Abstract

Frequency-resolved optical gating (FROG), a technique for measuring ultrashort laser pulses, involves producing a spectrogram of the pulse and then retrieving the pulse intensity and phase with an iterative algorithm. We study how several types of noise—multiplicative, additive, and quantization—affect pulse retrieval. We define a convergence criterion and find that the algorithm converges to a reasonable pulse field, even in the presence of 10% noise. Specifically, with appropriate filtering, 1% rms retrieval error is achieved for 10% multiplicative noise, 10% additive noise, and as few as 8 bits of resolution. For additive and multiplicative noise the retrieval errors decrease roughly as the square root of the amount of noise. In addition, the background induced in the wings of the pulse by additive noise is equal to the amount of additive noise on the trace. Thus the dynamic range of the measured intensity and phase is limited by a noise floor equal to the amount of additive noise on the trace. We also find that, for best results, a region of zero intensity should surround the nonzero region of the trace. Consequently, in the presence of additive noise, baseline subtraction is important. We also find that Fourier low-pass filtering improves pulse retrieval without introducing significant distortion, especially in high-noise cases. We show that the field errors in the temporal and the spectral domains are equal. Overall, the algorithm performs well because the measured trace contains N2 data points for a pulse that has only 2N degrees of freedom; FROG has built in redundancy.

© 1995 Optical Society of America

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References

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  1. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
    [CrossRef]
  2. 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]
  3. 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]
  4. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  5. 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]
  6. 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]
  7. B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. Delong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
    [CrossRef] [PubMed]
  8. 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]
  9. K. W. DeLong, R. Trebino, J. R. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
    [CrossRef]
  10. K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating using the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
    [CrossRef] [PubMed]
  11. W. Koenig, H. K. Dunn, and L. Y. Lacy, “The sound spectrograph,” J. Acoust. Soc. Am. 18, 19–49 (1946).
    [CrossRef]
  12. S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
    [CrossRef]
  13. R. A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 67, 1232–1246 (1980).
    [CrossRef]
  14. 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 using frequency-resolved optical gating,” Opt. Lett. 19, 1061–1063 (1994).
    [CrossRef] [PubMed]
  15. B. Kohler, V. V. Yakovlev, and K. R. Wilson, “Characterization of amplified pulses for quantum control,” in Generation, Amplification and Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2116, 360–364 (1994).
    [CrossRef]
  16. H. Stark, ed. Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).
  17. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  18. D. Israelevitz and 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]
  19. 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]
  20. R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
    [CrossRef]
  21. J. H. Seldin and J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
    [CrossRef]
  22. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  23. For real experimental data we prefer the following background-removal techniques. In a multishot experiment, in which spectra of the signal field are taken for various values of the relative delay time, we generally subtract off a background spectrum taken under dark conditions from all the measured spectra. Similarly, in a single-shot experiment we subtract the spectrum measured at large delay times to remove the contributions of incoherently scattered light.
  24. J. S. Lim, Two-Dimensional Signal and Image Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1990), Chap. 9.
  25. D. L. Donoho, “Wavelet shrinkage and W. V. D-A ten-minute tour,” (Stanford University, Stanford, Calif., January1993).

1995 (1)

1994 (5)

1993 (4)

1991 (1)

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

1990 (2)

1989 (1)

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

1987 (2)

D. Israelevitz and 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]

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]

1983 (1)

S. H. Nawab, T. F. Quatieri, and 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]

1946 (1)

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

Altes, R. A.

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

Cohen, L.

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

Delong, K. W.

Donoho, D. L.

D. L. Donoho, “Wavelet shrinkage and W. V. D-A ten-minute tour,” (Stanford University, Stanford, Calif., January1993).

Dunn, H. K.

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

Fienup, J. R.

Fittinghoff, D. N.

Fujimoto, J. G.

Hunter, J. R.

Ippen, E. P.

Israelevitz, D.

D. Israelevitz and 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]

Kane, D. J.

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.

Lacy, L. Y.

W. Koenig, H. K. Dunn, and 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]

Lim, J. S.

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

J. S. Lim, Two-Dimensional Signal and Image Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1990), Chap. 9.

Millane, R. P.

Nawab, S. H.

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

Paye, J.

Quatieri, T. F.

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

Ramaswamy, M.

Seldin, J. H.

Squier, J.

Taylor, A. J.

Trebino, R.

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. Delong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
[CrossRef] [PubMed]

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 using frequency-resolved optical gating,” Opt. Lett. 19, 1061–1063 (1994).
[CrossRef] [PubMed]

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

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

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]

K. W. DeLong, R. Trebino, J. R. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
[CrossRef]

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

White, W. E.

Wilson, K.

Wilson, K. R.

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. Delong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
[CrossRef] [PubMed]

B. Kohler, V. V. Yakovlev, and K. R. Wilson, “Characterization of amplified pulses for quantum control,” in Generation, Amplification and Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2116, 360–364 (1994).
[CrossRef]

Yakovlev, V. V.

