Abstract

We recently introduced frequency-resolved optical gating (FROG), a technique for measuring the intensity and phase of an individual, arbitrary, ultrashort laser pulse. FROG can use almost any instantaneous optical nonlinearity, with the most common geometries being polarization gate, self-diffraction, and second-harmonic generation. The experimentally generated FROG trace is intuitive, visually appealing, and can yield quantitative information about the pulse parameters (such as temporal and spectral width and chirp). However, the qualitative and the quantitative features of the FROG trace depend strongly on the geometry used. We compare the FROG traces for several common ultrashort pulses for these three common geometries and, where possible, develop scaling rules that allow one to obtain quantitative information about the pulse directly from the experimental FROG trace. We illuminate the important features of the various FROG traces for transform-limited, linearly chirped, self-phase modulated, and nonlinearly chirped pulses, pulses with simultaneous linear chirp and self-phase modulation, and pulses with simultaneous linear chirp and cubic phase distortion, as well as double pulses, pulses with phase jumps, and pulses with complex intensity and phase substructure.

© 1994 Optical Society of America

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  1. J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, and K. W. Wecht, "Two-photon excitation of fluorescence by picosecond light pulses," Appl. Phys. Lett. 11, 216–218 (1967).
    [CrossRef]
  2. E. P. Ippen and C. V. Shank, Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
    [CrossRef]
  3. S. M. Saltiel, K. A. Sankov, P. D. Yankov, and L. I. Telegin, "Realization of a diffraction-grating autocorrelator for single-shot measurement of ultrashort light pulse duration," Appl. Phys. B 40, 25–27 (1986).
    [CrossRef]
  4. J.-C. M. Diels, J. J. Fontain, I. C. McMichael, and F. Simoni, "Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond pulses," Appl. Opt. 24, 1270–1282 (1985).
    [CrossRef]
  5. K. Naganuma, K. Mogi, and H. Yamada, "General method for ultrashort pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
    [CrossRef]
  6. J. L. A. Chilla and O. E. Martinez, "Direct determination of the amplitude and the phase of femtosecond light pulses," Opt. Lett. 16, 39–41 (1991).
    [CrossRef] [PubMed]
  7. D. Marcuse and J. M. Weisenfeld, "Chirped picosecond pulses: evaluation of the time-dependent wavelength for semiconductor film lasers," Appl. Opt. 23, 74–82 (1984).
    [CrossRef] [PubMed]
  8. J.-C. Diels, X. M. Zhao, and S. Diddams, "Capturing electromagnetic fields with fs resolution," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 120–130 (1993).
    [CrossRef]
  9. V. Wong, J. Koshel, M. Beck, and I. Walmsley, "Measurement of the amplitude and phase of pulses from passively modelocked lasers," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 137–148 (1993).
    [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. W. Koenig, H. K. Dunn, and L. Y. Lacy, "The sound spectrograph," J. Acoust. Soc. Am. 18, 19–49 (1946).
    [CrossRef]
  14. L. Cohen, "Time-frequency distribution," Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  15. R. A. Altes, "Detection, estimation, and classification with spectrograms," J. Acoust. Soc. Am. 67, 1232–1246 (1980).
    [CrossRef]
  16. 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]
  17. E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848–3858 (1971).
    [CrossRef]
  18. A. Freiberg and P. Saari, "Picosecond spectrochronography," IEEE J. Quantum Electron. QE-19, 622–630 (1983).
    [CrossRef]
  19. J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262–2273 (1992).
    [CrossRef]
  20. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, "Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation," Opt. Lett. 18, 1946–1948 (1993).
    [CrossRef] [PubMed]
  21. K. DeLong, R. Trebino, and W. E. White, "Frequency-resolved optical gating using second-harmonic generation," submitted to J. Opt. Soc. Am. B.
  22. D. J. Kane and R. Trebino, U.S. Patent Application Serial Number 07/966,644 (October 26, 1992).
  23. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28–30 (1992).
    [CrossRef] [PubMed]
  24. The FROG trace of this transform-limited pulse is an ellipse (or circle) oriented along either axis, depending on the width of the pulse. The trace will be circular when the pulse width τFW = 0.79√N for PG and SD FROG and τFW = 0.66√N for SHG FROG, where N is the number of elements in the fast Fourier transform used to generate the FROG trace. The fast Fourier transform relates the scaling of the two axes through δƒ = 1/(Nδτ) [W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992)], where δƒ and δτ are the increments of frequency and delay in the fast Fourier transform.
  25. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.
  26. K. W. DeLong and J. Yumoto, "Chirped light and its characterization using the cross-correlation technique," J. Opt. Soc. Am. B 9, 1593–1604 (1992).
    [CrossRef]
  27. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989).
  28. D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, "Single-shot measurement of the intensity and phase of femtosecond UV laser pulse," Opt. Lett. 8, 1061–1063 (1994).
    [CrossRef]
  29. R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, "Compression of optical pulses to six femtoseconds by using cubic phase compensation," Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
  30. C. P. J. Barty, B. E. Lemoff, and C. L. Gordon III, "Generation, measurement, and amplification of 20-fsec high-peak-power pulses from a regeneratively initiated self-modelocked Ti:sapphire laser," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 6–30 (1993).
    [CrossRef]
  31. M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, M. M. Murname, and H. C. Kapteyn, "Generation and amplification of sub-20-femtosecond pulses in Ti:sapphire," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 37–41 (1993).
    [CrossRef]
  32. W. E. White, F. G. Patterson, L. D. V. Woerkom, D. F. Price, and R. L. Shepherd, "Limitations on the fidelity of 100 fsec pulses produced by chirped-pulse amplification," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 55–60 (1993).
    [CrossRef]
  33. M. Stern, J. P. Heritage, and E. W. Chase, "Grating compensation of the third-order fiber dispersion," IEEE J. Quantum Electron. 28, 2742–2748 (1992).
    [CrossRef]
  34. J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).
  35. J. T. Fourkas, W. L. Wilson, G. Wackerle, A. E. Frost, and M. D. Fayer, "Picosecond time-scale phase-related optical pulses: measurement of sodium optical coherence decay by observation of incoherent fluorescence," J. Opt. Soc. Am. B 6, 1905–1910 (1989).
    [CrossRef]
  36. S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
    [CrossRef] [PubMed]
  37. B. Kohler, J. L. Krause, F. Raski, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, and Y. J. Lam, "Quantum control with tailored femtosecond pulses," presented at OE/LASE 94 (Society of Photo-Optical Instrument Engineers, Los Angeles, 1994), presentation 2116-55.

