Abstract

The coexisting Rayleigh and Brillouin attosecond sum-frequency polarization beats (ASPBs) and the coexisting Raman, Rayleigh, and Brillouin ASPBs have also been investigated via homodyne and heterodyne detections, respectively. The Raman, Rayleigh, and Brillouin-enhanced four-wave mixing processes strongly compete with each other in ASPBs. The heterodyne-detected signal of ASPBs offers the phase dispersion of the third-order nonlinear susceptibility in the homogeneous broadening material.

© 2010 Optical Society of America

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  1. G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, 1981).
  2. A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
    [CrossRef] [PubMed]
  3. P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
    [CrossRef] [PubMed]
  4. Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
    [CrossRef]
  5. Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
    [CrossRef]
  6. Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
    [CrossRef]
  7. Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
    [CrossRef] [PubMed]
  8. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
    [CrossRef] [PubMed]
  9. P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
    [CrossRef] [PubMed]
  10. A. G. Jacobson and Y. R. Shen, “Coherent Brillouin spectroscopy,” Appl. Phys. Lett. 34, 464-467 (1979).
    [CrossRef]
  11. M. D. Skeldon, P. Narum, and R. W. Boyd, “Non-frequency-shifted, high-fidelity phase conjugation with aberrated pump waves by Brillouin-enhanced four-wave mixing,” Opt. Lett. 12, 343-345 (1987).
    [CrossRef] [PubMed]
  12. P. Narum and R. W. Boyd, “Non-frequency shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. 23, 1211-1216 (1987).
    [CrossRef]
  13. C. J. Randall and J. R. Albritton, “Chaotic nonlinear stimulated Brillouin scattering,” Phys. Rev. Lett. 52, 1887-1890 (1984).
    [CrossRef]
  14. B. Ya Zel'dovich and V. V. Shkunov, “Characteristics of stimulated scattering in opposite pump beams,” Sov. J. Quantum Electron. 12, 223-225 (1982).
    [CrossRef]
  15. C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
    [CrossRef]
  16. T. F. Schulz, P. P. Aung, L. R. Weisel, K. M. Cosert, M. W. Gealy, and D. J. Ulness, “Complete cancellation of noise by means of color-locking in nearly degenerate, four-wave mixing of noisy light,” J. Opt. Soc. Am. B 22, 1052-1061 (2005).
    [CrossRef]
  17. D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761-771 (1997).
    [CrossRef]
  18. A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034-2049 (1980).
    [CrossRef]

2009 (2)

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
[CrossRef] [PubMed]

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

2007 (1)

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

2005 (2)

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

T. F. Schulz, P. P. Aung, L. R. Weisel, K. M. Cosert, M. W. Gealy, and D. J. Ulness, “Complete cancellation of noise by means of color-locking in nearly degenerate, four-wave mixing of noisy light,” J. Opt. Soc. Am. B 22, 1052-1061 (2005).
[CrossRef]

2004 (1)

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[CrossRef]

2003 (1)

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

2002 (1)

P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
[CrossRef] [PubMed]

1997 (1)

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761-771 (1997).
[CrossRef]

1992 (1)

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

1987 (3)

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

M. D. Skeldon, P. Narum, and R. W. Boyd, “Non-frequency-shifted, high-fidelity phase conjugation with aberrated pump waves by Brillouin-enhanced four-wave mixing,” Opt. Lett. 12, 343-345 (1987).
[CrossRef] [PubMed]

P. Narum and R. W. Boyd, “Non-frequency shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. 23, 1211-1216 (1987).
[CrossRef]

1984 (1)

C. J. Randall and J. R. Albritton, “Chaotic nonlinear stimulated Brillouin scattering,” Phys. Rev. Lett. 52, 1887-1890 (1984).
[CrossRef]

1982 (1)

B. Ya Zel'dovich and V. V. Shkunov, “Characteristics of stimulated scattering in opposite pump beams,” Sov. J. Quantum Electron. 12, 223-225 (1982).
[CrossRef]

1980 (1)

A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034-2049 (1980).
[CrossRef]

1979 (1)

A. G. Jacobson and Y. R. Shen, “Coherent Brillouin spectroscopy,” Appl. Phys. Lett. 34, 464-467 (1979).
[CrossRef]

Ackerhalt, J. R.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

Albrecht, A. C.

