Abstract

We recently introduced a new technique, frequency-resolved optical gating (FROG), for directly determining the full intensity I(t) and phase φ(t) of a single femtosecond pulse. By using almost any instantaneous nonlinear-optical interaction of two replicas of the ultrashort pulse to be measured, FROG involves measuring the spectrum of the signal pulse as a function of the delay between the replicas. The resulting trace of intensity versus frequency and delay yields an intuitive display of the pulse that is similar to the pulse spectrogram, except that the gate is a function of the pulse to be measured. The problem of inverting the FROG trace to obtain the pulse intensity and phase can also be considered a complex two-dimensional phase-retrieval problem. As a result, the FROG trace yields, in principle, an essentially unique pulse intensity and phase. We show that this is also the case in practice. We present an iterative-Fourier-transform algorithm for inverting the FROG trace. The algorithm is unusual in its use of a novel constraint: the mathematical form of the signal field. Without the use of a support constraint, the algorithm performs quite well in practice, even for pulses with serious phase distortions and for experimental data with noise, although it occasionally stagnates when pulses with large intensity fluctuations are used.

© 1993 Optical Society of America

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  1. R. L. Fork, C. H. Brito-Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
  2. H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
    [CrossRef]
  3. J. H. Glownia, J. A. Misewich, P. P. Sorokin, “Femtosecond transition-state absorption spectroscopy of Bi atoms produced by photodissociation of gaseous Bi2molecules,”J. Chem. Phys. 92, 3335–3347 (1990).
    [CrossRef]
  4. E. P. Ippen, C. V. Shank, in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
    [CrossRef]
  5. J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
    [CrossRef]
  6. N. G. Basov, V. É. Pozhar, V. I. Pustovoit, “Measurement of the duration of high-power ultrashort optical pulses,” So. J. Quantum Electron. 15, 1429–1431 (1985).
    [CrossRef]
  7. J.-C. M. Diels, J. J. Fontaine, I. C. McMichael, F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond pulses,” Appl. Opt. 24, 1270–1282 (1985).
    [CrossRef] [PubMed]
  8. R. Trebino, C. C. Hayden, A. M. Johnson, W. M. Simpson, A. M. Levine, “Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses,” Opt. Lett. 15, 1079–1081 (1990).
    [CrossRef] [PubMed]
  9. J. T. Manassah, “Direct and second-harmonic interferometric determination of chirped pulse parameters,” Appl. Opt. 26, 2941–2942 (1987).
    [CrossRef]
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    [CrossRef]
  11. K. Naganuma, K. Mogi, H. Yamada, “General method for ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
    [CrossRef]
  12. T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
    [CrossRef]
  13. K. Naganuma, K. Mogi, H. Yamada, “Time direction determination of asymmetric ultrashort optical pulses from second-harmonic generation autocorrelation signals,” Appl. Phys. Lett. 54, 1201–1202 (1989).
    [CrossRef]
  14. J. Jansky, G. Corradi, “Full intensity profile analysis of ultrashort laser pulses using four-wave mixing or third-harmonic generation,” Opt. Commun. 60, 251–256 (1986).
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  15. N. G. Paulter, A. K. Majumdar, “A new triple correlator design for the measurement of ultrashort pulses,” Opt. Commun. 81, 96–100 (1991).
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  16. A. S. L. Gomes, V. L. da Silva, J. R. Taylor, “Direct measurement of nonlinear frequency chirp of Raman radiation in single-mode optical fibers using a spectral window method,” J. Opt. Soc. Am. B 5, 373–379 (1988).
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  18. J. E. Rothenberg, D. Grischkowsky, “Measurement of optical phase with subpicosecond resolution by time-domain interferometry,” Opt. Lett. 12, 99–101 (1987).
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  19. T. F. Albrecht, K. Seibert, H. Kurz, “Chirp measurement of large-bandwidth femtosecond optical pulses using two-photon absorption,” Opt. Commun. 84, 223–227 (1991).
    [CrossRef]
  20. K. W. DeLong, J. Yumoto, “Chirped light and its characterization using the cross-correlation technique,” J. Opt. Soc. Am. B 9, 1593–1604 (1992).
    [CrossRef]
  21. J. L. A. Chilla, O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991).
    [CrossRef] [PubMed]
  22. J. L. A. Chilla, O. E. Martinez, “Frequency-domain phase measurement of ultrashort light pulses. Effect of noise,” Opt. Commun. 89, 434–440 (1992).
    [CrossRef]
  23. J. L. A. Chilla, 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]
  24. A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).
  25. E. P. Ippen, C. V. Shank, “Dynamic spectroscopy and subpicosecond pulse compression,” Appl. Phys. Lett. 27, 488–490 (1975).
    [CrossRef]
  26. J. P. Heritage, A. M. Weiner, R. N. Thurston, “Fourier-transform picosecond pulse shaping and spectral-phase measurement in a grating pulse compressor,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 34–37.
    [CrossRef]
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    [CrossRef] [PubMed]
  28. J. P. Foing, L. P. Likforman, M. Joffre, “Femtosecond pulse phase measurement by spectrally resolved up-conversion,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).
  29. D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” J. Quantum Electron. 29, 571–579 (1993). This paper also contains the proof of uniqueness of the FROG-trace inversion, but for the nonlinear-optical interaction of self-diffraction.
    [CrossRef]
  30. D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond pulse,” in Ultrafast Phenomena IX, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).
  31. D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary femtosecond pulse using frequency-resolved optical gating,” Opt. Lett. (to be published).
  32. W. Koenig, H. K. Dunn, L. Y. Lacy, “The sound spectrograph,”J. Acoust. Soc. Am. 18, 19–49 (1946).
    [CrossRef]
  33. S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
    [CrossRef]
  34. R. A. Altes, “Detection, estimation, and classification with spectrograms,”J. Acoust. Soc. Am. 67, 1232–1246 (1980).
    [CrossRef]
  35. L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  36. We note here, as in Ref. 29, that the spectrogram is not considered to be the ideal time- and frequency-resolved quantity. The Wigner distribution and wavelet transform, for example, are much more popular at the moment. [See, e.g., C. E. Heil, D. F. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).] In our opinion, however, it is the spectrogram, or more precisely the FROG trace, that is the most easily experimentally measured time-and frequency-resolved quantity for ultrashort pulses. The above-mentioned alternative measures require such quantities as a time-reversed replica of the pulse or a variable-frequency and simultaneously variable-length window. Such quantities are quite difficult to obtain optically. On the other hand, the spectrogram’s main inadequacy is that its fixed-length window cannot simultaneously provide good measurements of both long-term and short-term variations. This is not a problem for ultrashort-pulse measurement, in which the variations of interest are all short term. It may also be the case that, because in FROG the gate has exactly those time scales as the pulse, FROG may actually avoid the above problem in general, but we have not investigated this issue as yet.
    [CrossRef]
  37. E. J. Akutowicz, “On the determination of the phase of a Fourier signal. I,” Trans. Am. Math. Soc. 83, 234–239 (1956).
  38. H. Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).
  39. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  40. R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
    [CrossRef]
  41. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
    [CrossRef]
  42. D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
    [CrossRef]
  43. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  44. J. H. Seldin, J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
    [CrossRef]
  45. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  46. R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
    [CrossRef]
  47. R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
    [CrossRef]
  48. A. M. Johnson, C. V. Shank, “Pulse compression in single-mode fibers—picoseconds to femtoseconds,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989). pp. 399–450.
    [CrossRef]
  49. Y. R. Shen, G.-Z. Yang, “Theory of self-phase modulation and spectral broadening,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), pp. 1–32.
    [CrossRef]

