Abstract

Based on three stochastic models, the subtle, Markovian field-correlation effects have been investigated in homodyne-detected attosecond sum-frequency Raman-enhanced polarization beating (REPB) and heterodyne-detected femtosecond difference-frequency REPB. The different roles of the amplitude fluctuations and the phase fluctuations have been pointed out in both time and frequency domains. Analytic closed forms of the sixth-order and fourth-order Markovian stochastic correlations are characterized for homodyne (quadratic) and heterodyne (linear) detection, respectively. The attosecond sum-frequency REPB signal shows resonant–nonresonant cross correlation, asymmetric temporal behavior, and drastic difference for three Markovian stochastic fields, and the autocorrelation of REPB exhibits hybrid-radiation-matter-detuning-terahertz damping oscillation. The phase-diffusion model predicts a damping oscillation of the signal around a constant value in the narrowband case. The heterodyne-detected signal of REPB potentially offers rich dynamic information about the homogeneous, broadening, material phase of the third-order nonlinear susceptibility.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. J. C. Kirkwood, D. J. Ulness, and A. C. Albrecht, "On the classification of the electric field spectroscopies: application to Raman scattering," J. Phys. Chem. A 104, 4167-4173 (2000).
    [CrossRef]
  4. M. A. Dugan and A. C. Albrecht, "Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory," Phys. Rev. A 43, 3877-3921 (1991).
    [CrossRef] [PubMed]
  5. 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]
  6. D. J. Ulness and A. C. Albrecht, "Theory of time-resolved coherent Raman scattering with spectrally tailored noisy light," J. Raman Spectrosc. 28, 571-578 (1997).
    [CrossRef]
  7. M. J. Stimson, D. J. Ulness, and A. C. Albrecht, "Time-resolved coherent spectroscopy controlled by spectrally tailored noisy light," J. Raman Spectrosc. 28, 579-587 (1997).
    [CrossRef]
  8. M. J. Stimson, D. J. Ulness, J. C. Kirkwood, and G. S. Boutis, "Noisy-light correlation functions by frequency resolved optical gating," J. Opt. Soc. Am. B 15, 505-514 (1998).
    [CrossRef]
  9. H. Ma, A. S. L. Gomes, and Cid B. de Araujo, "Raman-assisted polarization beats in time-delayed four-wave mixing," Opt. Lett. 17, 1052-1054 (1992).
    [CrossRef] [PubMed]
  10. X. Mi, Z. H. Yu, Q. Jiang, Y. B. Wang, L. J. Wang, and P. M. Fu, "Phase-sensitive detection of Raman-enhanced nondegenerate four-wave mixing," J. Opt. Soc. Am. B 17, 1543-1547 (2000).
    [CrossRef]
  11. Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
    [CrossRef]
  12. Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
    [CrossRef]
  13. A. T. Georges, "Resonance fluorescence in Markovian stochastic fields," Phys. Rev. A 21, 2034-2049 (1980).
    [CrossRef]
  14. R. Bratfalean and P. Ewart, "Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields," Phys. Rev. A 56, 2267-2279 (1997).
    [CrossRef]
  15. R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, "Two-photon spectroscopy: A technique for characterizing diode-laser noise," Phys. Rev. A 52, 3157-3169 (1995).
    [CrossRef] [PubMed]
  16. C. Chen, D. S. Elliott, and M. W. Hamilton, "Two-photon absorption from the real Gaussian field," Phys. Rev. Lett. 68, 3531-3534 (1992).
    [CrossRef] [PubMed]
  17. D. DeBeer, E. Usadi, and S. R. Hartmann, "Attosecond beats in sodium vapor," Phys. Rev. Lett. 60, 1262-1265 (1988).
    [CrossRef] [PubMed]
  18. V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).
  19. P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
    [CrossRef] [PubMed]
  20. Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
    [CrossRef]
  21. Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Fourth-order interference on polarization beats in a four-level system," J. Opt. Soc. Am. B 17, 690-696 (2000).
    [CrossRef]
  22. Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
    [CrossRef]
  23. Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, "Higher-order correlation on polarization beats in Markovian stochastic fields," Phys. Rev. A 63, 043802 (2001).
    [CrossRef]
  24. Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
    [CrossRef]
  25. Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
    [CrossRef]
  26. 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]
  27. Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
    [CrossRef]
  28. N. Morita and T. Yajima, "Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light," Phys. Rev. A 30, 2525-2536 (1984).
    [CrossRef]

2003

D. J. Ulness, "On the role of classical field time correlations in noisy light spectroscopy: color locking and a spectral filter analogy," J. Phys. Chem. A 107, 8111-8123 (2003).
[CrossRef]

2002

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
[CrossRef]

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]

2001

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, "Higher-order correlation on polarization beats in Markovian stochastic fields," Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

2000

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Fourth-order interference on polarization beats in a four-level system," J. Opt. Soc. Am. B 17, 690-696 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
[CrossRef]