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761-771 (1997).
[CrossRef]

Albritton, J. R.

C. J. Randall and J. R. Albritton, “Chaotic nonlinear stimulated Brillouin scattering,” Phys. Rev. Lett. 52, 1887-1890 (1984).
[CrossRef]

Anderson, B.

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
[CrossRef] [PubMed]

Aung, P. P.

Boyd, R. W.

P. Narum and R. W. Boyd, “Non-frequency shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. 23, 1211-1216 (1987).
[CrossRef]

M. D. Skeldon, P. Narum, and R. W. Boyd, “Non-frequency-shifted, high-fidelity phase conjugation with aberrated pump waves by Brillouin-enhanced four-wave mixing,” Opt. Lett. 12, 343-345 (1987).
[CrossRef] [PubMed]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

Brown, A. W.

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Cosert, K. M.

DeMott, D. C.

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761-771 (1997).
[CrossRef]

Eesley, G. L.

G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, 1981).

Fu, P. M.

P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

Gaeta, A. L.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

Gan, C. L.

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Ge, H.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Gealy, M. W.

Georges, A. T.

A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034-2049 (1980).
[CrossRef]

Jacobson, A. G.

A. G. Jacobson and Y. R. Shen, “Coherent Brillouin spectroscopy,” Appl. Phys. Lett. 34, 464-467 (1979).
[CrossRef]

Jiang, Q.

P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

Jiang, T.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Khadka, U.

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
[CrossRef] [PubMed]

Li, C. S.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Li, L.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Lu, K. Q.

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Ma, R. Q.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Mi, X.

P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

Milonni, P. W.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

Narum, P.

M. D. Skeldon, P. Narum, and R. W. Boyd, “Non-frequency-shifted, high-fidelity phase conjugation with aberrated pump waves by Brillouin-enhanced four-wave mixing,” Opt. Lett. 12, 343-345 (1987).
[CrossRef] [PubMed]

P. Narum and R. W. Boyd, “Non-frequency shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. 23, 1211-1216 (1987).
[CrossRef]

Nie, Z. Q.

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Randall, C. J.

C. J. Randall and J. R. Albritton, “Chaotic nonlinear stimulated Brillouin scattering,” Phys. Rev. Lett. 52, 1887-1890 (1984).
[CrossRef]

Saldana, J.

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Schulz, T. F.

Shen, Y. R.

A. G. Jacobson and Y. R. Shen, “Coherent Brillouin spectroscopy,” Appl. Phys. Lett. 34, 464-467 (1979).
[CrossRef]

Shkunov, V. V.

B. Ya Zel'dovich and V. V. Shkunov, “Characteristics of stimulated scattering in opposite pump beams,” Sov. J. Quantum Electron. 12, 223-225 (1982).
[CrossRef]

Si, J. H.

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Skeldon, M. D.

Song, J. P.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Ulness, D. J.

Weisel, L. R.

Wu, Y.

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[CrossRef]

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Xiao, M.

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
[CrossRef] [PubMed]

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Ya Zel'dovich, B.

B. Ya Zel'dovich and V. V. Shkunov, “Characteristics of stimulated scattering in opposite pump beams,” Sov. J. Quantum Electron. 12, 223-225 (1982).
[CrossRef]

Yang, X.

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[CrossRef]

Yu, X. J.

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Yu, Z. H.

P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

Zhang, Y. P.

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
[CrossRef] [PubMed]

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Zhang, Z. G.

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

Zhao, Y.

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Zhu, Y. F.

