Abstract

We report on significant improvements in the pulse-retrieval algorithm used to reconstruct the amplitude and the phase of ultrashort optical pulses from the experimental frequency-resolved optical gating trace data in the polarization-gate geometry. These improvements involve the use of an intensity constraint, an overcorrection technique, and a multidimensional minimization scheme. While the previously published, basic algorithm converged for most common ultrashort pulses, it failed to retrieve pulses with significant intensity substructure. The improved composite algorithm successfully converges for such pulses. It can now retrieve essentially all pulses of practical interest. We present examples of complex waveforms that were retrieved by the improved algorithm.

© 1994 Optical Society of America

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References

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  1. R. Trebino, 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]
  2. D. J. Kane, 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]
  3. R. A. Altes, “Detection, estimation, and classification with spectrograms,”J. Acoust. Soc. Am. 67, 1232–1246 (1980).
    [CrossRef]
  4. K. W. DeLong, R. Trebino, D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B (to be published).
  5. D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
    [CrossRef]
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  7. 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]
  8. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
    [CrossRef]
  9. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  10. A. J. Lichtenberg, M. A. Lieberman, Regular and Stochastic Motion, Vol. 38 of Applied Mathematical Sciences (Springer-Verlag, New York, 1983).
    [CrossRef]
  11. J. H. Seldin, J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
    [CrossRef]
  12. W. H. Press, W. T. Vetterling, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 420–425.

1994 (1)

D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
[CrossRef]

1993 (2)

1990 (2)

1987 (1)

1986 (1)

1982 (1)

1980 (1)

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

Altes, R. A.

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

DeLong, K. W.

D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
[CrossRef]

K. W. DeLong, R. Trebino, D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B (to be published).

Fienup, J. R.

Kane, D. J.

D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
[CrossRef]

R. Trebino, 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, 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]

K. W. DeLong, R. Trebino, D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B (to be published).

Kowalczyk, A. M.

Lichtenberg, A. J.

A. J. Lichtenberg, M. A. Lieberman, Regular and Stochastic Motion, Vol. 38 of Applied Mathematical Sciences (Springer-Verlag, New York, 1983).
[CrossRef]

Lieberman, M. A.

A. J. Lichtenberg, M. A. Lieberman, Regular and Stochastic Motion, Vol. 38 of Applied Mathematical Sciences (Springer-Verlag, New York, 1983).
[CrossRef]

Press, W. H.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 420–425.

Seldin, J. H.

Taylor, A. J.

D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 420–425.

Trebino, R.

D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
[CrossRef]

R. Trebino, 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, 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]

K. W. DeLong, R. Trebino, D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B (to be published).

Vetterling, W. T.

W. H. Press, W. T. Vetterling, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 420–425.

Wackerman, C. C.

Appl. Opt. (1)

J. Acoust. Soc. Am. (1)

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

J. Opt. Soc. Am. A (5)

Opt. Lett. (2)

D. J. Kane, A. J. Taylor, R. Trebino, K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating,” Opt. Lett. 18, 1061–1063 (1994).
[CrossRef]

D. J. Kane, 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]

Other (3)

K. W. DeLong, R. Trebino, D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B (to be published).

A. J. Lichtenberg, M. A. Lieberman, Regular and Stochastic Motion, Vol. 38 of Applied Mathematical Sciences (Springer-Verlag, New York, 1983).
[CrossRef]

W. H. Press, W. T. Vetterling, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 420–425.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the basic FROG algorithm. The equations used, starting clockwise from E(t), are Eq. (1) (Generate Signal), Eq. (3) (Apply Data), and Eq. (4) [Generate E(t)].

Fig. 2
Fig. 2

Error as a function of iteration number for two different pulses. Both pulses were computed on a 64-element array (FROG trace of 64 × 64 pixels). The basic FROG algorithm easily converges to the self-phase-modulated pulse, which has a FWHM of 10 and Q = 4 [see Eq. (7)]. The algorithm stagnates, however, for the double pulse, which has a FWHM of 6, with B = 1 and D = 12, as defined in Eq. (8), and no phase distortion (A = Q = 0).

Fig. 3
Fig. 3

Use of the intensity-constraint method enables the algorithm to make greater progress on the double pulse. The pulse was computed on a 64-element array, with a FWHM of 6 and B = 1, D = 12, and A = Q = 0, as defined in Eq. (8). The fluctuations are due to the method of implementation: two iterations with the intensity constraint followed by one iteration of the basic FROG algorithm.

Fig. 4
Fig. 4

Effect of different values of the exponent α in the over-correction method. Although raising the value of α increases the rate of convergence, too high a value can cause the algorithm to become unstable. The pulse was self-phase modulated, with a FWHM of 10 and Q = 4 on a 64-element grid.

