Measuring Ultrashort Laser Pulses III:
FROG tricks
Dealing with error in FROG measurements
Random error (noise) and how to suppress it; error bars
N pixels
FROG trac e--ex pa nd ed
60
50
40
30
20
10
10
20
30
40
50
60
Delay
remove
Nonrandom error (systematic error), how toRandomly
know when
it’s about half of
the datapoints from the trace.
there, and how to correct for it
The FROG marginals
Extremely simple FROG
beam geometry
Rick Trebino, Georgia Tech, rick.trebino@physics.gatech.edu
Random and Systematic Error in Pulse
Measurement
Consider an
autocorrelation
measurement.
Frequency
(or phase)
Intensity
The FROG trace overdetermines the pulse.
This has advantages.
Advantages:
1. Natural √N averaging occurs, reducing noise.
2. Can perform filtering operations to reduce noise further.
3. Can run algorithm with some points removed to determine error bars in the
intensity and phase—independent of the source of noise.
4. Can identify the presence of systematic error—independent of the source.
5. Can remove systematic error—independent of the source.
6. Can understand distortions in the autocorrelation due to systematic error.
Noise and its Suppression in FROG
3.00
Frequency (1/pulse width)
Fortunately, there are many techniques
for suppressing the noise with minimal
distortion to the retrieved pulse.
Frequency (1/pulse width)
Noise can corrupt a FROG trace and
yield an incorrect pulse measurement.
Without noise
0.00
-3.00
-3.00
With noise
3.00
0.00
-3.00
0.00
3.00
Time (pulse widths)
-3.00
0.00
3.00
Time (pulse widths)
1. Background subtraction
The FROG trace should be an island in a sea of zeroes. Otherwise,
data are missing. So we can subtract off any background.
2. Corner suppression
No data should be in the corners of the trace; what’s there can
only be noise, so set it to ~zero by multiplying by exp(-r4/d4).
3. Low-pass filtering
Noise varies from pixel to pixel, that is, with a high frequency.
The FROG trace has only slower variations.
Fittinghoff, et al., JOSA B, 12, 1955 (1995).
Noise and its Suppression in FROG: Example
0.00
Normalized Intensity
3.00
Intensity
1.2
0.8
0.4
0
-40
-5
-20
-2.5
00
20
2.5
40
5
Time (pulse widths)
-3.00
-3.00
0.00
3.00
Time (pulse widths)
This pulse has a narrow glitch in
its intensity vs. time, and it has
a phase jump of ~2 radians, a
difficult feature to reproduce.
We’ll add noise to this trace.
Phase
16
Phase (radians)
Frequency (1/pulse width)
FROG trace for
a complex pulse:
12
8
4
0
-40
-5
-20
-2.5
00
20
2.5
Time (pulse widths)
40
5
Corrupting a FROG Trace with Noise
3.00
Frequency (1/pulse width)
Frequency (1/pulse width)
Adding 10% additive noise turns this clear trace into this mess:
0.00
-3.00
-3.00
0.00
3.00
Time (pulse widths)
3.00
0.00
-3.00
-3.00
0.00
3.00
Time (pulse widths)
(Noise is Gaussian distributed with a mean of 10%.)
Note the resulting large background in the noisy trace.
Noise in the FROG trace can yield a
noisy retrieved intensity and phase.
3.00
-5
-2.5
0
2.5
5
Time (pulse widths)
0.00
1.2
-3.00
-3.00
Normalized Intensity
Frequency (1/pulse width)
Background at large delays yields
wings in the intensity. Background
at large frequency offsets yields
noise in those wings.
Actual
Retrieved
0.8
0.00
3.00
0.4
Time (pulse widths)
The retrieved pulse is very noisy!
0
It looks nothing like the actual pulse.
-5
-2.5
0
2.5
Time (pulse widths)
5
Frequency (1/pulse width)
Subtracting off the background improves
the retrieved intensity and phase.
