Abstract

We have employed chaotic and phase-diffusion models to study the effect of laser coherence on time-delayed laser-induced double gratings (TDLIDG). It is found that, although the temporal behavior of the four-wave mixing signal depends on the stochastic properties of the lasers and the relaxation time of the grating, the modulation of the signal is always damped out when the relative time delay between two pump beams is much longer than the laser coherence time. We performed a TDLIDG experiment mediated by thermal effects to study the temporal behavior of the modulation, and we obtained the frequency difference between two light beams with an accuracy limited by the laser linewidth. We also studied the influence of the phase change in the optical path on TDLIDG.

© 1993 Optical Society of America

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  1. N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525 (1984).
    [CrossRef]
  2. S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
    [CrossRef]
  3. H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
    [CrossRef]
  4. M. Fujiwara, R. Kuroda, and H. Nakatsuka, “Measurement of ultrafast dephasing time of Cresyl Fast Violet in cellulose by photon echoes with incoherent light,” J. Opt. Soc. Am. B 2, 1634 (1985).
    [CrossRef]
  5. R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467 (1985).
    [CrossRef] [PubMed]
  6. J. E. Golub and T. W. Mossberg, “Ultrahigh-frequency interference beats in transient, incoherent-light four-wave mixing,” Opt. Lett. 11, 431 (1986).
    [CrossRef] [PubMed]
  7. N. Morita, T. Tokizaki, and T. Yajima, “Time-delayed four-wave mixing using incoherent light for observation of ultra-fast population relaxation,” J. Opt. Soc. Am. B 4, 1269 (1987).
    [CrossRef]
  8. D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
    [CrossRef] [PubMed]
  9. X. Mi, Q. Jiang, Z. Yu, and P. Fu, “Observation of the beat between two independent light sources by a method of time-delayed laser-induced double gratings,” Opt. Lett. 16, 1526–1528 (1991).
    [CrossRef] [PubMed]
  10. Z. Yu, X. Mi, Q. Jiang, P. Ye, and P. Fu, “Distinguishing molecular reorientation gratings from thermal gratings by a time-delayed method,” Opt. Lett. 13, 117 (1988).
    [CrossRef]
  11. P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
    [CrossRef] [PubMed]
  12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 6.
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1992 (1)

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
[CrossRef] [PubMed]

1991 (1)

1988 (1)

1987 (2)

N. Morita, T. Tokizaki, and T. Yajima, “Time-delayed four-wave mixing using incoherent light for observation of ultra-fast population relaxation,” J. Opt. Soc. Am. B 4, 1269 (1987).
[CrossRef]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089 (1987).
[CrossRef]

1986 (2)

J. E. Golub and T. W. Mossberg, “Ultrahigh-frequency interference beats in transient, incoherent-light four-wave mixing,” Opt. Lett. 11, 431 (1986).
[CrossRef] [PubMed]

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
[CrossRef] [PubMed]

1985 (2)

1984 (3)

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
[CrossRef]

1983 (2)

1981 (1)

1976 (1)

1967 (1)

Asaka, S.

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
[CrossRef]

Beach, R.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467 (1985).
[CrossRef] [PubMed]

Beaty, E. C.

Boileau, E.

DeBeer, D.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467 (1985).
[CrossRef] [PubMed]

Drullinger, R. E.

Evenson, K. M.

Fu, P.

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
[CrossRef] [PubMed]

X. Mi, Q. Jiang, Z. Yu, and P. Fu, “Observation of the beat between two independent light sources by a method of time-delayed laser-induced double gratings,” Opt. Lett. 16, 1526–1528 (1991).
[CrossRef] [PubMed]

Z. Yu, X. Mi, Q. Jiang, P. Ye, and P. Fu, “Distinguishing molecular reorientation gratings from thermal gratings by a time-delayed method,” Opt. Lett. 13, 117 (1988).
[CrossRef]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089 (1987).
[CrossRef]

Fujiwara, M.

M. Fujiwara, R. Kuroda, and H. Nakatsuka, “Measurement of ultrafast dephasing time of Cresyl Fast Violet in cellulose by photon echoes with incoherent light,” J. Opt. Soc. Am. B 2, 1634 (1985).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
[CrossRef]

Golub, J. E.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 6.

Hall, J. L.

Hartmann, S. R.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467 (1985).
[CrossRef] [PubMed]

Hawkins, R. T.

Javan, A.

Jennings, D. A.

Jiang, Q.

Kelly, M. J.

