Abstract

We analyze the properties required of a linear interferometer composed of time-stationary and time-nonstationary filters and a square-law, integrating detector in order that it may be used for the complete characterization of ultrashort optical pulses. Some general conditions for measurements are formulated for various schemes, and, in particular, time-nonstationary filtering is shown always to be necessary. The roles of both phase-only time nonstationarity and amplitude time gating are investigated in the context of known measurement schemes, and the known nonlinear schemes are shown to be members of a larger class of methods that use amplitude time-nonstationary filters.

© 1995 Optical Society of America

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  1. C. Yan and J.-C. Diels, "Amplitude and phase recording of ultrashort pulses," J. Opt. Soc. Am. B 8, 1259–1263 (1991).
    [CrossRef]
  2. K. Naganuma, K. Mogi, and H. Yamada, "General method of ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
    [CrossRef]
  3. J. L. A. Chilla and O. E. Martinez, "Analysis of a method of phase measurement of ultrashort pulses in the frequency domain," IEEE J. Quantum Electron. 27, 1228–1235 (1991).
    [CrossRef]
  4. D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571–579 (1993); R. Trebino and D. J. Kane, "Using phase retrieval to measure the intensity and phase of ultrashort pulse: frequency-resolved optical gating," J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [CrossRef]
  5. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, "Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation," Opt. Lett. 18, 1946–1948 (1993).
    [CrossRef] [PubMed]
  6. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
    [CrossRef]
  7. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041–2043 (1993).
    [CrossRef] [PubMed]
  8. V. Wong and I. A. Walmsley, "Analysis of ultrashort pulse-shape measurement using linear interferometers," Opt. Lett. 19, 287–289 (1994).
    [CrossRef] [PubMed]
  9. I. A. Walmsley and V. Wong, "Pulse-shape measurement using linear interferometers," presented at the Optical Society of America Annual Meeting, Dallas, Tex., 1994.
  10. I. A. Walmsley and V. Wong are preparing a paper to be called "On the characterization of the electric field of ultrashort optical pulses."
  11. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).
  12. In fact, a transfer function of the form hĩ(ω) = exp[−(ω − ωc)2 τf2]exp(iωcτd) such that τdf » 1 is very nearly a causal filter, with the transmitted energy that is due to the noncausal part scaling as exp[−2(τdf)2].
  13. An EOPM that can achieve 57 rad of phase modulation at 16 GHz was recently reported: A. Morimoto, E. Saruwatari, and T. Kobayashi, "Quasi-velocity-matched electro-optic phase modulator with domain inversion for deep microwave modulation," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CME5.
  14. M. Beck and I. A. Walmsley, "The role of amplitude and phase shaping in the dispersive-pulse regime of a passively mode-locked dye laser," IEEE J. Quantum Electron. 28, 2274–2284 (1992).
    [CrossRef]
  15. M. Beck and I. A. Walmsley, "Measurement of group delay with high temporal and spectral resolution," Opt. Lett. 15, 492–494 (1990).
    [CrossRef] [PubMed]
  16. E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848–3858 (1971).
    [CrossRef]
  17. Y. Ishida, K. Naganuma, T. Yajima, and L. H. Lin, "Ultrafast self-phase modulation in a colliding pulse mode-locked ring dye laser," in Ultrafast Phenomena IV, D. H. Austin and K. B. Eisenthal, eds. (Springer-Verlag, New York, 1984), pp. 69–71; Y. Ishida, K. Naganuma, and T. Yajima, "Selfphase modulation in hybridly mode-locked cw dye lasers," IEEE J. Quantum Electron. QE21, 69–77 (1985).
    [CrossRef]

1994 (2)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

V. Wong and I. A. Walmsley, "Analysis of ultrashort pulse-shape measurement using linear interferometers," Opt. Lett. 19, 287–289 (1994).
[CrossRef] [PubMed]

1993 (3)

