Abstract

We demonstrate a simple method for complete characterization (of amplitudes and phases) of short optical pulses, using only a dispersive delay line and an oscilloscope. The technique is based on using a dispersive delay line to stretch the pulses and recording the temporal interference of two delayed replicas of the pulse train. Then, by transforming the time domain interference measurements to spectral interferometry, the spectral intensity and phase of the input pulses are reconstructed, using a Fourier-transform algorithm. In the experimental demonstration, mode-locked fiber laser pulses with durations of ∼1 ps were characterized with a conventional fast photodetector and an oscilloscope.

© 2005 Optical Society of America

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  1. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [CrossRef]
  2. J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
    [CrossRef]
  3. V. Wong, I. A. Walmsley, “Analysis of ultrashort pulse-shape measurement using linear interferometers,” Opt. Lett. 19, 287–289 (1994).
    [CrossRef] [PubMed]
  4. D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
    [CrossRef] [PubMed]
  5. C. Iaconis, I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [CrossRef]
  6. K. C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, A. Sullivan, “Direct measurement of the spectral phase of femtosecond pulses,” Opt. Lett. 20, 904–906 (1995).
    [CrossRef] [PubMed]
  7. S. Prein, S. Diddams, J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. N. K. Berger, B. Levit, B. Fischer, “Optical pulse intensity and phase measurement by time-to-space conversion based on time-lens operation,” Electron. Lett. 37, 1033–1034 (2001).
    [CrossRef]
  12. P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
    [CrossRef]
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    [CrossRef]
  15. M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
    [CrossRef]
  16. N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
    [CrossRef]
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    [CrossRef]

2005 (1)

N. K. Berger, B. Levit, B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251, 315–321 (2005).
[CrossRef]

2004 (2)

P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
[CrossRef]

C. Dorrer, “Chromatic dispersion characterization by direct instantaneous frequency measurement,” Opt. Lett. 29, 204–206 (2004).
[CrossRef] [PubMed]

2001 (1)

N. K. Berger, B. Levit, B. Fischer, “Optical pulse intensity and phase measurement by time-to-space conversion based on time-lens operation,” Electron. Lett. 37, 1033–1034 (2001).
[CrossRef]

2000 (1)

N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
[CrossRef]

1999 (2)

M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

1998 (1)

1997 (3)

D. Meshulach, D. Yelin, Y. Silberberg, “Real-time spatial-spectral interference measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B 14, 2095–2098 (1997).
[CrossRef]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Y. C. Tong, L. Y. Chan, H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[CrossRef]

1996 (2)

1995 (1)

1994 (2)

1987 (1)

1982 (1)

Andersen, J. K.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

Azaña, J.

P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
[CrossRef]

M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

Barry, L. P.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Bekker, A.

N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
[CrossRef]

Berger, N. K.

N. K. Berger, B. Levit, B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251, 315–321 (2005).
[CrossRef]

N. K. Berger, B. Levit, B. Fischer, “Optical pulse intensity and phase measurement by time-to-space conversion based on time-lens operation,” Electron. Lett. 37, 1033–1034 (2001).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
[CrossRef]

Bollond, P. G.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Bowie, J. L.

Carballar, A.

Chan, L. Y.

Y. C. Tong, L. Y. Chan, H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[CrossRef]

Chen, L. R.

P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
[CrossRef]

Chu, K. C.

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
[CrossRef] [PubMed]

Diddams, S.

S. Prein, S. Diddams, J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996).
[CrossRef]

Diels, J.-C.

S. Prein, S. Diddams, J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996).
[CrossRef]

Dienes, A.

Dorrer, C.

Dudley, J. M.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Fischer, B.

N. K. Berger, B. Levit, B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251, 315–321 (2005).
[CrossRef]

N. K. Berger, B. Levit, B. Fischer, “Optical pulse intensity and phase measurement by time-to-space conversion based on time-lens operation,” Electron. Lett. 37, 1033–1034 (2001).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
[CrossRef]

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
[CrossRef] [PubMed]

Fortenberry, R. M.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

Grant, R. S.

Grischkowsky, D.

Harvey, J. D.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Heritage, J. P.

Iaconis, C.

Ina, H.

Islam, M. N.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

Jennings, R. T.

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Kobayashi, S.

Kockaert, P.

P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
[CrossRef]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
[CrossRef] [PubMed]

LaRochelle, S.

