the resonator.

Periodic pulses with a peak intensity, which is proportional to the square of the number of modes times the amplitude of a single mode, can thus be created through the fixed phase relation between modes. Mode locking is based on this principle. By involving numerous modes, very high peak intensities can be achieved (see Figure 14: Equiphase superposition of 𝑁 = 100 modes).

Theoretical basics


Figure 14: Equiphase superposition of 𝑁 =100 modes

The pulse duration can be obtained analogously to N-beam-interference (see ap-pendix 9.4)

3.42

Δ𝑇 =

1

𝑁

2𝐿

𝑐=

1

𝑁𝑇

The pulse duration decreases proportionally to 1𝑁⁄ with a rising number of

modes 𝑁.

The number of available modes is limited by the spectrum of amplification of the active laser medium (see Figure 15: Relation of spectrum of amplification of the

Theoretical basics


active medium and the modes selected by the resonator), since only a limited number of modes can be sufficiently amplified by the medium.

Figure 15: Relation of spectrum of am-plification of the ac-tive medium and the modes selected by the resonator

Spectral amplitude and the temporal progression of the field strength can be transformed into each other by Fourier transformation (see appendix 9.5). As-suming a Gaussian spectrum with full width half maximum (FWHM) 𝛿𝜈 and a pulse with Gaussian pulse shape with pulse duration 𝜏 one obtains the constant time-bandwidth product

3.43

𝜏 ⋅ 𝛿𝜈 =

2 ln(2)

𝜋≈ 0,44.

This means the creation of a pulse with a given maximum length implies a mini-mal width of the spectrum. A wide spectrum however does not imply short pulses13. A pulse has its shortest length, if the spectral phase is constant (or varies only linearly with the frequency). Such a pulse is called bandwidth limited or (Fourier) transform limited.

Due to this duration-bandwidth relation the pulse width can be changed by var-ying the spectrum and vice versa. If the spectrum gets constricted by a filter for example, the pulse duration increases. You will conduct this kind of beam ma-nipulation in this experiment.

3.4.2 Active and passive modelocking, absorbers

The last section layed out the theoretical basis of pulse creation through super-position of many phase linked modes. This chapter will be about the experi-mental realisation. A prerequisit is a medium, which can amplify modes with a large bandwidth, in order to have many modes available for the mode locking process.

13 frequencies with statistical phases yield not a pulsed but a continuous intensity distribution; see above

Theoretical basics


All mechanisms used for triggering mode locking operation are based on the same approach: Resonator losses (or amplification respectively) are time-de-pendently modulated with the differential frequency Ω (see equation 3.33). The easiest way to illustrate this effect is achieved by assuming the reflectivity of a resonator mirror to be variable:

3.44

𝑟 = 𝑟0 + �̃� cos Ω𝑡 , �̃� < 𝑟0

Such a modulation of resonator losses causes an additional time dependence of the field strength of the resonator modes. Thus sidebands are created, which coincide with neighboring resonator modes (see Figure 16: Schema of the mod-ulated spectrum. The contribution of the mode E0,n after modulation is high-lighted. and appendix 9.6. for a mathematicly more detailed description). If the laser is in stationary operation, then any mode can be modelled as a forced os-cillation (by the frequency and amplitude of the neighboring modes). This leads to a synchronization of the phases of directly neighboring modes14.

Figure 16: Schema of the modulated spectrum. The con-tribution of the mode E0,n after mod-ulation is high-lighted.

The loss modulation can be achieved activly or passively. Electro- and acousto-optical components can be used as active modulators. By applying an alternating voltage (frequency Ω) to a Pockels cell15 succeeded by a polarisation filter for example, losses with a frequency of Ω can be induced. Saturable absorbers are used for passive modulation. This way the laser pulse traversing the resonator induces the loss modulation itself: Every time it passes the absorber the absorber is driven into saturation and the absorbtion losses are reduced. Since 𝑇 = 2𝜋

Ω⁄ is the time of circulation of the resonator, the process creates a loss modulation of period Ω. This method is called passive modelocking or selfinduced modelocking. The laser of this experiment utilises this mechanism.

14 Actual realization often have a more complicated modulation, which can be represented as a fourier series, which causes modes

not only to couple to their direct neighbors but also to neighbores of higher order. 15 The polarization rotates in dependance of the applied voltage.

Theoretical basics


3.4.3 Measuring the pulse durations

Ultra short laser pulses can have pulse durations shorter than those resolvable with oscilloscopes and photodiodes, since their response time is much longer than the pulse duration. Hence another method is utilised to determine the pulse duration.

The method applied to measure pulse durations of laser pulses in this experiment is based on interferometric autocorrelation (refer to [3] for more details). The pulse is compared to (convolved with) a reference of itself. The spatial exten-sion, which is linked to the pulse duration by the speed of light, is measured. The basic measurement setup is displayed in Figure 17: Principle of an autocor-relator.

Figure 17: Principle of an autocorrelator

The laser pulse is split into two equal parts by the beam splitter (BS). One partial pulse propagates to mirror M1 and the other one to M2. The mirror M2 is mov-able and hence the path length BS-M2 can be varied independently of the path length BS-M1. Thus the partial pulses can be delayed by a time 𝜏 relative to each other16.

After reflection by M1 and M2 respectively both pulses are sent towards the nonlinear crystal (NC) collinearly by the beam splitter. There the combined beam with both pulses is frequency doubled. The efficency of the frequency conver-sion is highly dependant on the intensity, hence the frequency conversion occures only efficiently, if both partial pulses traverse the crystal at the same time. The more they are delayed relative to each other, the less frequency conversion takes place. The time lag 𝜏 of the splitted pulses can be calculated from the de-

16 Light traverses roughly 300 µm in a picosecond!

Theoretical basics


fined varied path length of the path BS-M2 using the speed of light. The fre-quency-doubled radiation is separated from the original wavelength through a spectral filter (F) and the intensity is measured with a detector (D).

In a mathematic sense the intensity of the frequency doubled beam is propor-tional to the square of the intensity of the superposed waves:

3.45

𝐼(2𝜔, 𝑡, 𝜏)~[𝐸(𝑡) + 𝐸(𝑡 + 𝜏)]4

The detector integrates over the intensity, since its response time is higher then the pulses duration and puts out a signal 𝑆 (see [4]).

3.46

𝑆(2𝜔, 𝜏) ~ ∫ |𝐸2(𝑡) + 𝐸2(𝑡 + 𝜏) + 2𝐸(𝑡)𝐸(𝑡 + 𝜏)|2 𝑑𝑡

Figure 18: A typical result of measuring the pulse duration with an autocorrela-tor.

Complete constructive interference occurs for 𝜏 = 0. The signal oscillates at the doubled frequency of the fundamental light due to interference phenomena for 𝜏 ≠ 0 and starts to fade with increasing distance from 𝜏 ≠ 0 since the pulses stop to overlap (see17 Figure 18: A typical result of measuring the pulse duration with an autocorrelator.). The term 𝐸(𝑡)𝐸(𝑡 + 𝜏) vanishes for 1/𝜈𝑅𝑒𝑝 ≫ 𝜏 ≫ ∆𝑇 (no

overlap) and the signal takes a constant value, since both partial pulses contrib-ute to the signal, but the interference term vanishes. Plotting the signal (also called correlation function) over the time lag one can determine the temporal width of the correlation function.

