Abstract

Many ultrashort laser pulse measurement schemes such as frequency-resolved optical gating (FROG), interferometric FROG, dispersion scan, or time-domain ptychography require a specific, iterative algorithm to retrieve pulse amplitude and phase from the measurement. In this work, we present a common pulse retrieval algorithm (COPRA) that can be applied on a broad class of measurements, including but not limited to the aforementioned ones. We demonstrate its properties in comprehensive numerical tests and show that it is fast, reliable, and accurate in the presence of Gaussian noise. For FROG, we compare it to retrieval algorithms based on generalized projections and ptychography. Furthermore, we discuss the pulse retrieval problem as a nonlinear least squares problem and demonstrate the importance of obtaining a least squares solution for noisy data. In our tests, COPRA is shown to be faster and gives more accurate results in comparison to existing retrieval algorithms. Furthermore, it can be universally applied to measurements for which no specific retrieval algorithm was known before.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2018 (1)

2017 (7)

2016 (4)

2015 (2)

2014 (2)

2013 (1)

2012 (2)

2010 (1)

2009 (2)

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

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009).
[Crossref]

2008 (1)

A. Galler and T. Feurer, “Pulse shaper assisted short laser pulse characterization,” Appl. Phys. B 90, 427–430 (2008).
[Crossref]

2006 (1)

2005 (1)

2004 (2)

2003 (1)

2002 (1)

2001 (1)

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[Crossref]

1998 (2)

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[Crossref]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

1997 (1)

1994 (3)

1993 (1)

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[Crossref]

1987 (1)

1985 (1)

1984 (2)

D. Griffin and J. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoustics Speech Signal Process. 32, 236–243 (1984).
[Crossref]

A. Levi and H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[Crossref]

1970 (1)

R. Hegerl and W. Hoppe, “Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld,” Ber. Bunsen. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

Alonso, B.

Arnold, C.

Arnold, C. L.

Avnat, Z.

Barillot, T.

Beinert, R.

T. Bendory, R. Beinert, and Y. C. Eldar, “Fourier phase retrieval: uniqueness and algorithms,” in Compressed Sensing and its Applications, H. Boche, G. Caire, R. Calderbank, M. März, G. Kutyniok, and R. Mathar, eds. (Birkhäuser, 2017), pp. 55–91.

Bendory, T.

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

T. Bendory, R. Beinert, and Y. C. Eldar, “Fourier phase retrieval: uniqueness and algorithms,” in Compressed Sensing and its Applications, H. Boche, G. Caire, R. Calderbank, M. März, G. Kutyniok, and R. Mathar, eds. (Birkhäuser, 2017), pp. 55–91.

Brügmann, M.

Brügmann, M. H.

Canhota, M.

Chung, J.-H.

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[Crossref]

Clement, T. S.

Cohen, O.

Crespo, H.

Crespo, H. M.

Crozatier, V.

Cruz, J. M. D.

Dantus, M.

Davenport, J. J.

DeLong, K. W.

Diels, J.-C. M.

Dorrer, C.

Eldar, Y. C.

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

T. Bendory, R. Beinert, and Y. C. Eldar, “Fourier phase retrieval: uniqueness and algorithms,” in Compressed Sensing and its Applications, H. Boche, G. Caire, R. Calderbank, M. März, G. Kutyniok, and R. Mathar, eds. (Birkhäuser, 2017), pp. 55–91.

Escoto, E.

Feurer, T.

Fittinghoff, D. N.

Fontaine, J. J.

Fordell, T.

Forget, N.

Galatsanos, N. P.

Galler, A.

A. Galler and T. Feurer, “Pulse shaper assisted short laser pulse characterization,” Appl. Phys. B 90, 427–430 (2008).
[Crossref]

Gitzinger, G.

Greening, D.

Griffin, D.

D. Griffin and J. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoustics Speech Signal Process. 32, 236–243 (1984).
[Crossref]

Gunn, J. M.

Guo, C.

Habetler, G. J.

Harth, A.

Hegerl, R.

R. Hegerl and W. Hoppe, “Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld,” Ber. Bunsen. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

Heidt, A. M.

Hodgkinson, J.

Hoffmann, M.

Hoppe, W.

R. Hegerl and W. Hoppe, “Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld,” Ber. Bunsen. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

Hyyti, J.

J. Hyyti, E. Escoto, and G. Steinmeyer, “Pulse retrieval algorithm for interferometric frequency-resolved optical gating based on differential evolution,” Rev. Sci. Instrum. 88, 103102 (2017).
[Crossref]

J. Hyyti, E. Escoto, and G. Steinmeyer, “Third-harmonic interferometric frequency-resolved optical gating,” J. Opt. Soc. Am. B 34, 2367–2375 (2017).
[Crossref]

Iaconis, C.

Ippen, E. P.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer Berlin Heidelberg, 1977), pp. 83–122.