1993 (1)

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” J. Quantum Electron. 29, 571–579 (1993). This paper also contains the proof of uniqueness of the FROG-trace inversion, but for the nonlinear-optical interaction of self-diffraction.
[CrossRef]

1992 (3)

J. L. A. Chilla, O. E. Martinez, “Frequency-domain phase measurement of ultrashort light pulses. Effect of noise,” Opt. Commun. 89, 434–440 (1992).
[CrossRef]

A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).

K. W. DeLong, J. Yumoto, “Chirped light and its characterization using the cross-correlation technique,” J. Opt. Soc. Am. B 9, 1593–1604 (1992).
[CrossRef]

1991 (7)

C. Yan, J. C. Diels, “Amplitude and phase recording of ultrashort pulses,” J. Opt. Soc. Am. B 8, 1259–1263 (1991).
[CrossRef]

J. L. A. Chilla, O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991).
[CrossRef] [PubMed]

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

J. L. A. Chilla, 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]

N. G. Paulter, A. K. Majumdar, “A new triple correlator design for the measurement of ultrashort pulses,” Opt. Commun. 81, 96–100 (1991).
[CrossRef]

G. Szabo, A. Müller, Z. Bor, “A sensitive single-shot method to determine duration and chirp of ultrashort pulses with a streak camera,” Opt. Commun. 82, 56–62 (1991).
[CrossRef]

T. F. Albrecht, K. Seibert, H. Kurz, “Chirp measurement of large-bandwidth femtosecond optical pulses using two-photon absorption,” Opt. Commun. 84, 223–227 (1991).
[CrossRef]

1990 (5)

1989 (5)

H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, “General method for ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, “Time direction determination of asymmetric ultrashort optical pulses from second-harmonic generation autocorrelation signals,” Appl. Phys. Lett. 54, 1201–1202 (1989).
[CrossRef]

