Abstract

Frequency-resolved optical gating (FROG) is a technique that produces a spectrogram of an ultrashort laser pulse. The intensity and phase of the ultrashort laser pulse can be determined through solving for the phase of the spectrogram with an iterative, phase-retrieval algorithm. This work presents a new phase-retrieval algorithm that retrieves both the probe and the gate pulses independently by converting the FROG phase-retrieval problem to an eigenvector problem. The new algorithm is robust and general. It is tested theoretically by use of synthetic data sets and experimentally by use of single-shot, polarization-gate FROG. We independently and simultaneously characterize the electric field amplitude and phase of a pulse (probe) that was passed though 200 mm of BK7 glass and the amplitude of an unchanged pulse (gate) from an amplified Ti:sapphire laser. When the effect of the 200 mm of BK7 glass was removed mathematically from the probe, there was good agreement between the measured gate and the calculated, prechirped probe.

© 1997 Optical Society of America

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References

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  1. J.-C. Diels, J. J. Fontaine, and W. Rudolph, “Ultrafast diagnostics,” Revue Phys. Appl. 22, 1605–1611 (1987).
    [Crossref]
  2. V. Wong and I. A. Walmsley, “Linear filter analysis of methods for ultrashort-pulse-shape measurements,” J. Opt. Soc. Am. B 12, 1491–1499 (1995).
    [Crossref]
  3. E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
    [Crossref]
  4. J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991).
    [Crossref]
  5. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571 (1993).
    [Crossref]
  6. 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]
  7. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993).
    [Crossref] [PubMed]
  8. R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [Crossref]
  9. D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
    [Crossref] [PubMed]
  10. D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 19, 1061–1063 (1994).
    [Crossref] [PubMed]
  11. T. S. Clement, A. J. Taylor, and D. J. Kane, “Single-shot measurement of the amplitude and phase of ultrashort laser pulses in the violet,” Opt. Lett. 20, 70–72 (1995).
    [Crossref] [PubMed]
  12. K. W. DeLong, C. L. Ladera, R. Trebino, B. Kohler, and K. R. Wilson, “Ultrafast-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486–488 (1995).
    [Crossref] [PubMed]
  13. 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]
  14. K. W. DeLong and R. Trebino, “Improved ultrashort pulse-retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429–2437 (1994).
    [Crossref]
  15. 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 generalize projections,” Opt. Lett. 19, 2152–2154 (1994).
    [Crossref] [PubMed]
  16. E. Yudilevich, A. Levi, G. J. Habetler, and H. Stark, “Restoration of signals from their signed Fourier transform magnitude by the method of generalized projections,” J. Opt. Soc. Am. A 4, 236–246 (1987).
    [Crossref]
  17. Y. Yang, N. P. Galatsanos, and H. Stark, “Projection-based blind deconvolution,” J. Opt. Soc. Am. A 11, 2401–2409 (1994).
    [Crossref]
  18. A. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).
  19. W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. R. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  20. G. Rodriguez, J. P. Roberts, and A. J. Taylor, “Ultraviolet ultrafast pump-probe laser based on a Ti:sapphire laser system,” Opt. Lett. 19, 1146 (1994).
    [Crossref] [PubMed]
  21. D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort laser pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
    [Crossref]

1995 (5)

1994 (6)

1993 (4)

1991 (1)

J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[Crossref]

1987 (2)

1971 (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

Chilla, J. L. A.

J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[Crossref]

Clement, T. S.

DeLong, K. W.

Diels, J.-C.

J.-C. Diels, J. J. Fontaine, and W. Rudolph, “Ultrafast diagnostics,” Revue Phys. Appl. 22, 1605–1611 (1987).
[Crossref]

Fittinghoff, D. N.

Flannery, B. R.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. R. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Fontaine, J. J.

J.-C. Diels, J. J. Fontaine, and W. Rudolph, “Ultrafast diagnostics,” Revue Phys. Appl. 22, 1605–1611 (1987).
[Crossref]

Fujimoto, J. G.

Galatsanos, N. P.

Habetler, G. J.

Hunter, J.

Ippen, E.

Jain, A.

A. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

Kane, D. J.

Kohler, B.

Ladera, C. L.

Levi, A.

Martinez, O. E.

J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[Crossref]

Paye, J.

Press, W. H.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. R. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Ramaswamy, M.

Roberts, J. P.

Rodriguez, G.

Rudolph, W.

J.-C. Diels, J. J. Fontaine, and W. Rudolph, “Ultrafast diagnostics,” Revue Phys. Appl. 22, 1605–1611 (1987).
[Crossref]

Stark, H.

Taylor, A. J.

Teukolsky, S. A.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. R. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Treacy, E. B.

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

Trebino, R.