Johnson, D.

Jupé, M.

Kane, D. J.

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[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]

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[Crossref]

Keusters, D.

Kleinert, S.

Kohler, B.

L’Huillier, A.

Lahav, O.

Levi, A.

Li, P.

Lim, J.

D. Griffin and J. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoustics Speech Signal Process. 32, 236–243 (1984).
[Crossref]

Loriot, V.

Louisy, M.

Lozovoy, V. V.

Maiden, A.

Maiden, A. M.

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

Marangos, J. P.

Matia-Hernando, P.

McMichael, I. C.

Miranda, M.

Morgner, U.

Nagy, T.

Neoricic, L.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

O’Shea, P.

Ogilvie, J. P.

Oksenhendler, T.

Pastirk, I.

Penedones, J.

Ristau, D.

Rodenburg, J. M.

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

Rodriguez, G.

Rohwer, E.

Rohwer, E. G.

Saffell, J. R.

Seber, G.

G. Seber and C. Wild, Nonlinear Regression (Wiley, 2003).

Seifert, B.

Shank, C. V.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer Berlin Heidelberg, 1977), pp. 83–122.

Sidorenko, P.

Silva, F.

Simoni, F.

Spangenberg, D.

Spangenberg, D.-M.

Stark, H.

Steinmeyer, G.

Stibenz, G.

Stolz, H.

Tajalli, A.

Tan, H.-S.

Tarantola, A.

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).

Tasche, M.

Tatam, R. P.

Taylor, A. J.

Tisch, J. W. G.

Trebino, R.

Walke, D.

Walmsley, I. A.

Warren, W. S.

Weigand, R.

Weiner, A. M.

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[Crossref]

Wilcox, D. E.

Wild, C.

G. Seber and C. Wild, Nonlinear Regression (Wiley, 2003).

Willemsen, T.

Wilson, K.

Witting, T.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

Xu, B.

Yang, Y.

Yudilevich, E.

Zeek, E.

Adv. Opt. Photon. (1)

Appl. Opt. (3)

Appl. Phys. B (1)

A. Galler and T. Feurer, “Pulse shaper assisted short laser pulse characterization,” Appl. Phys. B 90, 427–430 (2008).
[Crossref]

Ber. Bunsen. Phys. Chem. (1)

R. Hegerl and W. Hoppe, “Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld,” Ber. Bunsen. Phys. Chem. 74, 1148–1154 (1970).
[Crossref]

IEEE J. Quantum Electron. (1)

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and the power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[Crossref]

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[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, 722–726 (2017).
[Crossref]

IEEE Trans. Acoustics Speech Signal Process. (1)

D. Griffin and J. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. Acoustics Speech Signal Process. 32, 236–243 (1984).
[Crossref]

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

J. Opt. Soc. Am. B (10)

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]

J. Hyyti, E. Escoto, and G. Steinmeyer, “Third-harmonic interferometric frequency-resolved optical gating,” J. Opt. Soc. Am. B 34, 2367–2375 (2017).
[Crossref]

M. Miranda, J. Penedones, C. Guo, A. Harth, M. Louisy, L. Neoričić, A. L’Huillier, and C. L. Arnold, “Fast iterative retrieval algorithm for ultrashort pulse characterization using dispersion scans,” J. Opt. Soc. Am. B 34, 190–197 (2017).
[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]

B. Xu, J. M. Gunn, J. M. D. Cruz, V. V. Lozovoy, and M. Dantus, “Quantitative investigation of the multiphoton intrapulse interference phase scan method for simultaneous phase measurement and compensation of femtosecond laser pulses,” J. Opt. Soc. Am. B 23, 750–759 (2006).
[Crossref]

N. Forget, V. Crozatier, and T. Oksenhendler, “Pulse-measurement techniques using a single amplitude and phase spectral shaper,” J. Opt. Soc. Am. B 27, 742–756 (2010).
[Crossref]

D. E. Wilcox and J. P. Ogilvie, “Comparison of pulse compression methods using only a pulse shaper,” J. Opt. Soc. Am. B 31, 1544–1554 (2014).
[Crossref]

E. Escoto, A. Tajalli, T. Nagy, and G. Steinmeyer, “Advanced phase retrieval for dispersion scan: a comparative study,” J. Opt. Soc. Am. B 35, 8–19 (2018).
[Crossref]

C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B 19, 1019–1029 (2002).
[Crossref]

D. Keusters, H.-S. Tan, P. O’Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20, 2226–2237 (2003).
[Crossref]

Opt. Express (5)

Opt. Lett. (8)

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

M. Canhota, F. Silva, R. Weigand, and H. M. Crespo, “Inline self-diffraction dispersion-scan of over octave-spanning pulses in the single-cycle regime,” Opt. Lett. 42, 3048–3051 (2017).
[Crossref]

