On the Uniqueness of FROG Methods

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by  Tamir Bendory

1
On the Uniqueness of FROG Methods
Tamir Bendory1, Pavel Sidorenko2, and Yonina C. Eldar, IEEE Fellow3
1 The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA
2Department of Physics and Solid State Institute, Technion-Israel Institute of Technology, Haifa, Israel
3Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel
Abstract—The problem of recovering a signal from its power structure of the signal, such as sparsity (e.g. [8], [12], [13])
spectrum, called phase retrieval, arises in many scientific fields. or knowledge on a portion of the signal (e.g. [2], [10],
Oneofmanyexamplesisultra-shortlaserpulsecharacterization
7 [11]). The second uses techniques that generate redundancy
in which the electromagnetic field is oscillating with ∼ 1015
1 intheacquireddatabytakingadditionalmeasurements.These
Hz and phase information cannot be measured directly due to
0
limitations of the electronic sensors. Phase retrieval is ill-posed measurementscanbeobtainedforinstanceusingradommasks
2
in most cases as there are many different signals with the same [14],[15]orbymultiplyingtheunderlyingsignalwithshifted

r Fouriertransformmagnitude.Toovercomethisfundamentalill- versions of a known reference signal, leading to short-time
p
posedness, several measurement techniques are used in practice.
A Fourier measurements [16], [17], [18].
One of the most popular methods for complete characterization
An important application for phase retrieval is ultra-short
of ultra-short laser pulses is the Frequency-Resolved Optical
1
Gating (FROG). In FROG, the acquired data is the power laser pulse characterization. Since the electromagnetic field is

spectrum of the product of the unknown pulse with its delayed oscillating at ∼ 1015 Hz, phase information cannot be mea-
]
T replica. Therefore the measured signal is a quartic function of sured directly due to limitations of the electronic sensors. To
the unknown pulse. A generalized version of FROG, where the
I overcome the fundamental ill-posedness of the phase retrieval
. delayed replica is replaced by a second unknown pulse, is called
s problem, a popular approach is to use Frequency-Resolved
blind FROG. In this case, the measured signal is quadratic with
c
[ respect to both pulses. In this letter we introduce and formulate Optical Gating (FROG). This technique measures the power-
FROG-type techniques. We then show that almost all band- spectrum of the product of the signal with a shifted version

3 limitedsignalsaredetermineduniquely,uptotrivialambiguities, of itself or of another unknown signal. The inverse problem
v by blind FROG measurements (and thus also by FROG), if in
of recovering a signal from its FROG measurements can be
1 addition we have access to the signals power spectrum.
thoughtofashigh-orderphaseretrievalproblem.Thefirstgoal
3
8 Index Terms—phase retrieval, quartic system of equations, of this letter is to introduce and formulate such FROG-type
ultra-short laser pulse measurements, FROG
2 methods.
0
Our second contribution is to derive a uniqueness result
1. I. INTRODUCTION for FROG-type models. Namely, conditions such that the
0 In many measurement systems in physics and engineering underlying signal is uniquely determined from the acquired
7
one can only acquire the power spectrum of the underlying data.Acommonstatementintheopticscommunity,supported
1
signal, namely, its Fourier transform magnitude. The problem by two decades of experimental measurements, is that a laser
:
v of recovering a signal from its power spectrum is called pulsecanbedetermineduniquelyfromFROGmeasurementsif
i
X phase retrieval and it arises in many scientific fields, such thepowerspectrumoftheunknownsignalisalsomeasured.To
as optics, X-ray crystallography, speech recognition, blind the best of our knowledge, the uniqueness of FROG methods
r
a channel estimation and astronomy (see for instance, [1], [2], was analyzed only in [19] under the assumption that we have
[3],[4],[5],[6]andreferencestherein).Phaseretrievalforone- accesstothefullcontinuousspectrum.Inthisletterweanalyze
dimensional (1D) signals is ill-posed for almost all signals. the discrete setup as it typically appears in applications.
Two exceptions are minimum phase signals [7] and sparse The letter is organized as follows. Section II introduces the
signalswithstructuredsupport[8],[9].Additionalinformation FROG problem and formulates it mathematically. Section III
on the sought signal can be used to guarantee uniqueness. For presents our uniqueness result, which is proved in Section IV.
instance, the knowledge of one signal entry or the magnitude Section V concludes the letter.
ofoneentryintheFourierdomain,inadditiontothethepower
spectrum, determines almost all signals [10], [11].
