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Measuring Ultrashort Laser Pulses
Rick Trebino
School of Physics
Georgia Institute of Technology
Atlanta, GA 30332 USA
1. Background, Phase Retrieval, and Autocorrelation
2. Frequency-Resolved Optical Gating
3. Interferometric Methods
These slides are available at http://public.me.com/ricktrebino.
The vast majority of humankind’s greatest
discoveries have resulted directly from
improved techniques for measuring light.
λ →Spectrometers led to quantum mechanics.
Interferometry led to relativity.
Microscopes led to biology.
Telescopes led
to astronomy.
X-ray crystallography
solved DNA.
And technologies, from medical imaging to
GPS, result from light measurement!
Most light is broadband, and hence
ultrafast, and also highly complex.
Ultrabroadband supercontinuum
Arbitrary waveforms
Ultrafast and complex in time
" in time and space
Nearly every pulse near a focus
Pulses emerging from almost any medium
We’ll learn much by measuring such light pulses.
Focusing pulse seen
from the side
Complex pulse
Time
Inte
nsity
Phase
To determine the temporal resolution of an experiment using it.
To determine whether a pulse can be
made even shorter.
To better understand the lasers that
emit them and to verify models
of ultrashort pulse generation.
To better study media: the better
we know the light in and light
out, the better we know the
medium we study with them.
To use shaped pulses to control
chemical reactions: Coherent control.
Because it’s there.
Why measure light with ultrafast variations?
As a molecule dissociates,
its emission changes color
(i.e., the phase changes),
revealing much about the
molecular dynamics, not avail-
able from the mere spectrum,
or even the intensity vs. time.
Excitation to excited state
Emission
Ground state
Excited
state
In order to measure
an event in time,
you need a shorter one.
To study this event, you need a
strobe light pulse that’s shorter.
But then, to measure the strobe light pulse,
you need a detector whose response time is even shorter.
And so on9
So, now, how do you measure the shortest event?
Photograph taken by Harold Edgerton, MIT
The Dilemma
The shortest events ever created are
ultrashort laser pulses.
So how do you measure the pulse itself?
You must use the pulse to measure itself.
But that isn’t good enough. It’s only as short as the pulse. It’s not shorter.
1 minute
10 fs light
pulse Age of universe
Time (seconds)
Computer
clock cycle
Camera
flash
Age of
pyramids
One
monthHuman existence
10-15 10-12 10-9 10-6 10-3 100 103 106 109 1012 1015 1018
1 femtosecond 1 picosecond
Intensity Autocorrelation1D Phase Retrieval
Single-shot autocorrelation
The Autocorrelation and SpectrumAmbiguities
Third-order Autocorrelation
Interferometric Autocorrelation
Measuring Ultrashort Laser Pulses I:
Background, Phase Retrieval, and
AutocorrelationThe dilemma
The goal: measuring the intensity and phase vs. time (or frequency)
Why?
The Spectrometer and Michelson Interferometer1D Phase Retrieval
Its electric field can be written:
Intensity Phase
[ ]{ }0Re exp ( (( ))( )) iE It tt tω φ= −
Alternatively, in the frequency domain:
Spectral
PhaseSpectrum
[ ]( ) exp ))( (SE iω ωω ϕ= −%
We need to measure both the temporal (or spectral) intensity and
phase.
A laser pulse has an intensity and phase
vs. time or frequency.
Sp
ectr
al
ph
ase
, ϕ(
ω)
FrequencySp
ectr
um
, S(ω
)
Ph
ase
, φ(t)
Time
Inte
nsity,
I(t
)
The instantaneous frequency:Example: Linear chirp P
hase, φ(t)
time
time
Fre
quency,
ω(t
)
time
The phase determines the pulse’s
frequency (i.e., color) vs. time.
0( ) /t d dtω φω= −
Time
Lig
ht
ele
ctr
ic f
ield
The group delay:Example: Linear chirp S
pectr
al
phase, ϕ(
ω)
frequency
frequency
Gro
up
dela
y, τg( ω
)
time
The group delay vs. frequency is
approximately the inverse of the
instantaneous frequency vs. time.
The spectral phase also yields a pulse’s
color evolution: the group delay vs. ωωωω.