1994

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 femtosecond UV laser pulse," Opt. Lett. 8, 1061–1063 (1994).
[CrossRef]

1993

1992

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

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

M. Stern, J. P. Heritage, and E. W. Chase, "Grating compensation of the third-order fiber dispersion," IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[CrossRef]

S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
[CrossRef] [PubMed]

1991

1989

1987

1986

S. M. Saltiel, K. A. Sankov, P. D. Yankov, and L. I. Telegin, "Realization of a diffraction-grating autocorrelator for single-shot measurement of ultrashort light pulse duration," Appl. Phys. B 40, 25–27 (1986).
[CrossRef]

1985

1984

1983

A. Freiberg and P. Saari, "Picosecond spectrochronography," IEEE J. Quantum Electron. QE-19, 622–630 (1983).
[CrossRef]

1980

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

1971

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

1967

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

1946

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

Van Stryland, E. W.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989).

Altes, R. A.

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

Asaki, M. T.

M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, M. M. Murname, and H. C. Kapteyn, "Generation and amplification of sub-20-femtosecond pulses in Ti:sapphire," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 37–41 (1993).
[CrossRef]

Barty, C. P. J.

C. P. J. Barty, B. E. Lemoff, and C. L. Gordon III, "Generation, measurement, and amplification of 20-fsec high-peak-power pulses from a regeneratively initiated self-modelocked Ti:sapphire laser," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 6–30 (1993).
[CrossRef]

Beck, M.

V. Wong, J. Koshel, M. Beck, and I. Walmsley, "Measurement of the amplitude and phase of pulses from passively modelocked lasers," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 137–148 (1993).
[CrossRef]

Becker, P. C.

Chase, E. W.

