Subject: [NEXTEL-1] Swamp Optics--Ultrashort Laser Pulse Measurement Tutorials

From: [email-masked]
Date: Sun, 05 Oct 2003 23:50:17 -0400

http://www.swampoptics.com/tutorials_FROG.htm
--

------------------------ Yahoo! Groups Sponsor ---------------------~-->
Buy Ink Cartridges or Refill Kits for your HP, Epson, Canon or Lexmark
Printer at MyInks.com. Free s/h on orders $50 or more to the US & Canada.
http://www.c1tracking.com/l.asp?cid=5511
http://us.click.yahoo.com/mOAaAA/3exGAA/qnsNAA/61FxlB/TM
---------------------------------------------------------------------~->

Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ 

HOME
>> TUTORIALS >> FROG

NEWS

Winner, 2003

R&D100
Award

Next
Conference:  See us at Frontiers in Optics, in Tucson,
Arizona, October 2003.

The
June 2001 issue of OPN featured GRENOUILLE on the cover and in the article:
Measuring Ultrashort Laser Pulses Just Got a Lot Easier

FROG Tutorial

If
you've read the section on autocorrelation,
you know that measuring an ultrashort pulse required using the pulse to
measure itself. But, in view of the little dilemma presented there, that
wasn't good enough.

Frequency-Resolved Optical Gating (FROG)
involves operating in a hybrid domain: the time-frequency domain.
Measurements in the time-frequency domain involve both temporal and
frequency resolution simultaneously. A well-known example of such a measurement
is the musical score, which is a plot of a sound wave's short-time
spectrum vs. time. Specifically, this involves breaking the sound wave
up into short pieces and plotting each piece's spectrum (vertically) as
a function of time (horizontally). So the musical score is a function
of time as well as frequency. See Fig. 1. In addition, there's information
on the top indicating intensity.

Fig. 1. The musical score is a plot
of an acoustic waveform's frequency vs. time, with information on
top regarding the intensity. Here the wave increases in frequency
with time. It also begins at low intensity (pianissimo), increases
to a high intensity (fortissimo), and then decreases again. Musicians
call this waveform a "scale," but ultrafast laser scientists
refer to it as a "linearly chirped pulse."

If you think about it, the musical score isn't a bad
way to look at a waveform. For simple waveforms containing only one note
at a time (we're not talking about symphonies here), it graphically shows
the waveform's instantaneous frequency, w,
vs. time, and, even better, it has additional information on the top indicating
the approximate intensity vs. time (e.g., fortissimo or pianissimo). Of
course, the musical score can handle symphonies, too.

A mathematically rigorous version of the musical score
is the spectrogram, Sg(w,t):

where g(t - t) is
a variable-delay gate function, and the subscript on the S
indicates that the spectrogram uses the gate function, g(t). Figure
2 is a graphical depiction of the spectrogram, showing a linearly chirped
Gaussian pulse and a rectangular gate function, which gates out a piece
of the pulse. For the case shown in Fig. 2, it gates a relatively weak,
low-frequency region in the leading part of the pulse. The spectrogram
is the set of spectra of all gated chunks of E(t) as the delay,
t, is varied.

Fig. 2. Graphical depiction of the
spectrogram. A gate function gates out a piece of the waveform (here
a linearly chirped Gaussian pulse), and the spectrum of that piece
is measured or computed. The gate is then scanned through the waveform
and the process repeated for all values of the gate position (i.e.,
delay).

The spectrogram is a highly intuitive display of a
waveform. Some examples of spectrograms are shown in Fig. 3, where you
can see that the spectrogram intuitively displays the pulse instantaneous
frequency vs. time. And pulse intensity vs. time is also evident in the
spectrogram. Indeed, acoustics researchers can easily directly measure
the intensity and phase of sound waves, which are many orders of magnitude
slower than ultrashort laser pulses, but they often choose to display
them using a time-frequency-domain quantity like the spectrogram. Importantly,
knowledge of the spectrogram of E(t) is sufficient to essentially
completely determine E(t) (except for a few unimportant ambiguities,
such as the absolute phase, which are typically of little interest in
optics problems).

Frequency-Resolved Optical Gating (FROG) measures a
spectrogram of the pulse.

Okay, so a spectrogram is a good idea. But recall the
dilemma of ultrashort pulse measurement: "In order to measure an
event in time, you need a shorter one." In the spectrogram, then,
isn't the gate function precisely that mythical shorter event, the one
we don't have?

Indeed, that is the case.

So, as in autocorrelation, we'll have to use the pulse
to measure itself. We must gate the pulse with itself. And to make a spectrogram
of the pulse, we'll have to spectrally resolve the gated piece of the
pulse.