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

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]

D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort laser pulses,” J. Opt. Soc. Am. B 12, 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 generalize projections,” Opt. Lett. 19, 2152–2154 (1994).
[Crossref] [PubMed]

K. W. DeLong and R. Trebino, “Improved ultrashort pulse-retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429–2437 (1994).
[Crossref]

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 19, 1061–1063 (1994).
[Crossref] [PubMed]

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]

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

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
[Crossref] [PubMed]

R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[Crossref]

Vetterling, W. T.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. R. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Walmsley, I. A.

White, W. E.

Wilson, K.

Wilson, K. R.

Wong, V.

Yang, Y.

Yudilevich, E.

IEEE J. Quantum Electron. (2)

J. L. A. Chilla and O. E. Martinez, “Analysis of a method of phase measurement of ultrashort pulses in the frequency domain,” IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[Crossref]

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

J. Appl. Phys. (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

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

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

Opt. Lett. (7)

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993).
[Crossref] [PubMed]

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
[Crossref] [PubMed]

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 19, 1061–1063 (1994).
[Crossref] [PubMed]

T. S. Clement, A. J. Taylor, and D. J. Kane, “Single-shot measurement of the amplitude and phase of ultrashort laser pulses in the violet,” Opt. Lett. 20, 70–72 (1995).
[Crossref] [PubMed]

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

G. Rodriguez, J. P. Roberts, and A. J. Taylor, “Ultraviolet ultrafast pump-probe laser based on a Ti:sapphire laser system,” Opt. Lett. 19, 1146 (1994).
[Crossref] [PubMed]

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 generalize projections,” Opt. Lett. 19, 2152–2154 (1994).
[Crossref] [PubMed]

Revue Phys. Appl. (1)

J.-C. Diels, J. J. Fontaine, and W. Rudolph, “Ultrafast diagnostics,” Revue Phys. Appl. 22, 1605–1611 (1987).
[Crossref]

Other (2)

A. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. R. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the FROG PCGPA presented in this paper. P(t) and G(t) refer to EProbe and EGate, respectively.

Fig. 2
Fig. 2

(a) FROG trace of the double pulse used for the algorithm test. (b) Solid curve, the original probe intensity; ×'s, retrieved probe intensity. (c) Solid curve, gate intensity; ×'s, retrieved gate intensity. (d) Solid curve, frequency deviation of the original probe pulse; ×'s, frequency deviation of the retrieved probe pulse.

Fig. 3
Fig. 3

Semi-log plot showing the error history of the PCGPA when retrieving the probe and gate from the spectrogram shown in Fig. 2(a).

Fig. 4
Fig. 4

(a) TREEFROG trace formed from the probe, depicted in parts (b) and (d), and the gate, depicted in part (c). (b) Solid curve, original probe intensity; ×'s, retrieved probe intensity. (c) Solid curve, gate intensity; ×'s, retrieved gate intensity. (d) Solid curve, frequency deviation of the original probe pulse; ×'s, frequency deviation of the retrieved probe pulse.

Fig. 5
Fig. 5

Schematic of the modified FROG device used in this work. Prisms were used instead of turning mirrors to add dispersion to the probe. The pump (dotted line) is sent to a delay line and is sent below the probe. Both beams are focused with a cylindrical lens and are combined in a quartz window acting as the nonlinear medium. The interaction region is imaged onto the slits of an imaging spectrometer.

Fig. 6
Fig. 6

(a) TREEFROG trace obtained in this experiment. (b) TREEFROG trace reconstructed from the probe and gate extracted from the experimental TREEFROG trace shown in (a).

Fig. 7
Fig. 7

(a) Intensity profiles of the pulses. The calculated gate is how the probe pulse would appear before the added BK7 glass with the dispersion removed mathematically. (b) Phases of the retrieved probe pulse and the calculated gate pulse.

Fig. 8
Fig. 8

Results of the error analysis for the probe and gate shown in Fig. 4.

Equations (10)

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IFROG(ω, τ)=-E(t)|E(t-τ)|2 exp(-iωt)dt2,
Z=t,τ=1N|Esig(t, τ)-EProbe(t)EGate(t-τ)|2
EProbe=[E1, E2, E3, E4, , EN],
EGate=[G1, G2, G3, G4, , GN].
E1G1E1G2E1G3E1G4E1GNE2G1E2G2E2G3E2G4E2GNE3G1E3G2E3G3E3G4E3GNE4G1E4G2E4G3E4G4E4GN...............ENG1ENG2ENG3ENG4ENGN.
E1G1E1G2E1G3E1GN-2E1GN-1E1GNE2G2E2G3E2G4E2GN-1E2GNE2G1E3G3E3G4E3G5E3GNE3G1E3G2E4G4E4G5E4G6E4G1E4G2E4G3.....................ENGNENG1ENG2ENGN-3ENGN-2ENGN-1τ=0τ=-1τ=-2τ=+3τ=+2τ=+1.
A=U×W×VT,
TF1N2i=1Nj=1N[ITF(ωi, τj)-IMEASURED(ωi, τj)]21/2,
2=i, j=1N|EOuteri, j-EProbeiEGatej|2
INOISYij=ITFij(1+mij)+max(ITFij)aij,

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