Frequency Resolved Optical Gating and Phase ... - MIT Media Lab

Reconstructing Ultrashort Pulses:
Frequency Resolved Optical Gating and
Phase Space Functions
Roarke Horstmeyer 1,*
1 The Media Lab, Massachusetts Institute of Technology, 75 Amherst Street, Cambridge, MA 02139 USA
* roarkeh@mit.edu
Abstract: In the following paper, I compare and contrast methods of measuring
the amplitude and phase of a short optical pulse over time with techniques that
attempt a similar measurement of a optical field over space. Both approaches
benefit from considering the energy of an optical wave in a phase space
representation, whether in a joint time-frequency representation for short pulses
or a space-spatial frequency representation for beam characterization. I will first
present a summary of the common pulse characterization method termed
frequency-resolved optical gating (FROG), examining it’s reconstruction
algorithm in the context of similar algorithms used in the spatial domain. I will
then connect the output of a FROG measurement, a spectrogram, with a common
output of spatial measurement setups, the Wigner distribution. Specifically, I will
show that a FROG’s spectrogram is equivalent to the convolution of the Wigner
distribution of the pulse with the Wigner distribution of the pulse’s gate. Finally,
I will apply the insights gained between temporal and spatial phase
measurements to modify FROG’s reconstruction algorithm, allowing for noise
reduction as well as the possibility of integrating prior knowledge into the
spectrogram’s decomposition each iteration.
1. Introduction and Background
Ultrafast optics has progressed over the past several decades into the extremely low
femtosecond regime, now producing pulses that contain only a few optical cycles within
their envelope. Due to the lack of other phenomena on such short time scales, direct
characterization of such fleeting events is quite difficult. However, several techniques
have been proposed that use the pulse itself to provide a method of backtracking its
amplitude and phase profile over time. One example is spectral phase interferometry for
direct electric-field reconstruction (SPIDER), which interferes a pulse and a delayed copy
of the same pulse with a chirped reference pulse, which eventually allows for the
extraction of spectral phase [1]. Other methods attempt to recover a pulse’s amplitude
and phase by relying on insights gained from a joint time-frequency representation of the
pulse [2]. One example of this genre of measurement is called chronocyclic tomography
[3], which combines a spectral filter with an ultrafast shutter to take many measurements
of a pulse with a quadratic temporal phase shift. Another example by Wong and
Walmsley also operates in the frequency domain by generating a pulse’s sonogram, the
Fourier dual of a spectrogram [4]. Probably the most common and reportedly robust [5,

6] method of measuring a pulse’s amplitude and phase, called frequency-resolved optical
gating (FROG), provides a method of gating a short pulse over time and then measuring
its spectrum. FROG will be the main focus of this paper. However, parallels to the
retrieval of the phase of a wavefront across space, primarily a concern in imaging
devices, can be drawn to each of the aforementioned mentioned techniques.
A large body of work has developed over the past several decades attempting to measure
an optical field’s spatial phase distribution. Phase retrieval algorithms like the Gerchberg-
Saxton [7] or Fienup [8] algorithms were a first attempt at iteratively solving for the
phase of a wavefront from two or more intensity measurements, but depend on a spatially
coherent wavefront. Since, more robust retrieval methods based on space-spatial
frequency functions, like the Wigner distribution, have been explored. Phase space
tomography [9,10] borrows tools like the Radon transform and filtered back-projection
from tomography to reconstruct a 2D phase space function from a set of its 1D slices.
Similarly, methods based on the transport-of-intensity equation [11,12] estimate a phase
space function from two or more closely spaced measurements.
While admittedly different than trying to characterize the shape of an ultrafast pulse,
work towards retrieving the phase of light in imaging setups has the potential to offer
unique insights into the algorithms used in FROG. This paper will consider a few of these
insights. After a brief summary of FROG, the first insight I will consider is how a FROG
trace can be decomposed into the commonly used Wigner distribution and ambiguity
function. I will then show how this decomposition can be useful in improving pulse
reconstruction in the presence of noise, and will further discuss its use both as a general
space-time representation and a method to gain insights into multi-pulse interaction.
Finally, I will consider the application of matrix decomposition techniques common in
imaging literature to FROG reconstruction.
