Addressing FROG with Nonlinear Compressive Sensing

This has been an on-going question in the back of my mind for a while now. If you recall this line of inquiry:

• Is FROG an instance of Nonlinear Compressed Sensing ?
• A small Q&A with Rick Trebino, the inventor of FROG.
• FROG, Nonlinear Compressive Sensing ? continued.

In those entries, I wondered (and eventually talked to the person behind the technique to see) if FROG could have a sparsity based approach. As you probably recall that FROG performs a certain kind of convolution/deconvolution so as to perform pulse measurement characterization, in Rick Trebino's words:

In order to measure an event in time, you must use a shorter one. But then, to measure the shorter event, you must use an even shorter one. And so on. So, now, how do you measure the shortest event ever created?

From Rick Trebino's FROG site.

I just came across (quite late) this paper on Sparsity-based super-resolution in instruments for diagnostics of short pulses by Zohar Avnat, Pavel Sidorenko, Yoav Shechtman, Yonina Eldar, Mordechai Segev and Oren Cohen 

We demonstrate experimentally algorithmic super-resolution for diagnostics of short pulses (amplitude and phase). Our approach is based on using the measured data for finding a mathematical basis in which the pulse is represented compactly. 

In the paper, the authors used X-FROG, a technique that uses the cross correlation of the signal (i.e. the experiment requires a previously measured reference pulse.) This is different from FROG or SPIDER which, you may recall, deals with a three way correlation measurement as explained in Rick's Light-Pulse-Measurement Tutorial. 

So there is an underlying set of studies that begins to put sparsity at the center of these optical experiments. I also noted the following recent paper from Rick's publication list: M. Rhodes, S. Rahaman, P. Bowlan, G. Steinmeyer, and R. Trebino, "From femtosecond to nanosecond, laser-pulse measurement is all about the single shot," Laser Focus World 50, 97-100 (2014). The abstract reads:

Recurring misinterpretations due to the infamous coherent artifact imply that pulse-measurement methods operate best using single shot. Fortunately, such methods have recently been extended to pulses about 1 ns long—filling a gaping technological hole that has existed for almost 50 years.

From the review:

Recently, the long-overdue study of the coherent artifact in FROG and SPIDER technique occurred, and the results were shocking for many. Simulations (see Fig. 2) involved one stable train of short pulses and two unstable trains of longer pulses, the latter comprising the stable train's short pulse plus longer random components.^3,4 It was found that neither method retrieves a typical pulse in the unstable trains, but FROG's pulse lengths are at least reasonable.

What Rick et al also point out is that if the technique is so sensitive to coherent artifacts in femto-second pulses, then there is solution (some care must be taken to perform pulse characterization experiments). However they also point that the technique doesn't work well for longer pulses because greater care means more money and space. In effect, while very short (and sparse) pulses can be very well characterized, longer pulses (and not so sparse) pulses fall through the cracks even though those are key to most of the Time of Flight cameras we have mentioned here on Nuit Blanche. Rick has a FROG solution for that that also depends on second order harmonics (SHG), i.e. a three term correlation, a far cry from the usual linear or cross correlation measurment we see in traditional and nonlinear compressive sensing literature. If you have followed this thread all the way to this line, here are the questions:

• can we use some of the recent nonlinear compressive sensing approaches to tackle the current FROG system of Rick et al ?
• if that can be done, is there a theoretical limit like the Donoho-Tanner phase transition for pulse reconstruction ?
• are there nonlinearities in the enconding that can help in the sparse reconstruction process ?
• are the coherent articfact mentioned by Rick, actual information as in here ?

Rick's group code Code for Retrieving a Pulse Intensity and Phase from Its FROG Trace and their recent publications:

• 2014
• M. Rhodes, S. Rahaman, P. Bowlan, G. Steinmeyer, and R. Trebino, "From femtosecond to nanosecond, laser-pulse measurement is all about the single shot," Laser Focus World 50, 97-100 (2014).
• 2013
• M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, "Pulse-shape instabilities and their measurement," Laser & Photonics Reviews, doi:10.1002/lpor.201200102 (2013).
• T. C. Wong and R. Trebino, "Single-frame measurement of complex laser pulses tens of picoseconds long using pulse-front tilt in cross-correlation frequency-resolved optical gating," J. Opt. Soc. Am. B 30, 2781-2786 (2013).
• T. C. Wong and R. Trebino, "Recent Developments in Experimental Techniques for Measuring Two Pulses Simultaneously," Appl. Sci. 3, 299-313 (2013).
• 2012
• T. C. Wong, J. Ratner, and R. Trebino, "Simultaneous measurement of two different-color ultrashort pulses on a single shot," J. Opt. Soc. Am. B 29, 1889-1893 (2012).
• J. Ratner, G. Steinmeyer, T. C. Wong, R. Bartels, and R. Trebino, "The coherent artifact in modern pulse measurements," Opt. Lett. 37, 2874-2876 (2012).
• T. C. Wong, J. Ratner, V. Chauhan, J. Cohen, P. M. Vaughan, L. Xu, A. Consoli, and R. Trebino, "Simultaneously measuring two ultrashort laser pulses on a single-shot using double-blind frequency-resolved optical gating," J. Opt. Soc. Amer. B 29, 1237-1244 (2012).
• P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari,"Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory," J. Opt. 14, 015701 (2012).
• J. Cohen, P. Bowlan, and R. Trebino, "Extending Femtosecond Metrology to Longer, More Complex Laser Pulses in Time and Space," IEEE J. Sel. Top. Quant. Electron. 18, 218-227 (2012).
• 2011
• P. Bowlan, and R. Trebino, "Using phase diversity for the measurement of the complete spatiotemporal electric field of ultrashort laser pulses," J. Opt. Soc. Amer. B 29, 244-248 (2012).
• P. M. Vaughan, and R. Trebino, "Optical-parametric-amplification imaging of complex objects," Opt. Expr. 19, 8920-8929 (2011).
• R. Trebino, "Optical complexity challenges pulse measurement methods," Laser Focus World (2011).
• R. Trebino, "Measuring the seemingly immeasurable," Nature Photonics 5, 189-192 (2011).
• J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, "Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses," Opt. Comm. 284, 3785-3794 (2011).
• P. Bowlan, and R. Trebino, "Complete single-shot measurement of arbitrary nanosecond laser pulses in time," Opt. Expr. 19, 1367-1377 (2011).
• 2010
• P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, "Time-and-Space-Domain Study or Diffracting and Non-Diffracting Light Pulses," Lith. J. Phys 50, 121-127 (2010).
• P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino,"Directly recording diffraction phenomena in time domain," Laser Phys. 20, 948-953 (2010).
• P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, "Basic diffraction phenomena in time domain," Opt. Expr. 18, 11083 (2010).
• M. Lõhmus, P. Bowlan, R. Trebino, H. Valtna-Lukner, P. Piksarv, and P. Saari,"Directly recording diffraction phenomena in the time domain," Lith J. Phys. 50, 69-74 (2010).
• J. Cohen, D. Lee, V. Chauhan, P. Vaughan, and R. Trebino, "Highly simplified device for measuring the intensity and phase of picosecond pulses," Opt. Expr. 18, 17484-17497 (2010).
• J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, "Measuring extremely complex pulses with time-bandwidth products exceeding 65,000 using multiple-delay crossed-beam spectral interferometry," Opt. Expr. 18, 24451-24460 (2010).
• J. Cohen, P. Bowlan, V. Chauhan, and R. Trebino, "Measuring temporally complex ultrashort pulses using multiple-delay crossed-beam spectral interferometry," Opt. Expr. 18, 6583-6597 (2010).
• V. Chauhan, J. Cohen, P. Vaughan, P. Bowlan, and R. Trebino, "Distortion-Free Single-Prism/Grating Ultrashort Laser Pulse Compressor," IEEE J. Quant. Electron. 46, 1726-1731 (2010).
• V. Chauhan, J. Cohen, and R. Trebino, "Simple dispersion law for arbitrary sequences of dispersive optics," Appl. Opt. 49, 6840 (2010).
• V. Chauhan, P. Bowlan, J. Cohen, and R. Trebino, "Single-diffraction-grating and grism pulse compressors," J. Opt. Soc. Amer. B 27, 619-624 (2010).
• P. Bowlan, and R. Trebino, "Extreme pulse-front tilt from an etalon," J. Opt. Soc. Amer. B 27, 2322-2327 (2010).
• S. Akturk, X. Gu, P. Bowlan, and R. Trebino, "Spatio-temporal couplings in ultrashort laser pulses," J. Opt. 12, 093001 (2010).

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