M. Stern, J. P. Heritage, and E. W. Chase, "Grating compensation of the third-order fiber dispersion," IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

Chilla, J. L. A.

Cohen, L.

L. Cohen, "Time-frequency distribution," Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Cruz, C. H. Brito

DeLong, K.

K. DeLong, R. Trebino, and W. E. White, "Frequency-resolved optical gating using second-harmonic generation," submitted to J. Opt. Soc. Am. B.

DeLong, K. W.

DeSalvo, R.

Diddams, S.

J.-C. Diels, X. M. Zhao, and S. Diddams, "Capturing electromagnetic fields with fs resolution," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 120–130 (1993).
[CrossRef]

Diels, J.-C.

J.-C. Diels, X. M. Zhao, and S. Diddams, "Capturing electromagnetic fields with fs resolution," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 120–130 (1993).
[CrossRef]

Diels, J.-C. M.

Dunn, H. K.

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

Fayer, M. D.

Fontain, J. J.

Fork, R. L.

Fourkas, J. T.

Freiberg, A.

A. Freiberg and P. Saari, "Picosecond spectrochronography," IEEE J. Quantum Electron. QE-19, 622–630 (1983).
[CrossRef]

Frost, A. E.

Fujimoto, J. G.

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

J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).

Garvey, D.

M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, M. M. Murname, and H. C. Kapteyn, "Generation and amplification of sub-20-femtosecond pulses in Ti:sapphire," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 37–41 (1993).
[CrossRef]

Giordmaine, J. A.

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

Gordon III, C. L.

C. P. J. Barty, B. E. Lemoff, and C. L. Gordon III, "Generation, measurement, and amplification of 20-fsec high-peak-power pulses from a regeneratively initiated self-modelocked Ti:sapphire laser," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 6–30 (1993).
[CrossRef]

Hagan, D. J.

Haus, H. A.

J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).

Heritage, J. P.

M. Stern, J. P. Heritage, and E. W. Chase, "Grating compensation of the third-order fiber dispersion," IEEE J. Quantum Electron. 28, 2742–2748 (1992).
[CrossRef]

Huang, C.-P.

M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, M. M. Murname, and H. C. Kapteyn, "Generation and amplification of sub-20-femtosecond pulses in Ti:sapphire," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 37–41 (1993).
[CrossRef]

Ippen, E. P.

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

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

Jacobsen, A. G.

J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).

Jacobsen, J. M.

J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).

Kane, D. J.

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, "Single-shot measurement of the intensity and phase of femtosecond UV laser pulse," Opt. Lett. 8, 1061–1063 (1994).
[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]

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]

D. J. Kane and R. Trebino, U.S. Patent Application Serial Number 07/966,644 (October 26, 1992).

Kapteyn, H. C.

M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, M. M. Murname, and H. C. Kapteyn, "Generation and amplification of sub-20-femtosecond pulses in Ti:sapphire," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 37–41 (1993).
[CrossRef]

Koenig, W.

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

Kohler, B.

B. Kohler, J. L. Krause, F. Raski, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, and Y. J. Lam, "Quantum control with tailored femtosecond pulses," presented at OE/LASE 94 (Society of Photo-Optical Instrument Engineers, Los Angeles, 1994), presentation 2116-55.

Koshel, J.

V. Wong, J. Koshel, M. Beck, and I. Walmsley, "Measurement of the amplitude and phase of pulses from passively modelocked lasers," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 137–148 (1993).
[CrossRef]

Krause, J. L.

B. Kohler, J. L. Krause, F. Raski, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, and Y. J. Lam, "Quantum control with tailored femtosecond pulses," presented at OE/LASE 94 (Society of Photo-Optical Instrument Engineers, Los Angeles, 1994), presentation 2116-55.

Lacy, L. Y.

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

Lam, Y. J.

B. Kohler, J. L. Krause, F. Raski, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, and Y. J. Lam, "Quantum control with tailored femtosecond pulses," presented at OE/LASE 94 (Society of Photo-Optical Instrument Engineers, Los Angeles, 1994), presentation 2116-55.

Lemoff, B. E.

