Abstract

We discuss the use of second-harmonic generation (SHG) as the nonlinearity in the technique of frequency-resolved optical gating (FROG) for measuring the full intensity and phase evolution of an arbitrary ultrashort pulse. FROG that uses a third-order nonlinearity in the polarization-gate geometry has proved extremely successful, and the algorithm required for extraction of the intensity and the phase from the experimental data is quite robust. However, for pulse intensities less than ~1 MW, third-order nonlinearities generate insufficient signal strength, and therefore SHG FROG appears necessary. We discuss the theoretical, algorithmic, and experimental considerations of SHG FROG in detail. SHG FROG has an ambiguity in the direction of time, and its traces are somewhat unintuitive. Also, previously published algorithms are generally ineffective at extracting the intensity and the phase of an arbitrary laser pulse from the SHG FROG trace. We present an improved pulse-retrieval algorithm, based on the method of generalized projections, that is far superior to the previously published algorithms, although it is still not so robust as the polarization-gate algorithm. We discuss experimental sources of error such as pump depletion and group-velocity mismatch. We also present several experimental examples of pulses measured with SHG FROG and show that the derived intensities and phases are in agreement with more conventional diagnostic techniques, and we demonstrate the high-dynamic-range capability of SHG FROG. We conclude that, despite the above drawbacks, SHG FROG should be useful in measuring low-energy pulses.

© 1994 Optical Society of America

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References

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  1. E. P. Ippen and C. V. Shank, "Techniques for measurement," in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
    [CrossRef]
  2. 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]
  3. S. M. Saltiel, K. A. Stankov, 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. Fontaine, I. C. McMichael, and F. Simoni, "Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy," Appl. Opt. 24, 1270–1282 (1985).
    [CrossRef] [PubMed]
  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. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041–2043 (1993).
    [CrossRef] [PubMed]
  8. V. Wong and I. A. Walmsley, "Analysis of ultrashort pulse-shape measurements using linear interferometers," Opt. Lett. 19, 287–289 (1994).
    [CrossRef] [PubMed]
  9. V. Wong and I. Walmsley, "Pulse-shape measurement using linear interferometers," in Generation, Amplification Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 254–267 (1994).
    [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. K. W. DeLong, R. Trebino, and D. J. Kane, "Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries," J. Opt. Soc. Am. B 11, 1595–1608 (1994).
    [CrossRef]
  15. D. J. Kane and R. Trebino, U.S. patent application 07/966,644 (October 26, 1992).
  16. 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]
  17. 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]
  18. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.
  19. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  20. A. Levi and H. Stark, "Restoration from phase and magnitude by generalized projections," in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 277–320.
  21. W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992), pp. 420–425.
  22. F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics (Wiley, New York, 1986), Vol. II.
  23. A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. 19, 1276–1283 (1983).
    [CrossRef]
  24. Schott Computer Glass Catalog 1.0 (Schott Glasswerke, Mainz, Germany, 1992).
  25. Y. Ishida, K. Naganuma, and T. Yajima, IEEE J. Quantum Electron. QE-21, 69–77 (1985).
    [CrossRef]

1994 (4)

1993 (5)

1991 (1)

1989 (1)

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

1986 (1)

S. M. Saltiel, K. A. Stankov, 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 (1)

1983 (1)

A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. 19, 1276–1283 (1983).
[CrossRef]

1982 (1)

1967 (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]

Beck, M.

Chilla, J. L. A.

DeLong, K. W.

Diels, J.-C. M.

Fienup, J. R.

Fontaine, J. J.

Fujimoto, J. G.

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]

Hopf, F. A.

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics (Wiley, New York, 1986), Vol. II.

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, "Techniques for measurement," in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[CrossRef]

Ishida, Y.

Y. Ishida, K. Naganuma, and T. Yajima, IEEE J. Quantum Electron. QE-21, 69–77 (1985).
[CrossRef]

Kane, D. J.

Levi, A.

A. Levi and H. Stark, "Restoration from phase and magnitude by generalized projections," in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 277–320.