Fig. 3. Spectrograms (bottom row) for
linearly chirped Gaussian pulses. The spectrogram, like the musical
score, reflects the pulse frequency vs. time. It also yields the pulse
intensity vs. time.

Will this work? It doesn't sound much better than autocorrelation,
which also involves gating the pulse with itself (but without any spectral
resolution). And autocorrelation isn't sufficient to determine even the
intensity of the pulse, never mind its phase, too. So how do we resolve
the dilemma?

And that's not the only problem. Even if this approach
does somehow resolve the fundamental dilemma of ultrashort pulse measurement,
spectrogram inversion algorithms assume that we know the gate function.
After all, who would've imagined gating a sound wave with itself when
it's so easy to do so electronically with detectors because acoustic time
scales are so slow? So no one ever considered a spectrogram in which the
unknown function gated itself—an idea, it would seem, that could
occur to only a seriously disturbed individual. Unfortunately, we have
no choice; we must gate the pulse with itself. But by gating the unknown
pulse with itself—i.e., a gate that is also unknown—we can't use available
spectrogram inversion algorithms. So all those nice things we said about
the spectrogram don't necessarily apply to what we're planning to do.
How will we avoid these problems?

Hang on. You'll see.

In its simplest form, FROG is any autocorrelation-type
measurement in which the autocorrelator signal beam is spectrally resolved.
Instead of measuring the autocorrelator signal energy vs. delay, which
yields an autocorrelation, FROG involves measuring the signal spectrum
vs. delay.

Fig. 4. FROG apparatus using the polarization-gate
beam geometry.

As an example, let's consider, not an SHG autocorrelator,
but a polarization-gate (PG) autocorrelation geometry. Ignoring constants,
as usual, this autocorrelator's signal field is Esig(t,t)
= E(t) |E(t-t)|2.
Spectrally resolving yields the Fourier Transform of the signal field
with respect to time, and we measure the squared magnitude, so the FROG
signal trace is given by:

Note that the (PG) FROG trace is a spectrogram in which
the pulse intensity gates the pulse field. In other words, the pulse gates
itself. The traces obtained by such a technique look just like the spectrograms
in Fig. 2. So making a FROG trace yields a very intuitive measure of the
pulse. But how do we retrieve the pulse intensity and phase from its spectrogram?

It turns out that this inversion problem is well known.
It is called the two-dimensional phase-retrieval problem.

Now, the two-dimensional phase-retrieval problem is
a close relative of the one-dimensional phase-retrieval problem, which
is well known to be unsolvable—many ambiguities exist, even in the presence
of an additional constraint that might limit the number of spurious solutions.
The one-dimensional phase-retrieval problem is bad news. It turns out
that the retrieval of the pulse form its spectrum is equivalent to the
one-dimensional phase-retrieval problem. And retrieving the intensity
from the intensity autocorrelation is also the one-dimensional phase-retrieval
problem. And those are unsolvable problems.

Almost certainly, the two-dimensional analog of a one-dimensional
piece of mathematical bad news can only be worse news.

Quite unintuitively, however, the two-dimensional phase-retrieval
problem has an essentially unique solution and is a solved
problem when certain additional information regarding Esig(t,t)
is available. This is in stark contrast to the one-dimensional problem,
where many solutions can exist, despite additional information. Indeed,
in the one-dimensional case, infinitely many additional solutions
typically exist. On the other hand, the two-dimensional phase-retrieval
problem, with a reasonable constraint, has only the usual "trivial"
ambiguities, such as an absolute phase or a translation in time. In addition,
there is an extremely small probability that another solution may exist,
but this is generally not the case for a given trace. This is what is
meant by essentially unique.

Okay, so the solution isn't totally unique, but it's
good enough for practical measurements, where we don't care about the
trivial ambiguities, and we probably won't be around long enough to do
enough experiments to bump into one of the highly improbable ambiguities.

Now what type of constraint allows FROG retrieval to
be essentially unique?

It's that Esig(t,t)
= E(t) |E(t-t)|2
, which is a very strong constraint on the mathematical form that
the signal field can have. There are other versions of FROG whose constraints
are slightly different. For example, in second-harmonic-generation (SHG)
FROG, Esig(t,t) = E(t)
E(t-t). They're sufficient, too.

Thus, the problem is solved. Indeed, it is solved in
a particularly robust manner, with many other advantageous features, such
as feedback regarding the validity of the data.

The two-dimensional phase-retrieval problem occurs
frequently in imaging problems, where the squared magnitude of the Fourier
transform of an image is often measured and where finite support is common.
The two-dimensional phase-retrieval problem and its solution are the basis
of an entire field, that of image recovery. If you're interested in reading
more on it, please check out Henry Stark's excellent book on this subject,
Image Recovery.