2. A Summary of FROG
Interfering a pulse with a shifted version of itself, as in a typical intensity autocorrelation
setup, does not provide enough information to backtrack the light pulse’s amplitude and
phase distribution over time. A host of additional modifications to this simple optical
concept, including the third order autocorrelation [13], slow third order autocorrelation
[14], a joint measure of the autocorrelation and spectrum [15], and fringe-resolved
autocorrelation [16], offered somewhat successful extensions. Many of these additional
experiments attempt to overcome two fundamental difficulties in pulse characterization.
The first is the availability only of intensity values with any detection technique. The
second is the fundamental theorem of algebra, which effectively proves that the retrieval
of phase with a one-dimensional input of data is ill-posed, without a unique solution [5].
FROG offers a method of pulse measurement that overcomes both of these issues. It
addresses the first issue through the use of a non-linear medium to generate higher
harmonic pulses or a non-linear polarization or diffraction response, and it addresses the
second by performing a two-dimensional time-frequency measurement, whose inverse
problem can almost always recover a unique solution. Following, I will present the

optical measurement setup used to create a FROG measurement trace, and then discuss
the specifics of this inversion process.
2.1 Optical Measurement
The evolution of FROG over the past two decades has led to the development of a
number of variants on it’s optical setup, For the purposes of this paper, I will be focusing
on the more basic variants, which can be summarized into four categories. However, it
should be noted that there are a number of other extensions. For example,
GRENOUILLE is able to capture a FROG trace in a single shot [17], XFROG simplifies
post-processing with a FROG correlation using a known gate (i.e., a cross-correlation)
[18], and Fiber-FROG is based on the χ (3) nonlinearity in an optical fiber [19]. Finally,
Blind FROG attempts to recover the signal of two unknown pulses that mix in a nonlinear
medium [20].
The fundamental principle of a FROG setup is to take a pulse E(t), create an exact copy
of the pulse with a set amount of time delay τ, E(t-τ), and then interfere these two pulses
in a non-linear medium. The spectrum of the output of the interference in the non-linear
medium is then measured using a spectrometer, yielding a 1D dataset of N pixels. This
process is repeated M times for an even range of delays τ. The data is then tiled together
to create an NxM time-frequency trace of the pulse. This trace can then be processed to
reveal the amplitude and phase of the original pulse with an almost infinite temporal
resolution [5]. The four main FROG variants differ in the non-linear mixture of E(t) and
E(t-τ) that they attempt to measure. In general, the signal output from the non-linear
crystal can be represented as,
E sig( t,τ)
= E( t)G(
t −τ)
(1)
Where G(t- τ) is a function that depends upon the non-linear reaction provided by the
crystal in the setup. The four main FROG variants use two pulses to create either secondharmonic
generation (SHG), € polarization gating (PG), self-diffraction (SD), or thirdharmonic
generation (THG) within the crystal. Ignoring proportionality constants,
Esig(t,τ) can be expressed for each of these setups as laid out in Table 1.
Non-linear process Esig(t, τ)
SHG
E(t)E(t −τ)
PG
E(t) E(t −τ)
€
2
SD
E(t) 2 E * (t −τ)
THG
E(t) 2 E(t −τ)
€
€
€
Table 1: The four main genres of FROG produce slightly different output signals

The optical configuration for each of the four main setups is shown in Fig. 1. In general,
the process of the two pulses mixing generates enough energy to produce a second or
third-order non-linear effect within a crystal, the strength of which is dependent upon the
pulse energy, length, crystal thickness, as well as many other factors. It is sometimes
helpful to think of the delayed pulse as a gate, overlapping with a portion of the original
pulse to produce the higher-order effect Esig(t, τ), which is then relayed to a spectrometer.
The spectral measurement captures the squared magnitude of the Fourier transform of
Esig(t, τ) with respect to t:
2
IFROG (ω,τ) = ∫ E sig( t,τ)exp(iωt)dt
Note that IFROG is a function of 2 variables – a time variable and a frequency variable.
Indeed, measuring the spectrum IFROG(ω) at multiple delays τ produces a spectrogram of
E(t), familiar in many
€
other disciplines involving signal processing [2].
A spectrogram provides a simultaneous view of two conjugate variables of a signal. An
example is in Fig. 2 for an ultrashort pulse in time-frequency coordinates. Other
examples include viewing a particle’s position and momentum [21], a wave’s space and
spatial frequency [22], and any electromagentic signal over time and frequency [2].
Besides providing an informative view of the signal, working with a 2D spectrogram also
avoids the pitfalls associated with phase retrieval over a 1D signal, as noted earlier.
Following, I will explain how this inversion process works.