II. MODELANDBACKGROUND
For general signals, many algorithms and measurement
techniques were suggested to make the problem well-posed. We consider two laser pulse characterization techniques,
These methods can be classified into two categories. The first called FROG and its generalized version blind FROG. These
utilizes some prior knowledge (if it exists) on the underlying methods are used to generate redundancy in ultra-short laser
pulse measurements. FROG is probably the most commonly
ThisworkwasfundedbytheEuropeanUnion’sHorizon2020researchand used approach for full characterization of ultra-short opti-
innovationprogramundergrantagreementNo.646804-ERCCOG-BNYQand
cal pulses due to its simplicity and good experimental per-
bytheIsraelScienceFoundationunderGrantno.335/14.T.B.waspartially
fundedbytheAndrewandErnaFinciViterbiFellowship. formance [20], [21]. A FROG apparatus produces a two-
2
Figure II.1: Illustration of the SHG FROG technique.
dimensional(2D)intensitydiagramofaninputpulsebyinter- withrandomphases.Arecentpapersuggeststoadoptptycho-
acting the pulse with delayed versions of itself in a nonlinear- graphic techniques where every power spectrum, measured at
optical medium, usually using a second harmonic generation each delay, is treated separately as a 1D problem [30]. In
(SHG)crystal[22].This2DsignaliscalledaFROGtraceand FigureII.2wepresentanexamplefortherecoveryofasignal
is a quartic function of the unknown signal. Hereinafter, we from its noisy FROG trace using this algorithm.
consider SHG FROG but other types of nonlinearities exist
forFROGmeasurements.AgeneralizationofFROGinwhich
two different unknown pulses gate each other in a nonlinear
medium is called blind FROG. This method can be used
to characterize simultaneously two signals [21], [23]. In this
case, the measured data is referred to as a blind FROG trace
and is quadratic in both signals. We refer to the problems
of recovering a signal from its blind FROG trace and FROG
trace as bivariate phase retrieval and quartic phase retrieval,
respectively. Note that quartic phase retrieval is a special case
of bivariate phase retrieval where both signals are equal. An
illustrationoftheSHGFROGmodelisdepictedinFigureII.1.
In bivariate phase retrieval we acquire, for each delay step
m, the power spectrum of
FigureII.2: Experimentalexampleofafemtosecond(fs)pulse
y [n]=x [n]x [n+mL], (II.1) reconstructionbySHG-FROG.Theexperimentwasconducted
m 1 2
with a delay step of 3 fs and 512 delay points. Hence, the
where L determines the overlap factor between adjacent sec-
complete FROG trace consists of 512×512 data points. The
tions. We assume that x ,x ∈ CN are periodic, namely,
1 2 laser pulse was produced by a typical ultrafast Ti-sapphire
x[i]=x[N(cid:96)+i] for all (cid:96)∈Z. The acquired data is given by
laser system (1KHz repetition rate, 2 Watt average power).
Z[k,m]=|Y[k,m]|2, (II.2) (a) measured FROG trace (b) recovered trace by alternating
projection algorithm for ptychography (Ptych.) proposed in
where [30] (c) recovered trace by the PCGPA algorithm [28] (d)
N−1 recovered amplitudes by PCGPA and Ptych. (e) recovered
(cid:88)
Y[k,m] = (Fy )[k]= y [n]e−2πjkn/N phases by PCGPA and Ptych.
m m
n=0
N(cid:88)−1 In the next section we present our main theoretical results.
= x [n]x [n+mL]e−2πjkn/N, (II.3)
1 2 First, in Proposition 1 we identify the trivial ambiguities of
n=0 blind FROG. Trivial ambiguities are the basic operations on
and F is the N ×N DFT matrix. Quartic phase retrieval is thesignalsx ,x thatdonotchangetheblindFROGtraceZ.