X. Mi, Z. H. Yu, Q. Jiang, Y. B. Wang, L. J. Wang, and P. M. Fu, "Phase-sensitive detection of Raman-enhanced nondegenerate four-wave mixing," J. Opt. Soc. Am. B 17, 1543-1547 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
[CrossRef]

J. C. Kirkwood and A. C. Albrecht, "Multi-dimensional time-resolved coherent Raman six-wave mixing: a comparison of the direct and cascaded processes with femtosecond excitation and noisy light interferometry," J. Raman Spectrosc. 31, 107-124 (2000).
[CrossRef]

J. C. Kirkwood, D. J. Ulness, and A. C. Albrecht, "On the classification of the electric field spectroscopies: application to Raman scattering," J. Phys. Chem. A 104, 4167-4173 (2000).
[CrossRef]

1998

1997

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]

D. J. Ulness and A. C. Albrecht, "Theory of time-resolved coherent Raman scattering with spectrally tailored noisy light," J. Raman Spectrosc. 28, 571-578 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, "Time-resolved coherent spectroscopy controlled by spectrally tailored noisy light," J. Raman Spectrosc. 28, 579-587 (1997).
[CrossRef]

R. Bratfalean and P. Ewart, "Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields," Phys. Rev. A 56, 2267-2279 (1997).
[CrossRef]

Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
[CrossRef]

1995

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, "Two-photon spectroscopy: A technique for characterizing diode-laser noise," Phys. Rev. A 52, 3157-3169 (1995).
[CrossRef] [PubMed]

1992

C. Chen, D. S. Elliott, and M. W. Hamilton, "Two-photon absorption from the real Gaussian field," Phys. Rev. Lett. 68, 3531-3534 (1992).
[CrossRef] [PubMed]

H. Ma, A. S. L. Gomes, and Cid B. de Araujo, "Raman-assisted polarization beats in time-delayed four-wave mixing," Opt. Lett. 17, 1052-1054 (1992).
[CrossRef] [PubMed]

1991

M. A. Dugan and A. C. Albrecht, "Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory," Phys. Rev. A 43, 3877-3921 (1991).
[CrossRef] [PubMed]

1989

V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).

1988

D. DeBeer, E. Usadi, and S. R. Hartmann, "Attosecond beats in sodium vapor," Phys. Rev. Lett. 60, 1262-1265 (1988).
[CrossRef] [PubMed]

1984

N. Morita and T. Yajima, "Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light," Phys. Rev. A 30, 2525-2536 (1984).
[CrossRef]

1980

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

Albrecht, A. C.

J. C. Kirkwood and A. C. Albrecht, "Multi-dimensional time-resolved coherent Raman six-wave mixing: a comparison of the direct and cascaded processes with femtosecond excitation and noisy light interferometry," J. Raman Spectrosc. 31, 107-124 (2000).
[CrossRef]

J. C. Kirkwood, D. J. Ulness, and A. C. Albrecht, "On the classification of the electric field spectroscopies: application to Raman scattering," J. Phys. Chem. A 104, 4167-4173 (2000).
[CrossRef]

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]

D. J. Ulness and A. C. Albrecht, "Theory of time-resolved coherent Raman scattering with spectrally tailored noisy light," J. Raman Spectrosc. 28, 571-578 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, "Time-resolved coherent spectroscopy controlled by spectrally tailored noisy light," J. Raman Spectrosc. 28, 579-587 (1997).
[CrossRef]

M. A. Dugan and A. C. Albrecht, "Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory," Phys. Rev. A 43, 3877-3921 (1991).
[CrossRef] [PubMed]

Blumel, R.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, "Two-photon spectroscopy: A technique for characterizing diode-laser noise," Phys. Rev. A 52, 3157-3169 (1995).
[CrossRef] [PubMed]

Bogdanov, V. L.

V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).

Boutis, G. S.

Bratfalean , R.

R. Bratfalean and P. Ewart, "Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields," Phys. Rev. A 56, 2267-2279 (1997).
[CrossRef]

Chen, C.

C. Chen, D. S. Elliott, and M. W. Hamilton, "Two-photon absorption from the real Gaussian field," Phys. Rev. Lett. 68, 3531-3534 (1992).
[CrossRef] [PubMed]

de Araujo, C. B.

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, "Higher-order correlation on polarization beats in Markovian stochastic fields," Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

de Araujo, Cid B.

DeBeer, D.

D. DeBeer, E. Usadi, and S. R. Hartmann, "Attosecond beats in sodium vapor," Phys. Rev. Lett. 60, 1262-1265 (1988).
[CrossRef] [PubMed]

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]

Dugan , M. A.

M. A. Dugan and A. C. Albrecht, "Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory," Phys. Rev. A 43, 3877-3921 (1991).
[CrossRef] [PubMed]

Elliott, D. S.