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

A. G. Jacobson and Y. R. Shen, “Coherent Brillouin spectroscopy,” Appl. Phys. Lett. 34, 464-467 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Narum and R. W. Boyd, “Non-frequency shifted phase conjugation by Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. 23, 1211-1216 (1987).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. A (7)

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761-771 (1997).
[CrossRef]

A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034-2049 (1980).
[CrossRef]

C. L. Gan, Y. P. Zhang, Z. Q. Nie, Y. Zhao, K. Q. Lu, J. H. Si, and M. Xiao, “Competition between Raman- and Rayleigh-enhanced four-wave mixings in attosecond polarization beats,” Phys. Rev. A 79, 023802 (2009).
[CrossRef]

Y. P. Zhang, C. L. Gan, L. Li, R. Q. Ma, J. P. Song, T. Jiang, X. J. Yu, C. S. Li, H. Ge, and K. Q. Lu, “Rayleigh-enhanced attosecond sum-frequency polarization beats via twin color-locking noisy lights,” Phys. Rev. A 72, 013812 (2005).
[CrossRef]

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[CrossRef]

P. M. Fu, Z. H. Yu, X. Mi, Q. Jiang, and Z. G. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530-1539 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, “Instabilities and chaos in the polarizations of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

P. M. Fu, Q. Jiang, X. Mi, and Z. H. Yu, “Rayleigh-type nondegenerate four-wave mixing: Ultrafast measurement and field correlation,” Phys. Rev. Lett. 88, 113902 (2002).
[CrossRef] [PubMed]

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Phys. Rev. Lett. 102, 013601 (2009).
[CrossRef] [PubMed]

C. J. Randall and J. R. Albritton, “Chaotic nonlinear stimulated Brillouin scattering,” Phys. Rev. Lett. 52, 1887-1890 (1984).
[CrossRef]

Sov. J. Quantum Electron. (1)

B. Ya Zel'dovich and V. V. Shkunov, “Characteristics of stimulated scattering in opposite pump beams,” Sov. J. Quantum Electron. 12, 223-225 (1982).
[CrossRef]

Other (1)

G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, 1981).

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Figures (13)

Fig. 1
Fig. 1

Phase-conjugation geometries of the coexisting Raman, Rayleigh, and Brillouin ASPBs. E 1 ( E 1 ) and E 2 ( E 2 ) are the fields with frequency ω 1 and ω 2 in beam 1 (beam 2), respectively. E 3 with frequency ω 3 is the field in beam 3. The enhanced FWM signals or the generated ASPB signals in beam 4 propagate almost along the opposite direction of beam 2.

Fig. 2
Fig. 2

The Rayleigh and Brillouin ASPB signals versus γ M τ for χ M / χ T = 10 , γ T / γ M = 2 × 10 6 , Δ 1 = 0 , r = 0 , ω 1 / γ M = 127450 , ω 2 / γ M = 120781 , α 1 / γ M = α 2 / γ M = α 3 / γ M = 1 , γ Br / γ M = 0.05 , ν Br / γ M = 0.33 , χ b / χ T = 3.33 , and η = 1 . Three curves are shown in CFM (solid curve), GAM (dotted curve), and PDM (dashed curve).

Fig. 3
Fig. 3

The Raman, Rayleigh, and Brillouin ASPB signals versus γ R τ with the same parameters as in Fig. 2.

Fig. 4
Fig. 4

The heterodyne detection spectra versus Δ 1 / γ M of the coexisting Raman, Rayleigh, and Brillouin ASPBs with (a) τ = ( 2 π + Δ k r ) / ( ω 1 + ω 2 ) for the real part of χ A 2 , while with (b) τ = ( 2 π + π / 2 + Δ k r ) / ( ω 1 + ω 2 ) for the imaginary part of χ A 2 when τ > 0 ; (c) τ = ( 2 π + Δ k r ) / ( ω 1 + ω 2 ) for the real part of χ A 2 , while with (d) τ = ( 2 π + π / 2 + Δ k r ) / ( ω 1 + ω 2 ) for the imaginary part of χ A 2 when τ < 0 . The other parameters are for χ M / χ T = 1000 , γ T / γ M = 1 , α 1 / γ M = α 2 / γ M = α 3 / γ M = 1 , ω 1 / γ M = 127450 , ω 2 / γ M = 120781 , γ Br / γ M = 0.1 , ν Br / γ M = 0.5 , χ b / χ T = 10 , and η = 10 . Theoretical curves are shown in CFM (dashed curve), PDM (dotted curve), GAM (dot-dashed curve), and the model with cw laser beams (dot-dot-dashed curve).