1 2
the special case in which x =x . Then, we derive a uniqueness result for the mapping between
1 2
CurrentFROGreconstructionprocedures[24],[25],[26]are the signals and their blind FROG trace. Particularly, suppose
based on 2D phase retrieval algorithms [2], [27]. One popular we can measure the power spectra of the unknown signals in
iterative algorithm is the principal components generalized addition to the blind FROG trace. We exploit recent advances
projections (PCGP) method [28]. In each iteration, PCGP in the theory of phase retrieval [11] and prove that in this
performs PCA (principal component analysis, see for instance case almost all band-limited signals are determined uniquely,
[29]) on a data matrix constructed by a previous estimation. up to trivial ambiguities. This result holds trivially for FROG
It is common to initialize the algorithm by a Gaussian pulse as well. The proof is based on the observation that given
3
the signal’s power spectrum, the problem can be reduced to IV. PROOFOFTHEOREM2
standard phase retrieval where both the temporal and spectral
The proof is based on the reduction of bivariate phase
magnitudes are known.
retrievaltoaseriesofmonovariatephaseretrievalproblemsin
which both temporal and spectral magnitudes are known [19].
III. UNIQUENESSRESULT
The latter problem is well-posed for almost all signals.
This letter aims at examining under what conditions the Let
measurementsZdeterminex andx uniquely.Insomecases,
1 2 N−1
there is no way to distinguish between two pairs of signals, x [n]= 1 (cid:88) xˆ [(cid:96)]e2πj(cid:96)n/N,
by any method, as they result in the same measurements. 1 N 1
(cid:96)=0
The following proposition describes four trivial ambiguities
N−1
of bivariate phase retrieval. The first three are similar to x [n]= 1 (cid:88) xˆ [(cid:96)]e2πj(cid:96)n/N,
equivalentresultsinphaseretrieval,seeforinstance[10].The 2 N 2
(cid:96)=0
proof follows from basic properties of the Fourier transform
and
and is given in the Appendix. (cid:40)
1 n=0,
Proposition 1. Let x1,x2 ∈ CN and let ym[n] := δ[n]:= 0 otherwise.
x [n]x [n+mL]forsomefixedL.Then,thefollowingsignals
1 2
have the same phaseless bivariate measurements Z[m,k] as Then we have
x1,x2: N−1
(cid:88)
1) multiplicationbyglobalphasesx1ejψ1,x2ejψ2 forsome Y[k,m] = x1[n]x2[n+m]e−2πjkn/N
ψ1,ψ2 ∈R, n=0
2) the shifted signal 1 N(cid:88)−1(cid:32)N(cid:88)−1 (cid:33)
= xˆ [(cid:96) ]e2πj(cid:96)1n/N
x [n−n ]x [n−n +mL]=y [n−n ] N2 1 1
1 0 2 0 m 0 n=0 (cid:96)1=0
for some n0 ∈Z, (cid:32)N(cid:88)−1xˆ [(cid:96) ]e2πjm(cid:96)2/Ne2πj(cid:96)2n/N(cid:33)e−2πjkn/N
3) the conjugated and reflected signal 2 2
(cid:96)2=0
x [−n]·x [−n+mL]=y [−n],
1 2 m 1 N(cid:88)−1N(cid:88)−1
4) modulation,x1[n]e−2πjk0n/N,x2[n]e2πjk0n/N forsome = N2 xˆ1[(cid:96)1]xˆ2[(cid:96)2]e2πjm(cid:96)2/N
k ∈Z. (cid:96)1=0(cid:96)2=0
0
N−1
(cid:88)
Assume that one of the signals is band-limited and that we e−2πj(k−(cid:96)1−(cid:96)2)n/N
have access to the power spectrum of the underlying signals n=0
|Fx |2, |Fx |2 as well as the blind FROG trace Z[m,k] (cid:124) (cid:123)(cid:122) (cid:125)
1 2 =Nδ[k−(cid:96)1−(cid:96)2]
of (II.2). In ultra-short pulse characterization experiments the
N−1
signals are indeed band-limited [31] and the power spectrum = 1 (cid:88) xˆ [k−(cid:96)]xˆ [(cid:96)]e2πjm(cid:96)/N.
of the pulse under investigation is often available, or it can be N 1 2
(cid:96)=0
easilymeasuredbyaspectrometer,whichisalreadyintegrated
in any FROG device. Inspired by [19], we show that in this Let us denote xˆi[(cid:96)] = |xˆi[(cid:96)]|ejφi[(cid:96)] for i = 1,2, I[k,(cid:96)] =
case, the bivariate problem can be reduced to a standard N1 |xˆ1[k−(cid:96)]||xˆ2[(cid:96)]| and P[k,(cid:96)]=φ1[k−(cid:96)]+φ2[(cid:96)]. Then2,
(monovariate)phaseretrievalproblemwhereboththetemporal
N−1
(cid:88)
and the spectral magnitudes are known. Consequently, we Y[k,−m]= I[k,(cid:96)]ejP[k,(cid:96)]e−2πjm(cid:96)/N.
derivethefollowingresultwhichisprovedinthenextsection.