Since the correlation function is always symmetrical (due to 𝐼(2𝜔, 𝑡, 𝜏) ~ [𝐸(𝑡) + 𝐸(𝑡 + 𝜏)]4), no information about possible assymetries of

17 A laser pulse at wavelength 1064 nm corresponds to 560 THz after being frequencydoubled. This means 560 oscillations per pico-

second.

Theoretical basics


the laser pulse can be extracted. The width of the correlation function does not equal the pulse duration, since the convolution of the partial pulses changes this. In order to obtain the pulse duration from the measured autocor-relation function a conversion factor, which depends on the shape of the pulse, is necessary. The conversion factors for converting the FWHM of the correlation function to the FWHM of the instantaneous power of Gaussian pulses can be seen in Table 2: Conversion factors for Gaussian pulses of FWHM of the correla-tion function to FWHM of the pulse as detailed in [3]..

Table 2: Conversion factors for Gaussian pulses of FWHM of the correlation func-tion to FWHM of the pulse as detailed in [3].

shape of the pulse formula Δ𝜏 Δ𝑇⁄ Δ𝑇 Δ𝜐⁄

Gaussian exp (−𝑡2/0,36Δ𝑇2) √2 0,441

3.5 Etalon

Etalons are used for reducing the width of the emission line of a laser and for tuning of the emitted wavelength. It consists of two partially reflective, co-planar surfaces, which enclose an optical medium and thus constitute a Fabry-Perot resonator18.

The field amplitude is split into reflected and transmitted parts for every pass through. The thickness 𝑑𝐸, the refraction index 𝑛 of the medium between the reflective surfaces as well as angle of incidence 𝜃𝑖 and wavelengh λ of the field define the phase difference 𝛥𝜙 between the reflected and transmitted beam after 𝑘 -th and 𝑘 + 1 -th pass through the etalon. The mathematic deduction of this relation is described in many text books for optics.

3.47

Δ𝜙 = 2𝜋 𝜆⁄ ∙ 𝑑𝐸 √𝑛2 − sin 𝜃𝑖

Due to the phase shift by 𝜋 for reflection at the interface to an optically denser medium, the destructive interference of the reflected beams coincides with the constructive interference of the transmitted beams and vice versa. For a plane wave, a lossless medium in the interspace and identical reflectivities 𝑅𝐸 for entry and exit surfaces the transmission 𝑇𝐸 of the etalon is an Airy-function:

18 The technical realisation of such an resonator is either made of a single transparent slab, where the boundary surfaces are consti-

tuted by the surfaces of the slab (called solid etalon) or of two transparent neighboring slabs enclosing a slip of air (air spaced eta-lon). The functional principle is the same for either.

Theoretical basics


3.48

𝑇𝐸(𝜃𝑖) = [1 + 4

𝑅𝐸

(1−𝑅𝐸)2 sin2(Δ𝜙/2)]−1

The transmission can reach 100% depending on the angle of incidence and the wavelength of the light. The transmission function of the Etalon is formed like a comb (see top of Figure 19: Calculated transmission of an Etalon with dE = 250 µm, made of fused silica (n = 1.5) with identical reflectivities of entry and exit surfaces RE = 50%, for both wavelength and angle of incidence.) in its fre-quency dependence due to the periodicity of the sin2-Term. The three defining properties for the characterization of an etalon are the free spectral range (FSR)

3.49

Δ𝜆𝐸 = 2𝜆2 ∙ 𝑑𝐸 √(𝑛2 − sin 𝜃𝑖)

the distance between neighboring orders of transmission, the width δλ of indi-vidual orders of transmission as well as the finess 𝐹∗ as the ratio of the former two. In a first order approximation the finess 𝐹∗ is only dependent on the reflec-tivity 𝑅𝐸 (see Figure 20: Calculated transmission for dE = 250µm of a fused silica solid etalon for various reflectivities.).

3.50

𝐹∗ = 𝜋 [2 ∙ arcsin (

1 − 𝑅𝐸

2 √𝑅𝐸

)]

−1

≈ 𝜋√𝑅𝐸

1 − 𝑅𝐸

Figure 19: Calcu-lated transmission of an Etalon with dE = 250 µm, made of fused silica (n = 1.5) with identical reflec-tivities of entry and exit surfaces RE = 50%, for both wave-length and angle of incidence.

Experimental Realisation


Figure 20: Calcu-lated transmission for dE = 250µm of a fused silica solid eta-lon for various re-flectivities.

A change in the angle of incidence 𝜃𝑖 leads to a change in the free spectral range, resulting in other wavelengths being transmitted.

Reducing the width of the emission line of a laser usually leads to an increase of the pulse duration (refer to section 3.4.1).

4 Experimental Realisation

In this experiment you will use a passivly mode locking laser of the type LYNX manufactured by Time Bandwidth Product. The laser is comprised of a diode laser as a pump source, a laser resonator and a beam shaping system (see Figure 21: Schematic of the laser source; pump radiation: blue; laser radiation: green; reso-nator endmirrors: red). The complete layout of the experiment additionally con-tains various measurement devices for characterising the laser properties (Figure 22: The USP lasers characterization track. Note the greenly tinted mirrors. Those are the only ones you need to adjust and are mounted on a green rail.).

The laser source

The ultra short pulse laser in this experiment emits diffraction limited radiation at the wavelength 𝜆𝐿 = 1064 nm, at an average power of up to 700 mW and at pulse durations in the order of magnitude of 10 ps.



It is a solid-state laser with Nd:YVO4 as an active laser medium. The mechanism of mode locking is achieved with a Semiconductor Saturable Absorber Mirror (in short: SESAM).The pump source is a laser diode.

The laser diode emits radiation at the wavelength 𝜆𝑃 = 808 nm. This radiation is not pulsed, but emitted continuously at an average power of up to 1.5 W. The active chip of the laser diode is a semiconductor component, consisting of mul-tiple emitting areas (emitters). The spatial beam properties are asymmetric. Whereas the epitaxial order of layers (height of the single emitters) is almost dif-fraction limited (𝑀2 ≈ 1), the superposition of the radiation of different emitters has a higher mode order (𝑀2 ≫ 1).

The laser light of the pump source passes through a dichroic mirror (DM) and is coupled longitudinally, i.e. collinearly to the resonator axis by a telescope optic (L1 and L2), into the active medium (the Nd:YVO4 crystal). The cubical crystal has an edge length of 𝑎 = 4 and is doped with a concentration of 𝜌𝑁𝑑 = 0,7 % Nd-ions.

The optical resonator comprises a system of mirrors (M0 through M7). The end mirrors are the SESAM (M7) and the partially reflective entry facing of the laser crystal acting as the outcoupling mirror (M0). The degree of outcoupling is 𝜂𝐴 =5 %. The light propagates through a system of 6 folding mirrors (M1 through M6) between the two planar end mirrors (M0, M7). The mirrors M2 and M6 have a radius of curvature of 𝑟𝑀2 = −500 mm and 𝑟𝑀6 = −250 mm respectively. The etalon, which is used for varying the amount of amplified modes in the resona-tor, is positioned exchangeably between the mirrors M5 and M6.