L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

We note here, as in Ref. 29, that the spectrogram is not considered to be the ideal time- and frequency-resolved quantity. The Wigner distribution and wavelet transform, for example, are much more popular at the moment. [See, e.g., C. E. Heil, D. F. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).] In our opinion, however, it is the spectrogram, or more precisely the FROG trace, that is the most easily experimentally measured time-and frequency-resolved quantity for ultrashort pulses. The above-mentioned alternative measures require such quantities as a time-reversed replica of the pulse or a variable-frequency and simultaneously variable-length window. Such quantities are quite difficult to obtain optically. On the other hand, the spectrogram’s main inadequacy is that its fixed-length window cannot simultaneously provide good measurements of both long-term and short-term variations. This is not a problem for ultrashort-pulse measurement, in which the variations of interest are all short term. It may also be the case that, because in FROG the gate has exactly those time scales as the pulse, FROG may actually avoid the above problem in general, but we have not investigated this issue as yet.
[CrossRef]

1988 (1)

1987 (6)

1986 (1)

J. Jansky, G. Corradi, “Full intensity profile analysis of ultrashort laser pulses using four-wave mixing or third-harmonic generation,” Opt. Commun. 60, 251–256 (1986).
[CrossRef]

1985 (2)

N. G. Basov, V. É. Pozhar, V. I. Pustovoit, “Measurement of the duration of high-power ultrashort optical pulses,” So. J. Quantum Electron. 15, 1429–1431 (1985).
[CrossRef]

J.-C. M. Diels, J. J. Fontaine, I. C. McMichael, F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond pulses,” Appl. Opt. 24, 1270–1282 (1985).
[CrossRef] [PubMed]

1984 (1)

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

1983 (1)

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

1982 (1)

1980 (1)

R. A. Altes, “Detection, estimation, and classification with spectrograms,”J. Acoust. Soc. Am. 67, 1232–1246 (1980).
[CrossRef]

1975 (1)

E. P. Ippen, C. V. Shank, “Dynamic spectroscopy and subpicosecond pulse compression,” Appl. Phys. Lett. 27, 488–490 (1975).
[CrossRef]

1967 (1)

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
[CrossRef]

1956 (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier signal. I,” Trans. Am. Math. Soc. 83, 234–239 (1956).

1946 (1)

W. Koenig, H. K. Dunn, L. Y. Lacy, “The sound spectrograph,”J. Acoust. Soc. Am. 18, 19–49 (1946).
[CrossRef]

Akutowicz, E. J.

E. J. Akutowicz, “On the determination of the phase of a Fourier signal. I,” Trans. Am. Math. Soc. 83, 234–239 (1956).

Albrecht, T. F.

T. F. Albrecht, K. Seibert, H. Kurz, “Chirp measurement of large-bandwidth femtosecond optical pulses using two-photon absorption,” Opt. Commun. 84, 223–227 (1991).
[CrossRef]

Altes, R. A.

R. A. Altes, “Detection, estimation, and classification with spectrograms,”J. Acoust. Soc. Am. 67, 1232–1246 (1980).
[CrossRef]

Barakat, R.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Basov, N. G.

N. G. Basov, V. É. Pozhar, V. I. Pustovoit, “Measurement of the duration of high-power ultrashort optical pulses,” So. J. Quantum Electron. 15, 1429–1431 (1985).
[CrossRef]

Bates, R. H. T.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Becker, P. C.

H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
[CrossRef]

R. L. Fork, C. H. Brito-Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

Bigot, J.-Y.

H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
[CrossRef]

Bor, Z.

G. Szabo, A. Müller, Z. Bor, “A sensitive single-shot method to determine duration and chirp of ultrashort pulses with a streak camera,” Opt. Commun. 82, 56–62 (1991).
[CrossRef]

Brito-Cruz, C. H.

Brun, A.

A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).

Chilla, J. L. A.

J. L. A. Chilla, O. E. Martinez, “Frequency-domain phase measurement of ultrashort light pulses. Effect of noise,” Opt. Commun. 89, 434–440 (1992).
[CrossRef]

J. L. A. Chilla, 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]

J. L. A. Chilla, O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991).
[CrossRef] [PubMed]

Cho, Y.

T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
[CrossRef]

Cohen, L.

L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Corradi, G.

J. Jansky, G. Corradi, “Full intensity profile analysis of ultrashort laser pulses using four-wave mixing or third-harmonic generation,” Opt. Commun. 60, 251–256 (1986).
[CrossRef]

da Silva, V. L.

DeLong, K. W.

Diels, J. C.

Diels, J.-C. M.

Dunn, H. K.

W. Koenig, H. K. Dunn, L. Y. Lacy, “The sound spectrograph,”J. Acoust. Soc. Am. 18, 19–49 (1946).
[CrossRef]

Fienup, J. R.

Foing, J. P.

J. P. Foing, L. P. Likforman, M. Joffre, “Femtosecond pulse phase measurement by spectrally resolved up-conversion,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

Fontaine, J. J.

