1. 11. Measuring Ultrashort Laser Pulses II: FROG
The Musical Score and the Spectrogram
Frequency-Resolved Optical Gating (FROG)
1D vs. 2D Phase Retrieval
FROG as a 2D Phase-retrieval Problem
Second-harmonic-generation (SHG) FROG (and other geometries)
Measuring the shortest event ever created
Single-shot FROG, XFROG, and TREEFROG

2. Consider, for example, acoustic waveforms.
Autocorrelation and related techniques yield little information about the pulse.
Perhaps it’s time to ask how researchers in other fields deal with their waveforms…
Consider, for example, acoustic waveforms.

3. Most people think of acoustic waves in terms of a musical score.
It’s a plot of frequency vs. time, with information on top about the intensity.
The musical score lives in the “time-frequency domain.”

4. If E(t) is the waveform of interest, its spectrogram is:
where g(t-t) is a variable-delay gate function and t is the delay.
Without g(t-t), SpE(w,) would simply be the spectrum.

5. Spectrogram

6. Spectrograms for Linearly Chirped Pulses
Frequency
Time
Frequency
Delay

7. Properties of the Spectrogram
Algorithms exist to retrieve E(t) from its spectrogram.
The spectrogram essentially uniquely determines the waveform intensity, I(t), and phase, (t).
There are a few ambiguities, but they are “trivial.”
The gate need not be—and should not be—significantly shorter than E(t).
Suppose we use a delta-function gate pulse:
= The Intensity.
No phase information!
The spectrogram resolves the dilemma! It doesn’t need the shorter event! It temporally resolves the slow components and spectrally resolves the fast components.

8. Frequency-Resolved Optical Gating (FROG)
“Polarization-Gate” Geometry
Spectro-
meter
Trebino, et al., Rev. Sci. Instr., 68, 3277 (1997) Kane and Trebino, Opt. Lett., 18, 823 (1993).

9. FROG

10. FROG Traces for Linearly Chirped Pulses
Frequency
Time
Frequency
Delay

11. FROG Traces for More Complex Pulses
Intensity
Frequency
Time
Frequency
Frequency
Delay
Delay

12. The FROG trace is a spectrogram of E(t).
Substituting for Esig(t,) in the expression for the FROG trace:
Esig(t,)  E(t) |E(t–)|2
yields:
g(t–) |E(t–)|2
where:
Unfortunately, spectrogram inversion algorithms
require that we know the gate function.

13. Instead, consider FROG as a two- dimensional phase-retrieval problem.
If Esig(t,), is the 1D Fourier transform with respect to delay t of some new signal field, Esig(t,W), then: 
The input pulse, E(t), is easily obtained from Esig(t,W): E(t)  Esig(t,) 
and 
So we must invert this integral equation and solve for Esig(t,W).
This integral-inversion problem is the 2D phase-retrieval problem,
for which the solution exists and is unique.
And simple algorithms exist for finding it.
Stark, Image Recovery, Academic Press, 1987.

14. 1D vs. 2D Phase Retrieval
1D Phase Retrieval: Suppose we measure S(w) and desire E(t), where:
We assume that
E(t) and E(x,y) are
of finite extent.
Given S(w), there are infinitely many solutions
for E(t). We lack the spectral phase.
2D Phase Retrieval: Suppose we measure S(kx,ky) and desire E(x,y):
Stark,
Image Recovery, Academic Press, 1987.
Given S(kx,ky), there is essentially one solution for E(x,y)!!!
It turns out that it’s possible to retrieve the 2D spectral phase! .
These results are related to the Fundamental Theorem of Algebra.

15. Phase Retrieval and the Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that all polynomials can
be factored:
fN-1 zN-1 + fN-2 zN-2 + … + f1 z + f0 = fN-1 (z–z1 ) (z–z2 ) … (z–zN–1)
The Fundamental Theorem of Algebra fails for polynomials of two variables.
Only a set of measure zero can be factored.
fN-1,M-1 yN-1 zM-1 + fN-1,M-2 yN-1 zM-2 + … + f0,0 = ?
Why does this matter?
The existence of the 1D Fundamental Theorem of Algebra implies that
1D phase retrieval is impossible.
The non-existence of the 2D Fundamental Theorem of Algebra implies that
2D phase retrieval is possible.

16. Phase Retrieval & the Fund Thm of Algebra 2
1D Phase Retrieval and the Fundamental Theorem of Algebra
The Fourier transform {F0 , … , FN-1} of a discrete 1D data set, { f0 , …, fN-1},
is:
where z = e–ik
polynomial!
The Fundamental Theorem of Algebra states that any polynomial,
fN-1zN-1 + … + f0 , can be factored to yield: fN-1 (z–z1 ) (z–z2 ) … (z–zN–1)
So the magnitude of the Fourier transform of our data can be written:
|Fk| = | fN-1 (z–z1 ) (z–z2 ) … (z–zN–1) |
where z = e–ik
Complex conjugation of any factor(s) leaves the magnitude unchanged,
but changes the phase, yielding an ambiguity! So 1D phase retrieval is
impossible!

17. Phase Retrieval & the Fund Thm of Algebra 2
2D Phase Retrieval and the Fundamental Theorem of Algebra
The Fourier transform {F0,0 , … , FN-1,N-1} of a discrete 2D data set, { f0.0 , …, fN-1,N-1}, is:
where y = e–ik and z = e–iq
Polynomial of 2 variables!
But we cannot factor polynomials of two variables. So we can only complex conjugate the entire expression (yielding a trivial ambiguity).
Only a set of polynomials of measure zero can be factored.
So 2D phase retrieval is possible! And the ambiguities are very sparse.

