Abstract

Frequency-resolved optical gating (FROG) is a very popular technique for complete characterization of ultrashort laser pulses. In FROG, a reconstruction algorithm retrieves the pulse amplitude and phase from a measured spectrogram; yet, current FROG reconstruction algorithms require and exhibit several restricting features that weaken FROG performances. For example, the delay step must correspond to the spectral bandwidth measured with a large enough SNR—a condition that limits the temporal resolution of the reconstructed pulse, obscures measurements of weak broadband pulses, and makes measurements of broadband mid-IR pulses hard and slow because the spectrograms become huge due to the geometrical time-smearing effect. We develop a new approach for FROG reconstruction based on ptychography (a scanning coherent diffraction imaging technique), that removes many of the algorithmic restrictions. The ptychographic reconstruction algorithm is significantly faster and more robust to noise than current FROG algorithms that are based on generalized projections (GP). We demonstrate, numerically and experimentally, that ptychographic reconstruction works well with very partial spectrograms, e.g., spectrograms with a reduced number of measured delays and spectrograms that have been substantially spectrally filtered. In addition, we implement the ptychographic approach to blind second harmonic generation (SHG) FROG and demonstrate robust and complete characterization of two unknown pulses from a single measured spectrogram and the power spectrum of only one of the pulses. We believe that the ptychography-based approach will become the standard reconstruction procedure in FROG and related diagnostics methods, allowing successful reconstructions from so far unreconstructable spectrograms.

© 2016 Optical Society of America

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Corrections

Pavel Sidorenko, Oren Lahav, Zohar Avnat, and Oren Cohen, "Ptychographic reconstruction algorithm for frequency resolved optical gating: super-resolution and extreme robustness: erratum," Optica 4, 1388-1389 (2017)
http://proxy.osapublishing.org/optica/abstract.cfm?uri=optica-4-11-1388
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References

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  24. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref]
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  32. B. Yellampalle, K. Kim, and A. J. Taylor, “Amplitude ambiguities in second-harmonic generation frequency-resolved optical gating,” Opt. Lett. 32, 3558–3560 (2007).
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    [Crossref]
  35. M.-A. Mycek, S. Weiss, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, and W. Schaefer, “Femtosecond time-resolved free-induction decay of room-temperature excitons in GaAs quantum wells,” Appl. Phys. Lett. 60, 2666–2668 (1992).
    [Crossref]
  36. A. Freiberg and P. Saari, “Picosecond spectrochronography,” IEEE J. Quantum Electron. 19, 622–630 (1983).
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    [Crossref]
  38. B. Seifert, H. Stolz, and M. Tasche, “Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness,” J. Opt. Soc. Am. B 21, 1089–1097 (2004).
    [Crossref]
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    [Crossref]
  40. A. Consoli, V. Chauhan, J. Cohen, L. Xu, P. Vaughan, F. J. Lopéz-Hernández, and R. Trebino, “Retrieving two pulses simultaneously and robustly using double-blind FROG,” in Frontiers in Optics 2010/Laser Science XXVI (OSA, 2010), paper FThD8.
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  42. C. Hernández-García, J. A. Pérez-Hernández, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, A. Jaron-Becker, A. Becker, and L. Plaja, “Zeptosecond high harmonic keV x-ray waveforms driven by midinfrared laser pulses,” Phys. Rev. Lett. 111, 033002 (2013).
    [Crossref]
  43. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13, 2617–2626 (2005).
    [Crossref]
  44. F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
    [Crossref]
  45. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
    [Crossref]
  46. A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
    [Crossref]
  47. P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6, 8209 (2015).
    [Crossref]

2015 (5)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

D. Spangenberg, P. Neethling, E. Rohwer, M. H. Brügmann, and T. Feurer, “Time-domain ptychography,” Phys. Rev. A 91, 021803(R) (2015).
[Crossref]

P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6, 8209 (2015).
[Crossref]

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref]

M. Lucchini, M. H. Brügmann, A. Ludwig, L. Gallmann, U. Keller, and T. Feurer, “Ptychographic reconstruction of attosecond pulses,” Opt. Express 23, 29502–29513 (2015).
[Crossref]

2013 (3)

J. R. Fienup, “Phase retrieval algorithms: a personal tour [Invited],” Appl. Opt. 52, 45–56 (2013).
[Crossref]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref]

C. Hernández-García, J. A. Pérez-Hernández, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, A. Jaron-Becker, A. Becker, and L. Plaja, “Zeptosecond high harmonic keV x-ray waveforms driven by midinfrared laser pulses,” Phys. Rev. Lett. 111, 033002 (2013).
[Crossref]

2012 (3)

2011 (1)

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331, 41–42 (2011).
[Crossref]

2010 (3)

2009 (4)

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).