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]

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]

Y. Ishida, K. Naganuma, and T. Yajima, IEEE J. Quantum Electron. QE-21, 69–77 (1985).
[CrossRef]

Paye, J.

Press, W. H.

W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992), pp. 420–425.

Ramaswamy, M.

Raymer, M. G.

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]

Saltiel, S. M.

S. M. Saltiel, K. A. Stankov, 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.

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

Shapiro, S. L.

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]

Siegman, A. E.

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

Simoni, F.

Stankov, K. A.

S. M. Saltiel, K. A. Stankov, 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]

Stark, H.

A. Levi and H. Stark, "Restoration from phase and magnitude by generalized projections," in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 277–320.

Stegeman, G. I.

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics (Wiley, New York, 1986), Vol. II.

Taylor, A. J.

Telegin, L. I.

S. M. Saltiel, K. A. Stankov, 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]

Teukolsky, S. A.

W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992), pp. 420–425.

Trebino, R.

Vetterling, W. T.

W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992), pp. 420–425.

Walmsley, I.

V. Wong and I. Walmsley, "Pulse-shape measurement using linear interferometers," in Generation, Amplification Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 254–267 (1994).
[CrossRef]

Walmsley, I. A.

Wecht, K. W.

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]

Weiner, A. M.

A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. 19, 1276–1283 (1983).
[CrossRef]

Wong, V.

V. Wong and I. A. Walmsley, "Analysis of ultrashort pulse-shape measurements using linear interferometers," Opt. Lett. 19, 287–289 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

V. Wong and I. Walmsley, "Pulse-shape measurement using linear interferometers," in Generation, Amplification Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 254–267 (1994).
[CrossRef]

Yajima, T.

Y. Ishida, K. Naganuma, and T. Yajima, IEEE J. Quantum Electron. QE-21, 69–77 (1985).
[CrossRef]

Yamada, H.

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

Yankov, P. D.

S. M. Saltiel, K. A. Stankov, 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]

Appl. Opt. (2)

Appl. Phys. B (1)

S. M. Saltiel, K. A. Stankov, 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]

Appl. Phys. Lett. (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]

IEEE J. Quantum Electron. (3)

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

A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. 19, 1276–1283 (1983).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Lett. (6)

Other (9)

Schott Computer Glass Catalog 1.0 (Schott Glasswerke, Mainz, Germany, 1992).

Y. Ishida, K. Naganuma, and T. Yajima, IEEE J. Quantum Electron. QE-21, 69–77 (1985).
[CrossRef]

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

D. J. Kane and R. Trebino, U.S. patent application 07/966,644 (October 26, 1992).

A. Levi and H. Stark, "Restoration from phase and magnitude by generalized projections," in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 277–320.

W. H. Press, W. T. Vetterling, and S. A. Teukolsky, Numerical Recipes in C: Second Edition (Cambridge U. Press, Cambridge, 1992), pp. 420–425.

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics (Wiley, New York, 1986), Vol. II.

V. Wong and I. Walmsley, "Pulse-shape measurement using linear interferometers," in Generation, Amplification Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2116, 254–267 (1994).
[CrossRef]

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

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Figures (13)

Fig. 1
Fig. 1

SHG FROG experimental setup. BS, Beam splitter.

Fig. 2
Fig. 2

Illustration of the method of projections. In a space consisting of all possible signal fields the fields that satisfy the constraints of Eqs. (1) and (2) form closed sets. The correct solution to the FROG problem is the intersection of these two sets. In the method of projections we close in on the correct solution as we iteratively move from the surface of one set to that of the other.

Fig. 3
Fig. 3

Schematic of the FROG pulse-retrieval algorithm.

Fig. 4
Fig. 4

SHG FROG trace of a laser pulse from a Ti:sapphire oscillator. This trace consists of 39 individual spectra of the autocorrelation of the pulse, each taken at a different time delay.

Fig. 5
Fig. 5

Intensity (solid curve with circles) and phase (dashed curve with diamonds) of the pulse retrieved from the FROG trace of Fig. 3. The pulse has a 90-fs FWHM. The final FROG error for this retrieved pulse was G = 0.00156.