FROG trace with 10% additive noise—after subtracting
the mean of the noise
3.0
0.0
-5
-2.5
0
2.5
5
Time (pulse widths)
-3.0
-3.0
0.0
3.0
Time (pulse widths)
Note the suppression of the wings
and of the noise in the wings of
the pulse.
-5
-2.5
0
2.5
Time (pulse widths)
5
Suppressing the corners of the trace also
improves the retrieved intensity and phase.
Frequency (1/pulse width)
FROG trace with 10% additive noise—after subtracting
off the background mean
and suppressing the corners
3.0
-5
0.0
-2.5
0
2.5
5
Time (pulse widths)
-3.0
-3.0
0.0
3.0
Time (pulse widths)
Trace was multiplied by a “superGaussian”: exp(-r4/d4), where r =
distance from trace center.
-5
Note further improvement in the wings.
-2.5
0
2.5
Time (pulse widths)
5
Wavelength (1/pulse width)
Low-pass filtering further improves the
retrieved intensity and phase.
FROG trace with 10% additive noise after mean subtraction, super-Gaussian
filtering and lowpass filtering
3.0
-5
0.0
-2.5
0
2.5
5
Time (pulse widths)
-3.0
-3.0
0.0
3.0
Time (pulse widths)
Fourier-transforming the trace, retaining only
the center region, and transforming back.
The resulting intensity and phase now
look very much like the actual curves!
-5
-2.5
0
2.5
Time (pulse widths)
5
Filtering summary: Always do it!
Without filtering
With filtering
Intensity:
-5
-2.5
0
2.5
5
Phase:
-5
-2.5
0
2.5
Time (pulse widths)
5 -5
-2.5
0
2.5
Time (pulse widths)
5
Dramatic improvements in the retrieval occur with little distortion. After
filtering,10% additive noise yields ~1% error; even less with multiplicative noise.
We can place error bars on the retrieved intensity
and phase using the “Bootstrap” method.
N pixels
Frequency
(or phase)
Intensity
Frequency
Frequency
FROG trac e--ex pa nd ed
60
50
40
30
20
10
10
20
30
40
50
60
Delay
Randomly remove about half of
the datapoints from the trace.
Munroe, et al., CLEO Proceedings, 1998.
Press, et al., Numerical Recipes
Intensity
Calculate the mean and standard
deviation of the intensity and phase
(or frequency) for each time.
Frequency
(or phase)
Repeat the above procedure several times, removing different points each time.
Time
Error Bars in the Intensity and Phase
Using the Bootstrap Method—Theory
Introducing 1% additive noise to the FROG trace:
Phase
Analytic intensity
Retrieved intensity
with noise
Time (arb. units)
Errors in the intensity are similar
everywhere (slightly larger at the
peak). Because the noise was additive, noise exists in the wings also.
4
Phase (radians)
Intensity (arb. units)
Intensity
Analytic phase
Retrieved phase
with noise
2
0
-2
-4
Time (arb. units)
Errors in the phase are much
larger in the wings, where the
intensity is near-zero and the
phase is necessarily undefined.
Error Bars in the Intensity and Phase
Using the Bootstrap Method—Exp’t
In practice, SHG FROG traces have mostly multiplicative noise:
Phase
Phase (radians)
Intensity (arb. units)
Intensity
2
1
0
-1
-2
-3
-400
-200
0
200
400
Time (fs)
Errors in the intensity are much
larger at the peak. Because the
noise was multiplicative, there is
almost no noise in the wings.
-400
-200
0
200
400
Time (fs)
The phase error is low, except
in the wings, where, as before,
the intensity is near-zero and the
phase is necessarily undefined.
Sources of Systematic Error in FROG
Source:
Check? Correct?
Variation in spectral response of optics
√
√
Variation in spectral response of camera
√
√
Dispersion of nonlinearity
√
√
Group-velocity mismatch/phase-matching bandwidth
√
√
Variable alignment of beam overlap
√
Unknown
√
Possibly!
It is possible, not only to check for systematic error, but also to correct
it in most pulse measurements using FROG, even when its origin in
unknown.