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. Delong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
[CrossRef] [PubMed]

B. Kohler, V. V. Yakovlev, and K. R. Wilson, “Characterization of amplified pulses for quantum control,” in Generation, Amplification and Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2116, 360–364 (1994).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

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

IEEE Trans. Acoust. Speech Signal Process. (2)

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

D. Israelevitz and 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]

J. Acoust. Soc. Am. (2)

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

R. A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 67, 1232–1246 (1980).
[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 (5)

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

Opt. Lett. (5)

Proc. IEEE (1)

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

Other (5)

B. Kohler, V. V. Yakovlev, and K. R. Wilson, “Characterization of amplified pulses for quantum control,” in Generation, Amplification and Measurement of Ultrashort Laser Pulses, R. P. Trebino and I. A. Walmsley, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2116, 360–364 (1994).
[CrossRef]

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

For real experimental data we prefer the following background-removal techniques. In a multishot experiment, in which spectra of the signal field are taken for various values of the relative delay time, we generally subtract off a background spectrum taken under dark conditions from all the measured spectra. Similarly, in a single-shot experiment we subtract the spectrum measured at large delay times to remove the contributions of incoherently scattered light.

J. S. Lim, Two-Dimensional Signal and Image Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1990), Chap. 9.

D. L. Donoho, “Wavelet shrinkage and W. V. D-A ten-minute tour,” (Stanford University, Stanford, Calif., January1993).

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Figures (19)

Fig. 1
Fig. 1

Intensities and phases of the five pulses used in this study: (a) Pulse 1, a flat-phase Gaussian, (b) Pulse 2, a linearly chirped Gaussian, (c) Pulse 3, a self-phase-modulated Gaussian, (d) Pulse 4, a double pulse with spectral cubic phase, and (e) Pulse 5, a double pulse with linear chirp, self-phase modulation, and spectral cubic phase.

Fig. 2
Fig. 2

Polarization-gate FROG trace of Pulse 5 [shown in Fig. 1(e)].

Fig. 3
Fig. 3

PG FROG trace of Pulse 5 after inclusion of 10% additive noise. The noise is Poisson distributed, with n = 5. No image processing has been used on the trace. The noise at large time delays leads to nonzero background intensity, and the noise at large frequency offsets leads to high-frequency fluctuations in the recovered intensity and phase.

Fig. 4
Fig. 4

Retrieved pulse for the FROG trace of Fig. 3 without mean subtraction. The retrieved intensity exhibits a large background intensity, and the secondary peak is unresolved. Both the retrieved intensity and phase exhibit high-frequency fluctuations. (a) The actual and the retrieved intensities. The rms intensity error defined in Eq. (8) is 15%. (b) The actual and the retrieved phases. The rms phase error defined in Eq. (9) is 0.65 rad.

Fig. 5
Fig. 5

PG FROG trace of the test pulse with 10% additive noise in the trace after subtraction of the mean of the noise. Subtracting the mean of the noise lowers the unphysical values at large time delays and frequencies, which is crucial for accurate pulse retrieval.

Fig. 6
Fig. 6

Retrieved pulse for the FROG trace of Fig. 5. Subtracting the mean greatly reduces the background intensity and high-frequency fluctuations. (a) The actual and the retrieved intensities. (b) The actual and the retrieved phases.

Fig. 7
Fig. 7

PG FROG trace of the test pulse with 10% additive noise after subtraction of the mean of the noise and corner suppression with d = 45 pixels. Super-Gaussian corner suppression forces the values at the perimeter, and especially at the corners, to zero.

Fig. 8
Fig. 8

Retrieved pulse for the FROG trace of Fig. 7. Note that the use of corner suppression lowers the background intensity in the retrieved pulse. Also, the reduction in the high-frequency noise allows the algorithm to resolve the two peaks of the pulse cleanly. (a) The actual and the retrieved intensities. (b) The actual and the retrieved phases.

Fig. 9
Fig. 9

PG FROG trace of the test pulse with 10% additive noise after subtraction of the mean of the noise, corner suppression with d = 45 pixels, and low-pass filtering with ρ = 0.5. The low-pass filtering removes the high-spatial-frequency noise from the entire trace. The resulting smoothing effect is apparent.

Fig. 10
Fig. 10

Retrieved pulse for the FROG trace of Fig. 9. The retrieved intensity is very good, clearly reproducing the peaks of the pulse with minimal background or high-frequency noise. The phase is also remarkably good for such an initially high amount of noise and does not strongly deviate from the actual phase, except, as expected, at times when the intensity is below 1% of the peak intensity. (a) The actual and the retrieved intensities. The rms intensity error is 4%. (b) The actual and the retrieved phases. The rms phase error is 0.14%.

Fig. 11
Fig. 11

Mean rms intensity and phase errors for PG FROG induced by low-pass filtering the FROG trace for the five test pulses. The data are plotted as a function of filter radius. The filter radius ρ in the transform space is given as a fraction of N/2, where N is the number of pixels in each of the two dimensions of the FROG trace. Error bars indicating one standard deviation from the mean error for the intensity and the phase are shown at filter radii of 0.4 and 0.3, respectively.