Fig. 5
Fig. 5

Flow chart showing the control of the composite FROG algorithm. P-R stands for Polak–Ribiere, and F-R stands for Fletcher–Reeves.

Fig. 6
Fig. 6

Intensity profiles of the field with the lowest error produced by the basic FROG algorithm and the intensity-constraint method in the case of a double pulse. The intensity-constraint method gets significantly closer to the correct answer. The pulse had a FWHM of 6, with B = 1 and D = 12 [see Eq. (8)], and no phase distortion.

Fig. 7
Fig. 7

Intensity and phase of three highly structured pulses that the composite algorithm succeeded in retrieving. All three pulses are the sum of four individual Gaussian pulses, with varying widths, heights, and amounts of phase distortion. In (a) there is no phase distortion, in (b) each pulse has a varying amount of self-phase modulation, and in (c) each pulse has a varying amount of linear chirp. These complicated pulses could not be retrieved from the basic FROG algorithm alone.

Tables (1)

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Table 1 Variation of the Error G for Various Pairs of Fieldsa

Equations (21)

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E sig ( t , τ ) = E ( t ) E ( t - τ ) 2 .
I FROG ( ω , τ ) = | - d t E sig ( t , τ ) exp ( i ω t ) | 2 = E sig ( ω , τ ) 2 .
E sig ( ω , τ ) = E sig ( ω , τ ) E sig ( ω , τ ) [ I FROG ( ω , τ ) ] 1 / 2 ,
- d τ E sig ( t , τ ) = - d τ E ( t ) E ( t - τ ) 2 = E ( t ) - d τ E ( t - τ ) 2 = C E ( t ) ,
G = { 1 N 2 ω , τ = 1 N [ I FROG ( ω , τ ) - E sig ( ω , τ ) 2 ] 2 } 1 / 2 ,
E ( t ) = E 1 ( t ) = E 2 ( t ) ,
E 1 ( t ) = exp { - 2 ( ln 2 ) ( t / t p ) 2 + i [ A t 2 + Q E 1 ( t ) 2 ] } ,
E 2 ( t ) = B exp { - 2 ( ln 2 ) [ ( t - D ) / t p ] 2 + i [ A ( t - D ) 2 + Q E 2 ( t ) 2 ] } ,
E sig ( ω , τ ) = - d t F ( t ) G ( t - τ ) exp ( i ω t ) ,
- d ω E sig ( ω , τ ) = F ( 0 ) G ( - τ ) .
E sig ( ω , τ ) = E sig ( ω , τ ) E sig ( ω , τ ) [ I FROG ( ω , τ ) ] 1 / 2 × [ 1 + I FROG ( ω , τ ) - E sig ( ω , τ ) 2 E sig ( ω , τ ) 2 ] .
E sig ( ω , τ ) = E sig ( ω , τ ) E sig ( ω , τ ) [ I FROG ( ω , τ ) ] 1 / 2 [ I FROG ( ω , τ ) E sig ( ω , τ ) 2 ] = E sig ( ω , τ ) { [ I FROG ( ω , τ ) ] 1 / 2 E sig ( ω , τ ) } 3 .
E sig ( ω , τ ) = E sig ( ω , τ ) | [ I FROG ( ω , τ ) ] 1 / 2 E sig ( ω , τ ) | α ,
H = ω , τ = 1 N [ I FROG ( ω , τ ) - E sig ( ω , τ ) 2 ] 2 ,
E sig ( ω , τ ) = t = 1 N E ( t ) E ( t - τ ) 2 exp ( i ω t ) ,
d H d E r ( t 0 ) = - 4 ω , τ N [ I FROG ( ω , τ ) - E sig ( ω , τ ) 2 ] × Re ( { E ( t 0 - τ ) 2 exp ( i ω t 0 ) + 2 E r ( t 0 ) E ( t 0 + τ ) exp [ i ω ( t 0 + τ ) ] } × E sig * ( ω , τ ) ) ,
d H d E i ( t 0 ) = - 4 ω , τ N [ I FROG ( ω , τ ) - E sig ( ω , τ ) 2 ] × Re ( { i E ( t 0 - τ ) 2 exp ( i ω t 0 ) + 2 E i ( t 0 ) E ( t 0 + τ ) exp [ i ω ( t 0 + τ ) ] } × E sig * ( ω , τ ) ) ,
d E ( t ) d E r ( t 0 ) = δ t , t 0             d E ( t ) d E i ( t 0 ) = i δ t , t 0 ,
H W = ω , τ N [ I FROG ( ω , τ ) - E sig ( ω , τ ) 2 I FROG ( ω , τ ) ] 2 .
α = ( 1.1 ) 1 + k / 5 .
E sig ( ω , τ ) = E sig ( ω , τ ) E sig ( ω , τ ) { 1 - β ) [ I FROG ( ω , τ ) ] 1 / 2 + β E sig ( ω , τ ) } ,

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