18. Generalized Projections
Esig(t,)  E(t) |E(t–)|2
Esig(t,)

19. Esig(t,) E(t) Find such that and is as close as possible to Esig(t,)
Algorithm
Esig(t,)
Find
such that
and is as close as
possible to Esig(t,)
E(t)
DeLong and Trebino, Opt. Lett., 19, 2152 (1994)
Code is available commercially from Femtosoft Technologies.

20. Applying the Signal Field Constraint
We must find
such that
and is as close as possible to Esig(t,).
The way to do this is to find the field, E(t), that minimizes:
Once we find the E(t) that minimizes Z, we write the new signal field as:
This is the new signal field in the iteration.
DeLong and Trebino, Opt. Lett., 19, 2152 (1994)

21. Kohler traces

22. Second-Harmonic-Generation FROG
Kane and Trebino, JQE, 29, 571 (1993).
DeLong and Trebino, JOSA B, 11, 2206 (1994).
Second-harmonic generation (SHG) is the strongest NLO effect.

23. SHG FROG traces are symmetrical with respect to delay.
Frequency
Time
Frequency
Delay

24. SHG FROG traces for complex pulses
Intensity
Frequency
Time
Frequency
Frequency
Delay
Delay

25. SHG FROG Measurements of a Free-Electron Laser
Richman, et al., Opt. Lett., 22, 721 (1997). 5 5 4 4 3 3
Intensity
Phase (rad)
Spectral Intensity
Spectral Phase (rad) 2 2 1 1
Time (ps)
Wavelength (nm)
SHG FROG works very well, even in the mid-IR and for difficult sources.

26. Shortest pulse vs. year
Plot prepared in 1994 (by Erich Ippen, MIT) reflecting the state of affairs at that time.
Shortest pulse length
Year

27. 10-fs spectra and autocorrs
The measured pulse spectrum had two humps,
and the measured autocorrelation had wings.
Two different theories emerged, and both agreed with the data.
From Harvey et. al, Opt. Lett.,
v. 19, p. 972 (1994)
From Christov et. al, Opt. Lett.,
v. 19, p (1994)
Data courtesy of Kapteyn
and Murnane, WSU
Despite different predictions for the pulse shape, both theories were
consistent with the data.

28. FROG distinguishes between the theories.
Taft, et al., J. Special Topics in Quant. Electron., 3, 575 (1996).

29. SHG FROG Measurements of a 4.5-fs Pulse!
Baltuska,
Pshenichnikov,
and Weirsma,
J. Quant. Electron.,
35, 459 (1999).

30. Using SHG FROG to study materials
The evolution of a pulse propagating through a fiber
Delay
Frequency
Frequency
Fatemi and coworkers, Optics Express 2002

31. Generalizing FROG to arbitrary nonlinear-optical interactions
Pulse retrieval remains equivalent
to the 2D phase-retrieval problem.
Many interactions have been used, e.g., polarization rotation in a fiber.

32. FROG traces of the same pulse for different geometries

33. FROG geometries: Pros and Cons
Second-
harmonic
generation
most sensitive;
most accurate
Third-
harmonic
generation
tightly focused
beams
useful for UV &
transient-grating
experiments
Transient-
grating
simple, intuitive,
best scheme for
amplified pulses
Polarization-
gate
Self-
diffraction
useful for UV

34. Generalized Projections with an Arbitrary Medium Response
DeLong, et al., Opt. Lett. 20, 486 (1995).

35. Single-shot FROG
Crossing beams at a large angle produces a range of delays across the nonlinear-optical medium and maps delay onto transverse position.

36. Single-Shot Polarization-Gate FROG
Kane and Trebino, Opt. Lett., 18, 823 (1993).

37. FROG allows a very simple imaging spectrometer.
We can use the focus of the beam in the nonlinear medium as the entrance slit for a home-made imaging spectrometer (in multi-shot and single-shot FROG measurements).
Signal pulse
Collimating Lens
Pulses from first half of FROG
Grating
Nonlinear Medium
Imaging Lens
Camera or
Linear Detector Array
This eliminates the need for a bulky expensive spectrometer as well as the need to align the beam through a tiny entrance slit (which would involve three sensitive alignment parameters)!

38. When a known reference pulse is available: Cross-correlation FROG (XFROG)
If a known pulse is available (it need not be shorter), then it can be used to fully measure the unknown pulse. In this case, we perform sum-frequency generation, and measure the spectrum vs. delay.
SFG
crystal
E(t)
Camera
Unknown pulse
Spectro-
meter
Eg(t–t)
Known pulse
Lens
The XFROG trace (a spectrogram):
XFROG completely determines the intensity and phase of the unknown pulse, provided that the gate pulse is not too long or too short.
If a reasonable known pulse exists, use XFROG, not FROG.
Linden, et al., Opt. Lett., 24, 569 (1999).

39. XFROG example: Ultrabroadband Continuum
Ultrabroadband continuum was created by propagating 1-nJ, 800-nm, 30-fs pulses through 16 cm of Lucent microstructure fiber. The 800-nm pulse was measured with FROG,
so it made an ideal known gate pulse.
Retrieved intensity
and phase
XFROG trace
Kimmel, Lin, Trebino, Ranka, Windeler, and Stentz, CLEO 2000.
This pulse has a time-bandwidth product of ~ 4000, and is the most complex ultrashort pulse ever measured.

40. Measuring two pulses simultaneously

41. 1D vs. 2D Blind Deconvolution

42. TREEFROG Example
DeLong, et al., JOSA B, 12, 2463 (1995).
TREEFROG is still under active study, and many variations exist.
It will be useful in excite-probe spectroscopic measurements, which involve crossing two pulses with variable relative delay at a sample.