C. Chen, D. S. Elliott, and M. W. Hamilton, "Two-photon absorption from the real Gaussian field," Phys. Rev. Lett. 68, 3531-3534 (1992).
[CrossRef] [PubMed]

Evdokimov, A. B.

V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).

Ewart, P.

R. Bratfalean and P. Ewart, "Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields," Phys. Rev. A 56, 2267-2279 (1997).
[CrossRef]

Eyler, E. E.

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, "Higher-order correlation on polarization beats in Markovian stochastic fields," Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Falnberg, B. D.

V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).

Farooqi, S. M.

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]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Fourth-order interference on polarization beats in a four-level system," J. Opt. Soc. Am. B 17, 690-696 (2000).
[CrossRef]

X. Mi, Z. H. Yu, Q. Jiang, Y. B. Wang, L. J. Wang, and P. M. Fu, "Phase-sensitive detection of Raman-enhanced nondegenerate four-wave mixing," J. Opt. Soc. Am. B 17, 1543-1547 (2000).
[CrossRef]

Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
[CrossRef]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

Gan, C. L.

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
[CrossRef]

Georges, A. T.

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

Gomes, A. S. L.

Hamilton, M. W.

C. Chen, D. S. Elliott, and M. W. Hamilton, "Two-photon absorption from the real Gaussian field," Phys. Rev. Lett. 68, 3531-3534 (1992).
[CrossRef] [PubMed]

Hartmann, S. R.

D. DeBeer, E. Usadi, and S. R. Hartmann, "Attosecond beats in sodium vapor," Phys. Rev. Lett. 60, 1262-1265 (1988).
[CrossRef] [PubMed]

Hou, X.

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
[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]

X. Mi, Z. H. Yu, Q. Jiang, Y. B. Wang, L. J. Wang, and P. M. Fu, "Phase-sensitive detection of Raman-enhanced nondegenerate four-wave mixing," J. Opt. Soc. Am. B 17, 1543-1547 (2000).
[CrossRef]

Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
[CrossRef]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

Kirkwood, J. C.

J. C. Kirkwood, D. J. Ulness, and A. C. Albrecht, "On the classification of the electric field spectroscopies: application to Raman scattering," J. Phys. Chem. A 104, 4167-4173 (2000).
[CrossRef]

Kirkwood , J. C.

J. C. Kirkwood and A. C. Albrecht, "Multi-dimensional time-resolved coherent Raman six-wave mixing: a comparison of the direct and cascaded processes with femtosecond excitation and noisy light interferometry," J. Raman Spectrosc. 31, 107-124 (2000).
[CrossRef]

Kirkwood, J. C.

Li, C. S.

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

Li, X. F.

Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
[CrossRef]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

Lu, K. Q.

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
[CrossRef]

Lukomskil, G. V.

V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).

Ma, H.

Metcalf, H. J.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, "Two-photon spectroscopy: A technique for characterizing diode-laser noise," Phys. Rev. A 52, 3157-3169 (1995).
[CrossRef] [PubMed]

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]

X. Mi, Z. H. Yu, Q. Jiang, Y. B. Wang, L. J. Wang, and P. M. Fu, "Phase-sensitive detection of Raman-enhanced nondegenerate four-wave mixing," J. Opt. Soc. Am. B 17, 1543-1547 (2000).
[CrossRef]

Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
[CrossRef]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

Morita , N.

N. Morita and T. Yajima, "Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light," Phys. Rev. A 30, 2525-2536 (1984).
[CrossRef]

Ryan, R. E.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, "Two-photon spectroscopy: A technique for characterizing diode-laser noise," Phys. Rev. A 52, 3157-3169 (1995).
[CrossRef] [PubMed]

Stimson, M. J.

M. J. Stimson, D. J. Ulness, J. C. Kirkwood, and G. S. Boutis, "Noisy-light correlation functions by frequency resolved optical gating," J. Opt. Soc. Am. B 15, 505-514 (1998).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, "Time-resolved coherent spectroscopy controlled by spectrally tailored noisy light," J. Raman Spectrosc. 28, 579-587 (1997).
[CrossRef]

Sun, L. Q.

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Fourth-order interference on polarization beats in a four-level system," J. Opt. Soc. Am. B 17, 690-696 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Tang, T. T.

Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Fourth-order interference on polarization beats in a four-level system," J. Opt. Soc. Am. B 17, 690-696 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Ulness, D. J.

D. J. Ulness, "On the role of classical field time correlations in noisy light spectroscopy: color locking and a spectral filter analogy," J. Phys. Chem. A 107, 8111-8123 (2003).
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J. C. Kirkwood, D. J. Ulness, and A. C. Albrecht, "On the classification of the electric field spectroscopies: application to Raman scattering," J. Phys. Chem. A 104, 4167-4173 (2000).
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M. J. Stimson, D. J. Ulness, J. C. Kirkwood, and G. S. Boutis, "Noisy-light correlation functions by frequency resolved optical gating," J. Opt. Soc. Am. B 15, 505-514 (1998).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, "Time-resolved coherent spectroscopy controlled by spectrally tailored noisy light," J. Raman Spectrosc. 28, 579-587 (1997).
[CrossRef]

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D. J. Ulness and A. C. Albrecht, "Theory of time-resolved coherent Raman scattering with spectrally tailored noisy light," J. Raman Spectrosc. 28, 571-578 (1997).
[CrossRef]

Ulness, D. J.