( ) /g d dτ ϕω ω=
We’d like to be able to measure,
not only linearly chirped pulses,
but also pulses with arbitrarily complex
phases and frequencies vs. time.
The spectrometer measures the spectrum, of course. Wavelength varies
across the camera, and the spectrum can be measured for a single pulse.
Pulse Measurement in the Frequency Domain:
The Spectrometer
Collimating
Mirror
“Czerny-Turner”
arrangement
Entrance
Slit
Camera or
Linear Detector Array
Focusing
Mirror
Grating
Broad-
band
pulse
One-dimensional phase retrieval
E.J. Akutowicz, Trans. Am. Math. Soc. 83, 179 (1956)
E.J. Akutowicz, Trans. Am. Math. Soc. 84, 234 (1957)
Retrieving it is called the 1D phase retrieval problem.
It’s more interesting than it appears
to ask what we lack when we know
only the pulse spectrum S(ωωωω).
Obviously, what we lack is the spectral phase ϕϕϕϕ(ωωωω).
Even with extra information,
it’s impossible.
Recall:
time
Sp
ectr
al
ph
ase
, ϕ(
ω)
FrequencySp
ectr
um
, S(ω
)
[ ]( ) exp ))( (SE iω ωω ϕ= −%
Pulse Measurement in the Time Domain: Detectors
Examples: Photo-diodes, Photo-multipliers
Detectors are devices that emit electrons in response to photons.
Detectors have very slow rise and fall times: ~ 1 nanosecond.
As far as we’re concerned, detectors have infinitely slow responses.
They measure the time integral of the pulse intensity from –∞ to +∞:
The detector output voltage is proportional to the pulse energy (or fluence).
By themselves, detectors tell us little about a pulse.
2( )detectorV E t dt
∞
−∞∝ ∫
Another symbol
for a detector:
Detector
Detector
Translation stage
Pulse Measurement in the Time Domain:
Varying the pulse delay
Since detectors are essentially infinitely slow, how do we make time-
domain measurements on or using ultrashort laser pulses?
We’ll delay a pulse in time.
And how will we do that?
By simply moving a mirror!
Since light travels 300 µm per ps, 300 µm of mirror displacement
yields a delay of 2 ps. This is very convenient.
Moving a mirror backward by a distance L yields a delay of:
τ = 2 L /cDo not forget the factor of 2!
Light must travel the extra distance
to the mirror—and back!
Input
pulse E(t)
E(t–τ)
Mirror
Output
pulse
We can also vary the delay using
a mirror pair or corner cube.
Mirror pairs involve two
reflections and displace
the return beam in space:
But out-of-plane tilt yields
a nonparallel return beam.
Corner cubes involve three reflections and also displace the return
beam in space. Even better, they always yield a parallel return beam:
Hollow corner cubes avoid propagation through glass.
Translation stage
Input
pulse
E(t)
E(t–τ)
MirrorsOutput
pulse
Apollo 11
Measuring the interferogram is equivalent to measuring the spectrum.
Pulse Measurement in the Time Domain:
The Michelson Interferometer
2 2 *( ) ( ) 2Re[ ( ) ( )]E t E t E t E t dtτ τ∞
−∞= + − − −∫
2( ) ( ) ( )MIV E t E t dtτ τ
∞
−∞∝ − −∫
2
*
( ) 2 ( )
2Re ( ) ( )
MIV E t dt
E t E t dt
τ
τ
∞
−∞
∞
−∞
∝
− −
∫
∫
∝ Pulse
energy
Field autocorrelation: Γ(2)(τ). Looks
interesting, but the Fourier transform of
ΓΓΓΓ(2)(ττττ) is just the spectrum!
Beam-splitter
Input
pulse
Delay
Slow
detector
Mirror
Mirror
E(t)
E(t–τ)
VMI(τ )
“Interfer-
ogram”
⇒VMI(τ)
0 Delay
The detected voltage will be:
Can we use these methods to measure a pulse?
V. Wong & I. A. Walmsley, Opt. Lett. 19, 287-289 (1994)
I. A. Walmsley & V. Wong, J. Opt. Soc. Am B, 13, 2453-2463 (1996)
Result: Using only time-independent, linear components, complete
characterization of a pulse is NOT possible with a slow detector.