Fork, R. L.

Fragnito, H. L.

H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
[CrossRef]

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

Georges, P.

A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).

Giordmaine, J. A.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
[CrossRef]

Glownia, J. H.

J. H. Glownia, J. A. Misewich, P. P. Sorokin, “Femtosecond transition-state absorption spectroscopy of Bi atoms produced by photodissociation of gaseous Bi2molecules,”J. Chem. Phys. 92, 3335–3347 (1990).
[CrossRef]

Gomes, A. S. L.

Grischkowsky, D.

Guo, F.-C

T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
[CrossRef]

Hayden, C. C.

Heil, C. E.

We note here, as in Ref. 29, that the spectrogram is not considered to be the ideal time- and frequency-resolved quantity. The Wigner distribution and wavelet transform, for example, are much more popular at the moment. [See, e.g., C. E. Heil, D. F. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).] In our opinion, however, it is the spectrogram, or more precisely the FROG trace, that is the most easily experimentally measured time-and frequency-resolved quantity for ultrashort pulses. The above-mentioned alternative measures require such quantities as a time-reversed replica of the pulse or a variable-frequency and simultaneously variable-length window. Such quantities are quite difficult to obtain optically. On the other hand, the spectrogram’s main inadequacy is that its fixed-length window cannot simultaneously provide good measurements of both long-term and short-term variations. This is not a problem for ultrashort-pulse measurement, in which the variations of interest are all short term. It may also be the case that, because in FROG the gate has exactly those time scales as the pulse, FROG may actually avoid the above problem in general, but we have not investigated this issue as yet.
[CrossRef]

Heritage, J. P.

J. P. Heritage, A. M. Weiner, R. N. Thurston, “Fourier-transform picosecond pulse shaping and spectral-phase measurement in a grating pulse compressor,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 34–37.
[CrossRef]

Ippen, E. P.

E. P. Ippen, C. V. Shank, “Dynamic spectroscopy and subpicosecond pulse compression,” Appl. Phys. Lett. 27, 488–490 (1975).
[CrossRef]

E. P. Ippen, C. V. Shank, in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[CrossRef]

Israelevitz, D.

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

Jansky, J.

J. Jansky, G. Corradi, “Full intensity profile analysis of ultrashort laser pulses using four-wave mixing or third-harmonic generation,” Opt. Commun. 60, 251–256 (1986).
[CrossRef]

Joffre, M.

J. P. Foing, L. P. Likforman, M. Joffre, “Femtosecond pulse phase measurement by spectrally resolved up-conversion,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

Johnson, A. M.

R. Trebino, C. C. Hayden, A. M. Johnson, W. M. Simpson, A. M. Levine, “Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses,” Opt. Lett. 15, 1079–1081 (1990).
[CrossRef] [PubMed]

A. M. Johnson, C. V. Shank, “Pulse compression in single-mode fibers—picoseconds to femtoseconds,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989). pp. 399–450.
[CrossRef]

Kane, D. J.

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” J. Quantum Electron. 29, 571–579 (1993). This paper also contains the proof of uniqueness of the FROG-trace inversion, but for the nonlinear-optical interaction of self-diffraction.
[CrossRef]

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond pulse,” in Ultrafast Phenomena IX, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary femtosecond pulse using frequency-resolved optical gating,” Opt. Lett. (to be published).

Kobayashi, T.

T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
[CrossRef]

Koenig, W.

W. Koenig, H. K. Dunn, L. Y. Lacy, “The sound spectrograph,”J. Acoust. Soc. Am. 18, 19–49 (1946).
[CrossRef]

Kurz, H.

T. F. Albrecht, K. Seibert, H. Kurz, “Chirp measurement of large-bandwidth femtosecond optical pulses using two-photon absorption,” Opt. Commun. 84, 223–227 (1991).
[CrossRef]

Lacy, L. Y.

W. Koenig, H. K. Dunn, L. Y. Lacy, “The sound spectrograph,”J. Acoust. Soc. Am. 18, 19–49 (1946).
[CrossRef]

Lane, R. G.

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

Le Saux, G.

A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).

Leaird, D. E.

Levine, A. M.

Likforman, L. P.

J. P. Foing, L. P. Likforman, M. Joffre, “Femtosecond pulse phase measurement by spectrally resolved up-conversion,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

Lim, J. S.

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

Majumdar, A. K.

N. G. Paulter, A. K. Majumdar, “A new triple correlator design for the measurement of ultrashort pulses,” Opt. Commun. 81, 96–100 (1991).
[CrossRef]

Manassah, J. T.

Martinez, O. E.

J. L. A. Chilla, O. E. Martinez, “Frequency-domain phase measurement of ultrashort light pulses. Effect of noise,” Opt. Commun. 89, 434–440 (1992).
[CrossRef]

J. L. A. Chilla, 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]

J. L. A. Chilla, O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991).
[CrossRef] [PubMed]

McMichael, I. C.