The light which is coupled out has a wavelength λ = 1064 nm and is separated from the pump light by the dichroic mirror (DM), collimated by two lenses and guided out of the laser housing. Since Nd:YVO4 is a birefringent crystal, the emitted radiation has a high linear polarisation. In this experiment the pre-dominant direction of the polarisation is parallel to the plane of the table. The polarisation extinction ratio (PER) is defined as the ratio of the power in hori-zontally and vertically polarised direction. The manufacturer states it to be higher than 450.



Figure 21: Schematic of the laser source; pump radiation: blue; laser radiation: green; resonator endmirrors: red

Characterisation track of the USP-laser

The laser light is guided towards a power head by the mirrors S1 through S3 in the subsequent experimental setup (see Figure 22: The USP lasers characteriza-tion track. Note the greenly tinted mirrors. Those are the only ones you need to adjust and are mounted on a green rail.). The reflectivities of the mirrors are less then one and chosen such that the transmitted beam has sufficient power for the subsequent characterization of spatial, spectral and temporal laser parame-ters (RS1 = 0,99, RS2 = 0,99, RS3 = 0,60). The rear sides of those mirrors have an antireflective coating in order to reduce negative effects.

Figure 22: The USP lasers characteriza-tion track. Note the greenly tinted mir-rors. Those are the only ones you need to adjust and are mounted on a green rail.



A half wave plate (HWP) and a thin film polariser (TFP) are mounted in front of the power head in order to measure the polarisation extinction ratio. The optical mount of the HWP can be rotated along the azimuthal angle. This varies the angle 𝛥𝜃 between the direction of polarisation of the incident light and the optical axis of the wave plate in order to rotate the direction of polarisation by 2∆𝜃. Only the horizontally polarised portion of radiation is transmitted by the subsequent TFP and enters the power head. The portion reflected at the TFP is guided into a beam dump.

The transmitted portion of the beam from mirrors S1 and S3 is guided by an additional set of two mirrors each in order to adjust the coupling into the meas-urement devices for the beam quality (M-square measurement device) and the pulse duration (autocorrelator) respectively. The beams have to be coupled into the entry apertures colinearly to the respective optical axis of the devices. The transmitted portion from S2 is used to measure the spectrum of the laser light.

Characterisation track of the diode laser.

Figure 23: Charac-terization setup for the pump diode. Greenly marked mir-rors are mounted on a green rail and sup-posed to be adjusted during the experi-ment.

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The pump radiation is guided out of the encased laser by putting a periscope through a prepared opening in the laser housing (see Figure 23: Characterization setup for the pump diode. Greenly marked mirrors are mounted on a green rail and supposed to be adjusted during the experiment.). The lense L4 collimates the pump radiation. The second periscope on the left has an output height that fits the height of the subsequent measurement devices. Mirror E1 reflects more than 99.8% of the pump radiation into a power head, which is used for meas-uring the power curve of the diode laser. The transmitted beam will be guided onto the CCD-camera by means of mirrors E2 through E4. The lens L5 focusses the beam to a measurable caustic. This caustic can be measured with the CCD-camera (beam analyser) by moving the camera along the rail. In order to not destroy the CCD-chip, the intensity of the radiation is further reduced by filters in the filter mount (FH) and mounted to the camera itself. The mirrors E3 and E4 will be used to adjust the beam parallel to the rail.

5 Conduction

Laser safety measures (section 2) have to be applied before starting up and while conducting the experiment. The entire optical system is within a flow box, which keeps away dust from the optics. Wearing a lab coat, diposable gloves and a disposable hood is mandatory, when working at the setup. Furthermore the flow speed has to be raised to level 5 (0.4 m/s) at the flow box panel before opening the flow box door.

At the beginning of the experiment your tutor will give you an introduction to the devices and the basic setup of the experiment. Only the etalon is accessible and adjustable within the laser source. The pumpradiation can be measured after inserting a periscope.

ATTENTION: Adjustments inside the laser of any kind (besides the above men-tioned adjustment of the Etalon) are forbidden under any circumstance. Mal-adjustment of any mirror leads to the resonator operating outside its specifica-tion or to the resonator not working at all. NEVER adjust the resonator with-out authorization. In the case that the laser is not working, contact your tutor. Misalignment of the resonator can only be fixed by an extensive adjusting pro-cedure. This would delay the procedure of the experiment severely. Achieving the goals of the experiment will then be impossible.

You can recognize the mirrors you are allowed to adjust by their position on or above a green rail. The possible positions for the powerhead are marked blue. If adjusting other mirrors becomes necessary in your opinion, first ask your tutor.

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Adjusting the wrong mirrors can lead to hazardous situations for the devices (e.g. by reflecting the beam back into the laser) and personnel.

You are not allowed to remove any filters (especially at the M-square meas-urement device)! Turn off the laser or block the beam physically beforehand for any larger change at the path of the beam (e.g. introducing new mirrors or lenses).

Make absolutely sure no objects fall into the beam source (especially when ad-justing the etalon or inserting the periscope)!

5.1 Characterising the laser diode (pump source)

You start the experiment by characterising the radiation emitted by the diode laser (pump source). It can be coupled out of the laser housing with a periscope (see Figure 24: left side: periscope and marked beam path (blue) and screw that you shall not loosen (red); right side: CCD-camera with marked filters (red) and cap (black)).

Figure 24: left side: periscope and marked beam path (blue) and screw that you shall not loosen (red); right side: CCD-camera with marked filters (red) and cap (black)

5.1.1 Implementing the beam path for the pump radiation

Open the round hole on the top of the laser housing. Put the periscope through the hole into the laser source. The lense on the bottom cube of the periscope should point towards the diode when you start lowering the periscope. The periscope must not be turned while inside of the laser housing. Fix the periscope to the laser housing by fastening the four screws (blue hexagons

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in Figure 24: left side: periscope and marked beam path (blue) and screw that you shall not loosen (red); right side: CCD-camera with marked filters (red) and cap (black) left). The periscope is already adjusted, thus there is no reason to losen the screws marked red in Figure 24: left side: periscope and marked beam path (blue) and screw that you shall not loosen (red); right side: CCD-camera with marked filters (red) and cap (black) (left).

Remove mirror E1 and the CCD-camera from their footings and put them to an unused place within the flowbox (optical surfaces should allways be oriented parallel to the direction of the air flow). Adjusting the beam path is done for low levels of beam intensity. Set the diode current to 0.38 A.19

Now turn on the laser and adjust E2 with the micrometer screws such that the beam hits E3 centricly. Check this with the detector card. Next you should adjust E3 such, that the beam hits E4 at the center.

Next put the iris mount into the footing of the ccd-camera and level the center of the iris to the beam height. Move the iris mount on the rail as close to the footing of the lens as possible and adjust the beam onto the center of the iris using mirror E3. Move the iris mount away from the lens footing and adjust the beam to the center of the iris using mirror E4. Repeat until the beam always hits the center of the iris when moving the filter mount.

Also take note of the approximate location of the beam waist (focus) by moving the detector card along the beam.

5.1.2 Measuring the power curve of the diode laser

Put the power head in the designated footing next to the laser housing. Set the wavelength sensitivity (button “Laser” on the Nova II power meter) to an appro-priate value. Put mirror E1 into its designated footing and adjust it to hit the sensitive area on the power head. Use the micrometer screws of E1 to measure as much power as possible.