C. P. J. Barty, B. E. Lemoff, and C. L. Gordon III, "Generation, measurement, and amplification of 20-fsec high-peak-power pulses from a regeneratively initiated self-modelocked Ti:sapphire laser," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 6–30 (1993).
[CrossRef]

Marcuse, D.

Martinez, O. E.

McMichael, I. C.

Mogi, K.

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

Murname, M. M.

M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, M. M. Murname, and H. C. Kapteyn, "Generation and amplification of sub-20-femtosecond pulses in Ti:sapphire," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 37–41 (1993).
[CrossRef]

Naganuma, K

J. M. Jacobsen, K Naganuma, H. A. Haus, J. G. Fujimoto, and A. G. Jacobsen, "Femtosecond pulse generation in a Ti:AlO laser by using second- and third-order intracavity dispersion," Opt. Lett. 17, 1609–1610 (1992).

Naganuma, K.

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

Patterson, F. G.

W. E. White, F. G. Patterson, L. D. V. Woerkom, D. F. Price, and R. L. Shepherd, "Limitations on the fidelity of 100 fsec pulses produced by chirped-pulse amplification," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 55–60 (1993).
[CrossRef]

Paye, J.

Price, D. F.

W. E. White, F. G. Patterson, L. D. V. Woerkom, D. F. Price, and R. L. Shepherd, "Limitations on the fidelity of 100 fsec pulses produced by chirped-pulse amplification," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 55–60 (1993).
[CrossRef]

Ramaswamy, M.

Raski, F.

B. Kohler, J. L. Krause, F. Raski, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, and Y. J. Lam, "Quantum control with tailored femtosecond pulses," presented at OE/LASE 94 (Society of Photo-Optical Instrument Engineers, Los Angeles, 1994), presentation 2116-55.

Rentzepis, P. M.

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

Rice, S. A.

S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
[CrossRef] [PubMed]

Rose-Petruck, C.

B. Kohler, J. L. Krause, F. Raski, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, and Y. J. Lam, "Quantum control with tailored femtosecond pulses," presented at OE/LASE 94 (Society of Photo-Optical Instrument Engineers, Los Angeles, 1994), presentation 2116-55.

Saari, P.

A. Freiberg and P. Saari, "Picosecond spectrochronography," IEEE J. Quantum Electron. QE-19, 622–630 (1983).
[CrossRef]

Saltiel, S. M.

S. M. Saltiel, K. A. Sankov, P. D. Yankov, and L. I. Telegin, "Realization of a diffraction-grating autocorrelator for single-shot measurement of ultrashort light pulse duration," Appl. Phys. B 40, 25–27 (1986).
[CrossRef]

Sankov, K. A.

S. M. Saltiel, K. A. Sankov, P. D. Yankov, and L. I. Telegin, "Realization of a diffraction-grating autocorrelator for single-shot measurement of ultrashort light pulse duration," Appl. Phys. B 40, 25–27 (1986).
[CrossRef]

Shank, C. V.

R. L. Fork, C. H. Brito Cruz, P. C. Becker, and 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 and C. V. Shank, Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[CrossRef]

Shapiro, S. L.

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[CrossRef]

V. Wong, J. Koshel, M. Beck, and I. Walmsley, "Measurement of the amplitude and phase of pulses from passively modelocked lasers," in Ultrafast Pulse Generation and Spectroscopy, T. R. Gosnell and A. J. Taylor, eds., Soc. Photo-Opt. Instrum. Eng. 1861, 137–148 (1993).
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The FROG trace of this transform-limited pulse is an ellipse (or circle) oriented along either axis, depending on the width of the pulse. The trace will be circular when the pulse width τFW = 0.79√N for PG and SD FROG and τFW = 0.66√N for SHG FROG, where N is the number of elements in the fast Fourier transform used to generate the FROG trace. The fast Fourier transform relates the scaling of the two axes through δƒ = 1/(Nδτ) [W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992)], where δƒ and δτ are the increments of frequency and delay in the fast Fourier transform.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989).

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Figures (14)

Fig. 1
Fig. 1

Schematic of the various experimental geometries for generating FROG traces. The nonlinear mixing signal is spectrally resolved as a function of delay time between the two replicas of the beam to be measured. The parametric conversion geometries use two crystals with a second-order nonlinearity, cascaded to produce an effective third-order nonlinearity.