Millane, R. P.

Misewich, J. A.

J. H. Glownia, J. A. Misewich, P. P. Sorokin, “Femtosecond transition-state absorption spectroscopy of Bi atoms produced by photodissociation of gaseous Bi2molecules,”J. Chem. Phys. 92, 3335–3347 (1990).
[CrossRef]

Mogi, K.

K. Naganuma, K. Mogi, H. Yamada, “General method for ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, “Time direction determination of asymmetric ultrashort optical pulses from second-harmonic generation autocorrelation signals,” Appl. Phys. Lett. 54, 1201–1202 (1989).
[CrossRef]

Morimoto, A.

T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
[CrossRef]

Müller, A.

G. Szabo, A. Müller, Z. Bor, “A sensitive single-shot method to determine duration and chirp of ultrashort pulses with a streak camera,” Opt. Commun. 82, 56–62 (1991).
[CrossRef]

Naganuma, K.

K. Naganuma, K. Mogi, H. Yamada, “General method for ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, “Time direction determination of asymmetric ultrashort optical pulses from second-harmonic generation autocorrelation signals,” Appl. Phys. Lett. 54, 1201–1202 (1989).
[CrossRef]

Nawab, S. H.

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

Newsam, G.

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Patel, J. S.

Paulter, N. G.

N. G. Paulter, A. K. Majumdar, “A new triple correlator design for the measurement of ultrashort pulses,” Opt. Commun. 81, 96–100 (1991).
[CrossRef]

Pozhar, V. É.

N. G. Basov, V. É. Pozhar, V. I. Pustovoit, “Measurement of the duration of high-power ultrashort optical pulses,” So. J. Quantum Electron. 15, 1429–1431 (1985).
[CrossRef]

Pustovoit, V. I.

N. G. Basov, V. É. Pozhar, V. I. Pustovoit, “Measurement of the duration of high-power ultrashort optical pulses,” So. J. Quantum Electron. 15, 1429–1431 (1985).
[CrossRef]

Quatieri, T. F.

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

Rentzepis, P. M.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
[CrossRef]

Rothenberg, J. E.

Salin, F.

A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).

Seibert, K.

T. F. Albrecht, K. Seibert, H. Kurz, “Chirp measurement of large-bandwidth femtosecond optical pulses using two-photon absorption,” Opt. Commun. 84, 223–227 (1991).
[CrossRef]

Seldin, J. H.

Shank, C. V.

H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
[CrossRef]

R. L. Fork, C. H. Brito-Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

E. P. Ippen, C. V. Shank, “Dynamic spectroscopy and subpicosecond pulse compression,” Appl. Phys. Lett. 27, 488–490 (1975).
[CrossRef]

E. P. Ippen, C. V. Shank, in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[CrossRef]

A. M. Johnson, C. V. Shank, “Pulse compression in single-mode fibers—picoseconds to femtoseconds,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989). pp. 399–450.
[CrossRef]

Shapiro, S. L.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
[CrossRef]

Shen, Y. R.

Y. R. Shen, G.-Z. Yang, “Theory of self-phase modulation and spectral broadening,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), pp. 1–32.
[CrossRef]

Simoni, F.

Simpson, W. M.

Sorokin, P. P.

J. H. Glownia, J. A. Misewich, P. P. Sorokin, “Femtosecond transition-state absorption spectroscopy of Bi atoms produced by photodissociation of gaseous Bi2molecules,”J. Chem. Phys. 92, 3335–3347 (1990).
[CrossRef]

Stark, H.

H. Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).

Sueta, T.

T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
[CrossRef]

Szabo, G.

G. Szabo, A. Müller, Z. Bor, “A sensitive single-shot method to determine duration and chirp of ultrashort pulses with a streak camera,” Opt. Commun. 82, 56–62 (1991).
[CrossRef]

Tan, D. G. H.

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Taylor, J. R.

Thurston, R. N.

J. P. Heritage, A. M. Weiner, R. N. Thurston, “Fourier-transform picosecond pulse shaping and spectral-phase measurement in a grating pulse compressor,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 34–37.
[CrossRef]

Trebino, R.

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” J. Quantum Electron. 29, 571–579 (1993). This paper also contains the proof of uniqueness of the FROG-trace inversion, but for the nonlinear-optical interaction of self-diffraction.
[CrossRef]

R. Trebino, C. C. Hayden, A. M. Johnson, W. M. Simpson, A. M. Levine, “Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses,” Opt. Lett. 15, 1079–1081 (1990).
[CrossRef] [PubMed]

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond pulse,” in Ultrafast Phenomena IX, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary femtosecond pulse using frequency-resolved optical gating,” Opt. Lett. (to be published).

Walnut, D. F.