Fig. 1: The optical setup for the four main variants of FROG (diagrams modified from [5]). While all
generally related, each setup differs slightly in the non-linear crystal used and the non-linear interaction that
is taken advantage of to create a spectrogram measurement.
(2)

€
€
Fig. 2: An example of the generation of a spectrogram, assuming a polarization gate (PG) FROG setup. (a)
A complex ultrafast pulse with a cubic phase component. (b) The gate function of the pulse, which is the
squared magnitude of (a). (c) The resulting FROG trace, which is a 2D function of frequency ω and delay
time τ. This trace was generated through the use of Eq. (2) assuming independent pulses in a SHG setp.
Note it is real and positive (with a maximum value of 1 here).
2.2 Reconstruction Algorithm
Over the course of development of the FROG technique, two main classes of algorithmic
reconstruction of the pulse from the data have emerged. The first proposed FROG setups
used a generalized projections algorithm to iteratively reconstruct the pulse and gate
function using two constraints [22, 23]. The first constraint requires that the estimate for
E sig( t,τ)',
which incorporates the estimate of pulse E( t)'
and gate G( t −τ)',
matches the
measured intensity data. This is simply accomplished by replacing the amplitude of
E sig( t,τ)'
with the measured amplitude IFROG( ω,τ)
, much like in common phase
retrieval algorithms [7, 8].
€
€
The second constraint is a mathematical one, requiring the estimate E sig( t,τ)'
to obey the
€
required outer-product of the pulse and gate, which depends upon the FROG setup used
(given from Table 1). This constraint is much more interesting and open to modification,
but generally focuses on finding E(t) and allowing the gate function G(t-τ) to progress
along. In the algorithm’s simplest form, E(t) is found by projecting
€
E sig( t,τ)'
along the τaxis:
E
k +1 ( ) t
(k) ( ) = ∫ E ( t,τ)dτ
(3)
sig €
This is based on the assumption that the gate is independent of τ, which is only valid for
PG-FROG. Alternatively, a generalized projections (GP) [25, 26] approach can be used
to induce a much quicker
€
convergence. With generalized projections, a Euclidean L2
distance is defined between the signal field from the current iteration and a new signal
field that obeys the mathematical constraints in Table 1. For example, if PG FROG is
used, then the distance Z can be defined as,

(k )
ZPG = E sig ( ti ,τ j)
− E (k +1) ( ti) E (k +1) N
∑
ti −τ j
i, j =1
( ) 2
where the second term in the summation is in the mathematically correct form of Esig(t,τ)
for PG FROG. During iteration, the goal is to minimize this distance, which then
becomes a
€
typical steepest descent or gradient descent problem, for which many solvers
exist [27]. There are a couple clear problems with the GP approach. To begin, there is no
guarantee that each constraint will produce a convex set to optimize over. Indeed, it is
clear that the overlap of the measurement and mathematical constraints is almost never
convex, leading to possible stagnations and solutions found at local, instead of global,
minima. However, shortcuts to speed up the process and a multidimensional approach
that attempts to offset stagnation have led to a useable algorithm for some circumstances
[28].
A second, more effective algorithm for pulse reconstruction is based on a singular-value
decomposition (SVD), and is called principle components generalized projections [29]. It
is a direct extension of generalized projections, based on an insight that the FROG trace
can be directly converted into a rank-1 2D function. From [29], we find that an outerproduct
state can be determined through a complex row shift and re-arrangement of the
outer product of a discrete pulse row vector E(t) and a gate column vector G(t). However,
the trace can also be reproduced by taking the outer product of E(t) and G(t), rescaling
the horizontal axis by 1/2, and rotating the matrix by 45 degrees:
E( t1)G( t2) ↔ E( t)G(
t −τ)
= E( t +τ /2)G(
t −τ /2),
(5)
where the center-difference coordinates τ=t1-t2 and t=(t1+t2)/2 have been used. Either
way, once any FROG trace estimate is expressed in outer product space, a direct
constraint € can be applied. Clearly, in (t1, t2) space, any valid FROG trace should be the
rank-1 outer product of a row and column vector – this is how it appears in the integral in
Eq. (2). If the estimate FROG trace is not, then the closest rank-1 estimate should be
selected and passed through to the next iterative step. In other words, the mathematical
constraint finding a minimum distance Z in Eq. (4) is equivalent to a rank-1 outer-product
constraint in (t1, t2) space of Eq. (5).