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571–579 (1993); R. Trebino and D. J. Kane, "Using phase retrieval to measure the intensity and phase of ultrashort pulse: frequency-resolved optical gating," J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[CrossRef]

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, "Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation," Opt. Lett. 18, 1946–1948 (1993).
[CrossRef] [PubMed]

1992 (1)

M. Beck and I. A. Walmsley, "The role of amplitude and phase shaping in the dispersive-pulse regime of a passively mode-locked dye laser," IEEE J. Quantum Electron. 28, 2274–2284 (1992).
[CrossRef]

1991 (2)

J. L. A. Chilla and O. E. Martinez, "Analysis of a method of phase measurement of ultrashort pulses in the frequency domain," IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[CrossRef]

C. Yan and J.-C. Diels, "Amplitude and phase recording of ultrashort pulses," J. Opt. Soc. Am. B 8, 1259–1263 (1991).
[CrossRef]

1990 (1)

1989 (1)

K. Naganuma, K. Mogi, and H. Yamada, "General method of ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

1971 (1)

E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848–3858 (1971).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Beck, M.

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Chilla, J. L. A.

J. L. A. Chilla and O. E. Martinez, "Analysis of a method of phase measurement of ultrashort pulses in the frequency domain," IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[CrossRef]

Diels, J.-C.

Fujimoto, J. G.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Ippen, E. P.

Ishida, Y.

Y. Ishida, K. Naganuma, T. Yajima, and L. H. Lin, "Ultrafast self-phase modulation in a colliding pulse mode-locked ring dye laser," in Ultrafast Phenomena IV, D. H. Austin and K. B. Eisenthal, eds. (Springer-Verlag, New York, 1984), pp. 69–71; Y. Ishida, K. Naganuma, and T. Yajima, "Selfphase modulation in hybridly mode-locked cw dye lasers," IEEE J. Quantum Electron. QE21, 69–77 (1985).
[CrossRef]

Kane, D. J.

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571–579 (1993); R. Trebino and D. J. Kane, "Using phase retrieval to measure the intensity and phase of ultrashort pulse: frequency-resolved optical gating," J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[CrossRef]

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Kobayashi, T.

An EOPM that can achieve 57 rad of phase modulation at 16 GHz was recently reported: A. Morimoto, E. Saruwatari, and T. Kobayashi, "Quasi-velocity-matched electro-optic phase modulator with domain inversion for deep microwave modulation," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CME5.

Lin, L. H.

Y. Ishida, K. Naganuma, T. Yajima, and L. H. Lin, "Ultrafast self-phase modulation in a colliding pulse mode-locked ring dye laser," in Ultrafast Phenomena IV, D. H. Austin and K. B. Eisenthal, eds. (Springer-Verlag, New York, 1984), pp. 69–71; Y. Ishida, K. Naganuma, and T. Yajima, "Selfphase modulation in hybridly mode-locked cw dye lasers," IEEE J. Quantum Electron. QE21, 69–77 (1985).
[CrossRef]

Martinez, O. E.

J. L. A. Chilla and O. E. Martinez, "Analysis of a method of phase measurement of ultrashort pulses in the frequency domain," IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[CrossRef]

Mogi, K.

K. Naganuma, K. Mogi, and H. Yamada, "General method of ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

Morimoto, A.

An EOPM that can achieve 57 rad of phase modulation at 16 GHz was recently reported: A. Morimoto, E. Saruwatari, and T. Kobayashi, "Quasi-velocity-matched electro-optic phase modulator with domain inversion for deep microwave modulation," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CME5.

Naganuma, K.

K. Naganuma, K. Mogi, and H. Yamada, "General method of ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

Y. Ishida, K. Naganuma, T. Yajima, and L. H. Lin, "Ultrafast self-phase modulation in a colliding pulse mode-locked ring dye laser," in Ultrafast Phenomena IV, D. H. Austin and K. B. Eisenthal, eds. (Springer-Verlag, New York, 1984), pp. 69–71; Y. Ishida, K. Naganuma, and T. Yajima, "Selfphase modulation in hybridly mode-locked cw dye lasers," IEEE J. Quantum Electron. QE21, 69–77 (1985).
[CrossRef]

Paye, J.