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]

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Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
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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).
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Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
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P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

Zhang, Y. P.

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Y. P. Zhang, C. L. Gan, S. M. Farooqi, K. Q. Lu, X. Hou, and T. T. Tang, "Four-level polarization beats with broadband noisy light," J. Opt. Soc. Am. B 19, 1204-1215 (2002).
[CrossRef]

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, "Higher-order correlation on polarization beats in Markovian stochastic fields," Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Fourth-order interference on polarization beats in a four-level system," J. Opt. Soc. Am. B 17, 690-696 (2000).
[CrossRef]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

J. Mod. Opt.

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, "Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system," J. Mod. Opt. 48, 549-564 (2001).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. A

D. J. Ulness, "On the role of classical field time correlations in noisy light spectroscopy: color locking and a spectral filter analogy," J. Phys. Chem. A 107, 8111-8123 (2003).
[CrossRef]

J. C. Kirkwood, D. J. Ulness, and A. C. Albrecht, "On the classification of the electric field spectroscopies: application to Raman scattering," J. Phys. Chem. A 104, 4167-4173 (2000).
[CrossRef]

J. Raman Spectrosc.

J. C. Kirkwood and A. C. Albrecht, "Multi-dimensional time-resolved coherent Raman six-wave mixing: a comparison of the direct and cascaded processes with femtosecond excitation and noisy light interferometry," J. Raman Spectrosc. 31, 107-124 (2000).
[CrossRef]

D. J. Ulness and A. C. Albrecht, "Theory of time-resolved coherent Raman scattering with spectrally tailored noisy light," J. Raman Spectrosc. 28, 571-578 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, "Time-resolved coherent spectroscopy controlled by spectrally tailored noisy light," J. Raman Spectrosc. 28, 579-587 (1997).
[CrossRef]

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V. L. Bogdanov, A. B. Evdokimov, G. V. Lukomskil, and B. D. Falnberg, "Subfemtosecond beats in an interference of the fields of Rayleigh scattering and four-wave mixing," JETP Lett. 49, 157-161 (1989).

Opt. Commun.

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, "Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry," Opt. Commun. 184, 265-276 (2000).
[CrossRef]

Y. P. Zhang, C. L. Gan, K. Q. Lu, C. S. Li, and X. Hou, "Raman-enhanced polarization beats in Markovian stochastic fields," Opt. Commun. 205, 163-186 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev. A

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[CrossRef] [PubMed]

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]

R. Bratfalean and P. Ewart, "Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields," Phys. Rev. A 56, 2267-2279 (1997).
[CrossRef]

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, "Two-photon spectroscopy: A technique for characterizing diode-laser noise," Phys. Rev. A 52, 3157-3169 (1995).
[CrossRef] [PubMed]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, "Ultrafast modulation spectroscopy in a cascade three-level system," Phys. Rev. A 52, 4867-4870 (1995).
[CrossRef] [PubMed]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, "Effects of field correlation on polarization beats," Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, "Effects of fourth-order coherence on ultrafast modulation spectroscopy," Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, "Higher-order correlation on polarization beats in Markovian stochastic fields," Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Z. H. Yu, X. Mi, Q. Jiang, X. F. Li, and P. M. Fu, "Field-correlation effects on Raman-enhanced nondegenerate four-wave mixing," Phys. Rev. A 55, 2334-2339 (1997).
[CrossRef]

N. Morita and T. Yajima, "Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light," Phys. Rev. A 30, 2525-2536 (1984).
[CrossRef]

Phys. Rev. Lett.

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]

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[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

(a) Phase-conjugation geometry of the attosecond REPB; (b) level configuration of PR1; (c) double-sided Feynman diagrams representing the Liouville pathways for P1, PR1, P2, respectively.

Fig. 2
Fig. 2

Normalized RENFWM signal intensity versus time delay γRτ for χ/χR=0.05, γ/γR=4, α3/γR=0.1. (a) Δ/γR=0, α1/γR=0.5 (dashed-dotted curve), 2 (dotted curve), 5 (dashed curve), 20 (solid curve). (b) α1/γR=2, Δ/γR=1 (dashed-dotted curve), 3 (dotted curve), 15 (dashed curve), 30 (solid curve).

Fig. 3
Fig. 3

Normalized RENFWM signal intensity versus Δ/γR for χ/χR=0.5, α3/γR=0.1, α1/γR=2, γRτ=10. γ/γR=0.001 (solid curve), 1 (dashed curve), 4 (dotted curve), 100 (dashed-dotted curve).

