Abstract

Previously we showed that the “bootstrap method,” a well-known statistical technique, can be used to automatically compute error bars in ultrashort pulse measurements using frequency-resolved optical gating (FROG) without the need for additional measurements or traditional error analysis. Here we extend the bootstrap method to pulse measurement in the presence of ambiguities, where traditional error bars would give misleading information. As a result, we provide a new approach to displaying this uncertainty, which nicely reveals the richness of information available (or perhaps unavailable) in a FROG trace (or other measurement) in the presence of ambiguities.

© 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 Publishers, Boston, 2002).
    [CrossRef]
  2. Jung-Ho Chung and Andrew M. Weiner, �??Ambiguity of Ultrashort Pulse Shapes Retrieved from the Intensity Autocorrelation and the Power Spectrum,�?? J. Special Top. Quantum Electron. 7, 656-666 (2001).
    [CrossRef]
  3. Y. M. Bruck and L. G. Sodin, �??On the ambiguity of the image reconstruction problem,�?? Opt. Commun. 30, 304�??308 (1979).
    [CrossRef]
  4. Z. Wang, E. Zeek, R. Trebino, P. Kvam, �??Determining error bars in measurements of ultrashort laser pulses,�?? J. Opt. Soc. Am B 20, 2400-2405 (2003).
    [CrossRef]
  5. R. P. Millane, �??Phase retrieval in crystallography and optics,�?? J. Opt.Soc. Am. A. 7, 394�??411 (1990).
    [CrossRef]
  6. A. Walther, �??The question of phase retrieval in optics,�?? Opt. Acta. 10, 1�??49 (1963).
    [CrossRef]
  7. Stark, H., ed. Image Recovery: Theory and Application. (Academic Press, Orlando, 1987).
  8. Brad. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman & Hall/CRC, Boca Raton, 1993).
  9. C. Davison and D. V. Hinkley, Bootstrap Methods and Their Application (Cambridge University Press, Cambridge, 1997).
  10. D. Keusters, H.S. Tan, P. O�??Shea, E. Zeek, R. Trebino, W. S.Warren, �??Relative-Phase Ambiguities in Measurements of Ultrashort Pulses with Well-Separated Multiple Frequency Components,�?? J. Opt. Soc. Am B 20, 2226-2237 (2003).
    [CrossRef]

J. Opt. Soc. Am B

Z. Wang, E. Zeek, R. Trebino, P. Kvam, �??Determining error bars in measurements of ultrashort laser pulses,�?? J. Opt. Soc. Am B 20, 2400-2405 (2003).
[CrossRef]

D. Keusters, H.S. Tan, P. O�??Shea, E. Zeek, R. Trebino, W. S.Warren, �??Relative-Phase Ambiguities in Measurements of Ultrashort Pulses with Well-Separated Multiple Frequency Components,�?? J. Opt. Soc. Am B 20, 2226-2237 (2003).
[CrossRef]

J. Opt.Soc. Am. A.

R. P. Millane, �??Phase retrieval in crystallography and optics,�?? J. Opt.Soc. Am. A. 7, 394�??411 (1990).
[CrossRef]

J. Special Top. Quantum Electron.

Jung-Ho Chung and Andrew M. Weiner, �??Ambiguity of Ultrashort Pulse Shapes Retrieved from the Intensity Autocorrelation and the Power Spectrum,�?? J. Special Top. Quantum Electron. 7, 656-666 (2001).
[CrossRef]

Opt. Acta.

A. Walther, �??The question of phase retrieval in optics,�?? Opt. Acta. 10, 1�??49 (1963).
[CrossRef]

Opt. Commun.

Y. M. Bruck and L. G. Sodin, �??On the ambiguity of the image reconstruction problem,�?? Opt. Commun. 30, 304�??308 (1979).
[CrossRef]

Other

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

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

Brad. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman & Hall/CRC, Boca Raton, 1993).

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

Supplementary Material (4)

» Media 1: AVI (1274 KB)     
» Media 2: AVI (922 KB)     
» Media 3: AVI (620 KB)     
» Media 4: AVI (944 KB)     

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Figures (8)

Fig. 1. (a).
Fig. 1. (a).

The intensity (green curve) and the two possible phase solutions (red and blue dashed curves) in an SHG FROG measurement of a linearly chirped pulse. Even in the absence of noise in the trace, half the bootstrap retrieved pulses would yield one phase solution and half would yield the other. Of course, only one is correct. (b). The retrieved intensity and phase using the bootstrap method for the same pulse (in the presence of 1% additive noise, although this is not important). Note that both the retrieved phase and its error bars are unacceptable, giving the impression that the most likely phase is approximately flat with increasingly large errors near the plot edges, rather than the correct result that the phase is quite accurately one parabola or the other.

Fig. 2.
Fig. 2.

The entire set of bootstrap solutions for the linearly chirped pulse in Fig. 1. Note that this display much better reveals the true uncertainty in the measured intensity and phase.

Fig. 3.
Fig. 3.

(1.3MB) Movie of two incorrect saddle-shaped phase curves that could be mistaken for the actual parabolic phase in Fig. 2. Such confusion occurs any ambiguous curves intersect.

Fig. 4.
Fig. 4.

(0.9MB) Movie of bootstrap solutions of linear chirped pulse from SHG FROG trace. It clearly reveals the ambiguity and noise of the result.

Fig. 5.
Fig. 5.

SHG FROG trace for a double-peaked pulse with a relative phase of π between two peaks.

Fig. 6.
Fig. 6.

(0.6MB) Movie of bootstrap solutions for the well-separated doubled-peaked pulse in time. In the plot, the first pulse’s phase is set to zero for all pulses.

Fig. 7.
Fig. 7.

(0.9MB) Movie of the bootstrap solutions of a pulse with well-separated frequency components. In this plot the phase of the first spectral component was set to zero for all retrieved pulses.

Fig. 8 (a).
Fig. 8 (a).

Bootstrap solutions for a noise-free FROG trace for a pulse with somewhat separated spectral components. Note that the solutions accurately determine the relative phase of the frequency components (although some uncertainty is beginning to appear in the phase of the second component). (b). Bootstrap solutions for a FROG trace for a pulse with somewhat separated spectral components, here with 1% additive noise added to the trace. Note that the solutions no longer accurately determine the relative phase of the frequency components (although the uncertainty is not yet 2π).

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