Ramaswamy, M.

Raymer, M. G.

Saruwatari, E.

An EOPM that can achieve 57 rad of phase modulation at 16 GHz was recently reported: A. Morimoto, E. Saruwatari, and T. Kobayashi, "Quasi-velocity-matched electro-optic phase modulator with domain inversion for deep microwave modulation," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CME5.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

Treacy, E. B.

E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848–3858 (1971).
[CrossRef]

Trebino, R.

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571–579 (1993); R. Trebino and D. J. Kane, "Using phase retrieval to measure the intensity and phase of ultrashort pulse: frequency-resolved optical gating," J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[CrossRef]

Walmsley, I. A.

V. Wong and I. A. Walmsley, "Analysis of ultrashort pulse-shape measurement using linear interferometers," Opt. Lett. 19, 287–289 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

M. Beck and I. A. Walmsley, "The role of amplitude and phase shaping in the dispersive-pulse regime of a passively mode-locked dye laser," IEEE J. Quantum Electron. 28, 2274–2284 (1992).
[CrossRef]

M. Beck and I. A. Walmsley, "Measurement of group delay with high temporal and spectral resolution," Opt. Lett. 15, 492–494 (1990).
[CrossRef] [PubMed]

I. A. Walmsley and V. Wong, "Pulse-shape measurement using linear interferometers," presented at the Optical Society of America Annual Meeting, Dallas, Tex., 1994.

I. A. Walmsley and V. Wong are preparing a paper to be called "On the characterization of the electric field of ultrashort optical pulses."

Wong, V.

V. Wong and I. A. Walmsley, "Analysis of ultrashort pulse-shape measurement using linear interferometers," Opt. Lett. 19, 287–289 (1994).
[CrossRef] [PubMed]

M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041–2043 (1993).
[CrossRef] [PubMed]

I. A. Walmsley and V. Wong are preparing a paper to be called "On the characterization of the electric field of ultrashort optical pulses."

I. A. Walmsley and V. Wong, "Pulse-shape measurement using linear interferometers," presented at the Optical Society of America Annual Meeting, Dallas, Tex., 1994.

Yajima, T.

Y. Ishida, K. Naganuma, T. Yajima, and L. H. Lin, "Ultrafast self-phase modulation in a colliding pulse mode-locked ring dye laser," in Ultrafast Phenomena IV, D. H. Austin and K. B. Eisenthal, eds. (Springer-Verlag, New York, 1984), pp. 69–71; Y. Ishida, K. Naganuma, and T. Yajima, "Selfphase modulation in hybridly mode-locked cw dye lasers," IEEE J. Quantum Electron. QE21, 69–77 (1985).
[CrossRef]

Yamada, H.

K. Naganuma, K. Mogi, and H. Yamada, "General method of ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

Yan, C.

Appl. Phys. Lett. (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, "Time-to-frequency converter for measuring picosecond optical pulses," Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

IEEE J. Quantum Electron. (4)

K. Naganuma, K. Mogi, and H. Yamada, "General method of ultrashort light pulse chirp measurement," IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[CrossRef]

J. L. A. Chilla and O. E. Martinez, "Analysis of a method of phase measurement of ultrashort pulses in the frequency domain," IEEE J. Quantum Electron. 27, 1228–1235 (1991).
[CrossRef]

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571–579 (1993); R. Trebino and D. J. Kane, "Using phase retrieval to measure the intensity and phase of ultrashort pulse: frequency-resolved optical gating," J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[CrossRef]

M. Beck and I. A. Walmsley, "The role of amplitude and phase shaping in the dispersive-pulse regime of a passively mode-locked dye laser," IEEE J. Quantum Electron. 28, 2274–2284 (1992).
[CrossRef]

J. Appl. Phys. (1)

E. B. Treacy, "Measurement and interpretation of dynamic spectrograms of picosecond light pulses," J. Appl. Phys. 42, 3848–3858 (1971).
[CrossRef]

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

Opt. Lett. (4)

Optik (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237–246 (1972).