You can now measure the power curve of the pump diode. Vary the diode cur-rent between 0 and 1.7 A and note the corresponding power values. Now turn the laser off.

19 The diode current can be changed by pushing the blue knob on the power supply and choose “Change Diode Current“ in the

subsequent menu. One further push lets you choose “Laser Diode Set Current”. The desired current can then be selected by turn-ing the knob. The current setting is changed by pressing the knob again.

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5.1.3 Preparing the beam profile measurement

Set the current back to 0.38 A (don’t turn on the laser yet!). Remove the detector card from the filter mount and put the filter mount into its original footing close to E4.

Put the ccd-camera into its original footing on the rail (don’t remove the cap yet). Take care not to place the camera in the focus of L5, since doing so exposes the ccd-chip and the cap (see Figure 24: left side: periscope and marked beam path (blue) and screw that you shall not loosen (red); right side: CCD-camera with marked filters (red) and cap (black)) to an unnecessary risk if the beam isn’t suf-ficiently attenuated.

After turning the laser on you can now check if the beam hits the cap centricly. Change height and orientation (perpendicular to the rail) of the camera if neces-sary.

Turn off the laser and put the VG9-filter into the filter mount.

5.1.4 Measuring the diode caustic

Remove the black cap+red filter (DON’T remove the blackfilter directly in front of the CCD chip! See Figure 24: left side: periscope and marked beam path (blue) and screw that you shall not loosen (red); right side: CCD-camera with marked filters (red) and cap (black)). The software “Beam Gage” controls the camera. First turn on the live image and conduct an “Ultracal” (this performs an offset subtraction in order to remove ambient light from the measurement). Additional information concerning the software can be obtained from instructions available in the lab room.

Make sure the exposure time is set to 120 ms and the diode current is set to I = 0.38 A. Now turn on the laser. Increase the diode current up to 0.45 A while take care not to overexpose the camera. Adjust the exposure time if necessary.

You can tell the camera is overexposed if some of the pixels are displayed white in BeamGage’s live image. If on the contrary the values displayed are on the lower side of the intensity scale, the exposure time can be increased againSearch the focus of the laser beam (only move the camera while live image is turned on in BeamGage and you therefore can prevent overexposure).Save an image of the beam profile in the focus. Make sure you choose a proper aperture, such that the BeamGage estimate of the beam width is reasonable. Detailed information for BeamGage settings will be available in the lab room.

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Measure the beam width for 7 different positions of the camera within ±3 cm of the focus position. Adjust the exposure time before saving each image. Also per-form an “Ultracal” (while the laser is turned off) prior to saving an image. Meas-ure the distance by which the camera is moved for each image.

5.1.5 Preparations for Measuring the USP laser

When you finished measuring the diode power curve and beam profile, turn off the laser and remove the periscope in the laser housing from the hole and close the hole with the cover.

The pump diode’s data sheet is available to you for comparison with your ob-tained data.

5.2 Beam characterisation of the ultra short pulse laser (USP laser)

The properties of the USP laser (Time-Bandwidth, LYNX series) will be measured in the second part of the experiment. This covers temporal spatial, spectral and energetic properties of the radiation.

5.2.1 Basic calibration of the measuring track and powercurve of the mode-locked laser

Mirrors S1 and S2 are adjusted. Don’t move these!

Place the power head between the laser source and mirror S1 first. The laser beam has to hit the power head centricly (The beam source should be turned off or blocked while placing the power head!). Set the wavelength sensibility (the setting labeled “Laser”) of the power head to an appropriate setting and note it down. Measure the power curve by noting the output power of the laser for various values of the pump current. Be carfeul not to lower the pump current for the diode beneath 0.8 A at the power supply! This is the lower power limit, at which mode-locked operation is possible20. Resonator optics (the SESAM in particular) can be damaged during operation outside of the mode-locking range. Measure the output power for diode currents from 0.85 to 1.7 A. Return the diode current to 1.4 A after taking the power curve!

Put the power head back to the place at which it is marked in the schematic (Figure 22: The USP lasers characterization track. Note the greenly tinted mirrors. Those are the only ones you need to adjust and are mounted on a green rail.).

20 This does not apply to meassurements for the pump diode, since the periscope keeps the beam from reaching the SESAM anyway.

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(Turn off the beam source or block it!). Readjust the power head’s orientation and heigth so that it is hitby the laser beam centricly.

Measurement of the other significant laser parameters is to be carrierd out now.

5.2.2 Measurement of the PER (polarisation extinction ratio)

Use the half wave plate and the polariser (TFP) to measure the PER by first tilt the laser beam’s direction of polarisation and then measuring the transmitted power behind the fixed polariser. In order to have the TFP work as a linear polariser, it must be tilted 55.4° to the propagation direction of the beam. The TFP is already adjusted, don’t move it. The direction of polarisation is changed by turning the half wave plate. Measure the transmitted power for various rotation angles be-

tween 0° and in reasonable intervals. Determine the angles of minimum and maximum transmission. Make sure you have a sufficient amount of data around these positions.

5.2.3 Measurement of the (pulse-) repetition rate

Make sure the 50 terminator is connected to the oscilloscope and connect the photodiode to the corresponding channel by the BNC-cable. Choose an appro-priate measuring range and identify proper trigger settings. Determine the fre-quency of the pulses. Save the results of the measurements on your flash drive. Turn off photo diode after the measurement!

5.2.4 Meassurement of the beam quality

The beam quality will be measured with the M-square measurement device „M2-200s-FW“by Spiricon (see Figure 25: Functional principle of the M2-200s-FW, which has a variable optical delay line.). It has a lense for creating an artificial beam waist and a CCD-camera for measuring the transverse intensity profile of the beam. The distance of the beam waist and the camera is automatically varied by a motorised delay line. The system is completely encased and the motorized mirrors are not accessible. A motorized wheel with different filters attenuate the laser power to levels unable to harm the camera. This filter wheel is located in the front turnable part of the device. Make sure those filters are always lo-cated in the path of the beam. If you are manually operating the device, e.g. while adjusting the beam (live display!), you have to use the control software to

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set the correct filters. Avoid overexposure of the camera under any circum-stances (white areas while viewing the live display). This is to avoid damage to the CCD chip of the camera.

Attention: Never point the laserbeam from the source directly into the device. Use the transmitted beam behind mirror S1 only. Else the filters are insufficient to avoid destruction of the CCD chip.

Figure 25: Functional principle of the M2-200s-FW, which has a variable optical de-lay line.

Start the control software „M2-200s-FW“ (Desktop). The device will now cali-brate automatically. Set the distance of the camera to minimum by hitting the „|<“-button.

5.2.4.1 Alignment of the M2-measuring device

Mount the threaded alignment tool (a tube with two blinds) to the M2-housing (see Figure 26: Alignment tool (green) fixed to a filter (orange) at the M2-device housing (blue)). Take care to never damage or take off the filter (orange). Set the internal filter setting to 3.8.

Figure 26: Alignment tool (green) fixed to a filter (orange) at the M2-device hous-ing (blue)

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Your objective is coupling the laser beam into the device along the measurement axis. Patience is required, becaus it is insufficient to align the beam to a single spot. The beam has to be aligned with the whole measuring axis. The beam has to hit the camera centrally for any position of the delay line. The alignment is sufficient, if the beam is inside the camera window all along the delay line. A perfect alignment is achieved if the position of the beam in the transverse plane does not change for changing camera distance.