Translation: If you don't have a detector or modulator that is fast
compared to the pulse width, you CANNOT measure the pulse
intensity and phase with only linear measurements, such as a
detector, interferometer, or a spectrometer.
We need a shorter event, and we don’t have one.
But we do have the pulse itself, which is a start.
And we can devise methods for the pulse to gate itself using
optical nonlinearities.
Pulse Measurement in the Time Domain:
The Intensity Autocorrelator
SHG
crystal
The Intensity
Autocorrelation:( ) ( )(2) ( )A I t I t dtτ τ
∞
−∞
≡ −∫
SHGcrystal
Pulse to be measured
Variable delay, τ
Detector
Beamsplitter
E(t)
E(t–τ)
Esig(t,τ)
The signal field is E(t) E(t-τ).So the signal intensity is I(t) I(t-τ)
Crossing beams in a nonlinear-optical crystal, varying the delay
between them, and measuring the signal pulse energy vs.
delay yields the Intensity Autocorrelation, A(2)(ττττ).
Gaussian Pulse and Its Autocorrelation
Pulse Autocorrelation
t τ
exp −2 ln2t
∆τ pFWHM
2
exp −2 ln2τ∆τA
FWHM
2
( )I t =
1.41 FWHM FWHM
p Aτ τ∆ = ∆
A2( ) τ( ) =
∆τ pFWHM
∆τ AFWHM
Pulse Autocorrelation
t τ
Sech2 Pulse and Its Autocorrelation
sech2 1.7627t
∆t pFWHM
3
sinh2 2.7196τ
∆τAFWHM
2.7196τ∆τA
FWHM coth2.7196τ∆τA
FWHM
−1
I t( ) =
1.54 FWHM FWHM
p Aτ τ∆ = ∆
A2( ) τ( ) =
∆τ pFWHM
∆τ AFWHM
Since theoretical models of ultrafast lasers often predict sech2 pulse
shapes, people usually simply divide the autocorrelation width by 1.54
and call it the pulse width. Even when the autocorrelation is Gaussian9
The Intensity Autocorrelation is always
symmetrical with respect to delay.
(2) (2)( ) ( )A Aτ τ= −
(2) (2)( ) ( ) ( ) ( ) ( ) ( )A I t I t dt I t I t dt Aτ τ τ τ′ ′ ′= − = + = −∫ ∫′ t = t − τ
This is easy to show:
⇒
This means that intensity autocorrelation cannot tell the direction of
time of a pulse. This is, however, a trivial ambiguity—not a big deal.
It’s known, not usually a problem, and easy to remove.
Of course, autocorrelation says nothing about the pulse phase either.
Autocorrelations of more complex intensities
-80 -60 -40 -20 0 20 40 60 80
Autocorrelation
AutocorrelationAmbiguous Autocorrelation
Delay
-40 -30 -20 -10 0 10 20 30 40
Intensity
IntensityAmbiguous Intensity
Time
Autocorrelations nearly always have considerably less structure
than the corresponding intensity.
An autocorrelation typically corresponds to many different intensities.
Thus the autocorrelation does not uniquely determine the intensity.
Autocorrelation
Even nice autocorrelations have ambiguities.
These complex intensities have nearly Gaussian autocorrelations.
-80 -60 -40 -20 0 20 40 60 80
IntensityAmbiguous Intensity
Time
Intensity
-150 -100 -50 0 50 100 150
Autocorrelation
AutocorrelationAmbig AutocorGaussian
Delay
Autocorrelation has many nontrivial ambiguities! They’re unknown,
usually a serious problem, and impossible to remove.
Retrieving the intensity from the intensity
autocorrelation is also equivalent to the
1D Phase-Retrieval Problem!
Applying the Autocorrelation Theorem:
2(2){ ( )} { ( )}A I tτ =F F
Thus, the autocorrelation yields only the magnitude of the Fourier
Transform of the Intensity. It says nothing about its phase!
It’s the 1D Phase-Retrieval Problem again!
We do have additional information: I(t) is always positive.
The positivity constraint removes many nontrivial ambiguities.
But many remain, and no one knows how to find them.