Fig. 4
Fig. 4

The attosecond, sum-frequency, REPB signal intensity versus relative time delay. The three curves represent the chaotic field (dashed curve), phase-diffusion field (solid curve), and Gaussian-amplitude field (dotted curve). The parameters are ω1=3.237 fs-1, ω3=3.358 fs-1, r=0, η1=η2=1, Δ=0, χ/χR=0.5, γR=5×10-5 fs-1, γ=2×10-4 fs-1, Δk=10.58 mm-1, α1=0.278 fs-1, α3=0.299 fs-1.

Fig. 5
Fig. 5

RENFWM spectrum with fixed time delay. The three curves represent the chaotic field (dashed curve), phase-diffusion field (solid curve), and Gaussian-amplitude field (dotted curve). The parameters are χ/χR=0.5, γR=γ=5×10-5 fs-1, τ=500 fs. (a) α1=2.78×10-5 fs-1, γR=1.077×10-4 fs-1. (b) α1=2.78×10-4 fs-1, γR=6.27×10-4 fs-1.

Fig. 6
Fig. 6

The heterodyne detection spectra of the difference-frequency REPB with (a) -Δk·r+(ω1-ω3)τ=0 and (b) -Δk·r+(ω1-ω3)τ=-π/2. Theoretical curves represent the chaotic field (dashed curve), phase-diffusion field (solid curve), and Gaussian-amplitude field (dotted curve) with parameters γR=1.077×10-4 fs-1, α1=2.78×10-5 fs-1, α2=2.75×10-5 fs-1, χ/χR=0.5, γR=γ=5×10-5 fs-1, Δk=1.449 mm-1, r=0, ω1-ω3=0.0166 fs-1, η=3.

Equations (64)