Aligning the beam into the M2-device works as follows: When the alignment tool is fixed to the housing at the start, use the precision screws at the mirror S 4.1 to point the beam through the first aperture of the tool and those of the mirror S 4.2 to align the beam through the second aperture. Iterate until both apertures are hit by the beam simultaneously. The beam is now coarsely aligned with the measurement axis. Close the shutter at the laser source and unmount the align-ment tool without removing the filter it is mounted on. Now use the “live display” of the control software to precisely align the beam onto the measuring axis. Similarily to the alignment before, use mirror S 4.1 for aligning the beam to the minimum z position and S 4.2 to align to the maximum z position. Alter the z position by the „|<“-button (for minimum) and “>|”-button (maximum). There is also a stop-button you can use, in case the beam leaves the camera window. The beam is considered aligned if the beam hits the camera centrally for all z po-sitions.

5.2.4.2 Starting the measurement

Now start the automated measurement („start“-button). The software saves the measured data automatically in the file „results.rlg“ (desktop->M2-Daten). Save this file before starting a new measurement or the data of the previous measurement will be overwritten. Also export a picture of the beam waist.

5.2.5 Measurement of the spectrum

The beam is coupled into the spectrum analyser by the fiber S6 which is mounted behind mirror M2. Start the software „Laser Spectrum Analyser 2059“. The fiber and the mirrors S1 and S2 are already aligned and should not be adjusted by you.

For software settings refer to the notes in the lab. The measured data can be saved as an ascii file by using the option “Datei->Speichern unter” with the for-mat “*.smx”. The datapoints of wavelength and associated relative intensity can be found in the last third of the file.

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5.2.6 Measurement of the pulse duration

Alignment of the autocorrelator is similar to the one for the M2-device. The radi-ation transmitted through mirror S3 is coupled into the autocorrelator by the mirrors S5.1 and S5.2.

Fold down the two alignment aides of the autocorrelator (at the entry aperture and at the mirror S5.2). Use the mirror S5.1 such that the beam is aligned cen-tricly on the autocorrelator mirror S5.2. Then use S5.2 to align the beam into the entry aperture. If the beam wanders of the center of the mirror S5.2, iterate the two steps until both alignment aides are hit centrally. Pay attention to the align-ment window slightly above the entry aperture (seen in Figure 27: Alignment window of the autocorrelator Pulse-Check; badly aligned laser beam (left) cor-rectly aligned laserbeam (right); source: [11]). In case you don’t see a spot of light in the alignment window, tilt mirror S5.2, while paying attention to the align-ment window. There should be some flickering at an edge of the window. Try moving the flickering into the window and proceed with aligning mirrors S5.1 and S5.2 as described above. Use the micrometer screw “Beamdistance” at the side of the housing, If you see two spots in the alignment window (see Figure 27: Alignment window of the autocorrelator Pulse-Check; badly aligned laser beam (left) correctly aligned laserbeam (right); source: [11]). If you see only one spot in the alignment window, adjust it onto the center of the cross hairs with the method described above. Contact your tutor if you encounter problems.

A reliable measurement can only be assured, if the beam is not cut off at the entry aperture and if the beam is aligned with the cross hairs. Keep in mind the autocorrelator function has to be symmetrical.

Figure 27: Alignment window of the auto-correlator Pulse-Check; badly aligned laser beam (left) cor-rectly aligned la-serbeam (right); source: [11]

Launch the APE Pulse Link software (desktop -> APE). Click the start button for starting a measurement. You can save the data by clicking (Datei -> Save Data). These files can be read and processed by common analysis software. Also pay attention to the notes in the lab room, which contain descriptions of additional functions of the software.

Analysis


Removing the small rectangular cover of the laser housing provides access to the mount of the etalon. The mount has a screw at the top, which is used for adjust-ing the angle of the etalon relative to the beam and therefore altering the inci-dent angle (compare Figure 19). Tilt the etalon by turning the screw with a hex-agon socket screw (Allen) key. First find the shortest pulse duration and measure spectrum and pulse duration. Afterwards tilt the etalon to increase the pulse duration. Measure the spectrum and the pulse duration each time you tilt the etalon. Iterate this process 4-5 times, until a noticeable increase in the pulse du-ration is observed (~50%).

6 Analysis

6.1 Power curve and laser threshold for laser diode and USP laser

Plot the power 𝑃𝑝𝑢𝑚𝑝 of the pump diode against the applied diode current and determine the laser threshold current of the laser source. Then plot the power curve of the USP-Laser 𝑃𝑙𝑎𝑠𝑒𝑟(𝑃𝑝𝑢𝑚𝑝). Determine the threshold power 𝑃𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 by fitting the power curve with a linear function and under the as-sumption of a linear dependence and an optimal resonator selectivity. Does the linear dependence hold?

6.2 Analysis of the repetition rate

Use the data of the oscilloscope with the photo diode in order to determine the repetition rate. Why is it not possible to observe the pulse shape with the oscilloscope? What do you see instead? Display the data of the oscilloscope graphicly and determine the repetition rate. Does it match manufacturer speci-fications? How long does the resonator have to be to send pulses with the ob-served repetition rate?

6.3 Analysis of the beam quality

The beam diameter 𝑑 at certain positions 𝑧 along the axis of propagation can be used to determine the M-square factor and other beam parameters by fit-ting a second order polynomial to the squared beam diameters (ISO 11146):

Analysis


𝑑2(𝑧) = 𝐴 + 𝐵 ∙ 𝑧 + 𝐶 ∙ 𝑧2. Data for the caustic of the USP laser is contained in the saved “results.rlg” file, which contains 8 columns including less relevant data like time and date. The last 3 columns contain the values of 𝑑𝑥(𝑧), 𝑑𝑦(𝑧) and 𝑧. The last row contains statistical values. Don’t include them in your fit. Perform mentioned polynomial fits to the squared diameters in dependance of the posi-tion for both the diode and the USP laser caustic. Explicitly write down the fitting parameters 𝐴, 𝐵 and 𝐶 with their uncertainties and derive the beam parameters 𝑧0, 𝑀2, 𝑤0, 𝑧𝑅 and 𝜃.

6.1

𝑧0 = −

𝐵

2𝐶

6.2

𝑀2 =

𝜋

4𝜆∙ √𝐴 ∙ 𝐶 − 𝐵

2

4⁄

6.3

2 𝑤0 = √𝐴 − 𝐵

2

4𝐶⁄

6.4

𝑧𝑅 =

1

𝐶√𝐴 ∙ 𝐶 − 𝐵2

4⁄

6.5

𝜃 = √𝐶 /2

Do the measurements satisfy the ISO specifications?

What values for 𝑀2 do you obtain? Compare the beam profiles and 𝑀2 of both lasers at the beam waist. What are the differences?

6.4 Analysis of pulse duration and spectrum

Display the measured pulse durations for different tilt angles of the etalon graphicly. Keep in mind, that the software for the autocorrelator displays the FWHM of the convolution of the pulse. Translate those into pulse durations. Plot the instantaneous power of the pulse against the time. Calculate the pulse peak power!