We note here, as in Ref. 29, that the spectrogram is not considered to be the ideal time- and frequency-resolved quantity. The Wigner distribution and wavelet transform, for example, are much more popular at the moment. [See, e.g., C. E. Heil, D. F. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).] In our opinion, however, it is the spectrogram, or more precisely the FROG trace, that is the most easily experimentally measured time-and frequency-resolved quantity for ultrashort pulses. The above-mentioned alternative measures require such quantities as a time-reversed replica of the pulse or a variable-frequency and simultaneously variable-length window. Such quantities are quite difficult to obtain optically. On the other hand, the spectrogram’s main inadequacy is that its fixed-length window cannot simultaneously provide good measurements of both long-term and short-term variations. This is not a problem for ultrashort-pulse measurement, in which the variations of interest are all short term. It may also be the case that, because in FROG the gate has exactly those time scales as the pulse, FROG may actually avoid the above problem in general, but we have not investigated this issue as yet.
[CrossRef]

Wecht, K. W.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
[CrossRef]

Weiner, A. M.

A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multi-element liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990).
[CrossRef] [PubMed]

J. P. Heritage, A. M. Weiner, R. N. Thurston, “Fourier-transform picosecond pulse shaping and spectral-phase measurement in a grating pulse compressor,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 34–37.
[CrossRef]

Wullert, J. R.

Yamada, H.

K. Naganuma, K. Mogi, H. Yamada, “General method for ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, “Time direction determination of asymmetric ultrashort optical pulses from second-harmonic generation autocorrelation signals,” Appl. Phys. Lett. 54, 1201–1202 (1989).
[CrossRef]

Yan, C.

Yang, G.-Z.

Y. R. Shen, G.-Z. Yang, “Theory of self-phase modulation and spectral broadening,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), pp. 1–32.
[CrossRef]

Yumoto, J.

Appl. Opt. (3)

Appl. Phys. Lett. (3)

E. P. Ippen, C. V. Shank, “Dynamic spectroscopy and subpicosecond pulse compression,” Appl. Phys. Lett. 27, 488–490 (1975).
[CrossRef]

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
[CrossRef]

K. Naganuma, K. Mogi, H. Yamada, “Time direction determination of asymmetric ultrashort optical pulses from second-harmonic generation autocorrelation signals,” Appl. Phys. Lett. 54, 1201–1202 (1989).
[CrossRef]

Chem. Phys. Lett. (1)

H. L. Fragnito, J.-Y. Bigot, P. C. Becker, C. V. Shank, “Evolution of the vibronic absorption spectrum in a molecule following impulsive excitation with a six fsec optical pulse,” Chem. Phys. Lett. 160, 101–104 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Naganuma, K. Mogi, H. Yamada, “General method for ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

J. L. A. Chilla, 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]

IEEE Trans. Acoust. Speech Signal Process. (3)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 520–525 (1987).
[CrossRef]

D. Israelevitz, J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 511–519 (1987).
[CrossRef]

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

J. Acoust. Soc. Am. (2)

R. A. Altes, “Detection, estimation, and classification with spectrograms,”J. Acoust. Soc. Am. 67, 1232–1246 (1980).
[CrossRef]

W. Koenig, H. K. Dunn, L. Y. Lacy, “The sound spectrograph,”J. Acoust. Soc. Am. 18, 19–49 (1946).
[CrossRef]

J. Chem. Phys. (1)

J. H. Glownia, J. A. Misewich, P. P. Sorokin, “Femtosecond transition-state absorption spectroscopy of Bi atoms produced by photodissociation of gaseous Bi2molecules,”J. Chem. Phys. 92, 3335–3347 (1990).
[CrossRef]

J. Math. Phys. (1)

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

J. Mod. Opt. (1)

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (3)

J. Quantum Electron. (1)

D. J. Kane, R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” J. Quantum Electron. 29, 571–579 (1993). This paper also contains the proof of uniqueness of the FROG-trace inversion, but for the nonlinear-optical interaction of self-diffraction.
[CrossRef]

Opt. Commun. (5)

J. Jansky, G. Corradi, “Full intensity profile analysis of ultrashort laser pulses using four-wave mixing or third-harmonic generation,” Opt. Commun. 60, 251–256 (1986).
[CrossRef]

N. G. Paulter, A. K. Majumdar, “A new triple correlator design for the measurement of ultrashort pulses,” Opt. Commun. 81, 96–100 (1991).
[CrossRef]

G. Szabo, A. Müller, Z. Bor, “A sensitive single-shot method to determine duration and chirp of ultrashort pulses with a streak camera,” Opt. Commun. 82, 56–62 (1991).
[CrossRef]

T. F. Albrecht, K. Seibert, H. Kurz, “Chirp measurement of large-bandwidth femtosecond optical pulses using two-photon absorption,” Opt. Commun. 84, 223–227 (1991).
[CrossRef]

J. L. A. Chilla, O. E. Martinez, “Frequency-domain phase measurement of ultrashort light pulses. Effect of noise,” Opt. Commun. 89, 434–440 (1992).
[CrossRef]

Opt. Lett. (5)

Proc. IEEE (1)

L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Brun, P. Georges, G. Le Saux, F. Salin, “Single-shot characterization of ultrashort light pulses,” Rev. Sci. Instrum. 27, 1225–1233 (1992).