Fig. 5
Fig. 5

The heterodyne detection spectra versus Δ 2 / γ R of the coexisting Raman, Rayleigh, and Brillouin ASPBs for (a) the real part of χ B , while with (b) the imaginary part of χ B when τ > 0 . (c) The real part of χ B , while with (d) the imaginary part of χ B when τ < 0 . Other parameters are for χ M / χ T = 1 , γ T / γ M = 1 × 10 6 , γ Br / γ M = 0.05 , ν Br / γ M = 0.33 , χ b / χ T = 3.33 , and η = 0.1 . Theoretical curves are shown in CFM (dashed curve), PDM (dotted curve), GAM (dot-dashed curve), and the model with cw laser beams (dot-dot-dashed curve).

Fig. 6
Fig. 6

The heterodyne detection spectra versus Δ 1 / γ R and Δ 2 / γ R of the coexisting ASPBs for η = 0.1 , χ M / χ R = 0.1 , γ R / γ M = 1 , χ T / χ R = 0.1 , γ T / γ M = 1 × 10 6 , α 1 / γ M = α 2 / γ M = α 3 / γ M = 1 , γ Br / γ M = 0.05 , ν Br / γ M = 0.3 , χ b / χ T = 3 with (a) τ = ( 2 n π Δ k r θ B ) / ( ω 1 + ω 2 ) for the real and (b) τ = ( 2 n π + π / 2 Δ k r θ B ) / ( ω 1 + ω 2 ) for the imaginary part of χ B , respectively.

Fig. 7
Fig. 7

The heterodyne detection spectra versus Δ 1 / γ M of the Brillouin-enhanced ASPBs with (a) τ = ( 2 π + Δ k r ) / ( ω 1 + ω 2 ) for the real part of χ Br , while with (b) τ = ( 2 π + π / 2 + Δ k r ) / ( ω 1 + ω 2 ) for the imaginary part of χ Br when τ > 0 ; (c) τ = ( 2 π + Δ k r ) / ( ω 1 + ω 2 ) for the real part of χ Br , while with (d) τ = ( 2 π + π / 2 + Δ k r ) / ( ω 1 + ω 2 ) for the imaginary part of χ Br when τ < 0 . Other parameters are for χ M / χ T = 1000 , γ T / γ M = 5 , α 1 / γ M = α 2 / γ M = α 3 / γ M = 1 , ω 1 / γ M = 127450 , ω 2 / γ M = 120781 , γ Br / γ M = 10 , ν Br / γ M = 2 , χ Br / χ T = 0.1 , and η = 10 . Theoretical curves are shown in CFM (solid curve), PDM (dashed curve), GAM (dot curve), and the model with cw laser beams (dot-dashed curve).

Fig. 8
Fig. 8

The heterodyne detection spectra versus Δ 2 / γ R of (a) the pure Raman ASPB, (b) the coexisting Raman and Rayleigh ASPBs, and (c) the coexisting Raman, Rayleigh, and Brillouin ASPBs with τ = ( ± 2 π + Δ k r ) / ( ω 1 + ω 2 ) for the real part of χ B . The parameters are χ M / χ T = 1 , γ T / γ M = 1 × 10 6 , α 1 / γ M = α 2 / γ M = α 3 / γ M = 1 , ω 1 / γ M = 127450 , ω 2 / γ M = 119924 , η = 0.1 , γ Br / γ M = 0.05 , ν Br / γ M = 0.3 , χ b / χ T = 3 , and χ R / χ T = 5 . Theoretical curves are shown in the model when τ > 0 (solid curve), τ < 0 (dashed curve), and the model with cw laser beams (dotted curve).

Fig. 9
Fig. 9

The heterodyne detection spectra versus Δ 2 / γ R of (a) the pure Raman ASPB, (b) the coexisting Raman and Rayleigh ASPBs, and (c) the coexisting of Raman, Rayleigh, and Brillouin ASPBs with τ = ( ± 2 π + π / 2 + Δ k r ) / ( ω 1 + ω 2 ) for the imaginary part of χ B . The parameters are the same as in Fig. 8. Theoretical curves are shown in the model when τ > 0 (solid curve), τ < 0 (dashed curve), and the model with cw laser beams (dotted curve).