Display the spectrum of the laser radiation graphicly for various tilt angles of the etalon and determine the individual FWHM. Determine the Signal-to-Noise-ratios (SNR) of the spectrum. Plot the logarithmic (log10) intensity against the fre-quency and determine the noise level. A signal below the noise level cannot be reliably measured. The SNR is typicly defined as the ratio of average signal power to average noise power or as average signal power to the standard deviation of the noise.

Analysis


Calculate the time-bandwidth product Δ𝜐 ∙ Δ𝑇 of the laser source for various etalon settings. Does the result fulfill your expectation (compare Table 2: Con-version factors for Gaussian pulses of FWHM of the correlation function to FWHM of the pulse as detailed in [3].)?

6.5 Analysis of the polarisation extinction ratio (PER)

Plot the power behind the polariser against the rotation angle of the half-wave plate. Fit a trigonometric function to the data. Determine the PER of the laser assuming an ideal polariser. The PER is defined as the ratio between the laser power polarised in horizontal and vertical direction. Discuss the effect a non-ideal polariser would have.

How much power is lost along the optical path? Compare this value to your expectation.

Questions for self examination


7 Questions for self examination

Which hazards can be encountered when handling laser radiation? Which safety measures have to be deployed? What kind of clothing is mandatory for the experiment?

Where is the intensity of the laser beam reduced and how is that achieved? What can happen to certain measurement devices, if the laser beam is not weakened appropriately?

Which mirrors may be turned? In what way do you adjust the laser reso-nator in this experiment?

What are the rate equations used for and what does each term refer to? Why is a laser operating in thermal equilibrium not possible?

Which advantage does a four-level laser medium have over a three-level laser medium?

How does mode-locking work? Which kind of mode-locking is deployed in this experiment? How is it implemented? What determines the mini-mum pulse duration and how short can a pulse be? How can the pulse duration be manipulated?

What are the important properties for describing laser radiation?

How does the measurement of the beam quality work? How many data points are required? What is a diffraction limited beam? What kind of curve do you expect to measure?

How does the autocorrelator work? What spatial size does a pulse of duration 1 ps have? What is a bandwidth limited pulse?

How do you determine the repetition rate of the laser? How does the repetition rate influences the emitted spectrum?

How do the half wave plate and the polarizer act on unpolarised radia-tion? How on linearly polarized radiation? What kind of fit function is appropriate to determine the PER?

Referenced documents


There is no manufacturer information for uncertainties of some measure-ments. Are there still options for you to obtain accurate error analysis for those?

8 Referenced documents

Literature

[1] POPRAWE, R.: Lasertechnik I, Skript zur Vorlesung, 3. Auflage, 2008

[2] LANGE,W.: Einführung in die Laserphysik, 2. Auflage, Wissenschaftliche Buchgesellschaft Darmstadt, 1994

[3] DEMTRÖDER,W.: Laserspektroskopie Band 2: Experimentelle Techniken, Springer-Verlag Berlin, 2013

[4] UNIVERSITÄT Kiel, Ultrakurze Lichtimpulse und THz Physik, http://www.phy-sik.uni-kl.de/fileadmin/beigang/Vorlesungen/WS_08_09/ UKP_WS08_09_V11_08_12_15.pdf, 11.02.2014

[5] DEMTRÖDER,W.: Experimentalphysik 2: Elektrizität und Optik, Springer-Ver-lag, 5. Auflage, 2009

Device manuals are available in the lab

[6] Time-Bandwidth, “User Manual, LYNX SERIES”

[7] Time-Bandwidth, „User Manual, 2DµP SERIES, Microprocessor Controlled Laser Heed Controller (Power Supply)“

[8] Spiricon, “M2-200/200s-FW User Guide”

[9] Tektronix, “TDS2000C and TDS1000C-EDU Series Digital Storage Oscillo-scopes User Manual”

[10] Ophir, “NOVA II, Laser Power/Energy Meter User Manual”

[11] A.P.E, “Autocorrelator with pulseLink Driver, pulseCheck USB, User Man-ual”

http://www.physik.uni-kl.de/fileadmin/beigang/Vorlesungen/WS_08_09/UKP_WS08_09_V11_08_12_15.pdf



Appendix


9 Appendix

9.1 Deriving the Gaussian beam from the wave equation

The scalar wave equation

9.1

Δ𝐸 + 𝑘2𝐸 = 0

has an infinite number of solutions. We are looking for solutions which propa-gate along an axis within a small solid angle. The simplest solution, which only propagates along one axis is the plane wave. But due to their infinite extension in the transverse plane they are not suited for the description of real radiation fields. In order to achieve spatial confinement, we will make the following an-satz:

9.2

𝐸(𝑥, 𝑦, 𝑧) = 𝐸0(𝑥, 𝑦, 𝑧)𝑒𝑖𝑘𝑧

This means the amplitude 𝐸0(𝑥, 𝑦, 𝑧), which were constant for a plane wave, is now dependant of the position. We now impose that the amplitude changes negligibly within the span of one wavelength, which is called the slowly-varying-envelope (SVE-) approximation. This is equivalent to the assumption that all parts of the wave propagate approximately paraxial to the the axis of propagation, hence it is also called the paraxial approximation. Plugging the ansatz into the wave equation yields:

9.3

𝜕2𝐸0

𝜕𝑥2+

𝜕2𝐸0

𝜕𝑦2+

𝜕

𝜕𝑧(

𝜕𝐸0

𝜕𝑧+ 2𝑖𝑘𝐸0) = 0

This is an exact solution exact. Applying the SVE-approximation means

9.4

𝜕𝐸0

𝜕𝑧≪ 𝑘𝐸0 = 2𝜋

𝐸0

𝜆

Thus we get the wave equation in paraxial approximation for Carthesian coordi-nates:

9.5

𝜕2𝐸0

𝜕𝑥2+

𝜕2𝐸0

𝜕𝑦2+ 2𝑖𝑘

𝜕𝐸0

𝜕𝑧= 0

Appendix


A rotation symmetry is often observed in laser resonators, hence we will trans-form into polar coordinates (𝐸0 being independant from 𝜑) and get the paraxial wave equation:

9.6

𝜕2𝐸0

𝜕𝑟2+

1

𝑟

𝜕𝐸0

𝜕𝑟+ 2𝑖𝑘

𝜕𝐸0

𝜕𝑧= 0

The solutions of equations 9.5 and 9.6 are a complete set of fundamental solu-tions (similarly to the wave equation of section 3.3).