(2)( ) ( ) ( )A I t I t dtτ τ= −∫
Autocorrelation of Very Complex Pulses
x intens ities with G aussian s low ly v arying
Intensity AutocorrelationAs the intensity
increases in
complexity, its
autocorrelation
approaches a
broad smooth
background and a
coherence spike.
2(2) (2)( ) ( ) ( ) ( )env envA I t I t dtτ τ τ
∞
−∞= Γ + −∫
Ienv(t)
ΓΓΓΓ(2)(t)
This shows why
retrieving the
intensity from the
autocorrelation is
fundamentally
impossible!
x
Geometrical distortions in autocorrelation
This effect causes a range of delays to occur at a given time and could
cause geometrical smearing, that is, a broadening of the
autocorrelation in multi-shot measurements.
When crossing beams at an angle, the delay varies across the beam.
Pulse #1
Pulse #2
Here, pulse #1 arrivesearlier than pulse #2
Here, pulse #1 and pulse #2arrive at the same time
Here, pulse #1 arriveslater than pulse #2
SHG crystal
xx
Single-shot autocorrelation
Use a large beam and a large beam crossing angle to achieve the
desired range of delays. Then image the crystal onto a camera.
So single-shot SHG AC has no geometrical smearing!
Crossing beams at an angle also maps delay onto transverse position.
( ) 2( / ) sin( / 2) /x x c x cτ θ θ= ≈
Long pulse Short pulse
SHSH
Thick crystal creates too narrow a SH
spectrum in a given direction and so
can’t be used for an autocorrelator.
Very thin crystal creates a broad SH
spectrum in the forward direction.
Autocorrelators require such very thin
crystals.
Very thin
SHG crystal
Thick SHG
crystal
We need to generate SH for all wavelengths in the pulse.
But the generated SH wavelength depends on angle.
And its angular width varies inversely with the crystal thickness.
Second harmonic generation bandwidth
The SHG crystal
bandwidth must
exceed that of the
pulse.
SHG efficiency vs. wavelength for
the nonlinear-optical crystal, beta-
barium borate (BBO) for different
crystal thicknesses (L):
L = 10 µm
L = 100 µm
L = 1000 µm
The SHG bandwidth scales as 1/L,
while the efficiency scales as L2.
A nasty trade-off.
The SHG bandwidth is usually called
the phase-matching bandwidth.
Third-order Autocorrelation
Nonlinearmedium (glass)
Pulse to be measured
Variable delay, τ
Beamsplitter
E(t)
E(t-τ)
Esig(t,τ) = E(t) |E(t-τ)|2
Some ambiguity problems in autocorrelation can be overcome by
using a third-order nonlinearity, such as the Optical Kerr effect.
45°
polarization
rotation
( ) ( )2(3) ( )A I t I t dtτ τ∞
−∞
≡ −∫
Isig(t,τ) = I(t) I(t-τ)2
The third-order autocorrelation is not
symmetrical, so it yields slightly more
information, but still not the full pulse.
This arrangement is called
Polarization Gating.
Note the 2
When a shorter reference pulse is available:
The Intensity Cross-Correlation
ESF (t,τ ) ∝ E(t)Eg(t −τ )
( , ) ( ) ( )SF gI t I t I tτ τ⇒ ∝ −
The Intensity Cross-correlation:
Delay
Unknown pulseSlow
detectorE(t)
Eg(t–τ)( ) ( )detV Cτ τ∝
SFGcrystal
LensReference gate pulse
C(τ) ≡ I(t) Ig (t− τ) dt−∞
∞
∫
If a shorter reference pulse is available (it need not be known), then it
can be used to measure the unknown pulse. In this case, we perform
sum-frequency generation, and measure the energy vs. delay.
If the reference pulse is much shorter than the unknown pulse, then the
intensity cross-correlation fully determines the unknown pulse intensity.
Michelson
Interferometer
Interferometric Autocorrelation
What if we use a collinear beam geometry, and allow the autocorrelator
signal light to interfere with the SHG from each individual beam?
