Abstract

We describe a new and simple method to aid in the analysis of retrieved pulses from inverted frequency-resolved optical gating (FROG) traces. The analysis can separate noise from distortion and shows that distortion is more deleterious to the retrieved pulse than is pure noise. The analysis relies on the fact that FROG traces can be constructed from a single outer product of two vectors, whereas distortion and noise require the sum of a series of outer products.

© 2000 Optical Society of America

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  1. D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
    [CrossRef]
  2. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
    [CrossRef]
  3. D. J. Kane, G. Rodriguez, A. J. Taylor, and T. Clement, J. Opt. Soc. Am. B 14, 935 (1997).
    [CrossRef]
  4. D. J. Kane, IEEE J. Sel. Top. Quantum Electron. 4, 278 (1998).
    [CrossRef]
  5. D. J. Kane, IEEE J. Quantum Electron. 35, 421 (1999).
    [CrossRef]
  6. T. I. Kuznetsova and I. A. Walmsley, Quantum Electron. 28, 728 (1998).
    [CrossRef]
  7. T. M. Shuman, M. E. Anderson, J. Bromage, C. Iaconis, L. Waxer, and I. A. Walmsley, Opt. Express 5, 134 (1999); http://epubs.osa.org/opticsexpress .
    [CrossRef] [PubMed]
  8. D. T. Reid, IEEE J. Quantum Electron. 35, 1584 (1999).
    [CrossRef]
  9. V. Wong and I. A. Walmsley, J. Opt. Soc. Am. B 14, 944 (1997).
    [CrossRef]
  10. J. L. A. Chilla and O. E. Martinez, Opt. Lett. 16, 39 (1991).
    [CrossRef] [PubMed]
  11. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

1999 (3)

1998 (2)

T. I. Kuznetsova and I. A. Walmsley, Quantum Electron. 28, 728 (1998).
[CrossRef]

D. J. Kane, IEEE J. Sel. Top. Quantum Electron. 4, 278 (1998).
[CrossRef]

1997 (3)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

D. J. Kane, G. Rodriguez, A. J. Taylor, and T. Clement, J. Opt. Soc. Am. B 14, 935 (1997).
[CrossRef]

V. Wong and I. A. Walmsley, J. Opt. Soc. Am. B 14, 944 (1997).
[CrossRef]

1993 (1)

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

1991 (1)

Anderson, M. E.

Bromage, J.

Chilla, J. L. A.

Clement, T.

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

Iaconis, C.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Kane, D. J.

D. J. Kane, IEEE J. Quantum Electron. 35, 421 (1999).
[CrossRef]

D. J. Kane, IEEE J. Sel. Top. Quantum Electron. 4, 278 (1998).
[CrossRef]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

D. J. Kane, G. Rodriguez, A. J. Taylor, and T. Clement, J. Opt. Soc. Am. B 14, 935 (1997).
[CrossRef]

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

Kuznetsova, T. I.

T. I. Kuznetsova and I. A. Walmsley, Quantum Electron. 28, 728 (1998).
[CrossRef]

Martinez, O. E.

Reid, D. T.

D. T. Reid, IEEE J. Quantum Electron. 35, 1584 (1999).
[CrossRef]

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

Rodriguez, G.

Shuman, T. M.

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

Taylor, A. J.

Trebino, R.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

Walmsley, I. A.

Waxer, L.

Wong, V.

IEEE J. Quantum Electron. (3)

D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993).
[CrossRef]

D. T. Reid, IEEE J. Quantum Electron. 35, 1584 (1999).
[CrossRef]

D. J. Kane, IEEE J. Quantum Electron. 35, 421 (1999).
[CrossRef]

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

D. J. Kane, IEEE J. Sel. Top. Quantum Electron. 4, 278 (1998).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (1)

Quantum Electron. (1)

T. I. Kuznetsova and I. A. Walmsley, Quantum Electron. 28, 728 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, Rev. Sci. Instrum. 68, 3277 (1997).
[CrossRef]

Other (1)

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

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Figures (5)

Fig. 1
Fig. 1

Steps required for SVD analysis of a FT. IFFT, inverse fast Fourier transform; FFT, fast Fourier transform.

Fig. 2
Fig. 2

Series of synthetic SHG FT’s (the square root of intensity versus frequency and time delay is presented to show more detail) with cubic spectral phase and various amounts of noise and distortion: 1, perfect FT with no distortions. 2, trace with shot noise added. (It has a per-pixel rms deviation of 12V1/2, where V is the pixel value, which varies from 0 to 62,500.) 3, only additive noise added (per-pixel rms deviation of 2% of the maximum pixel value). 4, both additive and shot noise. 5, modestly filtered version of the trace 4. 6, filtered version of trace 4 but visibly distorted as a result of filtering. After pulse retrieval the FT error and the retrieved pulse error are 1, 0%, 0%; 2, 0.8%, 0.14%; 3, 2.0%, 3.5%; 4, 2.2%, 2.7%; 5, 0.44%, 3.9%; 6, 0.3%, 8.5%, respectively.

Fig. 3
Fig. 3

Time and frequency marginals of FT’s 2–4 depicted in Fig. 2. The agreement among all frequency and time marginals is excellent.

Fig. 4
Fig. 4

SVD analysis of the FT depicted in Fig. 2. Any deviation from a straight line indicates either distortion or a lack of convergence of the algorithm. All plots are normalized to have a maximum value of 1.

Fig. 5
Fig. 5

(a) Experimental SHG FT (square root of the intensity). (b) Retrieved pulse. (c) SVD analysis. The SHG-PCGP algorithm was used to invert the FT. There is very little difference between the pulse and the gate, indicating that the FT is nearly symmetric. The slight curvature in the normalized weight plot indicates only a small amount of distortion. If we assume that the distortion is due only to the bandwidth limitation of the doubling crystal and compensation for it, we can achieve nearly noise-limited convergence [dashed curve in (c)].

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