Another way to look at this issue is that phase retrieval
is a type of de-convolution, which extracts information that's just beyond
the resolution of the device and that initially doesn't seem to be there.
For example, image de-convolution techniques can de-blur a photograph,
thus retrieving details smaller in size than the apparent resolution of
the camera that took the picture. After all, how else can CIA spy satellites
read your license plate on the ground?

Indeed, recall Fig. 2, in which a shorter rectangular
pulse gates the unknown longer pulse. This was the allegedly required
shorter pulse. At the time you first looked at that figure, you were probably
thinking, "Too bad we don't have an infinitely short gate pulse—a
delta-function in time. That'd really do a nice job of measuring the pulse."

But you'd be wrong. If it really were the case that
g(t - t) = d(t
- t), it's easy to do the integral and
see that the resulting spectrogram would be completely independent of
frequency. In fact, we would find that Sg(w,t)
= I(t). Thus, in this allegedly ideal case,
the spectrogram reduces to precisely the pulse intensity vs. time! All
phase-vs.-time information is lost! This is because the gated chunk of
the pulse will be infinitely short and hence have infinitely broad spectrum,
independent of the pulse color at the time.

So using too short a gate pulse is a bad idea. The
time-frequency domain is subtle. Having time- and frequency-domain information
simultaneously can be a bit unintuitive. Remember, you can't have perfect
time and frequency resolution at the same time, or you'd violate the uncertainty
principle. The better your time resolution the worse your frequency resolution.
In any case, having both temporal and frequency resolution on the order
of the pulse—which is what you have when you use the pulse to gate itself—is
the way to go, and that's what happens in FROG. And this resolves the
dilemma.

The pulse intensity and phase may be estimated simply
by looking at the experimental FROG trace, or the iterative algorithm
may be used to retrieve the precise intensity and phase vs. time or frequency.
Figure 5 shows a couple of pulses measured using PG FROG.

Fig. 5. Two pulses measured using PG
FROG. Left: a linearly chirped pulse. Right: a complex pulse. Traces
and figure provided by Prof. Bern Kohler, Ohio State University.

There are many different beam geometries for FROG.
Essentially any spectrally resolved autocorrelation works, and other geometries
do also. The most common and most sensitive FROG beam geometry is second-harmonic-generation
(SHG) FROG. (GRENOUILLE is a type of SHG FROG.) The SHG FROG beam geometry
is shown in Fig. 6. SHG FROG traces are shown in Fig. 7, which shows that
SHG FROG has symmetrical traces and hence has an ambiguity in the direction
of time. And Fig. 8 shows an SHG FROG measurement of one of the shortest
pulses ever created.

There are many nice features of FROG. FROG is very
accurate. Any known systematic error in the measurement can be modeled
in the algorithm, although this is not usually necessary, except for extremely
short pulses (< 10 fs) or for exotic wavelengths. Also, unlike other
ultrashort pulse measurement methods, FROG completely determines the pulse
with essentially infinite temporal resolution. It does this by using the
time domain to obtain long-time resolution and the frequency domain for
short-time resolution. As a result, if the pulse spectrogram is entirely
contained within the measured trace, then there can be no additional long-time
pulse structure (since the spectrogram is effectively zero for off-scale
delays), and there can be no additional short-time pulse structure (since
the spectrogram is essentially zero for off-scale frequency offsets).

Fig. 6. SHG FROG, the most common and
most sensitive version of FROG.

Fig. 7. SHG FROG traces
for linearly chirped pulses. Note that the traces are necessarily
symmetrical, so the direction of time is not determined. This and
a few "trivial" ambiguities are the only known undetermined
parameters in SHG FROG.

Interestingly, this extremely high temporal resolution
can be obtained by using delay increments that are as large as the time
scale of the structure. Again, this is because the short-time information
is obtained from large frequency-offset measurements. Thus, as long as
the measured FROG trace contains all the nonzero values of the pulse FROG
trace, the result is rigorous.

Another useful and important feature that's unique
to FROG is the presence of feedback regarding the validity of the measurement
data. FROG actually contains two different types of feedback. The first
is probabilistic, rather than deterministic, but it is still very helpful.
It results from the fact that the FROG trace is a time-frequency plot,
that is, an NxN array of points, which are then used to determine N intensity
points and N phase points, that is, 2N points. There is thus significant
over-determination of the pulse intensity and phase—there are many more
degrees of freedom in the trace than in the pulse. As a result, the likelihood
of a trace composed of randomly generated points corresponding to an actual
pulse is very small. Similarly, a measured trace that has been contaminated
by systematic error is unlikely to correspond to an actual pulse. Thus,
convergence of the FROG algorithm to a pulse whose trace agrees well with
the measured trace virtually assures that the measured trace is free of
systematic error. Conversely, non-convergence of the FROG algorithm (which
rarely occurs for valid traces) indicates the presence of systematic error.
To appreciate the utility of this feature, recall that intensity autocorrelations
have only three constraints: a maximum at zero delay, zero for large delays,
and even symmetry with respect to delay. These constraints do not limit
the autocorrelation trace significantly, and one commonly finds that the
autocorrelation trace can vary quite a bit in width during alignment while
still satisfying these constraints.

