Abstract

We present a simple and automatic method for determining the uncertainty in the retrieved intensity and phase versus time (and frequency) due to noise in a frequency-resolved optical-gating trace, independent of noise source. It uses the “bootstrap” statistical method and also yields an automated method for phase blanking (omitting the phase when the intensity is too low to determine it).

© 2003 Optical Society of America

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References

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  1. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Boston, Mass., 2002).
  2. E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I,” Trans. Am. Math. Soc. 83, 179–192 (1956).
  3. E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II,” Trans. Am. Math. Soc. 84, 234–238 (1957).
  4. J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
    [CrossRef]
  5. X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174–1176 (2002).
    [CrossRef]
  6. J. M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, R. Trebino, S. Coen, and R. S. Windeler, “Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and experiments,” Opt. Express 10, 1215–1221 (2002).
    [CrossRef] [PubMed]
  7. J.-Y. Zhang, A. P. Shreenath, M. Kimmel, E. Zeek, R. Trebino, and S. Link, “Measurement of the intensity and phase of attojoule femtosecond light pulses using optical-parametric-amplification cross-correlation frequency-resolved optical gating,” Opt. Express 11, 601–609 (2003).
    [CrossRef] [PubMed]
  8. D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216–1218 (2000).
    [CrossRef]
  9. W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, Cambridge, UK, 1995).
  10. B. Efron and R. J. Tibshirani, eds., An Introduction to the Bootstrap (CRC, Boca Raton, Fla., 1993).
  11. A. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Application (Cambridge University, Cambridge, UK, 1997).
  12. I. Buvat, “A non-paramteric bootstrap approach for analysing the statistical properties of SPECT and PET images,” Phys. Med. Biol. 47, 1671–1775 (2002).
    [CrossRef]
  13. M. Dahlbom, “Estimation of image noise in PET using the bootstrap method,” IEEE Trans. Nucl. Sci. 49, 2062–2066 (2002).
    [CrossRef]
  14. E. Zeek, A. P. Shreenath, P. O’Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal frequency-resolved optical gating,” Appl. Phys. B 74, S265–S271 (2002).
    [CrossRef]

2003 (1)

2002 (5)

I. Buvat, “A non-paramteric bootstrap approach for analysing the statistical properties of SPECT and PET images,” Phys. Med. Biol. 47, 1671–1775 (2002).
[CrossRef]

M. Dahlbom, “Estimation of image noise in PET using the bootstrap method,” IEEE Trans. Nucl. Sci. 49, 2062–2066 (2002).
[CrossRef]

E. Zeek, A. P. Shreenath, P. O’Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal frequency-resolved optical gating,” Appl. Phys. B 74, S265–S271 (2002).
[CrossRef]

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174–1176 (2002).
[CrossRef]

J. M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, R. Trebino, S. Coen, and R. S. Windeler, “Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and experiments,” Opt. Express 10, 1215–1221 (2002).
[CrossRef] [PubMed]

2001 (1)

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[CrossRef]

2000 (1)

1957 (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II,” Trans. Am. Math. Soc. 84, 234–238 (1957).

1956 (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I,” Trans. Am. Math. Soc. 83, 179–192 (1956).

Akutowicz, E. J.

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II,” Trans. Am. Math. Soc. 84, 234–238 (1957).

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I,” Trans. Am. Math. Soc. 83, 179–192 (1956).

Buvat, I.

I. Buvat, “A non-paramteric bootstrap approach for analysing the statistical properties of SPECT and PET images,” Phys. Med. Biol. 47, 1671–1775 (2002).
[CrossRef]

Chung, J.-H.

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[CrossRef]

Coen, S.

Dahlbom, M.

M. Dahlbom, “Estimation of image noise in PET using the bootstrap method,” IEEE Trans. Nucl. Sci. 49, 2062–2066 (2002).
[CrossRef]

Dudley, J. M.

Gu, X.

Kane, D. J.

Kimmel, M.

Link, S.

O’Shea, P.

Omenetto, F. G.

Shreenath, A. P.

Taylor, A. J.

Trebino, R.

Weiner, A. M.

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[CrossRef]

Windeler, R. S.

Xu, L.

Zeek, E.

Zhang, J.-Y.

Appl. Phys. B (1)

E. Zeek, A. P. Shreenath, P. O’Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal frequency-resolved optical gating,” Appl. Phys. B 74, S265–S271 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and power spectrum,” IEEE J. Sel. Top. Quantum Electron. 7, 656–666 (2001).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

M. Dahlbom, “Estimation of image noise in PET using the bootstrap method,” IEEE Trans. Nucl. Sci. 49, 2062–2066 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Med. Biol. (1)

I. Buvat, “A non-paramteric bootstrap approach for analysing the statistical properties of SPECT and PET images,” Phys. Med. Biol. 47, 1671–1775 (2002).
[CrossRef]

Trans. Am. Math. Soc. (2)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I,” Trans. Am. Math. Soc. 83, 179–192 (1956).

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II,” Trans. Am. Math. Soc. 84, 234–238 (1957).

Other (4)

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Boston, Mass., 2002).

W. H. Press, S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, Cambridge, UK, 1995).

B. Efron and R. J. Tibshirani, eds., An Introduction to the Bootstrap (CRC, Boca Raton, Fla., 1993).

A. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Application (Cambridge University, Cambridge, UK, 1997).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the bootstrap process for application to FROG measurement of ultrashort pulses. Each of the ∼100 resampled traces (obtained from a single measured trace as described in the text) is run through the FROG algorithm, and the mean and standard deviation of each point of the retrieved pulse intensity and phase versus time and frequency is calculated.

Fig. 2
Fig. 2

Polarization-gate (PG) FROG trace of the pulse used in these simulations.

Fig. 3
Fig. 3

Retrieved intensity and phase for the noise-free FROG trace in Fig. 2. Error bars have been computed with the bootstrap method as described in the text to determine whether error bars have zero length in the noise-free case, as required. Solid curves are the actual intensity and phase. In time, the integrated intensity error was 1.8×10-6 and the integrated intensity-weighted phase error was 2.6×10-6, and in frequency, these errors were 5.7×10-6 and 8.7×10-8, respectively. (In this and all other plots, error bars smaller than the size of the data point are omitted.)

Fig. 4
Fig. 4

Retrieved intensity and phase of a theoretical pulse with 1% additive noise introduced numerically to the FROG trace. The intensity error was 9.3×10-3 and the phase error was 1.2×10-2, and in frequency, the errors were 2.5×10-3 and 3.3×10-3.

Fig. 5
Fig. 5

Retrieved intensity and phase of a theoretical pulse with a different realization of 1% additive noise. Here the intensity error was 9.8×10-3, and the phase error was 1.1×10-3 (2.3×10-3 and 3.6×10-3 in frequency), essentially identical to the retrieval in Fig. 4.

Fig. 6
Fig. 6

Pulse retrieved from the same FROG trace, but now with 10% additive noise added. The error bars are about an order of magnitude larger, and the integrated errors are also larger, in time, 2.2×10-2 for intensity and 4.5×10-2 for phase (in frequency, the numbers are 5.9×10-3 and 1.6×10-2).

Fig. 7
Fig. 7

Error bars with the bootstrap method for an experimentally measured FROG trace.

Fig. 8
Fig. 8

Pulse from Fig. 6, but with phase blanking applied. Note how the removal of the extra (meaningless) phase points simplifies the plot.

Equations (4)

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

t0=itiI(ti)iI(ti),
Eˆ(t)=E(t)E(t0),
SI=i=1nσiInImax,
Sϕ=i=1nσiϕIinImax,

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