Signal Reconstruction from FROG Measurements · Frequency-resolved optical gating: the measurement of ultrashort laser pulses In Chapter 5, page 108, he writes: Goal: Conditions on

Signal Reconstruction from FROG Measurements

Tamir Bendory

Princeton UniversityThe Program in Applied and Computational Mathematics

Joint work with Dan Edidin and Yonina Eldar

Tamir Bendory August 17, 2017 1 / 16

Introduction

Motivation

In order to measure a short event, one must use a shorter pulse.


Introduction

Motivation


Many process in biology (e.g., protein folding), chemistry (e.g.,molecular vibration) and physics (e.g., photo-ionization) containevents in the femoseconds time scale.

In many of these experiments, the pulse structure plays role in theoutcome of the experiment.

How can we measure the shortest pulse?


Introduction

Motivation






Introduction

Motivation






Introduction

Motivation






Introduction

How to measure the shortest pulse?

The power spectrum of the signal can be measured with spectrometer.

But we cannot recover a 1D signal from its power spectrum

The Frequency-Resolved Optical Gating techniques were introducedin 1991 by Daniel J. Kane and Rick Trebino

Nowadays, FROG is a commonly–used method for fullcharacterization of ultra-short optical pulses due to its simplicity andgood experimental performance


Introduction







Introduction







Introduction







Introduction

The FROG technique

The FROG trace is the Fourier magnitude of product of the signal with atranslated version of itself, for several different translations.

|yk,m|2 =∣∣∣∣∣N−1∑n=0

xnxn+mLe−2πikn/N∣∣∣∣∣2

, m = 0, . . . , dN/Le − 1

It is a quartic intensity map CN 7→ RN×N/L.

Delay [fs]

Freq

uenc

y [P

Hz]

-400 -200 0 200 400

-0.5

0

0.5

Spectrometer𝑥𝑥𝑛𝑛+𝑚𝑚𝑚𝑚

𝑥𝑥𝑛𝑛Variabledelay, τ = mL

𝑦𝑦𝑛𝑛,𝑚𝑚 = 𝑥𝑥𝑛𝑛𝑥𝑥𝑛𝑛+𝑚𝑚𝑚𝑚


Introduction

The FROG technique

All the information about FROG can be found in Rick Trebino’s book:Frequency-resolved optical gating: the measurement of ultrashortlaser pulses

In Chapter 5, page 108, he writes:

Goal: Conditions on the number of samples required to determine asignal uniquely, up to trivial ambiguities, from its FROG trace


Introduction

The FROG technique





Introduction

The FROG technique





STFT Phase retrieval and XFROG

STFT phase retrieval and XFROG

In some cases, one can use a known reference pulse g :

|yk,m|2 =∣∣∣∣∣N−1∑n=0

xngn+mLe−2πikn/N∣∣∣∣∣2

, m = 0, . . . , dN/Le − 1

This technique is called XFROG.For L < N/2, unique mapping for almost all 1D signals, up to globalrotation [Jaganathan, Eldar and Hassibi, 2015]We examined two non-convex algorithms: least-squares minimizationand optimization over the manifold of phases [B., Eldar and Boumal,2017]).

Empirically, the basin of attraction of the non-convex programs isquite large. We derived some theoretical insights.Data–driven initialization based on approximating xx∗ with theoreticalanalysis





|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1

This technique is called XFROG.

For L < N/2, unique mapping for almost all 1D signals, up to globalrotation [Jaganathan, Eldar and Hassibi, 2015]We examined two non-convex algorithms: least-squares minimizationand optimization over the manifold of phases [B., Eldar and Boumal,2017]).






|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1

This technique is called XFROG.For L < N/2, unique mapping for almost all 1D signals, up to globalrotation [Jaganathan, Eldar and Hassibi, 2015]

We examined two non-convex algorithms: least-squares minimizationand optimization over the manifold of phases [B., Eldar and Boumal,2017]).






|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1







|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1


Empirically, the basin of attraction of the non-convex programs isquite large. We derived some theoretical insights.

Data–driven initialization based on approximating xx∗ with theoreticalanalysis





|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1




FROG

FROG symmetries

Similarly to phase retrieval, FROG has three symmetries or trivialambiguities:

ClaimThe following signals have the same FROG trace as x ∈ CN :

1 the rotated signal xeiψ for some ψ ∈ R;2 the translated signal x `n = xn−` for some ` ∈ Z;3 the reflected signal xn = x−n.