An SVD applied to the NxN matrix estimate E(t1)G(t2) will generate N singular values σi
with corresponding orthogonal eigenvectors ui and vj:
E( t1)G t2 ( ) = USV T T
= uiσiv i
This formulation expresses E(t1)G(t2) as a sum of rank-1 matrices uivi
€
T , each weighted
with its own singular value σi. From the Eckert-Young Theorem [30], we know that
summing the first n rank-1 matrices, where n

€
E(t1)G(t2) will yield it’s best rank-1 outer-product estimate in an L2-norm sense! This is
exactly what Eq. (4) attempts to find iteratively with gradient descent. In summary, an
SVD of E(t1)G(t2) can directly yield an optimal estimate for the pulse function E(t1) and
gate function G(t2) with,
E( t1)G t2 N
T ∑ →u1v 1
i=1
G t2
T
( ) = uiσiv i
E ˆ ( t1) = u1 , ˆ ( ) = v1 Here, all singular values € besides i =1 are set to zero to find the pulse and gate estimates
E ˆ ( t1) and G ˆ ( t2), which are carried through to the next iteration. One further benefit of
€
using the SVD is that a direct analysis of noise and a test of convergence can be obtained
by looking at the number and size of singular values greater than i = 1 [29]. A summary
of the generalized projection and principle component generalized projection iterative
algorithms € is in Fig. 3. In general, GP requires just an input guess for the pulse and
applies the gradient descent minimization of Z in Eq. (4), while principle components GP
takes a guess for the gate and pulse and uses the SVD constraint in Eq. (6) and Eq. (7) to
converge to a valid solution.
Fig. 3: Schematic flow of the two forms of algorithmic pulse reconstruction used with FROG. (a) GP takes
an initial guess for the pulse E(t) and generates a signal guess using a gate pulse from Table 1, depending
upon the FROG setup used. The simple data constraint is applied in the frequency-time domain, and the
mathematical constraint is applied using gradient descent with Eq. (3) and Eq. (4). (b) Principle
components GP instead takes a gate and pulse as input for an initial guess. The mathematical constraint in
this case is applied by taking the first singular value’s row and column vector as described in Eq. (6) and
Eq. (7).
3. The FROG Spectrogram, Wigner Distribution and Ambiguity Function
Attempting to determine the amplitude and phase of an optical pulse over time from
multiple spectral intensity measurements is quite similar to trying to find the amplitude
and phase of an optical beam at a given plane along its propagation from multiple images.
Indeed, both problems work in phase space to better pose the data for inversion. While
(7)

FROG is based in a time-frequency space, spatial phase retrieval techniques are viewed
in a space-spatial frequency space. The Wigner distribution function (WDF), as well as
Fig. 4: A 1D pulse has a 2D WDF and AF. (a) The example pulse with Gaussian amplitude (blue) and
cubic phase (green). (b) The pulse’s WDF is real, but can be positive or negative. (c) The AF is complex –
here its absolute value is shown.
it’s Fourier dual the ambiguity function (AF), are both very useful choices of functions
when trying to understand the spatial distribution of a beam’s phase [22, 38]. For a given
pulse E(t), its WDF and AF can be expressed in time-frequency as,
WDF( t,ω)
= E t + k ⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ E
⎝⎝ 2⎠⎠
* t − k ⎛⎛ ⎞⎞
∫ ⎜⎜ ⎟⎟ exp( ikω)dk
⎝⎝ 2⎠⎠
AF( k,ξ)
= E t +
€
€
k ⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ E
⎝⎝ 2⎠⎠
* t − k ⎛⎛ ⎞⎞
(8b)
∫ ⎜⎜ ⎟⎟ exp( itξ)dt
⎝⎝ 2⎠⎠
An example pulse with its 2D WDF and AF are shown in Fig. 4. Both can be thought of
as an “instantaneous” Fourier transform of the autocorrelation of a field E(t), with respect
to the dummy variable k for the WDF, and with respect to the frequency variable ξ for the
AF. Each is also commonly thought of as the Fourier transform of the mutual coherence
of a pulse, J(t,k), along the k and t axes, respectively. From a WDF, the original pulse can
be recovered up to a constant phase factor:
Thus, we can summarize that knowledge of the WDF of a signal effectively yields
knowledge of the pulse’s amplitude and phase up to a constant ambiguity, and can also
provide insights € into its mutual coherence state.
(8a)
E( t)E
* ( 0)
= ∫ WDF( t 2,ω)exp(
iωt)dω.