Geometrical time-smearing could yield
systematic error.
Avoiding Geometrical Time-Smearing
Wavelength-dependent SHG phase-matching
efficiency yields systematic error.
Group-velocity mismatch yields a wavelength-dependent SHG efficiency.
Usually, it’s a sinc2 curve, but even when two such curves fortuitously
overlap, there’s wavelength-dependent SHG efficiency:
Phase-matching efficiency vs. wavelength
Phasematched
wavelength
60-µm thick
KDP crystal
It’s impossible
to achieve the
desired flat
curve.
Taft, et al., J. Selected Topics
in Quant. Electron., 3, 575
(1996)
Even very thin SHG crystals may lack sufficient bandwidth for a 10-fs pulse.
The FROG Marginals
M ( ) 


50
40
30
20
10
10
I FR OG ( ,  ) d
The frequency marginal is the integral
of the FROG trace over all delays:
It is a function of frequency only.
M ( ) 
Frequency
The delay marginal is the integral
of the FROG trace over all frequencies.
It is a function of delay only.
SHG FROG trace--expanded
60
I F ROG ( ,  ) d
The marginals are essential in
checking for systematic error.
20
30
40
50
60
Delay
The FROG marginals can
be related to easily measured quantities:
M ( )  The Autocorrelation
M ( ) 
The Autoconvolution
of the Spectrum
DeLong, et al., JQE, 32, 1253 (1996).
Applications of the SHG FROG Marginals
Taft, et al., J. Selected Topics
in Quant. Electron., 3, 575
(1996)
Correcting for Systematic Error: Example
Wavelength (nm)
Attempts to measure a ~10-fs pulse produced this trace and pulse:
350
370
390
410
430
450
470
490
-40
FROG trace
measured 350
370
390
410
430
450
470
490
-20 0
20
Delay (fs)
40 -40
FROG trace
Retrieved pulse
retrieved
-20 0
20
Delay (fs)
40
Independently measured spectrum
Usually, systematic error yields poor
convergence. Here, however, despite good
convergence, the retrieved spectrum
disagrees with the independently measured
spectrum. (This is due to insufficient phasematching bandwidth in a 60-µm KDP crystal.)
Comparing the FROG frequency marginal
with the spectrum autoconvolution
1.2
FROG Marginal
Autoconvolution
Intensity
1
0.8
0.6
0.4
0.2
0
360
420 450 480
390
Wavelength (nm)
510
Although they should agree, they don’t! This is because the SHG
crystal did not phase-match the longer wavelengths of the pulse.
Forcing the frequency marginal to agree with the
spectrum autoconvolution yields an improved trace.
Wavelength (nm)
Multiplying the measured FROG trace by the ratio of the
spectrum autoconvolution and frequency marginal:
350
370
390
410
430
450
470
490
-40
FROG trace
350
corrected370
390
410
430
450
470
490
measured
-20 0
20
Delay (fs)
40 -40
1.2
FROG trace
retrieved
Retrieved pulse
3
2.5
1
0.8
2
0.6
1.5
0.4
1
0.2
0.5
0
680
765
850
935
0
1020
Wavelength (nm)
-20 0
20
Delay (fs)
The retrieved spectrum now agrees
with the measured spectrum.
The spectral phase has also changed.
40
Independently measured spectrum
The corrected pulse can now be used
in comparisons with theory.
The measured and predicted pulse vs. time:
Measured pulse
Intensity
0.6
0.4
0.8
6
0.6
4
0.4
2
0.2
0.2
0
0
-40
-20
0
Time (fs)
20
8
6
0.6
Intensity
Phase
0.4
0
-20
8
0.8
0
Time (fs)
4
Wavelength
-40
20
-20
40
0
6
4
2
0.2
0
-40
40
10
2
0
20
40
Time (fs)
This pulse measurement verifies that material dispersion is the pulselength-limiting effect in this laser.