2008 (3)

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008).
[Crossref]

J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, “The accurate FROG characterization of attosecond pulses from streaking measurements,” Appl. Phys. B 92, 25–32 (2008).
[Crossref]

D. J. Kane, “Principal components generalized projections: a review [Invited],” J. Opt. Soc. Am. B 25, A120–A132 (2008).
[Crossref]

2007 (1)

2005 (2)

G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13, 2617–2626 (2005).
[Crossref]

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

2004 (2)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

B. Seifert, H. Stolz, and M. Tasche, “Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness,” J. Opt. Soc. Am. B 21, 1089–1097 (2004).
[Crossref]

2000 (1)

1999 (2)

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999).
[Crossref]

D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
[Crossref]

1997 (2)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

D. J. Kane, G. Rodriguez, A. J. Taylor, and T. S. Clement, “Simultaneous measurement of two ultrashort laser pulses from a single spectrogram in a single shot,” J. Opt. Soc. Am. B 14, 935–943 (1997).
[Crossref]

1996 (2)

R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A 13, 725–734 (1996).
[Crossref]

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, “Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[Crossref]

1995 (2)

1994 (1)

1993 (3)

1992 (2)

M.-A. Mycek, S. Weiss, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, and W. Schaefer, “Femtosecond time-resolved free-induction decay of room-temperature excitons in GaAs quantum wells,” Appl. Phys. Lett. 60, 2666–2668 (1992).
[Crossref]

B. McCallum and J. Rodenburg, “Two-dimensional demonstration of Wigner phase-retrieval microscopy in the STEM configuration,” Ultramicroscopy 45, 371–380 (1992).
[Crossref]

1987 (1)

1983 (1)

A. Freiberg and P. Saari, “Picosecond spectrochronography,” IEEE J. Quantum Electron. 19, 622–630 (1983).
[Crossref]

1982 (1)

Arnold, C.

Balslev, I.

V. G. Lyssenko, J. Erland, I. Balslev, K.-H. Pantke, B. S. Razbirin, and J. M. Hvam, “Nature of nonlinear four-wave-mixing beats in semiconductors,” Phys. Rev. B 48, 5720–5723 (1993).
[Crossref]

Baltuska, A.

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Second-harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999).
[Crossref]

Bates, P. K.

Baumbach, T.

Bean, R.

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).

Becker, A.

C. Hernández-García, J. A. Pérez-Hernández, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, A. Jaron-Becker, A. Becker, and L. Plaja, “Zeptosecond high harmonic keV x-ray waveforms driven by midinfrared laser pulses,” Phys. Rev. Lett. 111, 033002 (2013).
[Crossref]

Berenguer, F.

Biegert, J.

Bigot, J.-Y.

M.-A. Mycek, S. Weiss, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, and W. Schaefer, “Femtosecond time-resolved free-induction decay of room-temperature excitons in GaAs quantum wells,” Appl. Phys. Lett. 60, 2666–2668 (1992).
[Crossref]

Brügmann, M. H.

Buades, A.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

Bullkich, E.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Chalus, O.

Chang, Z.

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chauhan, V.

A. Consoli, V. Chauhan, J. Cohen, L. Xu, P. Vaughan, F. J. Lopéz-Hernández, and R. Trebino, “Retrieving two pulses simultaneously and robustly using double-blind FROG,” in Frontiers in Optics 2010/Laser Science XXVI (OSA, 2010), paper FThD8.

Chemla, D. S.

M.-A. Mycek, S. Weiss, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, and W. Schaefer, “Femtosecond time-resolved free-induction decay of room-temperature excitons in GaAs quantum wells,” Appl. Phys. Lett. 60, 2666–2668 (1992).
[Crossref]

Chen, B.

Chini, M.

Clement, T. S.