S. Kleinert, A. Tajalli, T. Nagy, and U. Morgner, “Rapid phase retrieval of ultrashort pulses from dispersion scan traces using deep neural networks,” Opt. Lett. 44, 979–982 (2019).
[Crossref]

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[Crossref]

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Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. (a) Diagram of the discrete PNPS formalism. (b) First stage of COPRA: local iteration. (c) Second stage of COPRA: global iteration.
Fig. 2.
Fig. 2. (a) Synthetic SHG-FROG trace with N=M=64. (b) Trace error obtained by four different minimization algorithms plotted over the number of full evaluations of the trace T˜mn. Shown are the best out of 10 runs for every algorithm.
Fig. 3.
Fig. 3. Pulse retrieval from different PNPS measurements using COPRA. (a)–(d) Synthetic measurement traces with added Gaussian noise (σ=1%). (e)–(h) The retrieved pulses (blue, intensity; orange, phase) and the original pulse (in black). The retrieval error ε quantifies the retrieval accuracy. The test pulse has a time–bandwidth product of 2.
Fig. 4.
Fig. 4. Convergence behavior for σ=0%. (a) Comparison of algorithms, (b)–(d) comparison of PNPS methods when retrieving with COPRA. Shown is the median (bold line) and the interquartile range (shaded area) of the running minimum of R. Only the local iteration was used in COPRA.
Fig. 5.
Fig. 5. Convergence behavior for noisy measurements (σ=1%). (a) Comparison of algorithms; (b)–(d) comparison of PNPS methods when retrieving with COPRA (solid, local iteration; dashed, global iteration). Shown is the median (bold line) and the interquartile range (shaded area) of the running minimum of R.
Fig. 6.
Fig. 6. Robustness against additive Gaussian noise: (a) synthetic SHG-FROG trace of a pulse with TBP 2. A high level of Gaussian noise was added (σ=3%). (b)–(e) The pulses retrieved from the trace using different algorithms (blue, intensity; orange, phase) compared to the test pulse (black). ε quantifies the retrieval accuracy.
Fig. 7.
Fig. 7. Retrieval ratio (blue bar) and median retrieval error (inset number) of COPRA in dependence of the PNPS method and the noise level. A comparison with PCGPA and PIE for SHG-FROG is included. Successful retrieval was assumed if R<R0+1e4.
Fig. 8.
Fig. 8. Pulse retrieval from an incomplete THG-iFROG measurement. (a) Synthetic measurement trace with added noise (σ=1%). Twenty-five of N=256 spectral measurements were used for retrieval (indicated by color). (b) The retrieved pulse (blue, intensity; orange, phase) and the test pulse (black). ε quantifies the retrieval accuracy.

Tables (3)

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Table 1. Signal Operator for Selected Noncollinear Schemesa

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Table 2. Parametrization Filter for Selected Collinear Schemesa

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Table 3. Nonlinear Process Operators for Collinear Schemes

Equations (28)

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E˜(ω)=F[E](ω)=12πE(t)eiωtdt,
E(t)=F1[E˜](t)=E˜(ω)eitωdω.
T˜(δ,ω;E˜)=|F{Sδ[E˜](t)}(ω)|2.
Sδ[E˜](t)=N{F1[H˜δE˜]}(t).
tnt0+nΔt,n=0,,N1,
ωnω0+nΔω.
E˜n=FTkn(Ek)andEk=FTnk1(E˜n).
SmkSmk(E˜)Sδm[E˜](tk),m=0,,M1k=0,,N1.
S˜mnS˜mn(E˜)=FTkn(Smk),m=0,,M1n=0,,N1.
T˜mnT˜mn(E˜)=|S˜mn(E˜)|2T(δm,ωn;E˜).
rr(E˜)=m,n[T˜mnmeasμT˜mn(E˜)]2.
RR(E˜)=r1/2/[MN(maxm,nT˜mnmeas)2]1/2
μ=m,n[T˜mnmeasT˜mn(E˜)]/m,nT˜mn(E˜)2.
Smk=μ1/2FTnk1(S˜mn/|S˜mn|T˜meas).
Zm=k|SmkSmk(E˜)|2,
E˜n=E˜nγmjnZm.
γ=Zm/n|nZm|2.
gmj=max(gm1j,n|nZm|2)withg1j=0.
γmj=Zm/max(gmj,gM1j1),
Smk=Smkηrmkr,
ηr=α(r/lj|ljr|2),
mkr=4μΔt2πΔωFTnk1[(T˜mnmeasμT˜mn)S˜mn].
Z=mZm=mk|SmkSmk(E˜)|2,
nZ=mnZm.
E˜n=E˜nηznZ,
ηz=α(Z/k|kZ|2).
ε(E˜)[minρ,ϕ0,ϕ1n|E˜n0ρexp[i(ϕ0+ϕ1ω)]E˜n|2/(Nmaxn|E˜n0|2)]1/2.
R0R(E˜0).