Kowalski, F. V.

Kurnit, N. A.

Kuroda, R.

Layer, H. P.

Lee, P. H.

Matsuoka, M.

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
[CrossRef]

Mi, X.

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
[CrossRef] [PubMed]

X. Mi, Q. Jiang, Z. Yu, and P. Fu, “Observation of the beat between two independent light sources by a method of time-delayed laser-induced double gratings,” Opt. Lett. 16, 1526–1528 (1991).
[CrossRef] [PubMed]

Z. Yu, X. Mi, Q. Jiang, P. Ye, and P. Fu, “Distinguishing molecular reorientation gratings from thermal gratings by a time-delayed method,” Opt. Lett. 13, 117 (1988).
[CrossRef]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089 (1987).
[CrossRef]

Monchalin, J. P.

Morita, N.

N. Morita, T. Tokizaki, and T. Yajima, “Time-delayed four-wave mixing using incoherent light for observation of ultra-fast population relaxation,” J. Opt. Soc. Am. B 4, 1269 (1987).
[CrossRef]

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525 (1984).
[CrossRef]

Mossberg, T. W.

Nakatsuka, H.

M. Fujiwara, R. Kuroda, and H. Nakatsuka, “Measurement of ultrafast dephasing time of Cresyl Fast Violet in cellulose by photon echoes with incoherent light,” J. Opt. Soc. Am. B 2, 1634 (1985).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
[CrossRef]

Petersen, F. R.

Picinbono, B.

Pollock, C. R.

Schawlow, A. L.

Szoke, A.

Thomas, J. E.

Tokizaki, T.

Tomita, M.

H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
[CrossRef]

Van Wagenen, L. G.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
[CrossRef] [PubMed]

Wells, J. S.

Yajima, T.

N. Morita, T. Tokizaki, and T. Yajima, “Time-delayed four-wave mixing using incoherent light for observation of ultra-fast population relaxation,” J. Opt. Soc. Am. B 4, 1269 (1987).
[CrossRef]

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525 (1984).
[CrossRef]

Ye, P.

Z. Yu, X. Mi, Q. Jiang, P. Ye, and P. Fu, “Distinguishing molecular reorientation gratings from thermal gratings by a time-delayed method,” Opt. Lett. 13, 117 (1988).
[CrossRef]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089 (1987).
[CrossRef]

Yu, Z.

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
[CrossRef] [PubMed]

X. Mi, Q. Jiang, Z. Yu, and P. Fu, “Observation of the beat between two independent light sources by a method of time-delayed laser-induced double gratings,” Opt. Lett. 16, 1526–1528 (1991).
[CrossRef] [PubMed]

Z. Yu, X. Mi, Q. Jiang, P. Ye, and P. Fu, “Distinguishing molecular reorientation gratings from thermal gratings by a time-delayed method,” Opt. Lett. 13, 117 (1988).
[CrossRef]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089 (1987).
[CrossRef]

Zernike, F.

Zhang, Z.

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

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

J. Phys. (Paris) (1)

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089 (1987).
[CrossRef]

Opt. Commun. (1)

H. Nakatsuka, M. Tomita, M. Fujiwara, and S. Asaka, “Sub-picosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150 (1984).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (4)

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in Raman enhanced nondegenerate four-wave mixing spectrum by time-delayed method,” Phys. Rev. A 46, 1530 (1992).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467 (1985).
[CrossRef] [PubMed]

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286 (1984).
[CrossRef]

Phys. Rev. Lett. (1)

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128 (1986).
[CrossRef] [PubMed]

Other (1)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 6.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the basic principle of the TDLIDG.

Fig. 2
Fig. 2

Theoretical curves of the normalized FWM signal intensity versus α1τ for the chaotic model. Parameters are ωd/α1 = 20; γ/α1 = 0.01; α2/α1 = 1 in (a) and α2/α1 = 5 in (b). Here, η = 1 (solid curve), 0.2 (dashed curve).

Fig. 3
Fig. 3

Theoretical curves of the normalized FWM signal intensity versus α1τ for the chaotic model. Parameters: ωd/α1 = 20; α2/α1 = 1, η = 1 and γ/α1 = 1 (solid curve), 100 (dashed curve).

Fig. 4
Fig. 4

Theoretical curves of the normalized FWM signal intensity versus α1τ for the phase-diffusion model. The same set of parameters as in Fig. 3 are used.