FROG

FROG symmetries

Similarly to phase retrieval, FROG has three symmetries or trivialambiguities:

ClaimThe following signals have the same FROG trace as x ∈ CN :

1 the rotated signal xeiψ for some ψ ∈ R;2 the translated signal x `n = xn−` for some ` ∈ Z;3 the reflected signal xn = x−n.


FROG

FROG symmetries for bandlimited signals

The translation symmetry implies that signal with Fourier transfromxke−2πi`k/N for some ` ∈ Z has the same FROG trace as x

For bandlimited signals, the translation symmetry is continuous.Namely, signals with Fourier transform xkeiψk has the same FROGtrace as x

Example for signal with Fourier transform [1, i ,−i , 0, 0, 0, 0, 0, 0, i ,−i ]:

1 2 3 4 5 6 7 8 9 10 11

-0.2

-0.1

0

0.1

0.2

0.3

0.4

signalshifted by 3 entriesshifted by 1.5 entries


FROG





1 2 3 4 5 6 7 8 9 10 11

-0.2

-0.1

0

0.1

0.2

0.3

0.4



FROG





1 2 3 4 5 6 7 8 9 10 11

-0.2

-0.1

0

0.1

0.2

0.3

0.4



FROG

Uniqueness

TheoremLet x ∈ CN be a B–bandlimited signal for some B ≤ N/2.

If L ≤ N/4, then almost all signals are determined uniquely from theirFROG trace, modulo the trivial ambiguities, from 3B measurements.

If in addition we have access to the signal’s power spectrum and L ≤ N/3,then 2B measurements are sufficient.

FROG setup requires 3B measurements for B–bandlimited signalSTFT phase retrieval requires more than 2N measurementsRandom phase retrieval requires 4N − 4 measurements (to recover allsignals)


FROG

Uniqueness




FROG setup requires 3B measurements for B–bandlimited signal

STFT phase retrieval requires more than 2N measurementsRandom phase retrieval requires 4N − 4 measurements (to recover allsignals)


FROG

Uniqueness




FROG setup requires 3B measurements for B–bandlimited signalSTFT phase retrieval requires more than 2N measurements

Random phase retrieval requires 4N − 4 measurements (to recover allsignals)


FROG

Uniqueness




FROG setup requires 3B measurements for B–bandlimited signalSTFT phase retrieval requires more than 2N measurementsRandom phase retrieval requires 4N − 4 measurements (to recover allsignals)


FROG

Few words on the proof

The challenge: system of phaseless quartic equations (in contrast toquadratic system of equations in phase retrieval)

One can formulate the FROG measurements as

yk,m = 1N

N−1∑`=0

x`xk−`ω`m, ω = e2πiL/N ,

Because of the bandlimited assumption, one can form a pyramidstructure of x`xk−`

x20 , 0, . . . , 0

x0x1, x1x0, 0, . . . , 0x0x2, x2

1 , x2x0, . . . , 0 . . ....

xN/2−1x0, xN/2−2x1, . . . , xN/2−1x0, 0, . . . , 00, x1xN/2−1, x2xN/2−2, . . . , xN/2−1x1, 0, . . . , 0

...0, 0, . . . xN/2−1xN/2−1, . . . , 0, . . . , 0.

Because of the bandlimited assumption, one can form a pyramidstructureBecause of the two continuous symmetries, we can fix x0 and x1 to bereal and =x2 ≥ 0The rest of the coefficients are determined recursively. Given the first(k − 1) Fourier coefficients, we get a quadratic system for the kthcoefficient


FROG


The challenge: system of phaseless quartic equations (in contrast toquadratic system of equations in phase retrieval)One can formulate the FROG measurements as

yk,m = 1N

N−1∑`=0



x20 , 0, . . . , 0

x0x1, x1x0, 0, . . . , 0x0x2, x2

1 , x2x0, . . . , 0 . . ....


...0, 0, . . . xN/2−1xN/2−1, . . . , 0, . . . , 0.



FROG



yk,m = 1N

N−1∑`=0



x20 , 0, . . . , 0

x0x1, x1x0, 0, . . . , 0x0x2, x2

1 , x2x0, . . . , 0 . . ....


...0, 0, . . . xN/2−1xN/2−1, . . . , 0, . . . , 0.