Fig. 2
Fig. 2

FROG traces for a Gaussian-shaped transform-limited pulse. The FROG trace for all three geometries are ellipses for this pulse. (a) The PG and the SD FROG traces are identical for this case. (b) The SHG FROG trace is wider in frequency but narrower in delay than the PG and the SD FROG traces. The frequency axis is labeled in linear, not angular, frequency units.

Fig. 3
Fig. 3

(a)–(c) FROG traces for a linearly chirped pulse. The linear chirp parameter was chosen to broaden the spectrum by a factor of 3 over the unchirped spectral width. The PG and the SD FROG traces are tilted ellipses, whereas the SHG FROG trace is untilted. The nearly circular form of the SHG FROG trace in this case is an accident of the pulse parameters and scaling of the figure; the trace is, in general, elliptical. (d) Instantaneous frequency as a function of time, f(t), and the FODM’s for PG and SD FROG (in SHG FROG the FODM is flat). The FODM essentially measures the slope of the FROG trace. The slope of the SD FROG trace is twice that of the PG FROG trace for a linearly chirped pulse.

Fig. 4
Fig. 4

(a)–(c) FROG traces for a self-phase modulated pulse with Q = 3. The PG and the SD FROG traces show the characteristic features of SPM, with lower frequencies generated in the rising edge of the pulse and higher frequencies in the falling edge of the pulse.

Fig. 5
Fig. 5

(a)–(c) FROG traces for a self-phase modulated pulse with Q = 8. All the traces exhibit considerable structure. (d) f(t) for the pulse, as well as the FODM for PG and SD FROG traces (the FODM for SHG FROG is flat in this case). Here we see how SD FROG accentuates the features of SPM. The spectrum, seen in the inset, has three peaks, a fact most clearly reflected in the PG FROG trace (a).

Fig. 6
Fig. 6

(a)–(c) FROG traces of a pulse with temporal cubic phase distortion. All the traces exhibit a crescent shape. (d) f(t) for the pulse, as well as the FODM’s of the FROG traces. The slope of the SD FROG FODM is negative (see text). The spectrum of this pulse (inset) shows a beating phenomenon, a fact most clearly reflected in the SHG FROG trace.

Fig. 7
Fig. 7

(a)–(c) The FROG traces of a pulse with spectral cubic phase distortion mirrors the group time delay. The SHG FROG trace is symmetrical about the τ axis, and thus the sign of the cubic phase is ambiguous. (d) The group delay as a function of frequency t(f), and the first-order frequency marginal of the FROG traces. The intensity as a function of time (inset) exhibits a beating phenomenon.

Fig. 8
Fig. 8

The PG FROG traces of two pulses of equal amplitude separated by times Δt of (a) 2 and (b) 4 pulse widths. As the pulses move apart, the frequency of the fringes in the f direction becomes higher. The separation of the two pulses is encoded not only in this fringe spacing but also in the separation of the two lower-intensity spots at ±Δt. The SD and the SHG FROG traces are similar.

Fig. 9
Fig. 9

PG FROG trace for two pulses of equal amplitude separated by 4 pulse widths and out of phase. The trace is similar to that of Fig. 8(b) except that the fringes along the zero time delay axis are also shifted out of phase. The phase shift of the fringes can be used to measure the relative phase separation of the two pulses.

Fig. 10
Fig. 10

(a)–(c) FROG traces of a pulse after it has undergone SPM and positive GVD in a fiber. The pulse had an amplitude of 7, and the fiber was 0.2 unit long, both measured in soliton units. The spectrum (not shown) of the pulse has a four-peaked structure. The SD FROG trace has twice the slope as the PG FROG trace, as in linear chirp, while the SHG FROG trace again has no slope. (d) PG FROG trace of the same pulse as (a) after the pulse has undergone negative GVD compensation. The pulse width is roughly one sixth of its original length. The flaring out seen at high and low frequencies is evidence of imperfect compression and leads to the well-known satellite pulses in the time domain (see text).

