Abstract

In the coherent anti-Stokes Raman scattering process, the spectrum of the generated optical phonon depends on the degree of temporal correlation between the pump laser field and the Stokes field. When the two fields are strongly correlated, such as when the Stokes field is generated with stimulated Raman scattering (SRS), the spectral shape of the optical phonon is found experimentally and theoretically to be the same as the gain-narrowed Raman line shape because the laser phase fluctuations cancel out totally, leaving only the collisional noise in the SRS process. When the two fields are uncorrelated, the shape of the optical-phonon spectrum is found to be the same as the Raman line shape without gain narrowing. When two fields are partially correlated, then the two spectral components appear together. We provide a method to measure the degree of correlation between two optical fields that have different central frequencies. The theory developed to interpret the experimental results is an extension of the quantum theory of SRS to include anti-Stokes scattering. We show that only in the high-gain limit can the quantum fluctuations be thought of as arising from a classical noise process.

© 1988 Optical Society of America

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  1. R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
    [CrossRef]
  2. M. G. Raymer, J. Mostowski, J. L. Carlsten, Phys. Rev. A 19, 2304 (1979).
    [CrossRef]
  3. S. A. Akhamanov, Yu. E. D’yakov, L. I. Pavlov, Sov. Phys. JETP 39, 249 (1974); J. Eggleston, R. L. Byer, IEEE J. Quantum Electron. QE-16, 850 (1980); G. P. Dzhotyan, Yu. E. D’yakov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, Sov. Phys. JETP 46, 431 (1977); Sov. J. Quantum Electron. 7, 783 (1977).
    [CrossRef]
  4. E. A. Stappaerts, W. H. Long, H. Komine, Opt. Lett. 5, 4 (1980); J. Rifkin, M. L. Bernt, D. C. MacPherson, J. L. Carlsten, J. Opt. Soc. Am. B 5, 1607 (1988).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1988 (3)

S. Ya. Kilin, Europhys. Lett. 5, 419 (1988).
[CrossRef]

C. Radzewicz, Z. W. Li, M. G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Z. W. Li, C. Radzewicz, M. G. Raymer, Opt. Lett. 13, 491 (1988).
[CrossRef] [PubMed]

1987 (3)

Z. Y. Ou, C. K. Hong, L. Mandel, J. Opt. Soc. Am. B 4, 1574 (1987).
[CrossRef]

L. A. Westling, M. G. Raymer, Phys. Rev. A 32, 4835 (1987).
[CrossRef]

A. M. Bonch-Bruevich, S. G. Przhibel’skii, N. A. Chigir’, Sov. Phys. JETP 65, 439 (1987).

1986 (3)

1985 (2)

1984 (3)

M. Trippenbach, K. Rzazewski, M. G. Raymer, J. Opt. Soc. Am. B 1, 671 (1984).
[CrossRef]

L. A. Rahn, R. L. Farrow, R. P. Lucht, Opt. Lett. 9, 233 (1984); G. S. Agarwal, S. Singh, Phys. Rev. A 25, 3195 (1982); G. S. Agarwal, R. L. Farrow, J. Opt. Soc. Am. B 4, 1596 (1986); R. J. Hall, Opt. Commun. 56, 127 (1985).
[CrossRef]

T. A. Kennedy, S. Swain, J. Phys. B 17, L389 (1984).
[CrossRef]

1982 (2)

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

B. J. Dalton, P. L. Knight, Opt. Commun. 42, 411 (1982).
[CrossRef]

1981 (2)

J. Mostowski, M. G. Raymer, Opt. Commun. 36, 237 (1981); M. G. Raymer, J. Mostowski, Phys. Rev. A 24, 1980 (1981).
[CrossRef]

J. Perina, Opt. Acta 28, 1529 (1981).
[CrossRef]

1980 (1)

1979 (4)

M. G. Raymer, J. Cooper, Phys. Rev. A 20, 2238 (1979).
[CrossRef]

D. Polder, M. F. H. Schuurmans, Q. H. F. Vrehen, Phys. Rev. A 19, 1192 (1979).
[CrossRef]

M. G. Raymer, J. Mostowski, J. L. Carlsten, Phys. Rev. A 19, 2304 (1979).
[CrossRef]

M. A. Yuratich, Molec. Phys. 38, 625 (1979).
[CrossRef]

1978 (1)

A. Laubereau, W. Kaiser, Rev. Mod. Phys. 50, 607 (1978).
[CrossRef]

1975 (1)

M. Takatsuji, Phys. Rev. A 11, 619 (1975).
[CrossRef]

1974 (1)

S. A. Akhamanov, Yu. E. D’yakov, L. I. Pavlov, Sov. Phys. JETP 39, 249 (1974); J. Eggleston, R. L. Byer, IEEE J. Quantum Electron. QE-16, 850 (1980); G. P. Dzhotyan, Yu. E. D’yakov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, Sov. Phys. JETP 46, 431 (1977); Sov. J. Quantum Electron. 7, 783 (1977).
[CrossRef]

1970 (1)

R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
[CrossRef]

Akhamanov, S. A.