0
Phase (rad)
Wavelength
10
Phase (rad)
0.8
Intensity
Phase
1
8
Phase (rad)
Intensity
1
Predicted pulse
1.2
10
Intensity
1.2
Intensity
Phase
1
1.2
The FROG marginals can be used to understand
the effects of systematic error on autocorrelation
measurements.
Why pulse autocorrelations can appear
narrower when using a thick crystal
 = phase-matching bandwidth / pulse bandwidth
Thick crystal: 
Thicker crystal: 
SHG FROG trace
Cropped
SHG FROG trace
Very cropped
SHG FROG trace
Frequency
Thin crystal: 
Gaussian spectrum w/
cubic spectral phase
Delay
Delay
Delay
Autocorrelation
Incorrect
(thick-crystal)
Autocorrelations


Correct
(thin-crystal)
autocorrelation
=
Delay
Thick crystal
suppresses wings!
It’s difficult to know
if the crystal is thin
enough!
Can we simplify FROG?
Pulse to be
measured
FROG has 3 sensitive alignment degrees of
freedom (q,f of a mirror and also delay).
The thin crystal is also a pain.
Pulse to be
measured
SHG
crystal
Variable
delay
2 alignment q
parameters q
(q,f) q
Camera
Crystal must
be very thin,
which hurts
sensitivity.
1 alignment q
parameter q
(delay) q
Remarkably, we can design a FROG without these components!
SpecCamera
trometer
The angular width of second harmonic
varies inversely with the crystal thickness.
Suppose white light with a large divergence angle impinges on an SHG
crystal. The SH generated depends on the angle. And the angular width
of the SH beam created varies inversely with the crystal thickness.
Very thin crystal creates broad SH spectrum in all directions.
Standard autocorrelators and FROGs use such crystals.
Thin crystal creates narrower SH spectrum in
a given direction and so can’t be used
for autocorrelators or FROGs.
Very
Thin
SHG
crystal
Thick crystal begins to
separate colors.
Thin
SHG
crystal
Thick
Very thick crystal acts like
SHG crystal
a spectrometer! Why not replace the
spectrometer in FROG with a very thick crystal?
Very
thick crystal
GRating-Eliminated No-nonsense Observation
of Ultrafast Incident Laser Light E-fields
(GRENOUILLE)
Patrick O’Shea, Mark Kimmel, Xun Gu
and Rick Trebino, Optics Letters, 2001;
Trebino, et al., OPN, June 2001.
GRENOUILLE Beam Geometry
In GRENOUILLE, the GVM must be large!
This is the opposite
of the usual condition!
In GRENOUILLE, the GVD must still be small.
Putting it all together
GVM is usually much greater than GVD.
Testing GRENOUILLE
Compare a GRENOUILLE
measurement of a pulse
with a tried-and-true FROG
measurement of the same
pulse:
GRENOUILLE
Measured:
Retrieved:
Retrieved pulse in the time and frequency domains
FROG
Really Testing GRENOUILLE
GRENOUILLE
Even for highly structured
pulses, GRENOUILLE
allows for accurate
reconstruction of the
intensity and phase.
Measured:
Retrieved:
Retrieved pulse in the time and frequency domains
FROG
Advantages of GRENOUILLE
Disadvantages of GRENOUILLE
It currently only works for
pulses between ~ 40 fs and
~ 300 fs.
Like other single-shot
techniques, it requires
good spatial beam quality.
Improvements on the horizon:
Inclusion of GVD and GVM in FROG code to extend the range of
operation to shorter and longer pulses.
Folded beam geometry for even more compact arrangement.
Disadvantages of FROG and its relatives
FROG requires taking a lot of data. While this can be done
easily with a readily available camera, and it allows error
checking and correcting, multi-shot FROG measurements can
take minutes.
The algorithm can be slow, also taking minutes for complex
pulses. (There is, however, a new algorithm, based on singularvalue decomposition, which is much faster: < 1 sec.)
SHG FROG has an ambiguity in the direction of time.
FROG has a few advantages!
To learn more, see the FROG web site!
www.physics.gatech.edu/frog
Or read the cover story in
the June 2001 issue of OPN
Or read the book!