(cid:96)=0
Theorem 2. Let L = 1, and let xˆ1 := Fx1 and xˆ2 := Fx2 By assumption, |xˆ1| and |xˆ2| are known and therefore
be the Fourier transforms of x1 and x2, respectively. Assume I[k,(cid:96)] is known as well. Moreover, note that by assumption,
that xˆ1 has at least (cid:100)(N −1)/2(cid:101) consecutive zeros (e.g. for any fixed k, I[k,(cid:96)] has at least (cid:100)(N −1)/2(cid:101) consecutive
band-limitedsignal).Then,almostallsignals1 aredetermined zeros. Our problem is then reduced to that of recovering
uniquely, up to trivial ambiguities, from the measurements the signal S[k,(cid:96)] := I[k,(cid:96)]ejP[k,(cid:96)] from the knowledge of
Z[m,k] and the knowledge of |xˆ1| and |xˆ2|. By trivial ambi- Z[k,−m] and I[k,(cid:96)]. For fixed k, this is a standard phase
guities we mean that x1 and x2 are determined up to global retrieval problem with respect to the second variable where
phase, time shift and conjugate reflection. the temporal magnitudes are known. To proceed, we state the
finite-discrete version of Theorem 3.4 from [11]:
Corollary3. Thesameresultholdsforquarticphaseretrieval
in which x1 =x2. This model fits the FROG setup. Lemma 4. Let t ∈ [0,...,N −1]\{(N −1)/2} and let
u∈CN be such that u has at least (cid:100)(N −1)/2(cid:101) consecutive
Proof. The proof follows the proof technique of Theorem 2
zeros. Then, almost every complex signal u is determined
with x =x .
1 2
1Byalmostallsignals,wemeanthattheremaybeasetofmeasurezero 2Recall that all indices should be considered as modulo N. Hence,
forwhichthetheoremdoesnothold. Y[k,−m]isjustareorderingofY[k,m].
4
uniquely from the magnitude of its Fourier transform and thetranslationambiguities.Specifically,thetermejc1k reflects
|u[N −1−t]| up to to global phase. translation by c1 indices and the ejc2 product by a global
phase.Theconjugate-reflectnessambiguityarisesfromthefact
Lemma 4 implies that Z[k,−m] and I[k,(cid:96)] determine, for
thatboththeblindFROGtraceandthesignalspowerspectrum
fixed k, almost all P[k,(cid:96)] up to global phase. So, for all k,
are invariant to this property. This completes the proof.
P[k,(cid:96)] is determined up to an arbitrary function ψ[k]. We
note that while Lemma 4 requires only one sample of I[k,(cid:96)]
to determine S[k,(cid:96)] uniquely, I[k,(cid:96)] does not determine |xˆ | V. DISCUSSION
1
and |xˆ | uniquely. For this reason, we need the full power In this paper we analyzed the uniqueness of bivariate and
2
spectrum of the signals in addition to the blind FROG trace. quartic phase retrieval problems. Particularly, we proposed
Next, we will show that a uniqueness result showing that given the signals power
spectrum, blind FROG trace determines almost all signals up
P˜[k,(cid:96)]=P[k,(cid:96)]+ψ[k] (IV.1)
to trivial ambiguities for L = 1. Nevertheless, it was shown
=φ1[k−(cid:96)]+φ2[(cid:96)]+ψ[k], experimentallyandnumerically[30]thatstablesignalrecovery
is possible with L>1. It is therefore important to investigate
determines φ ,φ and ψ up to affine functions. Note that
1 2 the minimal number of measurements which can guarantee
generally (IV.1) may include additional terms of 2πs[k,(cid:96)]
for some integers s[k,(cid:96)] ∈ Z. However, phase wrapping is uniqueness for FROG and blind FROG.