S. A. Akhamanov, Yu. E. D’yakov, L. I. Pavlov, Sov. Phys. JETP 39, 249 (1974); J. Eggleston, R. L. Byer, IEEE J. Quantum Electron. QE-16, 850 (1980); G. P. Dzhotyan, Yu. E. D’yakov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, Sov. Phys. JETP 46, 431 (1977); Sov. J. Quantum Electron. 7, 783 (1977).
[CrossRef]

Bischel, W. K.

Bloembergen, N.

R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
[CrossRef]

Bonch-Bruevich, A. M.

A. M. Bonch-Bruevich, S. G. Przhibel’skii, N. A. Chigir’, Sov. Phys. JETP 65, 439 (1987).

Carlsten, J. L.

M. G. Raymer, J. Mostowski, J. L. Carlsten, Phys. Rev. A 19, 2304 (1979).
[CrossRef]

Carman, R. L.

R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
[CrossRef]

Chigir’, N. A.

A. M. Bonch-Bruevich, S. G. Przhibel’skii, N. A. Chigir’, Sov. Phys. JETP 65, 439 (1987).

Cooper, J.

M. G. Raymer, J. Cooper, Phys. Rev. A 20, 2238 (1979).
[CrossRef]

D’yakov, Yu. E.

S. A. Akhamanov, Yu. E. D’yakov, L. I. Pavlov, Sov. Phys. JETP 39, 249 (1974); J. Eggleston, R. L. Byer, IEEE J. Quantum Electron. QE-16, 850 (1980); G. P. Dzhotyan, Yu. E. D’yakov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, Sov. Phys. JETP 46, 431 (1977); Sov. J. Quantum Electron. 7, 783 (1977).
[CrossRef]

Dalton, B. J.

B. J. Dalton, P. L. Knight, Opt. Commun. 42, 411 (1982).
[CrossRef]

Dyer, M. J.

Ezekiel, S.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Farrow, R. L.

L. A. Rahn, R. L. Farrow, R. P. Lucht, Opt. Lett. 9, 233 (1984); G. S. Agarwal, S. Singh, Phys. Rev. A 25, 3195 (1982); G. S. Agarwal, R. L. Farrow, J. Opt. Soc. Am. B 4, 1596 (1986); R. J. Hall, Opt. Commun. 56, 127 (1985).
[CrossRef]

Friberg, S.

Hemmer, P. R.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Hong, C. K.

Injeyan, H.

Kaiser, W.

A. Laubereau, W. Kaiser, Rev. Mod. Phys. 50, 607 (1978).
[CrossRef]

Kennedy, T. A.

T. A. Kennedy, S. Swain, J. Phys. B 17, L389 (1984).
[CrossRef]

Kilin, S. Ya.

S. Ya. Kilin, Europhys. Lett. 5, 419 (1988).
[CrossRef]

Knight, P. L.

B. J. Dalton, P. L. Knight, Opt. Commun. 42, 411 (1982).
[CrossRef]

Komine, H.

Laubereau, A.

A. Laubereau, W. Kaiser, Rev. Mod. Phys. 50, 607 (1978).
[CrossRef]

Leiby, C. C.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Li, Z. W.

C. Radzewicz, Z. W. Li, M. G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Z. W. Li, C. Radzewicz, M. G. Raymer, Opt. Lett. 13, 491 (1988).
[CrossRef] [PubMed]

Lombardi, G. G.

Long, W. H.

Lucht, R. P.

L. A. Rahn, R. L. Farrow, R. P. Lucht, Opt. Lett. 9, 233 (1984); G. S. Agarwal, S. Singh, Phys. Rev. A 25, 3195 (1982); G. S. Agarwal, R. L. Farrow, J. Opt. Soc. Am. B 4, 1596 (1986); R. J. Hall, Opt. Commun. 56, 127 (1985).
[CrossRef]

Mandel, L.