(9)
Ties between the WDF and the simple ABCD matrices of geometrical optics allow for a
simple interpretation of beam propagation through free space and thin elements [22, 31,
32], and also extends nicely to explain partial coherence [33]. Furthermore, the AF
provides an informative visualization of an imaging system at any plane of defocus, as a
polar display of its optical transfer function (OTF) [34]. The AF is a nice function to take
advantage of when attempting to design a wavefront along its propagation axis [35].

Because of these useful properties of the WDF and the AF, I will first connect them to the
spectrogram that FROG creates. Then, I will provide a couple examples of how they can
help with ultrashort pulse analysis.
The general form of a FROG spectrogram is given in Eq. (2). For simplicity I will focus
on the SHG form of Esig(t, τ), although the following derivation can be extended to any of
the four main FROG categories with minor modifications. This derivation was motivated
by connections made in [36], which showed how a geometric light field and the Wigner
distribution are connected. Starting with the initial SHG FROG measurement,
€
∫
IFROG (ω,τ) = E( t)E
( t −τ )exp(iωt)dt
we can expand it into a product of complex conjugates,
€
2
,
(10)
E( t1)E ( t1 −τ )E * ( t2)E * ∫ ( t2 −τ )exp(iω(t 1 − t2))dt 1 dt2. (11)
Switching to center difference coordinates yields the expression,
E t + k ⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ E
⎝⎝ 2⎠⎠
* t − k ⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ E t +
⎝⎝ 2⎠⎠
k
2 −τ
⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ E
⎝⎝ ⎠⎠
* t − k
2 −τ
⎛⎛ ⎞⎞
∫ ⎜⎜ ⎟⎟ exp(iωk)dtdk
,
⎝⎝ ⎠⎠
where I have used the substitutions k=t1-t2 and t=t1+t2/2. It is clear that integral in Eq.
(12) is a multiplication of two mutual coherence functions – one for the field of the initial
pulse € Jp(t,k), and one for the field of the gate pulse, JG(t-τ,k):
(12)
IFROG (ω,τ) = ∫ JP ( t,k)J
G( t −τ,k )exp(iωk)dtdk
, (13)
where J(t,k) = E t +
€
€
k ⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ E
⎝⎝ 2⎠⎠
* t − k ⎛⎛ ⎞⎞
⎜⎜ ⎟⎟ . Using the convolution theorem, this Fourier transform
⎝⎝ 2⎠⎠
of a product can be expressed as a convolution of the Fourier transform of J(t,k) with
respect to the variable k. However, as we can see from Eq. (8a), the Fourier transform of
J(t,k) with respect to k is the WDF of E(t). Thus, Eq. (13) becomes,
IFROG (ω,τ) = ∫ WDFP ( t,k)
⊗ kWDFG
( t −τ,k )dt, (14)
€
where ⊗ represents convolution, here performed along the k-dimension. Noting the
integral in Eq. (14) is also a one-dimensional convolution along time, IFROG can be
expressed as, €
IFROG (ω,τ) = WDFP ( τ,k)
⊗WDFG ( −τ,k ) (15)
Eq. (15) shows that spectrogram of a FROG is a two dimensional convolution between
the Wigner distribution of the pulse and the flipped Wigner distribution of the gate pulse.
This offers a new
€
interpretation to the problem of FROG reconstruction. Inverting a trace

can alternatively be thought of as a coupled blind deconvolution problem. Given a
measurement IFROG, the goal is to determine two WDF’s, which are dependent upon one
another in a conventional SHG FROG setup, which convolve together to create IFROG.
Knowledge of the two WDF’s can directly yield the pulse and gate amplitude and phase
using Eq. (9), and can also provide information on the pulse’s mutual coherence. Another
interpretation of FROG pulse reconstruction is offered by the ambiguity function. Again
applying the convolution theorem and Eq. (8b) yields,
∫ IFROG (ω,τ) exp( iτξ + iωk)dωdτ
= AFP ( k,ξ)AFG
( −k,ξ)
(16)
In other words, the two-dimensional Fourier transform of the FROG trace is equivalent to
the product of two pulse and gate’s AF’s. This transfers the iterative deconvolution
problem € to an iterative multiplicative problem. An example of the equivalence of a
modeled FROG trace, the convolution of two WDF’s and the multiplication of two AF’s
are shown in Fig. 5. Note that the three FROG traces at the bottom of Fig. 5, generated
with Eq. (2), Eq. (15) and Eq. (16), respectively, are nearly identical.