2(2) 2( ) [ ( ) ( )]IA E t E t dtτ τ
∞
−∞≡ − −∫
2(2) 2 2( ) ( ) ( ) 2 ( ) ( )IA E t E t E t E t dtτ τ τ
∞
−∞≡ + − − −∫
Usual
Autocor-
relation
term
Newterms
Also called the Fringe-Resolved Autocorrelation
Filter Slow
detector
SHG
crystal
( ) ( )E t E t τ− −2[ ( ) ( )]E t E t τ− −
Lens
Beam-splitter
Input
pulse
Delay
Mirror
Mirror
E(t)
E(t–τ)
Interferometric Autocorrelation Math
The measured intensity vs. delay is:
(2) 2 2 *2 *2 * *( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( )IA E t E t E t E t E t E t E t E t dtτ τ τ τ τ∞
−∞
≡ + − − − + − − − ∫
{ 2(2) 2 2 *2 2 * *( ) ( ) ( ) ( ) 2 ( ) ( ) ( )IA E t E t E t E t E t E tτ τ τ
∞
−∞
= + − − − +∫Multiplying this out:
22 *2 2 2 * *( ) ( ) ( ) 2 ( ) ( ) ( )E t E t E t E t E t E tτ τ τ τ− + − − − − +
}2 2*2 *22 ( ) ( ) ( ) 2 ( ) ( ) ( ) 4 ( ) ( )E t E t E t E t E t E t E t E t dtτ τ τ τ− − − − − + −
{ 2 2 *2 *( ) ( ) ( ) 2 ( ) ( ) ( )I t E t E t I t E t E tτ τ∞
−∞
= + − − − +∫2 *2 2 *( ) ( ) ( ) 2 ( ) ( ) ( )E t E t I t I t E t E tτ τ τ τ− + − − − − +
}* *2 ( ) ( ) ( ) 2 ( ) ( ) ( ) 4 ( ) ( )I t E t E t I t E t E t I t I t dtτ τ τ τ− − − − − + −
where I(t) ≡ E(t)2
The Interferometric Autocorrelation is the
sum of four different quantities.
2 2( ) ( )I t I t dtτ∞
−∞= + −∫4 ( ) ( )I t I t dtτ
∞
−∞+ −∫
2 2*( ) ( ) . .E t E t dt c cτ∞
−∞+ − +∫
[ ] *2 ( ) ( ) ( ) ( ) .I t I t E t E t dt c cτ τ∞
−∞
− + − − +∫
Constant (uninteresting)
Sum-of-intensities-weighted ω“interferogram” of E(t) ω(oscillates at ω in delay)
Intensity autocorrelation
Interferogram of the SH;equivalent to the SH spectrum(oscillates at 2ω in delay)
The interferometric autocorrelation simply combines several measuresof the pulse into one (admittedly complex) trace. Conveniently, however,they occur with different oscillation frequencies: 0, ω, and 2ω.
Interferometric Autocorrelation: Examples
7-fs sech2 800-nm pulse
Double pulse
Pulse with cubic spectral
phase
Pulse #2
Phase tFWHM=
5.3 fs
-40 -20 0 20 40
Intensity
Despite very different pulse lengths, these pulses have nearly identical IAs!
Chung
and
Weiner,
IEEE
JSTQE,
2001.
Interferometric Autocorrelation also has ambiguities.
Interferometric
Autocorrelations
for Pulses
#1 and #2:
#1 and #2
Pulse #1
Intensity
Phase
tFWHM =
7.4 fs
-40 -20 0 20 40
Interferometric Autocorrelation of a
complex pulse
Complex pulse
Its Interferometric
Autocorrelation
Note that almost all the information in the pulse is missing from its
interferometric autocorrelation. Thus there are also many nontrivial
ambiguities in interferometric autocorrelation, and it is also
fundamentally impossible to retrieve a pulse from its IA.
-20 -10 0 10 200
0.2
0.4
0.6
0.8
1
Inte
nsity (a.u
)
Delay (fs)
-20
0
20
40
Phase (ra
d)
-10 0 100
2
4
6
8
Delay (fs)
Inte
nsity (
a.u
)
Quiz: Which is the most difficult?
A. Time travel
B. World peace
C. Human teleportation
D. Retrieving a pulse from its intensity autocorrelation
or interferometric autocorrelation
The correct answer is D. Only it has been proven to be
fundamentally impossible. The others are hard, but,
as far as we know, they may be possible.