Mostowski, J.

J. Mostowski, M. G. Raymer, Opt. Commun. 36, 237 (1981); M. G. Raymer, J. Mostowski, Phys. Rev. A 24, 1980 (1981).
[CrossRef]

M. G. Raymer, J. Mostowski, J. L. Carlsten, Phys. Rev. A 19, 2304 (1979).
[CrossRef]

Ou, Z. Y.

Pavlov, L. I.

S. A. Akhamanov, Yu. E. D’yakov, L. I. Pavlov, Sov. Phys. JETP 39, 249 (1974); J. Eggleston, R. L. Byer, IEEE J. Quantum Electron. QE-16, 850 (1980); G. P. Dzhotyan, Yu. E. D’yakov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, Sov. Phys. JETP 46, 431 (1977); Sov. J. Quantum Electron. 7, 783 (1977).
[CrossRef]

Perina, J.

J. Perina, Opt. Acta 28, 1529 (1981).
[CrossRef]

Picard, R. H.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Polder, D.

D. Polder, M. F. H. Schuurmans, Q. H. F. Vrehen, Phys. Rev. A 19, 1192 (1979).
[CrossRef]

Przhibel’skii, S. G.

A. M. Bonch-Bruevich, S. G. Przhibel’skii, N. A. Chigir’, Sov. Phys. JETP 65, 439 (1987).

Radzewicz, C.

Z. W. Li, C. Radzewicz, M. G. Raymer, Opt. Lett. 13, 491 (1988).
[CrossRef] [PubMed]

C. Radzewicz, Z. W. Li, M. G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Rahn, L. A.

L. A. Rahn, R. L. Farrow, R. P. Lucht, Opt. Lett. 9, 233 (1984); G. S. Agarwal, S. Singh, Phys. Rev. A 25, 3195 (1982); G. S. Agarwal, R. L. Farrow, J. Opt. Soc. Am. B 4, 1596 (1986); R. J. Hall, Opt. Commun. 56, 127 (1985).
[CrossRef]

Raymer, M. G.

C. Radzewicz, Z. W. Li, M. G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Z. W. Li, C. Radzewicz, M. G. Raymer, Opt. Lett. 13, 491 (1988).
[CrossRef] [PubMed]

L. A. Westling, M. G. Raymer, Phys. Rev. A 32, 4835 (1987).
[CrossRef]

M. G. Raymer, L. A. Westling, J. Opt. Soc. Am. B 2, 1417 (1985).
[CrossRef]

M. Trippenbach, K. Rzazewski, M. G. Raymer, J. Opt. Soc. Am. B 1, 671 (1984).
[CrossRef]

J. Mostowski, M. G. Raymer, Opt. Commun. 36, 237 (1981); M. G. Raymer, J. Mostowski, Phys. Rev. A 24, 1980 (1981).
[CrossRef]

M. G. Raymer, J. Mostowski, J. L. Carlsten, Phys. Rev. A 19, 2304 (1979).
[CrossRef]

M. G. Raymer, J. Cooper, Phys. Rev. A 20, 2238 (1979).
[CrossRef]

Rzazewski, K.

Schuurmans, M. F. H.

D. Polder, M. F. H. Schuurmans, Q. H. F. Vrehen, Phys. Rev. A 19, 1192 (1979).
[CrossRef]

Shimizu, F.

R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
[CrossRef]

Stappaerts, E. A.

Swain, S.

T. A. Kennedy, S. Swain, J. Phys. B 17, L389 (1984).
[CrossRef]

Takatsuji, M.

M. Takatsuji, Phys. Rev. A 11, 619 (1975).
[CrossRef]

Thomas, J. E.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Trippenbach, M.

Vrehen, Q. H. F.

D. Polder, M. F. H. Schuurmans, Q. H. F. Vrehen, Phys. Rev. A 19, 1192 (1979).
[CrossRef]

Wang, C. S.

R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
[CrossRef]

Westling, L. A.

L. A. Westling, M. G. Raymer, Phys. Rev. A 32, 4835 (1987).
[CrossRef]

M. G. Raymer, L. A. Westling, J. Opt. Soc. Am. B 2, 1417 (1985).
[CrossRef]

Willis, C. R.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Yuratich, M. A.