Fig. 5: The construction of a FROG trace three different ways. A pulse (a) and a gate (b), each with a
unique amplitude and phase as in a Blind FROG setup, create a FROG trace (c), which can be calculated
the “traditional” way through application of Eq. (2). From Eq. (15), another way to construct a FROG trace
is through a convolution of the WDF of the pulse and the WDF of the gate, shown in (d). From Eq. (16), a
third method of creating a FROG trace is by multiplication of the AF of the pulse and gate, and then an
inverse Fourier transform, shown in (f) and (g). Numerical error on the order of 2% is noticeable when
comparing the WDF and AF trace construction methods with the conventional trace in (c).

4. FROG reconstruction using the Wigner Distribution
The insights from Section 3 can be applied to the pulse reconstruction algorithms in
Section 2.3 to provide additional insight into the iterative process, as well as offer a
unique space in which constraints may be applied. As the WDF and AF of the pulse and
gate can be combined to directly yield a FROG trace, several modifications using either
phase space function could be imagined. As a demonstration of their utility, I will simply
modify the principle component GP algorithm to operate under a convolution of WDF’s
of the gate and pulse, as shown in Fig. 6, nicknamed Wigner-PCGP. This is equivalent to
a simplified implementation of a double-blind deconvolution algorithm [39]. Several
benefits of treating FROG pulse reconstruction in the Wigner domain are immediately
tangible. Following, I will first consider the benefit of using the WDF as a new space to
apply additional constraints in at each iterative step. Then, I will discuss several other
future modifications to Wigner-PCGP that could significantly extend the application
space of ultrafast pulse characterization, into both the spatial and multi-pulse domains.
Fig. 6: Modified Principle Components Generalized Projections algorithm using the WDF. Knowing that a
FROG trace can be generated from the convolution of two WDF’s from Section 3, we can replace the pulse
and gate distributions with this function during iteration. Eq. (9) can be used at the algorithm output to once
again establish the actual pulse and gate amplitude and phase distributions, up to a constant phase factor.
4.1 WDF Applied to Noise Reduction
As a basic demonstration, the pulse and gate WDFs that pass through the Wigner-PCGP
algorithm’s iteration chain can be used as a viewing space of noise performance. Let us
assume that an initial FROG trace measurement I0 (ω,t) is corrupted with some Gaussian
distribution of noise, and we wish to establish the original pulse and gate functions P(t)
and G(t-τ) to as high a fidelity as possible. An initial “guess” pulse and gate Wigner
distribution, WDFp and WDFG, can be generated from Gaussian pulses, for example,
using Eq. (8). The Wigner-PCGP algorithm can then be applied to create an estimate
trace E i Guess(t,τ) with correct amplitude values from the data constraint. The SVD
measurement constraint will yield two new Wigner distributions, WDFp i and WDFG i ,
which will each contain information about P(t) and G(t-τ), respectively. Each will also
exhibit some degree of error due to the presence of noise.

Examining the reconstruction problem in the Wigner’s time-frequency domain allows for
a direct view of noise performance. Noise will manifest itself primarily in the higher
frequency areas of the WDF, especially in areas where the WDF is commonly zero due to
an assumption of a relatively smooth pulse. This is clearly demonstrated in Fig. 7, where
areas away from the center of the pulse and gate WDFs exhibit a high degree of
fluctuation, even after 50 iterations. These fluctuations are a primary source of error in
the reconstructed FROG trace. The mean-square error (MSE) between actual and
expected FROG trace in the presence of Gaussian distributed noise with σ 2 =.02 is 3.80e-
5 after 5 iterations and 2.05e-5 after 50 iterations.
Fig. 7: The Wigner-PCGP algorithm helps convergence when a FROG trace with a certain degree of noise
is used as a data source. (a) The WDF can be used to view algorithm performance during iteration. The top
row shows the initial pulse and gate Gaussian Wigner distributions, WDFP and WDFG, which are used as
initial guesses, and the FROG trace data (with 2% noise). After 5 iterations, the WDFs show a large amount
of ringing and noise away from the center, which is still visible after 50 iterations. (b) A Gaussian filter
(top) can be applied to reduce off-axis WDF values, and hence suppress noise, each iteration.
