signal reconstruction from frog measurements · frequency-resolved optical gating: the measurement...
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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.
Tamir Bendory August 17, 2017 2 / 16
Introduction
Motivation
In order to measure a short event, one must use a shorter pulse.
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?
Tamir Bendory August 17, 2017 3 / 16
Introduction
Motivation
In order to measure a short event, one must use a shorter pulse.
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?
Tamir Bendory August 17, 2017 3 / 16
Introduction
Motivation
In order to measure a short event, one must use a shorter pulse.
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?
Tamir Bendory August 17, 2017 3 / 16
Introduction
Motivation
In order to measure a short event, one must use a shorter pulse.
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?
Tamir Bendory August 17, 2017 3 / 16
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
Tamir Bendory August 17, 2017 4 / 16
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
Tamir Bendory August 17, 2017 4 / 16
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
Tamir Bendory August 17, 2017 4 / 16
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
Tamir Bendory August 17, 2017 4 / 16
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
Tamir Bendory August 17, 2017 5 / 16
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
Tamir Bendory August 17, 2017 6 / 16
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
Tamir Bendory August 17, 2017 6 / 16
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
Tamir Bendory August 17, 2017 6 / 16
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
Tamir Bendory August 17, 2017 7 / 16
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
Tamir Bendory August 17, 2017 7 / 16
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
Tamir Bendory August 17, 2017 7 / 16
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
Tamir Bendory August 17, 2017 7 / 16
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
Tamir Bendory August 17, 2017 7 / 16
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
Tamir Bendory August 17, 2017 7 / 16
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.
Tamir Bendory August 17, 2017 8 / 16
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.
Tamir Bendory August 17, 2017 8 / 16
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
Tamir Bendory August 17, 2017 9 / 16
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
Tamir Bendory August 17, 2017 9 / 16
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
Tamir Bendory August 17, 2017 9 / 16
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)
Tamir Bendory August 17, 2017 10 / 16
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 signal
STFT phase retrieval requires more than 2N measurementsRandom phase retrieval requires 4N − 4 measurements (to recover allsignals)
Tamir Bendory August 17, 2017 10 / 16
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 measurements
Random phase retrieval requires 4N − 4 measurements (to recover allsignals)
Tamir Bendory August 17, 2017 10 / 16
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)
Tamir Bendory August 17, 2017 10 / 16
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
Tamir Bendory August 17, 2017 11 / 16
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
Tamir Bendory August 17, 2017 11 / 16
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
Tamir Bendory August 17, 2017 11 / 16
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 ≥ 0
The rest of the coefficients are determined recursively. Given the first(k − 1) Fourier coefficients, we get a quadratic system for the kthcoefficient
Tamir Bendory August 17, 2017 11 / 16
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
Tamir Bendory August 17, 2017 11 / 16
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?
Tamir Bendory August 17, 2017 12 / 16
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?
Tamir Bendory August 17, 2017 12 / 16
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?
Tamir Bendory August 17, 2017 12 / 16
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?
Tamir Bendory August 17, 2017 12 / 16
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?
Tamir Bendory August 17, 2017 12 / 16
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.
Tamir Bendory August 17, 2017 13 / 16
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.
Tamir Bendory August 17, 2017 13 / 16
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.
Tamir Bendory August 17, 2017 14 / 16
Open questions
New benchmark for crystallography problems
Tamir Bendory August 17, 2017 15 / 16
Open questions
Thanks for your attention!
Tamir Bendory August 17, 2017 16 / 16