Fig. 12
Fig. 12

Mean rms intensity and phase errors for PG FROG induced by the use of corner suppression on the FROG trace for the five test pulses. The data are plotted as a function of filter radius. The filter is centered in the center of the FROG trace and has a diameter given in pixels in each of the two dimensions of the FROG trace. Error bars indicating one standard deviation from the mean error for the intensity and the phase are shown at filter diameters of 40 and 60, respectively.

Fig. 13
Fig. 13

Mean rms intensity and phase errors for PG and SHG FROG for additive noise with n = 5. The mean of the noise background was subtracted from the FROG traces before retrieval, but no other filtering has been performed. The data are plotted as a function of noise fraction. The errors decrease roughly as the square root of the noise. An error bar indicating one standard deviation from the mean phase error for PG FROG is shown at a noise fraction of 10−2. The standard deviation for the PG intensity at a noise fraction of 10−3 is contained within the circular marker. These high noise levels in the retrieved intensity and phase reveal the importance of filtering (see Fig. 15 below).

Fig. 14
Fig. 14

Actual and retrieved intensities of Pulse 1 for additive noise with α = 0.001 and α = 0.0001 for PG FROG. The noise floor in the wings of the retrieved intensities is roughly equal to, and decreases linearly with, the amount of additive noise, α.

Fig. 15
Fig. 15

Intensity and phase errors for PG FROG for additive noise with n = 5 with and without optimized filtering. Filtering lowers the retrieved errors dramatically. Error bars indicating one standard deviation from the mean intensity error for the filtered and the unfiltered cases are shown at noise fractions of 10−2 and 10−3, respectively.

Fig. 16
Fig. 16

Intensity and phase errors for PG and SHG FROG for multiplicative noise. The traces were not filtered before the pulse was retrieved. The data are plotted as a function of noise fraction. Error bars indicating one standard deviation from the mean phase error for PG and SHG FROG are shown at noise fractions of 10−2 and 10−3, respectively.

Fig. 17
Fig. 17

Convergence ratio for PG and SHG FROG for additive and multiplicative noise without filtering. For additive noise n = 5. The solid horizontal line indicates the convergence limit where R = 2. The algorithm converges for all cases. Error bars indicating one standard deviation from the mean convergence ratio for multiplicative noise for PG and SHG FROG are shown at noise fractions of 10−2 and 10−3, respectively. The upper error bar for PG FROG is contained within the diamond marker.

Fig. 18
Fig. 18

Retrieved test pulse for PG FROG after quantization to 8 bits. The errors in the retrieved intensity and phase are small and become apparent mainly at intensities below 10−3 of the peak intensity. (a) The actual and the retrieved intensities. The rms intensity error is 1.1%. (b) The actual and the retrieved phases. The phase error is 0.0044 rad.

Fig. 19
Fig. 19

Retrieved test pulse for SHG FROG after quantization to 8 bits. The errors in the retrieved intensity and phase are small and become apparent mainly at intensities below 10−3 of the peak intensity. (a) The actual and the retrieved intensities. The rms intensity error is 0.63%. (b) The actual and the retrieved phases. The phase error is 0.00216 rad.

Equations (13)

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I FROG ( ω , τ ) | - E sig ( t , τ ) exp ( i ω t ) d t | 2 .
E sig ( t , τ ) E ( t ) E ( t - τ ) 2 .
E sig ( t , τ ) E ( t ) E ( t - τ ) .
FROG = { 1 N 2 i = 1 N j = 1 N [ I ˜ FROG ( ω i , τ j ) - I n FROG ( ω i , τ j ) ] 2 } 1 / 2 ,
I n FROG ( ω i , τ j ) = I FROG ( ω i , τ j ) ( 1 + m i j α ) .
I n FROG ( ω i , τ j ) = I FROG ( ω i , τ j ) + η i j ( α / n ) .
G ( ω i , τ j ) = exp { - β [ ( ω i - N / 2 ) 2 + ( τ j - N / 2 ) 2 ] 2 / d 4 } .
I = { 1 N j = 1 N [ I ˜ ( t j ) - I ( t j ) ] 2 } 1 / 2 ,
ϕ { 1 N j = 1 N I 2 ( t j ) [ ϕ ˜ ( t j ) - ϕ ( t j ) ] 2 } 1 / 2 / [ 1 N j = 1 N I 2 ( t j ) ] 1 / 2 .
t 2 = j = 1 N E ˜ ( t j ) - E ( t j ) 2 / j = 1 N E ( t j ) 2 .
ω 2 = j = 1 N E ˜ ( ω j ) - E ( ω j ) 2 / j = 1 N E ( ω j ) 2 .
R = [ I ˜ FROG ( ω i , τ j ) , I FROG ( ω i , τ j ] [ I n FROG ( ω i , τ j ) , I FROG ( ω i , τ j ) ] < 2 ,
[ C , D ] = [ 1 N 2 i = 1 N j = 1 N ( C i j - D i j ) 2 ] 1 / 2 / [ 1 N 2 i = 1 N j = 1 N D i j 2 ] 1 / 2

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