M. A. Yuratich, Molec. Phys. 38, 625 (1979).
[CrossRef]

Europhys. Lett. (1)

S. Ya. Kilin, Europhys. Lett. 5, 419 (1988).
[CrossRef]

J. Opt. Soc. Am. B (6)

J. Phys. B (1)

T. A. Kennedy, S. Swain, J. Phys. B 17, L389 (1984).
[CrossRef]

Molec. Phys. (1)

M. A. Yuratich, Molec. Phys. 38, 625 (1979).
[CrossRef]

Opt. Acta (1)

J. Perina, Opt. Acta 28, 1529 (1981).
[CrossRef]

Opt. Commun. (2)

B. J. Dalton, P. L. Knight, Opt. Commun. 42, 411 (1982).
[CrossRef]

J. Mostowski, M. G. Raymer, Opt. Commun. 36, 237 (1981); M. G. Raymer, J. Mostowski, Phys. Rev. A 24, 1980 (1981).
[CrossRef]

Opt. Lett. (3)

Z. W. Li, C. Radzewicz, M. G. Raymer, Opt. Lett. 13, 491 (1988).
[CrossRef] [PubMed]

L. A. Rahn, R. L. Farrow, R. P. Lucht, Opt. Lett. 9, 233 (1984); G. S. Agarwal, S. Singh, Phys. Rev. A 25, 3195 (1982); G. S. Agarwal, R. L. Farrow, J. Opt. Soc. Am. B 4, 1596 (1986); R. J. Hall, Opt. Commun. 56, 127 (1985).
[CrossRef]

E. A. Stappaerts, W. H. Long, H. Komine, Opt. Lett. 5, 4 (1980); J. Rifkin, M. L. Bernt, D. C. MacPherson, J. L. Carlsten, J. Opt. Soc. Am. B 5, 1607 (1988).
[CrossRef]

Phys. Rev. A (8)

L. A. Westling, M. G. Raymer, Phys. Rev. A 32, 4835 (1987).
[CrossRef]

D. Polder, M. F. H. Schuurmans, Q. H. F. Vrehen, Phys. Rev. A 19, 1192 (1979).
[CrossRef]

R. L. Carman, F. Shimizu, C. S. Wang, N. Bloembergen, Phys. Rev. A 2, 60 (1970).
[CrossRef]

M. G. Raymer, J. Mostowski, J. L. Carlsten, Phys. Rev. A 19, 2304 (1979).
[CrossRef]

C. Radzewicz, Z. W. Li, M. G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

W. K. Bischel, M. J. Dyer, Phys. Rev. A 33, 3113 (1986).
[CrossRef] [PubMed]

M. Takatsuji, Phys. Rev. A 11, 619 (1975).
[CrossRef]

M. G. Raymer, J. Cooper, Phys. Rev. A 20, 2238 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).
[CrossRef]

Rev. Mod. Phys. (1)

A. Laubereau, W. Kaiser, Rev. Mod. Phys. 50, 607 (1978).
[CrossRef]

Sov. Phys. JETP (2)

S. A. Akhamanov, Yu. E. D’yakov, L. I. Pavlov, Sov. Phys. JETP 39, 249 (1974); J. Eggleston, R. L. Byer, IEEE J. Quantum Electron. QE-16, 850 (1980); G. P. Dzhotyan, Yu. E. D’yakov, I. G. Zubarev, A. B. Mironov, S. I. Mikhailov, Sov. Phys. JETP 46, 431 (1977); Sov. J. Quantum Electron. 7, 783 (1977).
[CrossRef]

A. M. Bonch-Bruevich, S. G. Przhibel’skii, N. A. Chigir’, Sov. Phys. JETP 65, 439 (1987).

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Figures (4)

Fig. 1
Fig. 1

Illustration of the CARS process. The frequencies (in radians per second) of pump laser ωL1, generated Stokes field ωS, monochromatic probe field ωL2, and anti-Stokes field ωAS satisfy ωL1 + ωL2 = ωS + ωAS. The detuning from Raman resonance is Δ = ω21ωL1 + ωS, where ω21 is the Raman shift of the medium.

Fig. 2
Fig. 2

Experimental setup. Stokes light generated in Raman generator Cell(1) is phase correlated to pump-laser field EL1. The pump field is delayed by time td and combined with probe field EL2 in Cell(2). The anti-Stokes spectrum is measured with a Fabry–Perot interferometer and photodiode P: BS, beam splitter; DM, dichroic mirror, CF, colored-glass filter; ND, neutral-density filter; CC, corner cube; M1, mirror; L1–L4, lenses.