Fig. 5
Fig. 5

Experimental setup. BS, beam splitter; M, mirror; GP Glan prism; C, Soleil–Babinet compensator; λ/2, half-wave plate; ODL, optical delay line; S, sample.

Fig. 6
Fig. 6

Experimental data of the FWM signal intensity versus relative time delay. The solid curve is the theoretical curve with α1 = 5.6 × 1010 s−1, α2 = 2.8 × 1010 s−1, ωd/2π = 1.8 × 1011 s−1, and η = 3.0.

Fig. 7
Fig. 7

FWM signal intensity versus relative time delay for λ1 = 584 nm and λ2 = 582 nm. The three curves in the figure correspond to different settings of the calibrated screw of the compensator with x = 0, 0.25, and 0.5 cm in (a), (b), and (c), respectively.

Equations (24)

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E p 1 ( r , t ) = 1 u 1 ( t - τ ) exp [ i ( k 1 · r - ω 1 t + ω 1 τ ) ] + 2 u 2 ( t - τ ) exp [ i ( k 2 · r - ω 2 t + ω 2 τ ) ] , E p 2 ( r , t ) = 1 u 1 ( t ) exp [ i ( k 1 · r - ω 1 t + ϕ 1 ) ] + 2 u 2 ( t ) exp [ i ( k 2 · r - ω 2 t + ϕ 2 ) ] .
d Q d t + γ Q = ζ E p 1 ( r , t ) E p 2 * ( r , t ) .
d Q d t + γ Q = ζ { 1 ( 1 ) * u 1 ( t - τ ) u 1 * ( t ) exp i ( q 1 · r + ω 1 τ - ϕ 1 ) + 2 ( 2 ) * u 2 ( t - τ ) u 2 * ( t ) exp i ( q 2 · r + ω 2 τ - ϕ 2 ) + 1 ( 2 ) * u 1 ( t - τ ) u 2 * ( t ) × exp i [ ( k 1 - k 2 ) · r - ω d t + ω 1 τ - ϕ 2 ] + 2 ( 1 ) * u 2 ( t - τ ) u 1 * ( t ) × exp i [ ( k 2 - k 1 ) · r + ω d t + ω 2 τ - ϕ 1 ] } ,
Q ( r , t ) = ζ { 1 ( 1 ) * exp i ( q 1 · r + ω 1 τ - ϕ 1 ) × 0 d t u 1 ( t - t - τ ) u 1 * ( t - t ) exp ( - γ t ) + 2 ( 2 ) * exp i ( q 2 · r + ω 2 τ - ϕ 2 ) × 0 d t u 2 ( t - t - τ ) u 2 * ( t - t ) exp ( - γ t ) + 1 ( 2 ) * exp i [ ( k 1 - k 2 ) · r - ω d t + ω 1 τ - ϕ 2 ] × 0 d t u 1 ( t - t - τ ) u 2 * ( t - t ) exp [ - ( γ - i ω d ) t ] + 2 ( 1 ) * exp i [ ( k 2 - k 1 ) · r + ω d t + ω 2 τ - ϕ 1 ] × 0 d t u 2 ( t - t - τ ) u 1 * ( t - t ) exp [ - ( γ + i ω d ) t ] } .
E 3 ( r , t ) = 3 u 3 ( t ) exp [ i ( k 3 · r - ω 3 t ) ] ,
P ( r , t ) Q ( r , t ) E 3 ( r , t ) 1 ( 1 ) * 3 exp i [ ( k 1 - k 1 + k 3 ) · r - ω 3 t + ω 1 τ - ϕ 1 ] 0 d t u 1 ( t - t - τ ) u 1 * ( t - t ) u 3 ( t ) exp ( - γ t ) + 2 ( 2 ) * 3 × exp i [ ( k 2 - k 2 + k 3 ) · r - ω 3 t + ω 2 τ - ϕ 2 ] 0 d t u 2 ( t - t - τ ) u 2 * ( t - t ) u 3 ( t ) exp ( - γ t ) + 1 ( 2 ) * 3 × exp i [ ( k 1 - k 2 + k 3 ) · r - ( ω 3 + ω d ) t + ω 1 τ - ϕ 2 ] 0 d t u 1 ( t - t - τ ) u 2 * ( t - t ) u 3 ( t ) exp [ - ( γ - i ω d ) t ] + 2 ( 1 ) * 3 exp i [ ( k 2 - k 1 + k 3 ) · r - ( ω 3 - ω d ) t + ω 2 τ - ϕ 1 ] × 0 d t u 2 ( t - t - τ ) u 1 * ( t - t ) u 3 ( t ) exp [ - ( γ + i ω d ) t ] .
