Abstract

We use an algorithmic technique called “multi-grid” to improve the speed of convergence of the cross-correlation frequency-resolved-optical-gating (XFROG) pulse-retrieval algorithm for very complex pulses. The multi-grid approach uses a smaller trace (N/4 × N/4) drawn from the original N × N trace for initial iterations, yielding poorer resolution and range, but proceeding ~16 times faster for such iterations. The pulse field rapidly retrieved from this smaller array then provides the initial guess for the larger, full array, significantly reducing the number of iterations required on the full array. We first find that, for simple pulses and their resulting simple traces, the original generalized-projections FROG and XFROG algorithms already converge in less time than is required to plot the retrieved pulse, so speed improvements for them appear irrelevant in general. Considering therefore only complex pulses and their resulting complex traces, we adapted the multi-grid algorithm to XFROG, the technique used for complex pulses whenever possible. We show that extending multi-grid to even smaller arrays is not helpful, but intermediate-size arrays of N/2 × N/2 are, further reducing the number of iterations on the full array and further decreasing convergence time. We obtain a factor of ~7 improvement in speed for very complex pulses with time-bandwidth products of 50 to 90. This approach does not require modifications to the algorithm itself and so can be used in conjunction with essentially all FROG algorithms for improved speed. And it retains FROG’s ability to determine the pulse-shape stability in multi-shot measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Boston, 2002).
  2. K. W. Delong, C. L. Ladera, R. Trebino, B. Kohler, and K. R. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20(5), 486–488 (1995).
    [Crossref] [PubMed]
  3. H. G. S. Linden and J. Kuhl, “XFROG: new method for amplitude and phase characterization of ultraweak ultrashort pulses,” Phys. Status Solidi, B Basic Res. 206(1), 119–124 (1998).
    [Crossref]
  4. K. W. Delong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19(24), 2152–2154 (1994).
    [Crossref] [PubMed]
  5. T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the Uniqueness of FROG Methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
    [Crossref]
  6. D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B 12(10), 1955–1967 (1995).
    [Crossref]
  7. L. Xu, E. Zeek, and R. Trebino, “Simulations of frequency-resolved optical gating for measuring very complex pulses,” J. Opt. Soc. Am. B 25(6), A70–A80 (2008).
    [Crossref]
  8. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27(13), 1174–1176 (2002).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  15. P.-Y. Wu, H.-H. Lu, C.-Z. Weng, Y.-H. Chen, and S.-D. Yang, “Dispersion-corrected frequency-resolved optical gating,” Opt. Lett. 41(19), 4538–4541 (2016).
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  16. G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13(7), 2617–2626 (2005).
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  19. C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
    [Crossref]

2017 (2)

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the Uniqueness of FROG Methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and Multiple Pulsing Can Be Invisible to Ultrashort Pulse Measurement Techniques,” Appl. Sci. 7(1), 40 (2017).
[Crossref]

2016 (2)

2015 (1)

2013 (1)

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

2008 (1)

2007 (1)

2005 (1)

2002 (1)

1999 (2)

J. W. Nicholson, F. G. Omenetto, D. J. Funk, and A. J. Taylor, “Evolving FROGS: phase retrieval from frequency-resolved optical gating measurements by use of genetic algorithms,” Opt. Lett. 24(7), 490–492 (1999).
[Crossref] [PubMed]

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

1998 (2)

H. G. S. Linden and J. Kuhl, “XFROG: new method for amplitude and phase characterization of ultraweak ultrashort pulses,” Phys. Status Solidi, B Basic Res. 206(1), 119–124 (1998).
[Crossref]

M. J. Stimson, D. J. Ulness, J. C. Kirkwood, G. S. Boutis, and A. C. Albrecht, “Noisy-light correlation functions by frequency resolved optical gating,” J. Opt. Soc. Am. B 15(2), 505–514 (1998).
[Crossref]

1995 (2)

1994 (1)

Albrecht, A. C.

Avnat, Z.

Bendory, T.

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the Uniqueness of FROG Methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

Boutis, G. S.

Brügmann, M. H.

Chen, Y.-H.

Coen, S.

Cohen, O.

DeLong, K. W.

Dudley, J. M.

Eldar, Y. C.