Cohen, J.

A. Consoli, V. Chauhan, J. Cohen, L. Xu, P. Vaughan, F. J. Lopéz-Hernández, and R. Trebino, “Retrieving two pulses simultaneously and robustly using double-blind FROG,” in Frontiers in Optics 2010/Laser Science XXVI (OSA, 2010), paper FThD8.

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

P. Sidorenko, O. Kfir, Y. Shechtman, A. Fleischer, Y. C. Eldar, M. Segev, and O. Cohen, “Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects,” Nat. Commun. 6, 8209 (2015).
[Crossref]

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

Cohen-Hyams, T.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

Coll, B.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[Crossref]

Consoli, A.

A. Consoli, V. Chauhan, J. Cohen, L. Xu, P. Vaughan, F. J. Lopéz-Hernández, and R. Trebino, “Retrieving two pulses simultaneously and robustly using double-blind FROG,” in Frontiers in Optics 2010/Laser Science XXVI (OSA, 2010), paper FThD8.

Crespo, H.

Dana, H.

A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidorenko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, S. Shoham, M. Zibulevsky, I. Yavneh, Y. C. Eldar, O. Cohen, and M. Segev, “Sparsity-based single-shot subwavelength coherent diffractive imaging,” Nat. Mater. 11, 455–459 (2012).
[Crossref]

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, “Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[Crossref]

K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995).
[Crossref]

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
[Crossref]

Diaz, A.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref]

Donoho, D. L.

D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

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Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. (a) Mean angle (representing the error) between reconstructed and true signals as a function of the SNR for ptychographic-based algorithm (solid blue curve) and for the PCGPA method (dashed green curve). Error bars represent standard deviation of the average over 100 pulses. Plots (b)–(d) show a true pulse (blue solid curves), reconstructed by the ptychographic-based algorithm (red dotted curves) and by the PCGPA method (dashed black curve) for SNR=8, 20, and 40 dB, respectively.

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Fig. 2.
Fig. 2. Examples of pulse reconstructions from incomplete FROG traces using ptychographic-based reconstruction algorithm. Numerically simulated complete FROG trace without (a) and with 20 dB noise (b). All reconstructions below are from incomplete traces obtained by truncating the noisy trace in plot (b). Top horizontal panel (c)–(g) presents reconstruction from frequency filtered trace. Second horizontal panel (h)–(l) presents reconstruction from spectrally under sampled trace. Third panel (m)–(q) presents reconstruction from temporally filtered trace. Last bottom panel (r)–(v) presents reconstructions from temporally under-sampled trace. In each panel, the first (left) and second (left) plots show the incomplete FROG trace and its respective reconstructed trace. The third and fourth plots present the amplitude and phase of the original and reconstructed pulses, respectively. The fifth plot shows the angle between the reconstructed and original pulses (i.e., the reconstruction error) as a function of the incompleteness parameter. The red dashed circles correspond to the presented examples. All partial and reconstructed traces are presented with the same color map as in plot (a).

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Fig. 3.
Fig. 3. Demonstration of ptychographic-based reconstruction using experimental SHG FROG data. (a) Measured FROG trace. (b) and (c) are traces recovered by ptychgraphic-based algorithm and PCGPA algorithm, respectively. Amplitude (d) and phase (e) of the recovered pulse by ptychographic-based algorithm (solid blue curve) and PCGPA algorithm (red dashed curve), respectively.

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Fig. 4.
Fig. 4. Experimental pulse reconstructions from incomplete FROG traces. First horizontal panel (a)–(d) presents reconstruction from low-pass-filtered trace. Second horizontal panel (e)–(h) presents reconstruction from spectrally under-sampled trace. Third horizontal panel (i)–(l) presents reconstruction from a trace that was filtered in the delay axis. Forth horizontal panel (m)–(p) presents reconstruction from under-sampled trace in the delay axis. In each panel, the second plot from the left presents the reconstructed FROG traces from the corresponding incomplete measured trace to its left. The third and the fourth columns show the reconstructed amplitude and phase of the pulse from full (red dashed curve) and corresponding incomplete measured (blue solid curve) traces, respectively. All partial and reconstructed traces are presented with the same color map as in Fig. 3(a).