I P ( r , t ) 2 0 d s 0 d t u 1 ( t - t - τ ) u 1 ( t - s ) u 1 * ( t - t ) u 1 * ( t - s - τ ) u 3 ( t ) u 3 * ( t ) exp [ - γ ( t + s ) ] + η 2 0 d s 0 d t u 2 ( t - t - τ ) u 2 ( t - s ) u 2 * ( t - t ) u 2 * ( t - s - τ ) u 3 ( t ) u 3 * ( t ) exp [ - γ ( t + s ) ] + η exp ( - i ω d τ ) 0 d s 0 d t u 2 ( t - t - τ ) u 2 * ( t - t ) u 1 ( t - s ) u 1 * ( t - s - τ ) u 3 ( t ) u 3 * ( t ) exp [ - γ ( t + s ) ] + η * exp ( i ω d τ ) 0 d s 0 d t u 1 ( t - t - τ ) u 1 * ( t - t ) u 2 ( t - s ) u 2 * ( t - s - τ ) u 3 ( t ) u 3 * ( t ) exp [ - γ ( t + s ) ] .
u i ( t 1 ) u i ( t 2 ) u i * ( t 3 ) u i * ( t 4 ) = u i ( t 1 ) u i * ( t 3 ) u i ( t 2 ) u i * ( t 4 ) + u i ( t 1 ) u i * ( t 4 u i ( t 2 ) u i * ( t 3 ) .
u i ( t ) u i * ( t - τ ) = exp ( - α i τ ) .
u i ( t - t - τ ) u i ( t - s ) u i * ( t - t ) u i * ( t - s - τ ) = exp ( - 2 α i τ ) + exp ( - 2 α i t - s ) .
I ( γ γ + 2 α 1 ) + η 2 ( γ γ + 2 α 2 ) + exp ( - 2 α 1 τ ) + η 2 exp ( - 2 α 2 τ ) + η exp ( - i ω d τ ) exp [ - ( α 1 + α 2 ) τ ] + η * exp ( i ω d τ ) exp [ - ( α 1 + α 2 ) τ ] .
u i ( t - t ) u i ( t - s - τ ) u i * ( t - t - τ ) u i * ( t - s ) = exp [ - α i ( τ + t - s - t - s - τ ) ] .
I ( γ γ + 2 α 1 ) + η 2 ( γ γ + 2 α 2 ) + ( 2 α 1 γ + 2 α 1 ) exp [ - ( γ + 2 α 1 ) τ ] + η 2 ( 2 α 2 γ + 2 α 2 ) exp [ - ( γ + 2 α 2 ) τ ] + η exp ( i ω d τ ) exp [ - ( α 1 + α 2 ) τ ] + η * exp ( i ω d τ ) exp [ - ( α 1 + α 2 ) τ ] .
I exp ( - 2 α 1 τ ) + η 2 exp ( - 2 α 2 τ ) + η exp ( - i ω d τ ) exp [ - ( α 1 + α 2 ) τ ] + η * exp ( i ω d τ ) exp [ - ( α 1 + α 2 ) τ ] .
Δ ω d = 2 π c ( Δ x p / x p 2 ) .
Δ ω d = 2 π c x p ( Δ x t x t ) .
Δ ω d = 2 π c ξ / x t .
( x t ) max 2 c δ ω 1 .
Δ ω d π ξ δ ω 1 .
ϕ 1 - ϕ 2 ( ω d / c ) [ ( n 0 - 1 ) + ( ω 1 / ω d ) ( n e - n 0 ) ] x sin φ ,
Δ k = [ ( k 1 - k 3 ) 2 + k 2 2 - 2 k 2 ( k 1 - k 3 ) cos θ ] 1 / 2 - k 4 ( k 2 + k 3 - k 1 - k 4 ) - k 2 ( k 1 - k 3 ) 2 ( k 2 + k 3 - k 1 ) θ 2 .
Δ k 2 n c ( ω 2 - ω 1 ) - n ω 2 ( ω 1 - ω 3 ) 2 c ( ω 2 + ω 3 - ω 1 ) θ 2 .
I M / I S = γ 2 / ( γ 2 + ω d 2 ) .
u i ( t - t - τ ) u i ( t - s ) u i * ( t - t ) u j * ( t - s - τ ) = u i ( t - t - τ ) u i ( t - s ) u j * ( t - t ) u j * ( t - s - τ ) = 0

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