Other (5)

In fact, a transfer function of the form hĩ(ω) = exp[−(ω − ωc)2 τf2]exp(iωcτd) such that τdf » 1 is very nearly a causal filter, with the transmitted energy that is due to the noncausal part scaling as exp[−2(τdf)2].

An EOPM that can achieve 57 rad of phase modulation at 16 GHz was recently reported: A. Morimoto, E. Saruwatari, and T. Kobayashi, "Quasi-velocity-matched electro-optic phase modulator with domain inversion for deep microwave modulation," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CME5.

Y. Ishida, K. Naganuma, T. Yajima, and L. H. Lin, "Ultrafast self-phase modulation in a colliding pulse mode-locked ring dye laser," in Ultrafast Phenomena IV, D. H. Austin and K. B. Eisenthal, eds. (Springer-Verlag, New York, 1984), pp. 69–71; Y. Ishida, K. Naganuma, and T. Yajima, "Selfphase modulation in hybridly mode-locked cw dye lasers," IEEE J. Quantum Electron. QE21, 69–77 (1985).
[CrossRef]

I. A. Walmsley and V. Wong, "Pulse-shape measurement using linear interferometers," presented at the Optical Society of America Annual Meeting, Dallas, Tex., 1994.

I. A. Walmsley and V. Wong are preparing a paper to be called "On the characterization of the electric field of ultrashort optical pulses."

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Figures (6)

Fig. 1
Fig. 1

General interferometer for ultrashort pulse-shape measurement, consisting of four causal filters and a square-law, integrating detector.

Fig. 2
Fig. 2

Schematic diagram of the time–frequency converter. The EOPM is driven near the extremum of the sinusoidal modulation.

Fig. 3
Fig. 3

Schematic diagram of the spectral shearing interferometer for phase measurement. The EOPM is driven at the zero crossings of the modulation signal. It imposes a frequency shift of δω on the pulse in one arm of the interferometer.

Fig. 4
Fig. 4

Numerical simulation of the phase-reconstruction algorithm [Eq. (31)] for the spectral shearing interferometer. The input is a 51-fs Gaussian pulse with a quartic temporal phase structure. Simulation parameters are = 100 μm, τf = 2 ps, and δω = 2π(250) rad/s.

Fig. 5
Fig. 5

Schematic diagram of the FROG method using self-diffraction as the ANS time gate.

Fig. 6
Fig. 6

Schematic diagram of the FDPM method. The ANS time gate is constructed from a SHG crystal.

Equations (48)

