1. Measuring Ultrashort Laser Pulses III: FROG tricks
Dealing with error in FROG measurements
Random error (noise) and how to suppress it; error bars
Nonrandom error (systematic error), how to know when it’s
there, and how to correct for it
The FROG marginals
Extremely simple FROG
beam geometry
Rick Trebino, Georgia Tech,

2. Random and Systematic Error in Pulse Measurement
Consider an autocorrelation measurement.

3. The FROG trace overdetermines the pulse. This has advantages.
Frequency (or phase)
Intensity
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.

4. Noise and its Suppression in FROG
Without noise With noise
Noise can corrupt a FROG trace and yield an incorrect pulse measurement.
Fortunately, there are many techniques for suppressing the noise with minimal distortion to the retrieved pulse.
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.
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).
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).

5. Noise and its Suppression in FROG: Example
FROG trace for
a complex pulse:
Time (pulse widths)
Intensity
Time (pulse widths)
Phase
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.

6. Corrupting a FROG Trace with Noise
Adding 10% additive noise turns this clear trace into this mess:
(Noise is Gaussian distributed with a mean of 10%.)
Note the resulting large background in the noisy trace.

7. Noise in the FROG trace can yield a noisy retrieved intensity and phase.
Background at large delays yields wings in the intensity. Background at large frequency offsets yields noise in those wings.
Time (pulse widths)
The retrieved pulse is very noisy!
It looks nothing like the actual pulse.
Time (pulse widths)

8. Subtracting off the background improves the retrieved intensity and phase.
Time (pulse widths)
Note the suppression of the wings and of the noise in the wings of the pulse.
Time (pulse widths)

9. Suppressing the corners of the trace also improves the retrieved intensity and phase.
Time (pulse widths)
Trace was multiplied by a “super-
Gaussian”: exp(-r4/d4), where r =
distance from trace center.
Time (pulse widths)
Note further improvement in the wings.

10. Low-pass filtering further improves the retrieved intensity and phase.
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!
Time (pulse widths)

11. Filtering summary: Always do it!
Without filtering
With filtering
Intensity:
Phase:
Time (pulse widths)
Time (pulse widths)
Dramatic improvements in the retrieval occur with little distortion. After
filtering,10% additive noise yields ~1% error; even less with multiplicative noise.

12. We can place error bars on the retrieved intensity and phase using the “Bootstrap” method.
Frequency
Frequency (or phase)
Intensity
Frequency
Repeat the above procedure several times, removing different points each time.
Calculate the mean and standard deviation of the intensity and phase (or frequency) for each time.
Frequency (or phase)
Intensity
Munroe, et al., CLEO Proceedings, 1998.
Press, et al., Numerical Recipes

13. Error Bars in the Intensity and Phase Using the Bootstrap Method—Theory
Introducing 1% additive noise to the FROG trace:
Intensity
Phase
Analytic intensity
Retrieved intensity
with noise
Analytic phase
Retrieved phase
with noise 4 2
Intensity (arb. units)
Phase (radians) -2 -4
Time (arb. units)
Time (arb. units)
Errors in the intensity are similar everywhere (slightly larger at the peak). Because the noise was ad-ditive, noise exists in the wings also.
Errors in the phase are much
larger in the wings, where the
intensity is near-zero and the
phase is necessarily undefined.

14. Error Bars in the Intensity and Phase Using the Bootstrap Method—Exp’t
In practice, SHG FROG traces have mostly multiplicative noise:
Intensity
Phase
-400
-200
200
400
Intensity (arb. units)
Time (fs) -3 -2 -1 1 2
-400
-200
200
400
Phase (radians)
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.
The phase error is low, except
in the wings, where, as before,
the intensity is near-zero and the
phase is necessarily undefined.

15. Sources of Systematic Error in FROG
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.

16. Geometrical time-smearing could yield systematic error.

17. Avoiding Geometrical Time-Smearing

18. 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
60-µm thick
KDP crystal
Phase-matched wave-length
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.

19. The FROG Marginals The delay marginal is the integral10 20 30 40 50 60
SHG FROG trace--expanded
Frequency
Delay
The delay marginal is the integral
of the FROG trace over all frequencies.
It is a function of delay only.
The frequency marginal is the integral
of the FROG trace over all delays:
It is a function of frequency only.
The FROG marginals can
be related to easily meas-
ured quantities:
The Autocorrelation
The Autoconvolution
of the Spectrum
The marginals are essential in checking for systematic error.
DeLong, et al., JQE, 32, 1253 (1996).

20. Applications of the SHG FROG Marginals
Taft, et al., J. Selected Topics in Quant. Electron., 3, 575 (1996)

21. Correcting for Systematic Error: Example
Attempts to measure a ~10-fs pulse produced this trace and pulse:
FROG trace
FROG trace
Retrieved pulse
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 phase-matching bandwidth in a 60-µm KDP crystal.)

22. Comparing the FROG frequency marginal with the spectrum autoconvolution
Although they should agree, they don’t! This is because the SHG crystal did not phase-match the longer wavelengths of the pulse.

23. Forcing the frequency marginal to agree with the spectrum autoconvolution yields an improved trace.
Multiplying the measured FROG trace by the ratio of the spectrum autoconvolution and frequency marginal:
Retrieved pulse
FROG trace
corrected
Independently measured spectrum
The retrieved spectrum now agrees with the measured spectrum.
The spectral phase has also changed.

24. The corrected pulse can now be used in comparisons with theory.
The measured and predicted pulse vs. time:
Predicted pulse
Measured pulse
Wavelength
Wavelength
This pulse measurement verifies that material dispersion is the pulse-length-limiting effect in this laser.

25. The FROG marginals can be used to understand the effects of systematic error on autocorrelation measurements.

26. Why pulse autocorrelations can appear narrower when using a thick crystal
Incorrect
(thick-crystal)
Autocorrelations


Correct
(thin-crystal) autocorrelation
 = 
It’s difficult to know if the crystal is thin enough!
Delay

27. Can we simplify FROG?
SHG
crystal
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
Camera
Camera
SHG
crystal
Spec-
trom-
eter
Variable delay
2 alignment q parameters q
(q, f) q
Crystal must be very thin, which hurts sensitivity.
1 alignment q parameter q
(delay) q
Remarkably, we can design a FROG without these components!

28. 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
SHG crystal
Very thick crystal acts like
a spectrometer! Why not replace the
spectrometer in FROG with a very thick crystal?
Very
thick crystal

29. 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.

30. GRENOUILLE Beam Geometry

31. In GRENOUILLE, the GVM must be large!
This is the opposite of the usual condition!

32. In GRENOUILLE, the GVD must still be small.

33. Putting it all together

34. GVM is usually much greater than GVD.

35. Testing GRENOUILLE
GRENOUILLE
FROG
Compare a GRENOUILLE measurement of a pulse with a tried-and-true FROG measurement of the same pulse:
Measured:
Retrieved:
Retrieved pulse in the time and frequency domains

36. Really Testing GRENOUILLE
FROG
Measured:
Retrieved:
Even for highly structured pulses, GRENOUILLE allows for accurate reconstruction of the intensity and phase.
Retrieved pulse in the time and frequency domains

37. Advantages of GRENOUILLE

38. 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.

39. 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 singular-value decomposition, which is much faster: < 1 sec.)
SHG FROG has an ambiguity in the direction of time.

40. FROG has a few advantages!

41. Or read the cover story in the June 2001 issue of OPN
To learn more, see the FROG web site!
Or read the cover story in the June 2001 issue of OPN
Or read the book!