9.2 Deriving the longitudinal conditions in the resonator

As a basic model for confined light waves we will consider a one-dimensional area of length 𝐿, which is limited by two fully reflective mirrors. In this area the wave equation for vacuum applies and the fundamental solutions are:

9.7

𝐸 = (𝐴𝑒𝑖𝑘𝑧 + 𝐵𝑒−𝑖𝑘𝑧)𝑒−𝑖𝜔𝑡 mit 𝑘 =𝜔

𝑐0

The field amplitude has to vanish for two ideally conductive metallic surfaces 𝐸(𝑧 = 0) = 𝐸(𝑧 = 𝐿) = 0. The ansatz of plane waves results in:

9.8

𝐴 + 𝐵 = 0 𝐴𝑒𝑖𝑘𝐿 + 𝐵𝑒−𝑖𝑘𝐿 = 0

This linear system of equations is solvable, if the condition

9.9

𝑒2𝑖𝑘𝐿 = 1 ⇒ 𝑘𝐿 = 𝑛𝜋, 𝑛 = 0, 1, 2 …

is fullfilled; the possible wave numbers result from the solvability condition. Solv-ing the linear system of equations yields the coefficients 𝐴 and 𝐵. Since our case dictates that 𝐴 = −𝐵, we obtain the following eigenmodes:

9.10

𝐸𝑛 = 𝐸0,𝑛 sin(𝑘𝑛𝑧) 𝑒𝑛−𝑖𝜔𝑛𝑡

, 𝑘 = 𝑛𝜋

𝐿 , 𝜔𝑛 =

𝑐0

𝑘 , 𝑛 = 0,1,2 …

9.3 q-eigenparameter of a resonator

A Gaussian beam is described completely by the q-parameter 𝑞 = 𝑧 + 𝑖 𝑧𝑅. The propagation of Gaussian beams and their transformation by an optical element yields 𝑞2, which is calculated by

Appendix


9.11

𝑞2 =

𝐴𝑞1 + 𝐵

𝐶𝑞1 + 𝐷.

𝑞2 can be acquired by applying the ABCD-rule of corresponding beam transfer matrices on the original q-parameter 𝑞1. Refer to the literature for additional information. In order to determine the Eigenmodes of a resonator, one has to consider a full cycle of the beam within the resonator. It is constituted by:

1. propagation from mirror 1 to mirror 2 as a propagation over length L,

2. reflection at mirror 2,

3. propagation from mirror 2 to mirror 1 and

4. reflection at mirror 1 (see Figure 9: Scheme for a resonator round trip ).

The beam transfer matrices for propagation by a distance 𝑧 are 𝑴𝒑(𝑧), those for

reflection at a mirror with radius of curvature 𝑅 are 𝑴𝒔(𝑅) (see e.g. [5])

9.12

𝑴𝒑(𝑧) = (

1 𝑧0 1

) and 𝑴𝒔(𝑅) = (1 0

−2

𝑅1

)

For a full cylcle in a resonator of length 𝐿 and two mirrors with the radii of cur-vature 𝑅1 and 𝑅2 we obtain the transfer matrix by multiplying the transfer ma-trices of the four participant processes:

9.13

𝑴𝒓 = (

2𝑔2 − 1 2𝐿𝑔2

2

𝐿(2𝑔1𝑔2 − 𝑔1 − 𝑔2) 4𝑔1𝑔2 − 2𝑔2 − 1

)

We introduced the g-parameter 𝑔1,2 = 1 −𝐿

𝑅1,2 here. For the q parameter 𝑞 =

𝑧 + 𝑖 𝑧𝑅 after the j-th resonator cycle we obtain:

9.14

𝑞𝑗+1 =

𝐴𝑞𝑗 + 𝐵

𝐶𝑞𝑗 + 𝐷 with 𝑴𝒓 = (

𝐴 𝐵𝐶 𝐷

)

An eigensolution 𝑞𝐸 for the resonator is distinguished by 𝑞𝑗 = 𝑞𝑗+1 = 𝑞𝐸. This

means the q-parameter is not changed travelling a full cycle within the resonator. Using equation 9.14 and the general property det 𝑴 = 1 of beam transfer ma-trices 𝑴 we obtain:

9.15

𝑞𝐸 =

1

2𝐶(𝐴 − 𝐷 ± 𝑖√4 − (𝐴 + 𝐷)2)

Appendix


Thereby we can determine the Rayleigh length and position of the beam waist in dependency of the resonator parameters (see Figure 28: The Gaussian beam in a spherical resonator.). Since the Rayleigh length must be positive, we know which sign to choose in equation 9.15.

9.16

𝑧𝑅 = 𝐿

√𝑔1𝑔2(1 − 𝑔1𝑔2)

𝑔1 + 𝑔2 − 2𝑔1𝑔2 and 𝑧 ≔ 𝑧1 = −

𝐿𝑔2(1 − 𝑔1)

𝑔1 + 𝑔2 − 2𝑔1𝑔2

Figure 28: The Gaussian beam in a spherical resonator.

Due to the definition of the Rayleigh length (see section 3.2) we can derive an expression for the beam waist:

9.17

𝑤0 = √𝜆𝐿

𝜋⋅

[𝑔1𝑔2(1 − 𝑔1𝑔2)]14

√𝑔1 + 𝑔2 − 2𝑔1𝑔2

From this yields that

9.18

0 < 𝑔1𝑔2 < 1

is required in order to have a real and positive beamwaist. This condition is called the stability condition for spherical resonators. Resonators which fulfill it are called stable resonators. In the stable case, the radiation field of the eigensolu-tions remains concentrated around the optical axis. The resonator is called un-stable, if this criterium is not met.

The calculations of the last section are only applicable for stable resonators, since otherwise the beam radius will not remain small compared to the dimensions of the mirrors and hence the negligence of diffraction is not justified. In an unstable

Appendix


resonator, however, the beam will expand until it is limited by the radiation leav-ing the resonators at one of the two mirrors. This can be utilized for beam out-coupling: The beam gets coupled out by passing the sides of the smaller mirror. For stable resonators however, the use of partially transmissive mirrors is re-quired.

Figure 29: Stability diagram for spherical resonators. depicts a stability diagram for spherical resonators: 𝑔1 is plotted against 𝑔2 and stable regions are marked. Obviously only parameter pairs within the 1st and 3rd quadrants are stable. Some parameter pairs corresponding to special resonator configurations are marked.

Figure 29: Stability diagram for spherical resonators.

9.4 Pulse width of the interference of N equiphase waves

In order to estimate the width of the created maxima, we will examine the su-perposition of equiphase waves in more detail. For a given time, e.g. t = 0, the superposition of modes corresponds exactly to the interference of N plane waves. For the intensity distribution of N interfering beams after (screen at a distance D) a lattice with lattice constant 𝑔 applies:

Appendix


9.19

𝐼(𝑧) = 𝐼0

sin (12

𝑁𝑘𝑧𝑧)2

sin (12 𝑘𝑧𝑧)

2 , 𝑘𝑧 = 𝑘𝑔

𝐷

Replacing 𝑘𝑧 with the wave number difference Δ𝑘 converts this to N-mode in-terference. Similarly we can consider the temporal intensity curve at a given po-sition, e.g. z = 0,

9.20

𝐼(𝑧) = 𝐼0

sin (𝑁Ω2

𝑡)2

sin (Ω2

𝑡)2 , Ω =

𝜋𝑐

𝐿

From equation 3.20 we then obtain the FWHM Δ𝑇 of the pulse as discussed in section 3.4.1

9.21

Δ𝑇 =

1

𝑁

2𝐿

𝑐=

1

𝑁𝑇

9.5 Derivation of the time-bandwidth product

The spectral amplitude 𝐴(𝜈) and the temporal shape of the field strength 𝐸(𝑡) are linked by Fourier-transformation. Thus for a Gaussian spectral amplitude

𝐴(𝜈) = 𝐴0 exp(−(𝜈 − 𝜈0)2/𝛿𝜈′2) we can obtain the electrical field strength:

9.22

𝐸(𝑡) =

1

2𝜋∫ 𝑑𝜈 𝐴(𝜈) exp(−𝑖2𝜋𝜈𝑡)

∞

−∞

.