Fig. 10
Fig. 10

The heterodyne detection spectra with τ = ( ± 2 π + Δ k r ) / ( ω 1 + ω 2 ) versus Δ 1 / γ M of (a) the pure Rayleigh ASPB, (b) the coexisting Raman and Rayleigh ASPBs for the real part of χ A 1 , and (c) the coexisting Raman, Rayleigh, and Brillouin ASPBs for the real part of χ A 2 . The parameters are χ M / χ T = 1 , γ T / γ M = 1 × 10 6 , α 1 / γ M = α 2 / γ M = α 3 / γ M = 1 , γ Br / γ M = 0.05 , ν Br / γ M = 0.3 , χ b / χ T = 3 , χ R / χ T = 5 , ω 1 / γ M = 127450 , ω 2 / γ M = 119924 , and η = 10 . Theoretical curves are shown in the model when τ > 0 (solid curve), τ < 0 (dashed curve), and the model with cw laser beams (dotted curve).

Fig. 11
Fig. 11

The heterodyne detection spectra with τ = ( ± 2 π + π / 2 + Δ k r ) / ( ω 1 + ω 2 ) versus Δ 1 / γ M of (a) the pure Rayleigh ASPB, (b) the coexisting Raman and Rayleigh ASPBs for the real part of χ A 1 , and (c) the coexisting Raman, Rayleigh, and Brillouin ASPBs for the real part of χ A 2 . The parameters are the same as in Fig. 10. Theoretical curves are shown in the model when τ > 0 (solid curve), τ < 0 (dashed curve), and the model with cw laser beams (dotted curve).

Fig. 12
Fig. 12

Phase angle θ B in the pure Raman enhanced-FWM versus Δ 2 / γ R for (a) χ M / χ R = χ T / χ R = 0.2 (solid curve), 0.3 (dashed curve), and 0.5 (dotted curve), and for (c) α 1 / γ M = α 2 / γ M = α 3 / γ M = 0.3 (solid curve), 0.6 (dashed curve), and 0.9 (dotted curve). θ B in the coexisting Raman, Rayleigh, and Brillouin-enhanced FWMs versus Δ 2 / γ R for (b) χ M / χ R = χ T / χ R = 0.3 (solid curve), 0.6 (dashed curve), and 0.9 (dotted curve), and for (d) α 1 / γ M = α 2 / γ M = α 3 / γ M = 0.9 (solid curve), 1.2 (dashed curve), and 2 (dotted curve).

Fig. 13
Fig. 13

Phase angle θ A 1 in the pure Rayleigh-enhanced FWM versus Δ 1 / γ M for (a) χ T / χ M = 0.1 (solid curve), 0.5 (dashed curve), and 1 (dotted curve), and for (d) α 1 / γ M = α 3 / γ M = 0.1 (solid curve), 0.5 (dashed curve), and 1 (dotted curve). Phase angle θ A 2 in the coexisting Rayleigh and Brillouin-enhanced FWMs versus Δ 1 / γ M for (b) χ T / χ M = 0.1 (solid curve), 0.5 (dashed curve), and 1 (dotted curve) and for (e) α 1 / γ M = α 3 / γ M = 0.1 (solid curve), 0.5 (dashed curve), and 1 (dotted curve). Phase angle θ A 2 in the coexisting Raman, Rayleigh, and Brillouin-enhanced FWMs versus Δ 1 / γ M for (c) χ M / χ R = 0.05 (solid curve), 0.1 (dashed curve), and 0.3 (dotted curve), and for (f) α 1 / γ M = α 2 / γ M = α 3 / γ M = 0.1 (solid curve), 0.5 (dashed curve), and 1 (dotted curve).