Fig. 3
Fig. 3

Measured average power spectra of (curve 1) pump laser and (curve 2) the generated anti-Stokes field, for the case of zero Raman detuning (both cells at 48 atm). The narrow width of the anti-Stokes field illustrates the cancellation of the laser and Stokes phase fluctuations. The curves are normalized to a peak height of unity, and the horizontal scale is with respect to the center frequency of each field.

Fig. 4
Fig. 4

Power spectra of anti-Stokes field for detuning of Δ = 3.8 GHz and for various values of time delay td between pump and Stokes fields. The frequency scale is labeled with respect to the location of the narrow peak, which is predicted to occur at ωL1 + ωL2ωS. (a), (b), (c) Measured anti-Stokes spectra corresponding to delay td equal to 3.3, 37, and 160 psec, respectively. (d), (e), (f) Theoretical predictions for anti-Stokes spectra, evaluated by combining Eqs. (8) and (15), corresponding to the optical delay td in (a), (b), and (c), respectively. (g), (h), (i): Solid curves are the power spectra Se(ω) of the effective driving field, evaluated using Eq. (15), corresponding to optical delay td in (a), (b), and (c), respectively. The dashed curves of (g), (h), and (i) are the Lorentzian Raman line shapes, which, when multiplied by the solid curve Se(ω), give the corresponding spectra in (d), (e), and (f). Γ, Γ1, ΓL1, Δ, and gl are taken to be the same as the experimental parameters. All curves are normalized to a peak height of unity.

Equations (37)