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Ep1=E1(r, t)+E2(r, t)=A1(r, t)exp(-iω1t)+A2(r, t)exp(-iω3t)=1u1(t)exp[i(k1·r-ω1t)]+2u3(t-τ)exp[i(k2·r-ω3t+ω3τ)],
Ep2=E1(r, t)+E2(r, t)=A1(r, t)exp(-iω1t)+A2(r, t)exp(-iω3t)=1u1(t-τ)exp[i(k1·r-ω1t+ω1τ)]+2u3(t)exp[i(k2·r-ω3t)].
E3(r, t)=A3(r, t)exp(-iω3t)=3u3(t)exp[i(k3·r-ω3t)].
dQ1dt+γQ1=χγE1(r, t)[E1(r, t)]*,
dQ2dt+γQ2=χγE2(r, t)[E2(r, t)]*.
dQR1dt+(γR-iΔ)QR1=iαR4[A1(r, t)]*A3(r, t).
P1=Q1(r, t)E3(r, t)=χγ1(1)*3×exp{i[(k1-k1+k3)·r-ω3t-ω1τ]}×0u1(t-t)u1*(t-t-τ)u3(t)×exp(-γt)dt,
P2=Q2(r, t)E3(r, t)=χγ2(2)*3×exp{i[(k2-k2+k3)·r-ω3t+ω3τ]}×0u3*(t-t)u3(t-t-τ)u3(t)×exp(-γt)dt,
PR1=12NαRQR1(r, t)E1(r, t)×exp[i(ω1-ω3)t-iω1τ]=iχRγR1(1)*3 exp{i[(k1-k1+k3)·r-ω3t-ω1τ]}0u1(t)u1*(t-t-τ)u3×(t-t)exp[-(γR-iΔ)t]dt,
I(Δ, τ)|P(3)|2=P(3)(P(3))*=(P1+P2+PR1)[(P1)*+(P2)*+(PR1)*]
ui(t1)ui(t2)ui(t3)ui*(t4)ui*(t5)ui*(t6)=ui(t1)ui*(t4)ui(t2)ui(t3)ui*(t5)ui*(t6)+ui(t1)ui*(t5)ui(t2)ui(t3)ui*(t4)ui*(t6)+ui(t1)ui*(t6)ui(t2)ui(t3)ui*(t4)ui*(t5),
i=1,2,3,
ui(t1)ui(t2)ui*(t3)ui*(t4)
=ui(t1)ui*(t3)ui(t2)ui*(t4)
+ui(t1)ui*(t4)ui(t2)ui*(t3).
ui(t1)ui*(t2)=exp(-αi|t1-t2|).
I(Δ, τ)
B1+χ2+χR2γR2γR(γR+α1)(γR2+Δ2)+2χχRγRΔγR2+Δ2exp(-2α1|τ|)+η2χ2(4γ2+5γα3+2α32)(γ+α3)(γ+2α3)×exp(-2α3|τ|)+2ηχγ+2α3 2χ(γ+α3)+γχRγRΔ(2α3-γR)2+Δ2-γχRγRΔγR2+Δ2×cos[Δk·r-(ω1+ω3)τ]+γχRγRγRγR2+Δ2-γχRγR(2α3-γR)(2α3-γR)2+Δ2sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|],
I(Δ, τ)
B1+χ2+χR2γR2(2α1-γR)(α1-γR)[(2α1-γR)2+Δ2]-2χχRγRΔ(γR-2α1)2+Δ2exp(-2α1|τ|)+η2χ2(4γ-3α3)γ-α3exp(-2α3|τ|)+6η2χ2γ2α32(γ2-α32)(γ2-4α32) exp(-2γ|τ|)+χR2γR2γR-2γR(γR-2γR)2+Δ2 1α1-γR+η22α3-γR+γRγR2+Δ2 1α1+γR+η22α3+γR-2(1+η22)[γR2+Δ2-(α1+α3)2](γR2+Δ2)[(γR-2γR)2+Δ2]×exp(-2γR|τ|)+4ηχ2-χχRγRΔ(2α1-γR)2+Δ2cos[Δk·r-(ω1+ω3)τ]-χχRγR(2α1-γR)(2α1-γR)2+Δ2 sin[Δk·r-(ω1+ω3)τ]×exp[-(α1+α3)|τ|].
B1=χ2γ1γ+2α1+η2(2γ+α3)(α3+γ)(2α3+γ)+χR2γRγRγR2+Δ2-2χRχγRγΔ×(γR+γ+2α1)(γR+γ)+2α1γR+Δ2(γ+2α1)(γR2+Δ2)[(γR+γ)2+Δ2].
exp(-αi|t1-t2|)2αiδ(t1-t2),i=1,2,3.
I(Δ, τ)B2+χ2+χR2γR2α3(α1+γR)-2χχRγRΔγR2+Δ2exp(-2α1|τ|)+η2χ2(γ3+2α33+5γ2α3+5α32γ)α3(γ+α3)(γ+2α3)×exp(-2α3|τ|)+2ηχγ+2α3×2χ(γ+α3)+γχRγRΔ(2α3-γR)2+Δ2-γχRγRΔγR2+Δ2cos[Δk·r-(ω1+ω3)τ]+γχRγRγRγR2+Δ2-γχRγR(2α3-γR)(2α3-γR)2+Δ2sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|],
I(Δ, τ)B2+χ2-χR2γR2α3(α1-γR)-2χχRγRΔ(γR-2α1)2+Δ2exp(-2α1|τ|)+η2χ2[3α3(α3-γ)-γ2]α3(α3-γ)×exp(-2α3|τ|)+2η2χ2γ2α32-γ2 exp(-2γ|τ|)+2χR2γR2α1α3(α12-γR2) exp(-2γR|τ|)+4ηχ2-χχRγRΔ(α1-α3)2+Δ2×cos[Δk·r-(ω1+ω3)τ]-χχRγR(α1-α3)(α1-α3)2+Δ2 sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|].