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the Uniqueness of FROG Methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

Feurer, T.

Fittinghoff, D. N.

Funk, D. J.

Genty, G.

Gu, X.

Guang, Z.

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and Multiple Pulsing Can Be Invisible to Ultrashort Pulse Measurement Techniques,” Appl. Sci. 7(1), 40 (2017).
[Crossref]

Kimmel, M.

Kirkwood, J. C.

Kohler, B.

Kuhl, J.

H. G. S. Linden and J. Kuhl, “XFROG: new method for amplitude and phase characterization of ultraweak ultrashort pulses,” Phys. Status Solidi, B Basic Res. 206(1), 119–124 (1998).
[Crossref]

Ladera, C. L.

Lahav, O.

Linden, H. G. S.

H. G. S. Linden and J. Kuhl, “XFROG: new method for amplitude and phase characterization of ultraweak ultrashort pulses,” Phys. Status Solidi, B Basic Res. 206(1), 119–124 (1998).
[Crossref]

Lu, H.-H.

Nicholson, J. W.

O’Shea, P.

Omenetto, F. G.

J. W. Nicholson, F. G. Omenetto, D. J. Funk, and A. J. Taylor, “Evolving FROGS: phase retrieval from frequency-resolved optical gating measurements by use of genetic algorithms,” Opt. Lett. 24(7), 490–492 (1999).
[Crossref] [PubMed]

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Ratner, J.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

Rhodes, M.

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and Multiple Pulsing Can Be Invisible to Ultrashort Pulse Measurement Techniques,” Appl. Sci. 7(1), 40 (2017).
[Crossref]

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

Rohwer, E.

Shreenath, A. P.

Siders, C. W.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Siders, J. L. W.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Sidorenko, P.

Spangenberg, D.

Steinmeyer, G.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

G. Stibenz and G. Steinmeyer, “Interferometric frequency-resolved optical gating,” Opt. Express 13(7), 2617–2626 (2005).
[Crossref] [PubMed]

Stibenz, G.

Stimson, M. J.

Taylor, A. J.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

J. W. Nicholson, F. G. Omenetto, D. J. Funk, and A. J. Taylor, “Evolving FROGS: phase retrieval from frequency-resolved optical gating measurements by use of genetic algorithms,” Opt. Lett. 24(7), 490–492 (1999).
[Crossref] [PubMed]

Trebino, R.

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and Multiple Pulsing Can Be Invisible to Ultrashort Pulse Measurement Techniques,” Appl. Sci. 7(1), 40 (2017).
[Crossref]

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

L. Xu, E. Zeek, and R. Trebino, “Simulations of frequency-resolved optical gating for measuring very complex pulses,” J. Opt. Soc. Am. B 25(6), A70–A80 (2008).
[Crossref]

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27(13), 1174–1176 (2002).
[Crossref] [PubMed]

K. W. Delong, C. L. Ladera, R. Trebino, B. Kohler, and K. R. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20(5), 486–488 (1995).
[Crossref] [PubMed]

D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B 12(10), 1955–1967 (1995).
[Crossref]

K. W. Delong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19(24), 2152–2154 (1994).
[Crossref] [PubMed]

Ulness, D. J.

Weng, C.-Z.

Wilson, K.

Wilson, K. R.

Windeler, R. S.

Wu, P.-Y.

Xu, L.

Yang, S.-D.

Zeek, E.

Appl. Sci. (1)

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and Multiple Pulsing Can Be Invisible to Ultrashort Pulse Measurement Techniques,” Appl. Sci. 7(1), 40 (2017).
[Crossref]

IEEE J. Quantum Electron. (1)

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

IEEE Signal Process. Lett. (1)

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the Uniqueness of FROG Methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

J. Opt. Soc. Am. B (4)

Laser Photonics Rev. (1)

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (6)

Optica (1)

Phys. Status Solidi, B Basic Res. (1)

H. G. S. Linden and J. Kuhl, “XFROG: new method for amplitude and phase characterization of ultraweak ultrashort pulses,” Phys. Status Solidi, B Basic Res. 206(1), 119–124 (1998).
[Crossref]

Other (2)

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Boston, 2002).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, Boston, 1995).