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z E AS ( z , t ) = - i κ 2 A E L 2 ( t ) Q ( t ) exp ( - i Δ k · z ) ,
t Q ( t ) = ( i Δ - Γ ) Q ( t ) - i κ 1 * E L 1 ( t ) E S * ( t ) ,
E AS ( z , t ) = - i κ 2 A E L 2 ( t ) Q ( t ) z .
E L 1 ( t ) = A L 1 exp [ - i ϕ L 1 ( t ) ] ,
E S ( t ) = ξ ( t - t d ) E L 1 ( t - t d ) ,
E e ( t ) = A L 1 2 ξ ( t - t d ) exp [ - i ϕ L 1 ( t ) + i ϕ L 1 ( t - t d ) ] .
Q ( t ) = - i κ 1 * 0 t exp [ ( i Δ - Γ ) ( t - t ) ] E e ( t ) d t .
S Q ( ω ) = κ 1 2 1 Γ 2 + ( ω - Δ ) 2 S e ( ω ) ,
S F ( ω ) = - + F ( t + τ ) F * ( t ) e - i ω τ d τ ,
S e ( ω ) = - + E L 1 ( t + τ ) E S * ( t + τ ) E L 1 * ( t ) E S ( t ) e - i ω τ d t ,
S ξ ( ω ) = I S I L 1 2 ( π g l ) 1 / 2 Γ 1 exp ( - g l Γ 1 2 ω 2 ) ,
S e ( ω ) = - + E L 1 ( t + τ ) E L 1 * ( t - t d + τ ) E L 1 * ( t ) E L 1 ( t - t d ) × e - i ω τ d τ S ξ ( ω ) .
E L 1 ( t + τ ) E L 1 * ( t ) = E L 1 2 exp ( - Γ L 1 τ ) ,
E L 1 ( t + τ ) E L 1 * ( t - t d + τ ) E L 1 * ( t ) E L 1 ( t - t d ) = I L 1 2 exp [ Γ L 1 ( t d + τ + t d - τ - 2 t d - 2 τ ) ] .
S e ( ω ) + I L 1 I S 2 ( π g l ) 1 / 2 Γ 1 { exp ( - 2 Γ L 1 t d ) [ 2 π δ ( ω ) - L ( ω ) × cos ( ω t d ) - 2 Γ L 1 L ( ω ) sin ( ω t d ) ω ] + L ( ω ) } exp ( - g l Γ 1 2 ω 2 ) ,
S Q ( ω , t d 0 ) = κ 1 2 I L 1 I S 4 π ( π g l ) 1 / 2 Γ 1 1 Γ 2 + Δ 2 exp ( - g l Γ 1 2 ω 2 ) ,
S Q ( ω , t d ± ) = κ 1 2 I L 1 I S ( 2 ) 3 / 2 π L ( Δ ) 1 Γ 2 + ( ω - Δ ) 2
E ^ S ( - ) ( l , t ) = E ^ S ( - ) ( 0 , t ) + ξ ^ ( l , t ) E L 1 ( t ) ,
ξ ( l , t ) = ( κ 1 κ 2 l ) 1 / 2 0 t d t exp [ - Γ 1 ( t - t ) ] E L 1 * ( t ) E ^ S ( - ) ( 0 , t ) × I 1 ( { 4 κ 1 κ 2 l [ ( p ( t ) - p ( t ) ] } 1 / 2 ) [ p ( t ) - p ( t ) ] 1 / 2 - i κ 2 exp ( - Γ 1 t ) 0 l d z Q ^ ( z , 0 ) I 0 ( [ 4 κ 1 κ 2 ( l - z ) p ( t ) ] 1 / 2 ) - i κ 2 0 t d t 0 l d z exp [ - Γ 1 ( t - t ) ] F ^ ( z , t ) × I 0 ( { 4 κ 1 κ 2 ( l - z ) [ p ( t ) - p ( t ) ] } 1 / 2 ) ,
p ( t ) = 0 t E L 1 ( t ) 2 d t .
Q ^ ( z , 0 ) Q ^ ( z , 0 ) = ( A N ) - 1 δ ( z - z ) ,
F ^ ( z , t ) F ^ ( z , t ) = 2 Γ 1 ( A N ) - 1 δ ( z - z ) δ ( t - t ) ,
Q ^ ( z , 0 ) Q ^ ( z , 0 ) = F ^ ( z , t ) F ^ ( z , t ) = 0 ,
Q ^ ( z , 0 ) F ^ ( z , t ) = Q ^ ( z , 0 ) F ^ ( z , t ) = 0 ,
E ^ S ( + ) ( 0 , t ) F ^ ( z , t ) = F ^ ( z , t ) E ^ S ( - ) ( 0 , t ) = 0 ,
E ^ S ( - ) ( l , t ) = ξ ^ ( l , t ) E L ( t ) ,
E ^ AS ( - ) ( z , t ) = E ^ AS ( - ) ( 0 , t ) - i κ 2 A E L 2 ( t ) z 0 t exp [ ( i Δ - Γ ) ( t - t ) ] × E L 1 ( t ) E ^ S ( + ) ( l , t ) d t .
ξ ^ ( t ) ξ ^ ( t + τ ) = I S I L 1 exp ( - Γ 1 2 4 g l τ 2 ) ,
I S = 2 π ω S A c Γ 1 2 ( π g l ) 1 / 2 e g l ,
ξ ^ ( t + τ ) ξ ^ ( t ) = κ 1 κ 2 z 0 t + τ d t 0 t d t exp [ - Γ 1 ( 2 t + τ - t - t ) ] × E L ( t ) E L * ( t ) E ^ S ( + ) ( 0 , t ) E ^ S ( - ) ( 0 , t ) × I 1 ( { 4 κ 1 κ 2 l [ p ( t + τ ) - p ( t ) ] } 1 / 2 ) [ p ( t + τ ) - p ( t ) ] 1 / 2 × I 1 ( { 4 κ 1 κ 2 l [ p ( t ) - p ( t ) ] } 1 / 2 ) [ p ( t ) - p ( t ) ] 1 / 2 .
[ E ^ S ( + ) ( 0 , t ) , E ^ S ( - ) ( 0 , t ) ] = 2 π ω S A c δ ( t - t ) .
ξ ^ ( t + τ ) ξ ^ ( t ) = 2 π ω S A c f ( τ ) I L 1 ,
f ( τ ) = ( α z ) 2 4 exp ( - Γ 1 τ ) 0 d t exp ( - 2 Γ 1 t ) I 1 ( [ α l t ] 1 / 2 ) [ α l t ] 1 / 2 × I 1 ( [ α l ( t + τ ) ] 1 / 2 ) [ α l ( t + τ ) ] 1 / 2 ,
P ( ω ) = 1 π Re 0 f ( τ ) e - i ω τ d τ .
P ( ω ) = 1 4 π 2 Re ( α l ) 2 4 | 0 d t e i ω t e - Γ t exp [ ( α l t ) 1 / 2 ] ( α l t ) 3 / 4 | 2 .
P ( ω ) 1 2 π e g l exp ( - g l Γ 1 2 ω 2 ) ,
ξ ^ ( t + τ ) ξ ^ ( t ) = I S I L 1 exp ( - Γ 1 2 4 g l τ 2 ) .

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