Fig. 6
Fig. 6

Measured autocorrelation (solid curve) of the same laser that created the FROG trace of Fig. 3 and numerical autocorrelation (circles) of the retrieved pulse of Fig. 4. The agreement between the laboratory measurement and the pulse retrieved through FROG is excellent.

Fig. 7
Fig. 7

Comparison of the experimentally measured spectrum (solid curve) of the laser that created the FROG trace of Fig. 3 with the numerically generated spectrum (circles) of the retrieved pulse of Fig. 4. The agreement between the laboratory measurement and the pulse retrieved through FROG is excellent. The phase of the spectrum (dashed curve with diamonds) as retrieved from the FROG pulse-retrieval algorithm is also shown.

Fig. 8
Fig. 8

SHG FROG trace of a pulse from a Ti:sapphire oscillator with excess glass in the cavity. The horseshoe characteristic is indicative of spectral cubic phase distortion.

Fig. 9
Fig. 9

Electric-field intensity (solid curve with circles) and phase (dashed curve with diamonds) associated with the SHG FROG trace of Fig. 7. The π-phase-shifted satellite pulse is indicative of spectral cubic phase distortion. The convergence parameter was G = 0.00350.

Fig. 10
Fig. 10

Spectral intensity (solid curve with circles) and phase (dashed curve with diamonds) of the pulse of Fig. 8. Here we see clearly the cubic dependence of the phase in the spectral domain.

Fig. 11
Fig. 11

Intensity (solid curve with circles) and phase (dashed curve with diamonds) pulse from a Ti:sapphire laser retrieved through SHG FROG (G = 0.00312) before propagation through BK7 glass. The pulse, centered near 750 nm, is 90 fs wide (FWHM).

Fig. 12
Fig. 12

Intensity (solid curve with circles) and phase (dashed curve with diamonds) of the pulse of Fig. 11 after propagation through 6.5 cm of BK7 glass. The pulse has broadened to 179 fs and has acquired a substantial quadratic phase. The width and the phase curvature of this pulse (G = 0.00270) are extremely close to predictions from theory (see the text).

Fig. 13
Fig. 13

SHG FROG can achieve a large dynamic range in the retrieved pulse. Here we see the (a) temporal and the (b) spectral intensities of a pulse retrieved with SHG FROG (G = 0.00108) plotted on a logarithmic scale. The dynamic range of the retrieved pulse is limited by the dynamic range of the data, which had a maximum of 30,226 photoelectron counts and a noise background of 1–3 counts.

Tables (1)

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Table 1 Comparison of the Various SHG FROG Algorithmsa

Equations (12)

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E sig ( t , τ ) = E ( t ) E ( t - τ ) ,
I FROG ( ω , τ ) = | - d t E sig ( t , τ ) exp ( i ω t ) | 2 = E sig ( ω , τ ) 2 .
E ( t ) = exp ( - a t 2 + i b t 2 ) .
I FROG ( ω , τ ) = exp [ - 4 ( a 3 + a b 2 ) τ 2 - a ω 2 4 ( a 2 + b 2 ) ] .
G = { 1 N 2 ω , τ = 1 N [ I FROG ( ω , τ ) - E sig ( ω , τ ) 2 ] 2 } 1 / 2 .
E sig ( ω , τ ) = E sig ( ω , τ ) E sig ( ω , τ ) [ I FROG ( ω , τ ) ] 1 / 2 .
Z = t , τ = 1 N E sig ( t , τ ) - E ( t ) E ( t - τ ) 2 .
E SHG ( t ) E ( t ) tanh [ α E ( t ) ] ,
t w = ( 1 v g SHG - 1 v g fund ) L ,
F ( ω ) = [ sin ( t w ω / 2 ) t w ω / 2 ] 2
M τ ( τ ) = - d ω I FROG ( ω , τ ) ,
τ 0 = - d τ τ M τ ( τ ) - d τ M τ ( τ )

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