It is clear that a naïve noise constraint can be applied to the pulse and gate’s WDF to
reduce values in certain areas of the pulse’s time-frequency representation. This simple
constraint is demonstrated in Fig. 7(b), where a Gaussian multiplicative mask is applied
in the time-frequency domain to limit energy both in the high-frequency areas as well as
along the pulse envelope where we assume the pulse has tailed off. This simple constraint
allows for a much quicker convergence to a trace estimate. After this simple constraint in
the WDF domain, the MSE between actual and expected FROG trace becomes 1.307e-5
after 5 iterations and 1.269 after 50 iterations. In other words, masking the WDF at large t
and ω values helps speed up algorithm convergence for a noisy data set to achieve a more
accurate pulse recovery.

€
4.2 Future Directions of the WDF FROG Trace Solver
While not within the scope of this paper, it is interesting to consider what additional
benefits ultrashort pulse analysis may receive by using phase space functions like the
WDF and the AF during reconstruction. Following, I will briefly cover a couple
scenarios when thinking about a FROG trace in terms of these popular functions may be
beneficial.
4.2.1 Joint space-time Wigner distribution for pulse analysis
As mentioned earlier, the Wigner distribution is a popular choice for analyzing the spatial
distribution and spatial frequency content of a beam at different planes along its direction
of propagation [38]. Thus, if we wish to extend an ultrashort pulse analysis to include the
spatial extent of the beam, an all-encompassing Wigner distribution can be generated.
This Wigner distribution will include the spatial, spatial frequency, temporal and
frequency content of a pulse in a single function. For a pulse in 2D, this space-time
Wigner distribution will be a 6D function:
WDF( x,u, y,v,t,ω)
=
E x + x' ⎛⎛ y' k ⎞⎞
∫∫∫ ⎜⎜ , y + ,t + ⎟⎟ E
⎝⎝ 2 2 2⎠⎠
* x − x' ⎛⎛ y' k ⎞⎞
⎜⎜ ,y − ,t − ⎟⎟ exp[ i( x'u + y'v + kω)
]dx'dy'dk
⎝⎝ 2 2 2⎠⎠
Propagation of this pulse will exhibit a number of interesting characteristics, which can
be visualized through viewing the function’s marginals. A 6D WDF will have 15 unique
marginals, while a 4D WDF (assuming one spatial dimension) will have 6 marginals,
shown for an example pulse in Fig. 8. For first order systems like free-space or a pulseshaper,
this WDF can be treated with linear canonical matrix transforms, in this case with
6x6 matrices that can transform it through any linear change. A more detailed
examination of space-time WDFs of pulses can be found in [40].
Fig. 8: A visualization of a space-time WDF, in this case a 4D function WDF(x,u,t,ω), borrowed from [40].
Note that each marginal shows a unique structure, which shear and mix as the pulse moves through space
and/or linear materials and transformations like lenses, prisms, etc.
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4.2.2 Multiple Pulse Analysis
A second direction where the WDF and AF can lend additional insights into ultrashort
pulse analysis is when more than two pulses are used to generate a non-linear signal for
measurement. With conventional FROG measurement setups, this is rarely performed –
at most two completely independent pulses are used to generate a non-linear effect, as in
Blind FROG [20]. However, some measurement scenarios, like those that extend FROG
into the interferometric domain to assist in complex spectroscopic measurements, may
benefit from a multi-pulse analysis [41, 42, 5]. Furthermore, generating higher-order nonlinear
optical effects with more than two pulses is simply itself an interesting
experimental endeavor. As a simple example, let’s consider three pulses (E1(t), E2(t),
E3(t)) instead of two interacting in a χ (3) non-linear medium. The third-harmonic
generation from these three pulses will generate a polarization of the form [5],
[ ]
Ρ1 = 3
4 ε0χ (3) *
E1E 2E
3 exp i( ω1 −ω 2 +ω 3)t
− i( k1 − k2 + k3)⋅ r
This type of three-wave effect is typically referred to as four-wave mixing. However, out
of the many terms this type of mixing will produce (most of them not desirable
experimentally) € one term will be of the form,
E sig( t,τ1 ,τ2) ∝ E1( t)E
2
* ( t −τ1)E
3 t −τ 2
Depending upon the optical setup and delays chosen, this output signal from the nonlinear
medium may be used to create a FROG trace. If each delay is treated
independently, € then the trace will be a 3D measurement of the form,
2
IFROG (ω,τ1 ,τ2 ) = ∫ E sig( t,τ1 ,τ2)exp(iωt)dt (18)
( ). (19)
This type of measurement may be quite difficult to treat directly in the traditional FROG
formalism. However, Eq. (8) can be applied to separate the contributions to the FROG
trace into three € AFs, for example:
∫ IFROG (ω,τ1 ,τ2 ) exp( iτ1ξ + iτ 2ψ + iωk)dωdτ1dτ2
( )
= AF1( k,ξ)
AF2( −k,ξ)
⊗ k AF3( k,ψ)
This form of separation may lead to an easier analysis, given that it partially decouples
the three contributions from each wave. However, if the three input waves are not
independent,
€
than the associated AFs will not be independent either, in which case
separation may not help at all. In general, the WDF and AF can provide an additional
window through which the complications of four-wave mixing can be viewed, although
as of yet I do not have a concrete solution to suggest.