Fig. 11
Fig. 11

(a)–(c) FROG traces of a pulse with linear chirp and spectral cubic phase. The traces resemble the traces for pure spectral cubic phase but with an asymmetry induced by the presence of linear chirp. (d) Intensity and phase profiles of a pulse with a small amount of spectral cubic phase and with a combination of spectral cubic phase and linear chirp. The amount of spectral cubic phase is the same in both pulses. The addition of linear chirp amplifies the effect of the spectral cubic phase considerably.

Fig. 12
Fig. 12

PG FROG trace of a pulse with a phase jump of π at pulse center and a Gaussian intensity profile. The SD FROG trace of this pulse is identical.

Fig. 13
Fig. 13

(a) Pulse with a complicated intensity and phase profile. (b) PG FROG trace of the same pulse. The SD and the SHG FROG traces show similar levels of complexity.

Fig. 14
Fig. 14

(a) PG FROG trace of a pulse exhibiting positive linear chirp and temporal cubic phase. (b) Pulse intensity and phase derived from the FROG trace of (a), as well as an estimate (dashed curve) of the phase generated from a visual inspection of the trace. Agreement is good, considering the complicated structure of the pulse.

Equations (28)

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E sig PG ( t , τ ) = E ( t ) E ( t - τ ) 2 ,
E sig SD ( t , τ ) = E 2 ( t ) E * ( t - τ ) .
E sig SHG ( t , τ ) = E ( t ) E ( t - τ ) .
I FROG ( ω , τ ) = | - d t E sig ( t , τ ) exp ( i ω t ) | 2 .
S g [ ω , τ ; E ( t ) ] = | - d t E ( t ) g ( t - τ ) exp ( i ω t ) | 2 ,
M τ ( τ ) - d ω I FROG ( ω , τ ) ,
M τ PG ( τ ) = M τ SD ( - τ ) = - d t I ( t ) I 2 ( t - τ ) .
M τ SHG ( τ ) = - d t I ( t ) I ( t - τ ) ,
M ω ( ω ) - d τ I FROG ( ω , τ ) .
M ω PG ( ω ) = I ( ω ) * F { A ( 2 ) ( τ ) } ,
M ω SD ( ω ) = I ( - ω ) * F { E 2 ( t ) [ E * ( t ) ] 2 } ,
M ω SD ( ω ) = I ( - ω ) * I SH ( ω ) ,
M ω SHG ( ω ) = 2 I ( ω ) * I ( ω ) .
f ( t ) = - 1 2 π d ϕ ( t ) d t ,
f ¯ ( τ ) = 1 2 π - d ω ω I FROG ( ω , τ ) - d ω I FROG ( ω , τ ) .
Ω sig PG ( τ ) = ω ( 2 / 3 τ ) .
Ω sig SD ( τ ) = n = 0 [ 2 - ( - 2 ) n 3 n ] ω ( n ) ( 0 ) τ n n ! ,
Ω sig SD ( τ ) = ω ( 0 ) + ( 4 3 ) ω ( 1 ) ( 0 ) τ - ( 2 9 ) ω ( 2 ) ( 0 ) 2 τ 2 + ( 10 27 ) ω ( 3 ) ( 0 ) 6 τ 3 + ,
Ω sig SHG ( τ ) = n = 0 ω ( n ) ( 0 ) τ n n ! [ ( 1 2 ) n + ( - 1 2 ) n ] ,
Ω sig SD ( τ ) = 2 ω ( 0 ) + ( 1 2 ) ω ( 2 ) ( 0 ) 2 τ 2 + ( 1 8 ) ω ( 4 ) ( 0 ) 24 τ 4 + .
E ( t ) = exp ( - a t 2 ) ,
τ FW = [ 2 ln ( 2 ) a ] 1 / 2
f FWHM = 2 ln ( 2 ) π τ FW .
E ( t ) = exp [ ( - a + i b ) t 2 ] .
E ( t ) = exp [ - a t 2 + i Q E ( t ) 2 ] .
E ( t ) = exp ( - a t 2 + i c t 3 ) .
E ˜ ( ω ) = exp ( - ω 2 4 a + i d ω 3 ) .
E ( t ) = exp ( - a t 2 ) + B exp [ - a ( t - Δ t ) 2 + i ω 0 Δ t ] ,

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