SIAM Rev. (1)

We note here, as in Ref. 29, that the spectrogram is not considered to be the ideal time- and frequency-resolved quantity. The Wigner distribution and wavelet transform, for example, are much more popular at the moment. [See, e.g., C. E. Heil, D. F. Walnut, “Continuous and discrete wavelet transforms,” SIAM Rev. 31, 628–666 (1989).] In our opinion, however, it is the spectrogram, or more precisely the FROG trace, that is the most easily experimentally measured time-and frequency-resolved quantity for ultrashort pulses. The above-mentioned alternative measures require such quantities as a time-reversed replica of the pulse or a variable-frequency and simultaneously variable-length window. Such quantities are quite difficult to obtain optically. On the other hand, the spectrogram’s main inadequacy is that its fixed-length window cannot simultaneously provide good measurements of both long-term and short-term variations. This is not a problem for ultrashort-pulse measurement, in which the variations of interest are all short term. It may also be the case that, because in FROG the gate has exactly those time scales as the pulse, FROG may actually avoid the above problem in general, but we have not investigated this issue as yet.
[CrossRef]

So. J. Quantum Electron. (1)

N. G. Basov, V. É. Pozhar, V. I. Pustovoit, “Measurement of the duration of high-power ultrashort optical pulses,” So. J. Quantum Electron. 15, 1429–1431 (1985).
[CrossRef]

Trans. Am. Math. Soc. (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier signal. I,” Trans. Am. Math. Soc. 83, 234–239 (1956).

Other (10)

H. Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond pulse,” in Ultrafast Phenomena IX, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary femtosecond pulse using frequency-resolved optical gating,” Opt. Lett. (to be published).

T. Kobayashi, F.-C Guo, A. Morimoto, T. Sueta, Y. Cho, “Novel method of waveform evaluation of ultrashort optical pulses,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 93–95.
[CrossRef]

E. P. Ippen, C. V. Shank, in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[CrossRef]

J. P. Foing, L. P. Likforman, M. Joffre, “Femtosecond pulse phase measurement by spectrally resolved up-conversion,” in Ultrafast Phenomena VIII, J. L. Martin, A. Migus, eds. (Springer-Verlag, to be published).

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

A. M. Johnson, C. V. Shank, “Pulse compression in single-mode fibers—picoseconds to femtoseconds,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989). pp. 399–450.
[CrossRef]

Y. R. Shen, G.-Z. Yang, “Theory of self-phase modulation and spectral broadening,” in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), pp. 1–32.
[CrossRef]

J. P. Heritage, A. M. Weiner, R. N. Thurston, “Fourier-transform picosecond pulse shaping and spectral-phase measurement in a grating pulse compressor,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 34–37.
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

(a) FROG involves splitting the pulse and overlapping the two resulting pulse replicas E(t) and E(tτ) in an instantaneously responding nonlinear-optical medium. In this polarization-gating arrangement the probe pulse E(t) passes through crossed polarizers and is gated at the nonlinear-optical sample medium by the gate pulse E(tτ). The signal pulse is then spectrally resolved, and its intensity is measured versus frequency ω and delay τ. A single-shot beam geometry is shown here: each replica of the pulse is focused with a cylindrical lens to a line in the sample medium. A large beam angle is used so that delay varies spatially across the sample. The line focus at the sample medium is then imaged onto the entrance slit of the spectrometer with a spherical lens. Thus delay varies along the slit, and, after dispersion by the spectrometer, frequency varies along the dimension perpendicular to the slit. The FROG trace is the output intensity versus wavelength and delay as seen by the camera. (b) Schematic of the polarization-optical-gate FROG interaction of two Gaussian pulses. The signal pulse is also Gaussian and is centered at the time 2τ/3. Because the signal pulse’s phase is contributed entirely by E(t) [E(tτ) appears as the squared magnitude], the signal pulse will reflect the instantaneous frequency of E(t) at the time 2τ/3. For complex phase dependences, when the instantaneous frequency is an inappropriate description of the pulse phase, the signal pulse reflects the short-time spectrum of the probe pulse.

Fig. 2
Fig. 2

FROG traces for negatively chirped, unchirped, and positively chirped Gaussian pulses. The top figures show the instantaneous frequency versus delay curves for the pulses. Note that the FROG trace reflects the instantaneous frequency of the pulse in all cases. Here the FROG traces are shown as density plots, with black indicating high intensity and white indicating low intensity.