FROG



yk,m = 1N

N−1∑`=0



x20 , 0, . . . , 0

x0x1, x1x0, 0, . . . , 0x0x2, x2

1 , x2x0, . . . , 0 . . ....


...0, 0, . . . xN/2−1xN/2−1, . . . , 0, . . . , 0.

Because of the bandlimited assumption, one can form a pyramidstructureBecause of the two continuous symmetries, we can fix x0 and x1 to bereal and =x2 ≥ 0

The rest of the coefficients are determined recursively. Given the first(k − 1) Fourier coefficients, we get a quadratic system for the kthcoefficient


FROG



yk,m = 1N

N−1∑`=0



x20 , 0, . . . , 0

x0x1, x1x0, 0, . . . , 0x0x2, x2

1 , x2x0, . . . , 0 . . ....


...0, 0, . . . xN/2−1xN/2−1, . . . , 0, . . . , 0.



Blind FROG

Blind FROG

In some experimental setups, two pulses are necessary: one to excite amedium and the other to probe it.

Estimating two pulses simultaneously from the blind FROG trace

|yk,m|2 =∣∣∣∣∣N−1∑n=0

unvn+mLe−2πikn/N∣∣∣∣∣2

, m = 0, . . . , dN/Le − 1.

Additional continuous symmetry: the pair (uneinφ, vne−inφ) has thesame blind FROG trace as (u, v) for any φ ∈ R.When L = 1, the two signals are determined uniquely, up tosymmetries [B., Sidorenko and Eldar, 2017].

How many measurements do we need to determine a generic pair ofsignals from their blind FROG trace?


Blind FROG

Blind FROG



|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1.




Blind FROG

Blind FROG



|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1.

Additional continuous symmetry: the pair (uneinφ, vne−inφ) has thesame blind FROG trace as (u, v) for any φ ∈ R.

When L = 1, the two signals are determined uniquely, up tosymmetries [B., Sidorenko and Eldar, 2017].



Blind FROG

Blind FROG



|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1.




Blind FROG

Blind FROG



|yk,m|2 =∣∣∣∣∣N−1∑n=0


, m = 0, . . . , dN/Le − 1.




Open questions

Additional open questions

Many FROG non-linearities to consider. For example, a setup forcharacterization of Attosecond pulses, called FROG for CompleteReconstruction of Attosecond Bursts (FROG CRAB), is modeled as:

|yk,m|2 =∣∣∣∣∣N−1∑n=0

uneivn+mLe−2πikn/N∣∣∣∣∣2

, m = 0, . . . , dN/Le − 1

Analysis of FROG algorithms. Currently, the most popular algorithmis the Principal Component Generalized Projections which alternatesbetween the known intensities and the form of the non-linearinteraction.


Open questions

Additional open questions

Many FROG non-linearities to consider. For example, a setup forcharacterization of Attosecond pulses, called FROG for CompleteReconstruction of Attosecond Bursts (FROG CRAB), is modeled as:

|yk,m|2 =∣∣∣∣∣N−1∑n=0

uneivn+mLe−2πikn/N∣∣∣∣∣2

, m = 0, . . . , dN/Le − 1

Analysis of FROG algorithms. Currently, the most popular algorithmis the Principal Component Generalized Projections which alternatesbetween the known intensities and the form of the non-linearinteraction.


Open questions

References

1 Trebino, R.. Frequency-resolved optical gating: the measurement ofultrashort laser pulses. Springer Science & Business Media, 2012.

2 Bendory, T., Eldar. Y. and Boumal N. ”Non-convex phase retrievalfrom STFT measurements”. To appear in IEEE Transactions onInformation Theory, 2017.

3 Bendory, T., Sidorenko P. and Eldar Y. ”On the uniqueness of FROGmethods.” IEEE Signal Processing Letters, vol. 24, issue 5, pp.722-726, 2017.

4 Bendory, T., Edidin D., and Eldar Y. ”On Signal Reconstruction fromFROG Measurements.” arXiv preprint arXiv:1706.08494, 2017.


Open questions

New benchmark for crystallography problems


Open questions

Thanks for your attention!


Documents

Signal Reconstruction from FROG Measurements · Frequency-resolved optical gating: the measurement of ultrashort laser pulses In Chapter 5, page 108, he writes: Goal: Conditions on