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Fig. 5.
Fig. 5. Pulse reconstructions from incomplete FROG traces and pulse power spectrum. We use the same exemplary pulse that was used in Fig. 2. The structures of (a)–(t) are the same as the structures of Figs. 2(c)–2(v). All partial and reconstructed traces are presented with same color map as in Fig. 2(a).

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Fig. 6.
Fig. 6. Statistical investigation of pulse reconstructions from incomplete FROG traces. Reconstruction mean angle as a function of the incompleteness parameter for frequency filtered traces (a) and for delay under-sampling (b) with spectral prior information (red solid curve) and without any prior information (blue dashed curve). Error bars correspond to standard deviation over 100 pulses. Four exemplary pulse reconstructions: Spectrogram is spectrally low pass filtered, with [plots (c)–(g), η=0.07, δ=0.096] and without [plots (h)–(l), η=0.102, δ=0.093] power spectrum prior information. Spectrogram is delay under-sampled, with [plots (m)–(q), η=0.031, δ=0.093] and without [plots (r)–(v), η=0.125, δ=0.085] power spectrum prior information. Each panel shows the complete FROG traces (left column plots), filtered trace, reconstructed trace, original (dashed red), and reconstructed (solid blue) pulse amplitudes and pulse phases in the right column plots. All partial and reconstructed traces are presented with same color map as in Fig. 2(a).

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Fig. 7.
Fig. 7. Numerical characterization of the ptychographic blind FROG reconstruction algorithm. Mean angle (representing the error) between reconstructed and original gate (a) and probe (b) pulses as a function of the SNR. Error bars represent standard deviation of the average over 100 pairs. Reconstruction examples are shown in plots (c)–(h). The upper and lower panels show the original (blue solid curves) and reconstructed (red dotted curves) gate and probe pulses, respectively. The SNR is 3 dB in plots (c) and (d), 12 dB in plots (e) and (f), and 20 dB in plots (g) and (h).

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Fig. 8.
Fig. 8. Experimental blind SHG FROG. (a) Measured blind FROG trace. (b) Trace recovered by ptychographic-based blind FROG algorithm (NMSE=0.092). Amplitude (c) and phase (d) of the first pulse recovered from blind FROG (red dotted curve) and FROG (solid blue curve). Amplitude (e) and phase (f) of the second pulse recovered from blind FROG (red dotted curve) and FROG (solid blue curve).

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Equations (18)

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IFROGSHG(ω,Δt)=|∫−∞∞E(t)E(t−Δt)e−iωtdt|2,
Ij(ω)=|F[χj(t)]|2,
ψj(t)=Ej(t)Ej(t−s(j)Δt).
ΦjΩ(ω)=IS(j)Ω(ω)F[ψj(t)]|F[ψj(t)]|.
ΦjΩC(ω)=fγST(Re{F[ψj(t)]})+ifγST(Im{F[ψj(t)]}),
fγST(x)={0if  x<γx−γ sign(x)if  x≥γ,
ψj′(t)=F−1[Φj(ω)].
Ej+1(t)=Ej(t)+αEj*(t−s(j)Δt)|Ej(t−s(j)Δt)|max2(ψj′(t)−ψj(t)).
δ(E,E^)=arccos(|⟨E^(t)|E(t)⟩|⟨E^(t)|E^(t)⟩⟨E(t)|E(t)⟩),
η=#  of pixels in the incomplete trace#  of pixels in the complete trace.
E^j+1(ω)=I^(ω)F[Ej+1(t)]|F[Ej+1(t)]|,
E^j+1(t)=F−1[E^j+1(ω)].
Iblind  FROGSHG(ω,Δt)=|∫−∞∞P(t)G(t−Δt)e−iωtdt|2,
ψj(t)=Pj(t)Gj(t−s(j)Δt).
Pj+1(t)=Pj(t)+αGj*(t−s(j)Δt)|Gj(t−s(j)Δt)|max2(ψj′(t)−ψj(t))
Gj+1(t−s(j)Δt)=Gj(t)+αPj*(t−s(j)Δt)|Pj(t−s(j)Δt)|max2(ψj′(t)−ψj(t)).
G^j+1(ω)=I^G(ω)F[Gj+1(t)]|F[Gj+1(t)]|,
G^j+1(t)=F−1[G^j+1(ω)].

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