Equations (44)

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E 1 ( r , t ) = A 1 ( r , t ) exp ( i ω 1 t ) = ε 1 u 1 ( t ) exp [ i ( k 1 r ω 1 t ) ] ,
E 1 ( r , t ) = A 1 ( r , t ) exp [ i ω 1 ( t τ ) ] = ε 1 u 1 ( t τ ) exp [ i ( k 1 r ω 1 t + ω 1 τ ) ] ,
E 3 ( r , t ) = A 3 ( r , t ) exp ( i ω 3 t ) = ε 3 u 3 ( t ) exp [ i ( k 3 r ω 3 t ) ] .
d Q M 1 / d t + γ M Q M 1 = χ M γ M E 1 ( r , t ) [ E 1 ( r , t ) ] ,
d Q T 1 / d t + γ T Q T 1 = χ T γ T E 1 ( r , t ) [ E 1 ( r , t ) ] ,
d Q R M / d t + γ M Q R M = χ M γ M [ E 1 ( r , t ) ] E 3 ( r , t ) ,
d Q R T / d t + γ T Q R T = χ T γ T [ E 1 ( r , t ) ] E 3 ( r , t ) .
P M 1 = Q M 1 ( r , t ) E 3 ( r , t ) = χ M γ M S 1 ( r ) ε 1 ( ε 1 ) ε 3 0 u 1 ( t t ) u 1 ( t t τ ) u 3 ( t ) exp ( γ M t ) d t ,
P T 1 = Q T 1 ( r , t ) E 3 ( r , t ) = χ T γ T S 1 ( r ) ε 1 ( ε 1 ) ε 3 0 u 1 ( t t ) u 1 ( t t τ ) u 3 ( t ) exp ( γ T t ) d t ,
P R M = Q M 1 ( r , t ) E 1 ( r , t ) = χ M γ M S 1 ( r ) ( ε 1 ) ε 3 ε 1 0 u 1 ( t t τ ) u 3 ( t t ) u 1 ( t ) exp [ ( γ M i Δ 1 ) t ] d t ,
P R T = Q T 1 ( r , t ) E 1 ( r , t ) = χ T γ T S 1 ( r ) ( ε 1 ) ε 3 ε 1 0 u 1 ( t t τ ) u 3 ( t t ) u 1 ( t ) exp [ ( γ T i Δ 1 ) t ] d t .
d 2 Q Br / d t 2 + γ Br d Q Br / d t + ν Br 2 Q Br = χ b [ E 2 ( r , t ) ] E 3 ( r , t ) .
P Br = 0 χ b γ Br 2 ν Br 2 Δ 1 2 i Δ 1 γ Br ε 1 ε 2 ε 3 S 1 ( r ) u 1 ( t τ ) u 3 ( t ) u 1 ( t ) d t .
d Q M 2 / d t + γ M Q M 2 = χ M γ M E 2 ( r , t ) [ E 2 ( r , t ) ] ,
d Q T 2 / d t + γ T Q T 2 = χ T γ T E 2 ( r , t ) [ E 2 ( r , t ) ] ,
d Q R / d t + ( γ R i Δ 2 ) Q R = χ R γ R [ E 2 ( r , t ) ] E 3 ( r , t ) .
E p 1 = E 2 ( r , t ) = A 2 ( r , t ) exp [ i ω 2 ( t τ ) ] = ε 2 u 2 ( t τ ) exp [ i ( k 2 r ω 2 t + ω 2 τ ) ] ,
E p 2 = E 2 ( r , t ) = A 2 ( r , t ) exp ( i ω 2 t ) = ε 2 u 2 ( t ) exp [ i ( k 2 r ω 2 t ) ] .
P M 2 = Q M 2 ( r , t ) E 3 ( r , t ) = χ M γ M S 2 ( r ) ( ε 2 ) ε 2 ε 3 0 u 2 ( t t ) u 2 ( t t τ ) u 3 ( t ) exp ( γ M t ) d t ,
P T 2 = Q T 2 ( r , t ) E 3 ( r , t ) = χ T γ T S 2 ( r ) ( ε 2 ) ε 2 ε 3 0 u 2 ( t t ) u 2 ( t t τ ) u 3 ( t ) exp ( γ T t ) d t ,
P R = Q R ( r , t ) E 2 ( r , t ) = i χ R γ R S 2 ( r ) ( ε 2 ) ε 3 ε 2 0 u 2 ( t t ) u 3 ( t t ) u 2 ( t τ ) exp [ ( γ R i Δ 2 ) t ] d t .