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Figures (6)

Fig. 1
Fig. 1 Multi-grid algorithm used in GP XFROG. The parameter, l, corresponds to the reduced size array. In this work, l = 0 corresponds to an N×N trace; l = 1 corresponds to N/2×N/2, and l = 2 corresponds to N/4×N/4.
Fig. 2
Fig. 2 For implementation of the GP algorithm, the FROG trace should be “an island in a sea of zeros.” However, due to the uncertainty principle, the trace cannot go to exactly zero on the perimeters. Here, the orange and yellow colors represent the regions where Isig(ω,τ) is greater than 10−4, and 10−3 of the maximum value of the trace, respectively. Here each column corresponds to the trace of a different pulse. Each row represents N×N, N/2×N/2, N/4×N/4, and N/8×N/8 XFROG traces for that column’s pulse that would be used for the multi-grid algorithm. Left to right: 1st column, N = 256 (TBPrms = 10), 3rd column N = 512 (TBPrms = 30), and 5th column N = 1024 (TBPrms = 55). Notice the change in delay and frequency ranges in the smaller traces. The corresponding regions where data points are greater than 10−4 and 10−3 of the maximum of trace are shown for all the traces on the adjacent column. Thus, N/4 × N/4 traces are generally acceptable, but N/8 × N/8 traces are too cropped and hence usually unacceptable.
Fig. 3
Fig. 3 Comparison of the standard GP XFROG algorithm (a), the Siders, et al., multi-grid GP XFROG algorithm (which uses only N/4 × N/4 and N × N traces) (b), and our approach using N/4 × N/4, N/2 × N/2, and N × N traces on a 1024×1024 XFROG trace for a pulse with TBPrms = 55. Time on the horizontal axis corresponds to real time. Note that our multi-grid GP approach is almost ten times faster than the standard GP approach for this pulse. The parameters kl indicate the number of iterations on the N/2l × N/2l array. In other words, k0 corresponds to the full N × N array, k1 corresponds to the N/2 × N/2 array, and k2 corresponds to the coarsest N/4 × N/4 array. The same initial guess is used for the retrieval on N/4 × N/4 traces in (b) and (c). The difference in convergence behaviors is due to the fact that many more iterations are required for a given value of l when fewer array sizes are used. For example, when the N/2 × N/2 array is not used, as in (b), more iterations are required on both the coarsest and finest arrays. Note that the time per iteration is the same for all approaches and depends only on the value of l.
Fig. 4
Fig. 4 (a) The temporal intensity and phase of the actual theoretical pulse are shown by orange and cyan colors, respectively. The temporal intensity and phase retrieved from the N/2 × N/2 trace to be used for the initial guess for the full N × N (512×512) trace are represented by the dashed red and blue lines, respectively. Additional points have been interpolated for comparison with the full N × N array. (b) The retrieved electric field obtained from the full N × N trace with G error = 1.4 × 10−3. The temporal intensity and phase of the retrieved pulse (indistinguishable from the actual theoretical pulse, shown in (a)) are represented by red and blue curves, respectively. (c) The difference between the intensities (dark red) of the actual field and the field retrieved from the N/2 × N/2 trace and used as the initial guess for the retrieval on the fine-grid. (d) The difference between the phases (dark blue) of the actual field and the field retrieved from the N/2 × N/2 trace and used as the initial guess for the retrieval on the fine-grid. Note the significant discrepancies in (c) and (d), confirming the need for at least a few iterations on the full N × N trace.
Fig. 5
Fig. 5 Average number of iterations on a set of 50 pulses with (a) TBP = 20 and a 256 × 256 trace, (b) TBP = 30 and a 512 × 512 trace, and (c) TBP = 55 and a 1024 × 1024 trace. kl corresponds to average number of iterations for the array of size reduced by the factor 2l. kMG-GP and kGP correspond to equivalent number of iterations of kls in terms of iterations on the fine grid, and average number of iterations required in GP XFROG, respectively.
Fig. 6
Fig. 6 The average retrieval times for sets of 50 pulses in the presence of noise. The improvement in speed is the same whether or not 1% additive and multiplicative noise is present.

Equations (2)

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I FROG (ω,τ) | E sig (t,τ)exp(iωt)dt | 2 ,
N= 1 δτδν =ΔτΔν

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