It is worth noting different FROG nonlinearities. Two ex-
physicallymeaninglesssinceitwillnotchangethelightpulse
amples are third-harmonic generation FROG and polarization
[21, Section 2].
gating FROG. In these techniques, the measured signal is
Therelation(IV.1)canbewrittenusingmatrixnotation.Let
P˜ ∈RN2 be a column stacked version of P˜ and let modeled as the power spectrum of ym[n]=x2[n]x[n−mL]
vec
and y [n] = x[n]|x[n−mL]|, respectively [32], [20]. It is
  m
φ
1 interestingtoexaminetheuniquenessofthesehighpolynomial
v:=φ2∈R3N. degree phase retrieval problems in different FROG imple-
ψ
mentations. Another important application is the so called
Then we obtain the over-determined linear system Frequency-ResolvedOpticalGatingforCompleteReconstruc-
tion of Attosecond Bursts (FROG CRAB), which is based on
P˜ =Av, (IV.2)
vec the photoionization of atoms by the attosecond field, in the
where A ∈ RN2×3N is the matrix that relates v and P˜ presence of a dressing laser field. In this setup, the signal is
vec
according to (IV.1). modeled as the power spectrum of ym[n]=x1[n]ejx2[n−mL]
We aim at identifying the null space of the linear operator [33].
A. To this end, suppose that there exists another triplet
φ˜ ,φ˜ ,ψ˜ that solves the linear system, i.e. ACKNOWLEDGEMENT
1 2
P˜[k,(cid:96)]=φ˜ [k−(cid:96)]+φ˜ [(cid:96)]+ψ˜[k], We would like to thank Kishore Jaganathan for many
1 2
insightful discussions, Oren Cohen for his advice on ultra-
forallkand(cid:96).Letusdenotethedifferencefunctionsbyd1 := fast laser pulse measurement methods and Robert Beinert for
φ1 −φ˜1,d2 := φ2 −φ˜2 and d3 := ψ −ψ˜. Then, we can helpful discussions about [11].
directly conclude that for all k,(cid:96) we have
d [k−(cid:96)]+d [(cid:96)]+d [k]=0. (IV.3) APPENDIX
1 2 3
Proof of Proposition 1
Particularly, for k =0 and (cid:96)=0 we obtain the relations
The proof is based on basic properties of the DFT matrix.
d [−(cid:96)]+d [(cid:96)]+d [0]=0,
1 2 3 Recall that y [n]:=x [n]x [n+mL].
(IV.4) m 1 2
d1[k]+d2[0]+d3[k]=0. 1) Let ψ1,ψ2 ∈ R and define xψ1 := x1ejψ1, xψ2 :=
Plugging (IV.4) into (IV.3) (and replace −(cid:96) by (cid:96)) we have x2ejψ2 and ymψ[n]:=xψ1[n]xψ2[n+mL]. Hence, ymψ =
ymej(ψ1+ψ2) and it is then clear that Z is independent
d [k+(cid:96)]=d [(cid:96)]+d [k]+d [0]+d [0].
1 1 1 3 2 of ψ ,ψ .
1 2
Hence, we conclude that d1 is an affine function of the form 2) Let n0 ∈Z and define y˜m[n]:=ym[n−n0]. Then, by
d [k] = ak−d [0]−d [0] for some scalar a. We can also standard Fourier properties we get
1 3 2
derivethatd [k]=ak+d [0]andd [k]=−ak+d [0].This
2 2 3 3 (Fy˜ )[k]=(Fy )[k]e−2πjkn0/N,
m m
impliesthatthenullspaceofAcontainsthoseaffinefunctions.
We can compute the phases by v=A†P˜ , where A† is the and consequently |Fy˜ |=|Fy |.
vec m m
Moore-Penrose pseudoinverse. 3) BystandardFourierpropertieswehave|Fy` |=|Fy |.
m m
To complete the proof, we recall that φi, i = 1,2, are the 4) Let k0 ∈ Z and define xk10[n] := x1[n]−2πjk0n/N,
phases of the Fourier transforms of xi. As we can estimate xk20[n] := x2[n]2πjk0n/N and ymk0[n] := xk10[n]xk20[n+
the phases up to affine functions, we can only determine mL]. Then, ymk0[n] = ym[n]e2πjmLk0/N. According to
xˆi[k] = |xˆi[k]|ej(φi[k]+c1k+c2) for some constants c1 and c2. theglobalphaseambiguity,Zisindependentofk0.This
This unknown affine function reflects the global phase and completes the proof.
5
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