| P A 2 | 2 = P M 1 P M 1 + P M 1 P T 1 + P M 1 P R M + P M 1 P R T + P M 1 P Br + P T 1 P M 1 + P T 1 P T 1 + P T 1 P R M + P T 1 P R T + P T 1 P Br + P R M P M 1 + P R M P T 1 + P R M P R M + P R M P R T + P R M P Br + P R T P M 1 + P R T P T 1 + P R T P R M + P R T P R T + P R T P Br + P Br P M 1 + P Br P T 1 + P Br P R M + P Br P R T + P Br P Br .
P R M P M 1 = i χ M 2 γ M 2 ( ε 1 ) ε 3 ε 1 S 1 ( r ) ε 1 ( ε 1 ) ( ε 3 ) S 1 ( r ) 0 d t 0 d s u 1 ( t τ ) u 1 ( t s ) u 1 ( t t ) u 1 ( t s τ ) u 3 ( t t ) u 3 ( t ) exp [ ( γ M i Δ 1 ) t γ M s ] .
| P C | 2 = P M 2 P M 2 + P M 2 P T 2 + P T 2 P M 2 + P T 2 P T 2 ,
P A 2 P C + P A 2 P C = P M 1 P M 2 + P T 1 P M 2 + P R M P M 2 + P R T P M 2 + P Br P M 2 + P M 1 P T 2 + P T 1 P T 2 + P R M P T 2 + P R T P T 2 + P Br P T 2 + P M 2 P M 1 + P T 2 P M 1 + P M 2 P R M + P M 2 P R T + P M 2 P Br + P T 2 P M 1 + P T 2 P T 1 + P T 2 P R M + P T 2 P R T + P T 2 P Br .
u i ( t 1 ) u i ( t 2 ) u i ( t 3 ) u i ( t 4 ) PDM = exp [ α i ( | t 1 t 3 | + | t 1 t 4 | + | t 2 t 3 | + | t 2 t 4 | ) ] exp [ α i ( | t 1 t 2 | + | t 3 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 1 ) u i ( t 2 ) u i ( t 3 ) u i ( t 4 ) .
u i ( t 1 ) u i ( t 2 ) u i ( t 3 ) u i ( t 4 ) GAM = u i ( t 1 ) u i ( t 2 ) u i ( t 3 ) u i ( t 4 ) CFM + u i ( t 1 ) u i ( t 2 ) u i ( t 3 ) u i ( t 4 ) = exp [ α i ( | t 1 t 3 | + | t 2 t 4 | ) ] + exp [ α i ( | t 1 t 4 | + | t 2 t 3 | ) ] + exp [ α i ( | t 1 t 2 | + | t 3 t 4 | ) ] .
| P B | 2 = P M 2 P M 2 + P M 2 P T 2 + P M 2 P R + P T 2 P M 2 + P T 2 P T 2 + P T 2 P R + P R P M 2 + P R P T 2 + P R P R ,
P A 2 P B + P A 2 P B = P M 1 P M 2 + P T 1 P M 2 + P R M P M 2 + P R T P M 2 + P Br P M 2 + P M 1 P T 2 + P T 1 P T 2 + P R M P T 2 + P R T P T 2 + P Br P T 2 + P M 1 P R + P T 1 P R + P R M P R + P R T P R + P Br P R + P M 2 P M 1 + P T 2 P M 1 + P M 2 P R M + P M 2 P R T + P M 2 P Br + P T 2 P M 1 + P T 2 P T 1 + P T 2 P R M + P T 2 P R T + P T 2 P Br + P R P M 1 + P R P T 1 + P R P R M + P R P R T + P R P Br .
χ A 2 = χ M + χ T + χ M γ M / [ ( α 1 + α 3 + γ M ) i Δ 1 ] + χ T γ T / [ ( α 1 + α 3 + γ T ) i Δ 1 ] + χ b γ Br 2 / ( ν Br 2 Δ 1 2 i Δ 1 γ Br ) .
χ A 2 = χ M + χ T + γ M ( α 1 + α 3 + γ M ) χ M / [ ( α 1 + α 3 + γ M ) 2 + Δ 1 2 ] + γ T ( α 1 + α 3 + γ T ) χ T / [ ( α 1 + α 3 + γ T ) 2 + Δ 1 2 ] + χ b γ Br 2 ( ν Br 2 Δ 1 2 ) / [ ( ν Br 2 Δ 1 2 ) 2 + ( Δ 1 γ Br ) 2 ] ,