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E out ( t ) = d t H i ( t , t ) E in ( t )             ( for the i th filter ) .
F ˜ ( ω ) = d t F ( t ) exp ( i ω t ) ,
F ( t ) = 1 2 π d ω F ˜ ( ω ) exp ( - i ω t ) .
H ˜ i ( t , ω ) = d t H i ( t , t ) exp ( i ω t ) .
E out ( t ) = d t h i ( t - t ) E in ( t ) ,
H i ( t , t ) = h i ( t - t ) ,
H ˜ i ( t , ω ) = h ˜ i ( - ω ) exp ( i ω t ) ,
E ˜ out ( ω ) d ω h ˜ i ( ω - ω ) E ˜ in ( ω ) .
H i ( t , t ) = h i ( t ) δ ( t - t ) ,
H ˜ i ( t , ω ) = h i ( t ) exp ( i ω t ) .
S ( τ ) = d t E f 1 ( t ) + E f ( t + τ ) 2
= d t { E f 1 ( t ) 2 + E f 2 ( t + τ ) 2 + [ E f 1 ( t ) E f 2 * ( t + τ ) + c . c . ] }
= S 1 + S 2 + S I ,
E f 1 ( t ) = d t d t H 2 ( t , t ) H 1 ( t , t ) E in ( t )
E f 1 ( t ) = d ω 1 d ω 2 E ˜ in ( ω 2 ) H ˜ 2 ( t , ω 1 ) × [ d t H ˜ 1 ( t , - ω 2 ) exp ( - i ω 1 t ) ] ,
S 1 = d ω 1 d ω 2 d ω 3 d ω 4 E ˜ in ( ω 3 ) E ˜ in * ( ω 4 ) × [ d t H ˜ 2 ( t , ω 1 ) H ˜ 2 * ( t , ω 2 ) ] × [ d t H ˜ 1 ( t , - ω 3 ) exp ( - i ω 1 t ) ] × [ d t H ˜ 1 * ( t , - ω 4 ) exp ( i ω 2 t ) ] .
S I ( τ ) = 2 Re d ω 1 d ω 2 d ω 3 d ω 4 E ˜ in ( ω 3 ) E ˜ in * ( ω 4 ) × [ d t H ˜ 2 ( t , ω 1 ) H ˜ 4 * ( t + τ , ω 2 ) ] × [ d t H ˜ 1 ( t , - ω 3 ) exp ( - i ω 1 t ) ] × [ d t H ˜ 3 * ( t , - ω 4 ) exp ( i ω 2 t ) ] .
S I ( τ ) = 2 Re d ω E ˜ in ( ω ) 2 H ˜ 1 ( ω ) H ˜ 2 ( ω ) H ˜ 3 * ( ω ) H ˜ 4 * ( ω ) × exp ( i ω τ ) .
h ( t ) = exp [ i Φ ( t ) ] .
h ˜ 1 ( ω ) = exp ( - i ϕ ω 2 ω 2 ) .
h 2 ( t ) = exp ( i ϕ t 2 t 2 ) .
E f ( t ) = d t H 3 ( t , t ) { d t H 2 ( t , t ) × [ d t H 1 ( t , t ) E in ( t ) ] } .
E f ( t ) = d ω 1 d ω 2 E ˜ i n ( ω 2 ) h ˜ 3 ( - ω 1 , ω c ) exp ( i ω 1 t ) × exp [ - i 1 2 ( ϕ ω + 1 ϕ t ) ω 2 2 ] × exp ( - i 1 2 ϕ t ω 1 2 ) exp ( - i 1 ϕ t ω 1 ω 2 ) .
S ( ϕ ω , ϕ t , ω c ) = d t E f ( t ) 2
= d ω 1 d ω 2 E ˜ in ( ω 2 ) E ˜ in * ( ω 1 ) × exp [ - i 1 2 ( ϕ ω + 1 ϕ t ) ( ω 2 2 - ω 1 2 ) ] × d ω 3 h ˜ 3 ( ω 3 , ω c ) 2 × exp [ i 1 ϕ t ( ω 2 - ω 1 ) ω 3 ] .
S ( ϕ ω , ϕ t , ω c ) | d ω E ˜ in ( ω ) × exp [ - i 1 2 ( ϕ ω + 1 ϕ t ) ω 2 ] × exp ( i ω c ϕ t ω ) | 2 .