Transforming yields:

9.23

𝐸(𝑡) = 𝐸0 exp(−𝑖2𝜋𝜈0𝑡) exp(−

𝑡2

𝜏′2)

with 𝐸0 =𝛿𝜈′

√2𝐴0 and 𝜏′ =

1

𝜋𝛿𝜈′. We also used the identity

∫ 𝑑𝑥 exp(−1

2

(𝑥−𝑥0)2

𝜎2 )∞

−∞= √2𝜋𝜎.

Appendix


Hence the bandwidth limited pulse has a Gaussian distribution in the time do-main, if the spectrum is a Gaussian function of the frequency. Due to 𝜏′ ⋅ 𝛿𝜈′ =1

𝜋 we obtain a value for the time-bandwidth product

9.24

𝜏 ⋅ 𝛿𝜈 =

2 ln(2)

𝜋≈ 0,44.

9.6 Locking of Modes

A simple application of trigonometric theorems helps understanding how the loss modulation of the electric field works. We will first consider the electric field 𝐸(𝑡) as a composition of modes with different frequencies 𝜔𝑛 and phases 𝜑𝑛 before passing the saturable absorber:

9.25

𝐸(𝑡) = ∑ 𝐸0,𝑛 cos(𝜔𝑛𝑡 + 𝜑𝑛) = ∑ 𝐸𝑛(𝑡)

∞

𝑛

∞

𝑛

with 𝜔𝑛 = 𝑛Ω

After passing the absorber, the wave train is modulated by the absorber. This proces is mathematically described by multiplying with the loss modulation:

9.26

𝑉(𝑡) = (𝐴 + 𝐵 cos(Ωt)) , 𝐴 + 𝐵 = 1

which yields the mode transformed by the loss modulation:

9.27

�̂�𝑛(𝑡) = 𝑉(𝑡)𝐸𝑛(𝑡) = 𝐴 𝐸𝑛 + 𝐵 𝐸0,𝑛 cos (𝜔𝑛𝑡 + 𝜑𝑛)cos (Ω𝑡)

By using the trigonometric theorem21

9.28

cos(𝑎) cos(𝑏) = 0.5 (cos(𝑎 − 𝑏) + cos (𝑎 + 𝑏))

and the relation 𝜔𝑛 = 𝑛 ∙ Ω one obtains:

9.29

�̂�𝑛(𝑡) = 𝑉(𝑡)𝐸𝑛(𝑡) = 𝐴 𝐸𝑛(𝑡) +

𝐵

2𝐸0,𝑛(cos(𝜔𝑛−1𝑡 + 𝜑𝑛) + cos (𝜔𝑛+1𝑡 + 𝜑𝑛))

Note that the last two terms now have the same phase 𝜑𝑛 as the original fre-quency mode but frequencies 𝜔𝑛±1, which are shifted by ±Ω. These terms are the sidebands visible in Figure 12. Due to this mechanism the phases of different frequency modes interact with each other.

21 Which is easily derived by Euler-decomposition of the cosine.

Appendix


The total modulated electrical field can now be written as:

9.30

�̂�(𝑡) = ∑ 𝐸0,𝑛 [𝐴 cos(𝜔𝑛𝑡 + 𝜑𝑛) +

𝐵

2cos(𝜔𝑛−1𝑡 + 𝜑𝑛) +

𝐵

2cos (𝜔𝑛+1𝑡 + 𝜑𝑛)]

∞

𝑛

This sum can be reorganized such that the summands contain similar frequen-cies instead of similar phases:

9.31

�̂�(𝑡) = ∑ [𝐴 𝐸0,𝑛 cos(𝜔𝑛𝑡 + 𝜑𝑛) +

𝐵 𝐸0,𝑛−1

2cos(𝜔𝑛𝑡 + 𝜑𝑛−1)

∞

𝑛

+𝐵 𝐸0,𝑛+1

2cos(𝜔𝑛𝑡 + 𝜑𝑛+1)]

The spectral amplitudes behind the absorber now do not only depend on the properties gained by the spontaneous emission (𝐸0,𝑛 and 𝜑𝑛) but furthermore on the properties of the adjacent modes (𝜑𝑛+1 and 𝜑𝑛−1). The implication this means can further be elaborated by the transformation

9.32

𝐴 cos(𝜔𝑡 + 𝜌) = 𝑎0 cos(𝜔𝑡) + 𝑎1 cos(𝜔𝑡 + 𝜑1) + 𝑎2cos (𝜔𝑡 + 𝜑2)

The time variable is implicitly chosen such that the phase in the 𝑎0-term disap-pears and thus the phases of the other two terms can be interpreted as phase differences. This is an equation with two unknown variables (𝜌 and 𝐴) and is solvable making one simplifying assumption. It can be transformed by using the trigonometric equation in formula 9.28:

9.33

𝐴 cos(𝜔𝑡) cos(𝜌) − 𝐴 sin(𝜔𝑡) sin(𝜌)

= 𝑎0 cos(𝜔𝑡) + 𝑎1 cos(𝜔𝑡) cos(𝜑1) − 𝑎1 sin(𝜔𝑡) sin(𝜑1)+ 𝑎2 cos(𝜔𝑡) cos(𝜑2) − 𝑎2 sin(𝜔𝑡) sin(𝜑2)

Our simplifying assumption is to treat sin(𝜔𝑡) cos(𝜔𝑡) as two independent equations:

9.34

𝐴 cos(𝜔𝑡) cos(𝜌) = 𝑎0 cos(𝜔𝑡) + 𝑎1 cos(𝜔𝑡) cos(𝜑1) + 𝑎2 cos(𝜔𝑡) cos(𝜑2)

and

Appendix


9.35

𝐴 sin(𝜔𝑡) sin(𝜌) = 𝑎1 sin(𝜔𝑡) sin(𝜑1) + 𝑎2 sin(𝜔𝑡) sin(𝜑2)

The solutions to these two equations are:

9.36

𝜌 = arcsin (

𝑎1 𝑠𝑖𝑛(𝜑1) + 𝑎2 𝑠𝑖𝑛(𝜑2)

𝐴)

and

9.37

𝐴2 = 𝑎02 + 𝑎1

2 + 𝑎22 + 2𝑎1𝑎2 sin(𝜑1) sin (𝜑2) + 2𝑎0𝑎1cos (𝜑1) + 2𝑎0𝑎2cos (𝜑2)

+ 2𝑎1𝑎2 cos(𝜑1) cos (𝜑2)

These equations describe a new constant phase 𝜌 and a new amplitude 𝐴. The new amplitude now depends on the respective phase differences between those terms (the 𝑎1𝑎2–terms can be brought into a form which shows the difference between 𝜑1 and 𝜑2). Depending on their phase relationship these terms can contribute positively (e.g. 𝑎0 + 𝑎1 + 𝑎2 for 𝜑1 = 𝜑2 = 0) or negatively (e.g. 𝐴 =𝑎0 − 𝑎1 − 𝑎2 for 𝜑1 = 𝜑2 = 𝜋) to the total amplitude.

This shows how adjacent modes can either amplify each other or extinguish each other depending on whether their phases are matched or not. Since the ampli-tude is amplified during the next circulation in the resonator, modes with well aligned phases will be amplified more by the medium.

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