P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
[CrossRef]

Leonhardt, R.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Levit, B.

N. K. Berger, B. Levit, B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251, 315–321 (2005).
[CrossRef]

N. K. Berger, B. Levit, B. Fischer, “Optical pulse intensity and phase measurement by time-to-space conversion based on time-lens operation,” Electron. Lett. 37, 1033–1034 (2001).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
[CrossRef]

Lin, H.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

Liu, K. X.

Meshulach, D.

Muriel, M. A.

Newton, S. A.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

Papoulis, A.

Prein, S.

S. Prein, S. Diddams, J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996).
[CrossRef]

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Rothenberg, J. E.

Silberberg, Y.

Sorin, W. V.

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

Sullivan, A.

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
[CrossRef] [PubMed]

Takeda, M.

Thomsen, B. C.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Thomson, M. D.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

Tong, Y. C.

Y. C. Tong, L. Y. Chan, H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[CrossRef]

Trebino, R.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
[CrossRef] [PubMed]

Tsang, H. K.

Y. C. Tong, L. Y. Chan, H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[CrossRef]

Walmsley, I. A.

White, W. E.

Wong, V.

Yelin, D.

Electron. Lett. (3)

N. K. Berger, B. Levit, B. Fischer, “Optical pulse intensity and phase measurement by time-to-space conversion based on time-lens operation,” Electron. Lett. 37, 1033–1034 (2001).
[CrossRef]

Y. C. Tong, L. Y. Chan, H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[CrossRef]

N. K. Berger, B. Levit, A. Bekker, B. Fischer, “Real-time optical spectrum analyser based on chirped fibre Bragg gratings,” Electron. Lett. 36, 1189–1191 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35, 441–450 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. Kockaert, J. Azaña, L. R. Chen, S. LaRochelle, “Full characterization of uniform ultrahigh-speed trains of optical pulses using fiber Bragg gratings and linear detectors,” IEEE Photon. Technol. Lett. 16, 1540–1542 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

N. K. Berger, B. Levit, B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251, 315–321 (2005).
[CrossRef]

S. Prein, S. Diddams, J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123, 567–573 (1996).
[CrossRef]

Opt. Lett. (7)

Rev. Sci. Instrum. (1)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time–frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Other (1)

R. M. Fortenberry, W. V. Sorin, H. Lin, S. A. Newton, J. K. Andersen, M. N. Islam, “Low-power ultrashort optical pulse characterization using linear dispersion,” in Optical Fiber Communication Conference, Vol. 6 of 1997 OSA Technical Digest Series (Optical Society of America, 1997), pp. 290–291, paper ThL3.

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of the measurement setup. EDFA, erbium-doped fiber amplifier.

Fig. 2
Fig. 2

Oscilloscope trace of the interference between two replicas of the stretched pulse reflected from the two surfaces of a glass plate. The time and frequency scales are related by Eq. (2).

Fig. 3
Fig. 3

Absolute value of the Fourier transform of the interference pattern shown in Fig. 2.

Fig. 4
Fig. 4

Spectral intensity I(ω) (solid curve) and spectral phase φ(ω) (dashed curve) of the input pulse, reconstructed from Fig. 3. The time and frequency scales are related by Eq. (2).

Fig. 5
Fig. 5

Reconstructed temporal intensity I(t) (solid curve) and phase φ(t) (dashed curve) of the fiber laser pulse.

Fig. 6
Fig. 6

Measurements of the phase response of a 9 m long conventional fiber obtained by our method (dashed curve) as well as by measuring the frequency dependence of the group delay (solid curve).

Fig. 7
Fig. 7

Autocorrelation of the laser pulse, measured (solid curve) and calculated (dashed curve) for the reconstructed pulse intensity shown in Fig. 5.

Equations (6)

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

E out ( t ) exp [ i t 2 / ( 2 β 2 L ) ] F ( t / β 2 L ) ,
t = β 2 L ω ,
I int ( t ) = I ( ω ) + I ( ω + Δ ω ) + 2 [ I ( ω ) I ( ω + Δ ω ) ] 1 / 2 cos [ Δ ω t + Δ φ ( ω ) ] ,
Δ φ ( ω ) = φ ( ω + Δ ω ) φ ( ω ) ( d φ / d ω ) Δ ω
φ ( ω ) = ( 1 / Δ ω ) Δ φ ( ω ) d ω .
δ t s 2 π β 2 L / τ p .

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