E ˜ out ( ω ) = d ω δ ( ω + β - ω ) E ˜ in ( ω )
= E ˜ in ( ω + β ) .
h 1 ( t ) = exp ( i δ ω t )
h 3 ( t ) = 1 ,
h ˜ 4 ( ω ) = h ˜ 2 ( ω ) = h ˜ ( ω ) .
S I ( τ ) = 2 Re d ω E ˜ in ( ω + δ ω ) E ˜ in * ( ω ) h ˜ ( ω ) 2 exp ( i ω τ ) .
h ˜ ( ω ) = h ˜ ( ω , ω c ) = exp [ - ( ω - ω c ) 2 τ f 2 ] ,
S I ( ω c , τ ) 2 E ˜ in ( ω c + δ ω ) E ˜ in ( ω c ) × Re { exp [ i ( ϕ ω c + δ ω - ϕ ω c ) ] × d ω exp [ - 2 ( ω - ω c ) 2 τ f 2 ] exp ( i ω τ ) } .
S I ( ω c , τ ) 2 E ˜ in ( ω c + δ ω ) E ˜ in ( ω c ) exp ( - τ 2 8 τ f 2 ) × cos ( ω c τ + δ ω ϕ ω c ) ,
ϕ ω c + δ ω - ϕ ω c δ ω ϕ ω | ω c = δ ω ϕ ω c .
S ( ω c , τ ) E ˜ in ( ω c ) 2 + E ˜ in ( ω c + δ ω ) 2 + 2 E ˜ in ( ω c ) E ˜ in ( ω c + δ ω ) × exp ( - τ 2 8 τ f 2 ) cos ( ω c τ + δ ω ϕ ω c ) .
2 π τ - 2 π τ + Δ τ | δ ω ( ϕ ω c 1 - ϕ ω c 2 ) τ | ,
Δ τ τ δ ω ( ϕ ω c 1 - ϕ ω c 2 ) .
ϕ ω c 1 δ ω { cos - 1 [ S ( ω c , τ ) - E ˜ in ( ω c ) 2 - E ˜ in ( ω c + δ ω ) 2 2 E ˜ in ( ω c + δ ω ) E ˜ in ( ω c ) exp ( - τ 2 8 τ f 2 ) ] - ω c τ } .
h 1 ( t ) = h 1 ( t ; T ) = E in ( t ) E in * ( t - T ) .
S ( T , ω c ) = d ω 1 d ω 2 d ω 3 d ω 4 E ˜ in ( ω 3 ) E ˜ in * ( ω 4 ) × [ d t h ˜ 2 ( - ω 1 ) exp ( i ω 1 t ) h ˜ 2 * ( - ω 2 ) exp ( - i ω 2 t ) ] × [ d t h 1 ( t ) exp ( - i ω 3 t ) exp ( - i ω 1 t ) ] × [ d t h 1 * ( t ) exp ( i ω 4 t ) exp ( i ω 2 t ) ] .
S ( T , ω c ) = d t d t E in ( t ) E in * ( t ) h 1 ( t , T ) h 1 * ( t , T ) × d ω h ˜ 2 ( ω , ω c ) 2 exp [ i ω ( t - t ) ] .
S ( T , ω c ) | d t E in ( t ) h 1 ( t , T ) exp ( i ω c t ) | 2 .
h 2 ( t ) = h 2 ( t ; T ) = E in ( t - T ) .
S ( T , ω c ) = d ω 1 d ω 2 d ω 3 d ω 4 E ˜ in ( ω 2 ) E ˜ in * ( ω 4 ) × { d t h 2 ( t , T ) 2 exp [ i ( ω 1 - ω 2 ) t ] } × [ d t h ˜ 1 ( ω 3 , ω c ) exp ( - i ω 3 t ) exp ( - i ω 1 t ) ] × [ d t h ˜ 1 * ( ω 4 , ω c ) exp ( i ω 4 t ) exp ( i ω 2 t ) ] .
S ( T , ω c ) = d ω 3 d ω 4 E ˜ in ( ω 3 ) E ˜ in * ( ω 4 ) × h ˜ 1 ( ω 3 , ω c ) h ˜ 1 * ( ω 4 , ω c ) d t h 2 ( t , T ) 2 × exp [ - i ( ω 3 - ω 4 ) t ] .
S ( T , ω c ) | d ω E ˜ in ( ω ) h ˜ 1 ( ω , ω c ) exp ( - i ω T ) | 2 .

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