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5. Constrained FROG reconstruction using Principle Components
Apart from the Wigner distribution, a final example of the benefit of considering prior
work in spatial phase retrieval during FROG reconstruction is clear when prior
constraints on the input pulse and gate distributions are known. For example, the current
FROG recovery process already allows the experimentalist to define an initial starting
condition for the iteration procedure, assuming they may have some idea of the original
pulse’s amplitude and phase. However, the algorithm is not able to easily restrict the
search space of possible functions – an SVD simply provides the closest Euclidean rank-
1 approximation for any complex 2D matrix. Recent work in matrix factorization has
examined the problem of error minimization for matrices with certain constraints. A
simple example is non-negative matrix factorization (NMF), which simply determines the
closest positive Euclidean rank-n approximation of a given matrix [43]. Applied to the
problem of ultrashort pulse reconstruction, this is equivalent to determining the two
pulses than created a given FROG trace, assuming either one or both of the pulses does
not have any phase (i.e., a constant phase).
While there are numerous formulations, a simple NMF factorization reduces an input
matrix E(t1)G(t2), which is the FROG trace estimate each iteration, into a sum of outerproduct
matrices:
E( t1)G t2 n
∑
( ) ≈ w i h i = WH
i=1
W ,H > 0
n < N
In other words, we can replace the SVD in Eq. (9) in the FROG reconstruction algorithm
€
with the above NMF factorization, setting n=1 to find the largest rank-1 contribution to
the trace.
€
€ This will allow us to find the closest pulse and gate that have a constant phase
profile:
E ˆ ( t1) = w1, G ˆ ( t2)
= h1 This is useful if the experiment provides prior knowledge of constant phase, or if one
wishes to see the closest two phase-free pulses that could approximate a given FROG
trace. If the original pulse € and gate functions E(t) and G(t-τ) actually have a constant
phase, then the algorithm iterates to a solution comprised of real positive numbers, which
is not guaranteed if using the SVD. An example is shown in Fig. 9. In summary, there is
no solid argument to use an NMF approach over and SVD approach unless it is known apriori
that either the pulse or gate has constant phase. Instead, this discussion is simply a
final example of how ultrashort pulse reconstruction could possibly benefit in the future
by considering reconstruction techniques from the spatial phase retrieval community.
6. Conclusion and Future Work
In conclusion, a variety of additional insights and possible benefits are available when the
time-frequency methods of determining the amplitude and phase of a short pulse are
mixed with the space-spatial frequency distribution. Those included in this document are
noise reduction, a joint representation of a pulse in space and time, the possibility of a
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Fig. 9: FROG trace reconstruction from a noisy measurement using an SVD approach and the postive
matrix factorization approach with NMF. (a) The original pulse and gate, here independent as in a blind
FROG setup, don’t have any phase – this a unique, but demonstrative situation. (b) The measured trace
with noise degredation. (c) The SVD-based reconstruction of the trace after 30 iterations, using the
constraint in Eq. (9). (d) The NMF-based reconstruction of the trace after 60 iterations (due to slower
convergence) shows a similar structure. Both reconstructions offer an MSE on the order of 3e-5.
phase space-based method of viewing multi-pulse interactions, and the use of alternative
matrix decomposition methods to assist in algorithm development if a-priori pulse
information is known. In the future, I would like to continue to think about how the fields
of ultrashort pulse estimation and spatial phase retrieval could benefit from one another. I
think a clear area ultrashort pulse could also benefit from is with compressive sensing,
which could dramatically reduce the number of measurements required to construct a
trace. In general, though, I believe the spatial domain could greatly benefit from insights
found in the temporal domain, which I would like to concentrate on for my future
research.
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