χ A 2 = χ M γ M Δ 1 / [ ( α 1 + α 3 + γ M ) 2 + Δ 1 2 ] + χ T γ T Δ 1 / [ ( α 1 + α 3 + γ T ) 2 + Δ 1 2 ] + χ b γ Br 3 Δ 1 / [ ( ν Br 2 Δ 1 2 ) 2 + ( Δ 1 γ Br ) 2 ] .
I ASPB ( Δ 1 , τ ) η 2   exp ( 2 i Δ k r ) { [ 1 + n a   exp ( 2 α 2 | τ | ) ] χ M 2 + L 4 } + η L 1 { exp [ i Δ k r i ( ω 1 + ω 2 ) | τ | ] χ A 2 + exp [ i Δ k r + i ( ω 1 + ω 2 ) | τ | ] χ A 2 }
I ASPB ( Δ 1 , τ ) I C ( τ ) + 2 L 1 η | χ A 2 | cos [ Δ k r ( ω 1 + ω 2 ) | τ | + θ A 2 ] .
I ASPB ( Δ 1 , τ ) I C ( τ ) + 2 L 1 η | χ A 2 | cos   θ A 2 I C + 2 L 1 η χ A 2 .
I ASPB ( Δ 1 , τ ) I C ( τ ) + 2 L 1 η | χ A 2 | sin   θ A 2 I C ( τ ) + 2 L 1 η χ A 2 .
I ASPB ( Δ 1 , Δ 2 , τ ) η 2   exp ( 2 i Δ k r ) { [ 1 + n a   exp ( 2 α 2 | τ | ) ] L 5 + L 6 } + η   exp [ ( α 1 + α 2 ) | τ | ] { exp [ i Δ k r i ( ω 1 + ω 2 ) | τ | ] χ B χ A 2 + exp [ i Δ k r + i ( ω 1 + ω 2 ) | τ | ] χ B χ A 2 } .
I ( Δ 1 , Δ 2 , τ ) I B ( Δ 2 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ B | | χ A 2 | cos [ Δ k r ( ω 1 + ω 2 ) | τ | θ B + θ A 2 ] .
I ( Δ 1 , Δ 2 , τ ) I B ( Δ 2 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ B | | χ A 2 | cos   θ A 2 = I B ( Δ 2 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ B | χ A 2 .
I ( Δ 1 , Δ 2 , τ ) I B ( Δ 2 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ B | | χ A 2 | sin   θ A 2 = I B ( Δ 2 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ B | χ A 2 .
I ( Δ 1 , Δ 2 , τ ) { [ 1 + n a   exp ( 2 α 1 | τ | ) ] L 2 + L 3 } + η   exp [ ( α 1 + α 2 ) | τ | ] { exp [ i Δ k r i ( ω 1 + ω 2 ) | τ | ] χ B χ A 2 + exp [ i Δ k r + i ( ω 1 + ω 2 ) | τ | ] χ B χ A 2 } .
I ( Δ 1 , Δ 2 , τ ) I A 2 ( Δ 1 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ B | | χ A 2 | cos [ Δ k r ( ω 1 + ω 2 ) | τ | θ B + θ A 2 ] .
I ( Δ 1 , Δ 2 , τ ) I A 2 ( Δ 1 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ A 2 | | χ B | cos   θ B = I A 2 ( Δ 1 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ A 2 | χ B .
I ( Δ 1 , Δ 2 , τ ) I A 2 ( Δ 1 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ A 2 | | χ B | sin   θ B = I A 2 ( Δ 1 , τ ) + 2 η   exp [ ( α 1 + α 2 ) | τ | ] | χ A 2 | χ B .

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