Fig. 8. One of the shortest events
ever measured, a 4.5 fs pulse, measured using SHG FROG. Baltuska,
Pshenichnikov, and Weirsma, J. Quant. Electron., 35, 459 (1999).

Another feedback mechanism in FROG has proven extremely
effective in revealing systematic error in SHG FROG measurements of ~10
fs pulses, where crystal phase-matching bandwidths are insufficient for
the massive bandwidths of the pulses to be measured. It involves computing
the marginals of the FROG trace, that is, integrals of the trace
with respect to delay or frequency. The marginals can be compared to the
independently measured spectrum or autocorrelation, and expressions have
been derived relating these quantities. Comparison with the spectrum is
especially useful. Marginals can even be used to correct an erroneous
trace.

In practice, FROG has been shown to work very well
in the IR, visible, and UV. Work is underway to extend FROG to other wavelength
ranges, such as the x-ray. It has been used to measure pulses from a few
fs to many ps in length. It has measured pulses from fJ to mJ in energy.
And it can measure simple near-transform-limited pulses to extremely complex
pulses with time-bandwidth products in excess of 1000. It can use nearly
any fast nonlinear-optical process that might be available. FROG has proven
to be a marvelously general technique that works. If an autocorrelator
can be constructed to measure a given pulse, then making a FROG is straightforward
since measuring the spectrum of it is usually easy.

Fig. 9. Measurements of the spectrum
of a broadband continuum pulse. The FROG measurement (left) reveals
the spectral structure, which washes out in the spectrometer measurement
(right).

FROG has other advantages. Figure 9 shows two different
measurements of the spectrum of a very broadband light pulse ("continuum").
On the left is a FROG measurement (accumulated over ~109 laser
shots), and on the right is a simple spectrometer measurement (accumulated
over 106 laser shots). The continuum spectrum contained much
fine-scale structure that fluctuated greatly form pulse to pulse, and
which averaged out in the spectrometer spectrum. FROG, on the other hand,
because it has both time and frequency resolution, sees the structure.
This structure was confirmed by single-shot spectral measurements.

What FROG Doesn't Measure

We've been saying that FROG measures the complete intensity
and phase vs. time or frequency. Actually, there are a few aspects of
the intensity and phase that FROG does not measure (the "trivial"
ambiguities). First, since FROG is a magnitude-squared quantity, it doesn't
measure the absolute phase, j0,
in the Taylor expansion of the spectral phase. Also, because FROG involves
the pulse gating itself, there is no absolute time reference, so FROG
doesn't measure the pulse arrival time, which corresponds in the frequency
domain to j1, the first-order term
coefficient in the spectral-phase Taylor series. In other words, the linear
component of the slope of the spectral phase will vary randomly, but this
is reasonable. So j0 and j1
are the only two parameters not measured in FROG, although a few versions
of FROG have their own unmeasured parameters in specific situations, and
these are discussed in Frequency-Resolved
Optical Gating: The Measurement of Ultrashort Laser Pulses. There
is, however, a direction-of-time ambiguity in SHG FROG, which means that
a pulse and its time-reversed replica are both possible, but this ambiguity
can be removed by having some (almost any) additional information available.

In any case, it is common to see the phase jump around
apparently randomly due to these undetermined, but not very important,
quantities. Please don't interpret this to mean that the FROG algorithm
isn't operating properly. Also, by definition, the phase becomes undetermined
when the intensity goes to zero. So you'll see the phase jumping around
in the pulse wings, where the intensity is nearly zero, too. This is also
as it should be.

Summary

Frequency-resolved-optical-gating (FROG) is the
foremost technique for completely characterizing an ultrashort laser pulse
in time. Since its introduction about a decade ago, FROG has evolved into
a marvelously general and powerful technique for measuring ultrashort
laser pulses. FROG is any autocorrelation-type measurement in which the
autocorrelator signal beam is spectrally resolved. Instead of measuring
the autocorrelator signal energy vs. delay, which yields an autocorrelation,
FROG involves measuring the signal spectrum vs. delay.

This extensive tutorial introduces the FROG technique,
its capabilities, and its applications for characterizing modern ultrashort
laser pulses.

Copyright
© 2003 Swamp Optics, LLC. All Rights Reserved.