B2=χ2γ212α1γ+η2(3γ+α3)2α3γ(α3+γ)+χR2γRα1-4χγχRγRΔα1[(γR+γ)2+Δ2].
ui(t1)ui(t2)ui(t3)ui*(t4)ui*(t5)ui*(t6)=exp[-αi(|t1-t4|+|t1-t5|+|t1-t6|+|t2-t4|+|t2-t5|+|t2-t6|+|t3-t4|+|t3-t5|+|t3-t6|)]exp[αi(|t1-t2|+|t1-t3|+|t2-t3|+|t4-t5|+|t4-t6|+|t5-t6|)],
ui(t1)ui(t2)ui*(t3)ui*(t4)
=exp[-αi(|t1-t3|+|t1-t4|+|t2-t3|+|t2-t4|)]exp[αi(|t1-t2|+|t3-t4|)].
I(Δ, τ)B3+4χRχγRα1Δ-2χRχγRΔ(γ+2α1)(γ+2α1)(γR2+Δ2)+2ηχγχRγRα3γR2+Δ2 χ(γR2+Δ2)γχRγRα3-Δγα3+i(γR-iΔ)(2α3-γR+iΔ)(γR+γ-iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]-i(γR+iΔ)(2α3-γR-iΔ)(γR+γ+iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]×cos[Δk·r-(ω1+ω3)τ]+γRγα3-(γR-iΔ)(2α3-γR+iΔ)(γR+γ-iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]-(γR+iΔ)(2α3-γR-iΔ)(γR+γ+iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]×sin[Δk·r-(ω1+ω3)τ]×exp[-(α1+α3)|τ|],
I(Δ, τ)
B3+2iα1χγχRγR×(2α1-γR-iΔ)(γR+iΔ)(γR+γ+iΔ)[(2α1-γR)2+Δ2](γR2+Δ2)[(γR+γ)2+Δ2]-(2α1-γR+iΔ)(γR-iΔ)(γR+γ-iΔ)[(2α1-γR)2+Δ2](γR2+Δ2)[(γR+γ)2+Δ2]+2iΔγ(2α1+γ)[(2α1-γR)2+Δ2]+2χ2η-4ηχχRγRα3Δ(2α3+γ)[(α1-3α3-γR-γ)2+Δ2]×cos[Δk·r-(ω1+ω3)τ]-2ηχγχRγR(2α1-γR)(2α3+γ)[(2α1-γR)2+Δ2]×sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|].
I(Δ, τ)B4+2ηχγχRγRα3γR2+Δ2 χ(γR2+Δ2)γχRγRα3-Δγα3+i(γR-iΔ)(2α3-γR+iΔ)(γR+γ-iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]-i(γR+iΔ)(2α3-γR-iΔ)(γR+γ+iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]×cos[Δk·r-(ω1+ω3)τ]+γRγα3-(γR-iΔ)(2α3-γR+iΔ)(γR+γ-iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]-(γR+iΔ)(2α3-γR-iΔ)(γR+γ+iΔ)[(2α3-γR)2+Δ2][(γR+γ)2+Δ2]×sin[Δk·r-(ω1+ω3)τ]×exp[-(α1+α3)|τ|],
I(Δ, τ)B4+2χ2η-4ηχχRγRα3Δ(2α3+γ)[(α1-3α3-γR-γ)2+Δ2]×cos[Δk·r-(ω1+ω3)τ]-2ηχγχRγR(2α1-γR)(2α3+γ)[(2α1-γR)2+Δ2] sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|].
B3=χ2γ1γ+2α1+η22α3+γ+χR2γRγRγR2+Δ2,
B4=χ2γ2(1+η2)2α3γ+χR2γRα1-4χRχγγRΔα1[(γR+γ)2+Δ2].
ui(t1)ui(t2)ui(t3)ui(t4)ui(t5)ui(t6)=ui(t1)ui(t4)ui(t2)ui(t3)ui(t5)ui(t6)+ui(t1)ui(t5)ui(t2)ui(t3)ui(t4)ui(t6)+ui(t1)ui(t6)ui(t2)ui(t3)ui(t4)ui(t5)+ui(t1)ui(t2)ui(t3)ui(t4)ui(t5)ui(t6)+ui(t1)ui(t3)ui(t2)ui(t4)ui(t5)ui(t6),
ui(t1)ui(t2)ui(t3)ui(t4)=ui(t1)ui(t3)ui(t2)ui(t4)+ui(t1)ui(t4)×ui(t2)ui(t3)+ui(t1)ui(t2)ui(t3)ui(t4).
I(Δ, τ)B5-2χ2+χR2γR2γR(γR+α1)(γR2+Δ2)-2χχRγRΔγR2+Δ2exp(-2α1|τ|)+η2χ2(11γ2+14γα3+4α32)(γ+α3)(γ+2α3)×exp(-2α3|τ|)+2ηχγ+2α3×χ(3γ+2α3)+γχRγRΔ(2α3-γR)2+Δ2+χRγRΔ(2α3-γ)γR2+Δ2cos[Δk·r-(ω1+ω3)τ]+χRγRγR(γ-2α3)γR2+Δ2-γχRγR(2α3-γR)(2α3-γR)2+Δ2×sin[Δk·r-(ω1+ω3)τ]×exp[-(α1+α3)|τ|],
I(Δ, τ)