Fig. 3
Fig. 3

Theoretical instantaneous frequencies versus time, spectra, and FROG traces for self-phase-modulated pulses: (a) a weakly self-phase-modulated pulse (Q = 3), (b) a strongly self-phase-modulated pulse (Q = 8). Note that for the weakly self-phase-modulated the FROG trace visually displays the pulse’s instantaneous frequency versus time. For the strongly self-phase-modulated pulse the pulse trace also visually displays the pulse’s instantaneous frequency versus time if the mean wavelength is computed for each delay. FROG The additional structure in the latter trace indicates the breakup of the spectrum because of self-phase modulation (SPM). Spectrograms of such pulses are similar to these traces when similar-length windows are used.

Fig. 4
Fig. 4

Iterative-Fourier-transform algorithm for inverting a FROG trace to obtain an ultrashort pulse’s intensity and phase.

Fig. 5
Fig. 5

Results of the algorithm after ten iterations for a squarish pulse: the exact pulse intensity, the initial guess intensity, and the derived pulse intensity.

Fig. 6
Fig. 6

Results of the algorithm after 20 iterations for a linearly chirped Gaussian pulse: the exact pulse phase, the initial guess phase, and the derived pulse phase.

Fig. 7
Fig. 7

Results of the algorithm for a Gaussian-intensity pulse with a trailing satellite pulse each with some SPM (Q = 2.5): (a) the exact pulse intensity and the derived pulse intensity, (b) the exact pulse phase and the derived pulse phase.

Fig. 8
Fig. 8

Results of the algorithm for a severely self-phase-modulated Gaussian-intensity pulse (Q = 8): (a) the exact pulse intensity and the derived pulse intensity, (b) the exact pulse phase and the derived pulse phase.

Fig. 9
Fig. 9

Comparison of the two measures of error for the severely self-phase-modulated pulse. Note that the FROG-trace rms error decreases steadily, whereas the E-field rms error appears to stagnate. This occurs because the derived E field is slightly translated in time compared with the exact E field (see Fig. 8). In all the cases in which such discrepancies occurred we verified that the cause is this translation. In general, because the FROG trace essentially uniquely determines the E field, the FROG trace error is a more reliable measure of error than is the E-field error. (The discontinuities in the E-field error between iterations 20 and 30 are also the result of temporal translations.)

Fig. 10
Fig. 10

Plot of the FROG-trace error for the iteration for the pulse with satellite and SPM.

Fig. 11
Fig. 11

Plot of FROG-trace rms error versus iteration number for the severely self-phase-modulated Gaussian-intensity pulse for various initial guesses. Observe that noise yields the fastest convergence.

Fig. 12
Fig. 12

Experimental single-shot FROG trace for a linearly chirped pulse.

Fig. 13
Fig. 13

Derived intensity evolution for the pulse whose FROG trace is shown in Fig. 12.

Fig. 14
Fig. 14

Derived phase evolution for the pulse whose FROG trace is shown in Fig. 12. The inverted parabolic shape of the phase evolution indicates positive chirp, i.e., linearly increasing frequency versus time [ω(t) = −dφ/dt]. (Phase behavior for large positive and large negative times is indeterminate because the intensity at these times is zero.)

Fig. 15
Fig. 15

Experimental and derived third-order intensity autocorrelation for the pulse of Figs. 12 and 14.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

E sig ( t , τ ) E ( t ) E ( t - τ ) 2 .
I FROG ( ω , τ ) | - E sig ( t , τ ) exp ( - i ω t ) d t | 2 .
S E ( ω , τ ) | - E ( t ) g ( t - τ ) exp ( - i ω t ) d t | 2
I FROG ( ω , τ ) | - - E sig ( t , Ω ) exp ( - i ω t - i Ω τ ) d t d Ω | 2 ,
E ( t ) - E sig ( t , τ ) d τ ,
E ( k + 1 ) ( t ) - E sig ( k ) ( t , τ ) d τ .
E sig ( k + 1 ) ( t , τ ) E ( k + 1 ) ( t ) E ( k + 1 ) ( t - τ ) 2 .
E ( k ) [ 1 N j = 1 N E ( k ) ( t j ) - E ( t j ) 2 ] 1 / 2 ,
FROG ( k ) { 1 N 2 i = 1 N j = 1 N [ I FROG ( k ) ( ω i , τ j ) - I FROG ( ω i , τ j ) ] 2 } 1 / 2 ,
E ( k + 1 ) ( t ) = β [ C - E sig ( k ) ( t , τ ) d τ ] + ( 1 - β ) E ( k ) ( t ) ,
I FROG SHG ( ω , τ ) = | - E ( t ) E ( t - τ ) exp ( - i ω t ) d t | 2 .
I FROG SHG ( ω , τ ) = | - E ( t ) E ( t + τ ) exp ( - i ω t ) d t | 2 .

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