B5+2χ2+χR2γR2(2α1-γR)(α1-γR)[(2α1-γR)2+Δ2]+2χχRγRΔ(γR-2α1)2+Δ2exp(-2α1|τ|)+10η2χ2×exp(-2α3|τ|)+12η2χ2γ2α32(γ2-α32)(γ2-4α32)×exp(-2γ|τ|)+2χR2γR2×γR-2γR[(γR-2γR)2+Δ2](α1-γR)+γR(γR2+Δ2)(α1+γR)-2[γR2+Δ2-(α1+α3)2](γR2+Δ2)[(γR-2γR)2+Δ2]×exp(-2γR|τ|)+4η3χ2-2χχRγRΔ(2α1-γR)2+Δ2cos[Δk·r-(ω1+ω3)τ]-χχRγR(2α1-γR)(2α1-γR)2+Δ2 sin[Δk·r-(ω1+ω3)τ]×exp[-(α1+α3)|τ|]+4ηχχRγRα3(2α3+γ)(2α3-γ)×Δ(2α3+γ)(γR-γ-2γR)2+Δ2-Δ(2α3-γ)(γR-2γR)2+Δ2+Δ(2α3-γ)(γR+γ)2+Δ2-Δ(2α3+γ)γR2+Δ2cos[Δk·r-(ω1+ω3)τ-iΔτ]+(γR-γ-2γR)(2α3+γ)(γR-γ-2γR)2+Δ2-(γR-2γR)(2α3-γ)(γR-2γR)2+Δ2-(γR+γ)(2α3-γ)(γR+γ)2+Δ2+γR(2α3+γ)γR2+Δ2sin[Δk·r-(ω1+ω3)τ-iΔτ]exp[-(γ+γR)|τ|].
B5=χ2γ1γ+2α1+η2(3γ+α3)(α3+γ)(2α3+γ)+χR2γRγRγR2+Δ2-2χRχγRγΔ×(γR+γ+2α1)(γR+γ)+2α1γR+Δ2(γ+2α1)(γR2+Δ2)[(γR+γ)2+Δ2].
I(Δ, τ)B6+2χ2-χR2γR2(γR+α1)α3-2χχRγRΔγR2+Δ2exp(-2α1|τ|)+η2χ2(γ3+5γ2α3+5γα32+2α33)α3(γ+α3)(γ+2α3)+η2χ22γ+2α3γ+2α3+γ2α3(γ+α3)×exp(-2α3|τ|)+2ηχγ+2α3 χ(3γ+2α3)+γχRγRΔ(2α3-γR)2+Δ2-χRγγRΔ(5α3+2γ)α3(γR2+Δ2)cos[Δk·r-(ω1+ω3)τ]+χRγγRγR(2γ+5α3)α3γR2+Δ2-γχRγR(2α3-γR)(2α3-γR)2+Δ2sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|],
I(Δ, τ)B6+2χ2-χR2γR2(α1-γR)α3-2χχRγRΔ(γR-2α1)2+Δ2exp(-2α1|τ|)+η2χ2(10γα3+10α32-2γ2)α3(α3-γ)×exp(-2α3|τ|)+4η2χ2γ2α32-γ2 exp(-2γ|τ|)+4χR2γR2α1α3(α12-γR2) exp(-2γR|τ|)+4ηχ2-χχRγRΔ(2α1-γR)2+Δ2×cos[Δk·r-(ω1+ω3)τ]-χχRγR(2α1-γR)(2α1-γR)2+Δ2 sin[Δk·r-(ω1+ω3)τ]exp[-(α1+α3)|τ|].
B6=χ2γ212α1γ+η2(3γ+α3)2α3γ(2α3+γ)+η2α3(α3+γ)+χR2γRα1-4χRγχγRΔα1[(γR+γ)2+Δ2].
Ep1=E1(r, t)+F2(r, t)=A1(r, t)exp(-iω1t)+B2(r, t)exp(-iω3t)=1u1(t)exp[i(k1·r-ω1t)]+2u2(t)×exp[i(k2·r-ω3t)],
EP2=E1(r, t)+F2(r, t)=A1(r, t)exp(-iω1t)+B2(r, t)exp(-iω3t)=1u1(t-τ)exp[i(k1·r-ω1t+ω1τ)]+2u2(t-τ)exp[i(k2·r-ω3t+ω2τ)].
dQ2dt+γQ2=χγF2(r, t)[F2(r, t)]*.
P2=Q2(r, t)E3(r, t)=χγ2(2)*3×exp{i[(k2-k2+k3)·r-ω3t-ω3τ]}×0u2(t-t)u2*(t-t-τ)u3(t)×exp(-γt)dt.
I(Δ, τ)|P(3)|2=P(3)(P(3))*=(P1+P2+PR1)[(P1)*+(P2)*+(PR1)*]
A=exp[-(α2+α3+γR)|τ|]×2χRχγηγR[α12-(α3+γR)2-Δ2]4Δ2(α3+γR)2+[α12-(α3+γR)2+Δ2]2
IP2=(χγη)21γ(γ+2α2)+exp(-2α2|τ|)γ2
θR=tan-1[χRγRγR/χ(γR2+Δ2)-χRΔγR].
 I(Δ, τ)IP2+χ2γγ+2α1+χR2γRγRγR2+Δ2-2χγχRγRΔ(Δ2+2γα1+γ2+4γRα1+2γRγ+γR2)(γ+2α1)(γR2+Δ2)[(γR+γ)2+Δ2]+χ2+χR2γR2γR-2χχRγRΔ(γR+α1)(γR+α1)(γR2+Δ2)exp(-2α1|τ|)+2|χ(3)|χη exp[-(α1+α2)|τ|]×cos[-Δk·r+(ω1-ω3)τ-θR],
 I(Δ, τ)IP2+χ2γγ+2α1+χR2γRγRγR2+Δ2-2χγχRγRΔ(Δ2+2γα1+γ2+4γRα1+2γRγ+γR2)(γ+2α1)(γR2+Δ2)[(γR+γ)2+Δ2]+χ2+χR2γR2γR-2χχRγRΔ(γR+α1)(γR+α1)(γR2+Δ2)exp(-2α1|τ|)+4χRα1×χγR 2γRΔ cos(Δ|τ|)+[(α1+α3)2-γR2+Δ2]sin(Δ|τ|)(γR2+Δ2)[(α1+α3-γR)2+Δ2]+χRγR2Δ×[γR2-α12+Δ2-2(γR+α1)α3-3α32]sin(Δ|τ|)-[γR2+(γR-α1+α3)(γR-α1-α3)+γRΔ]cos(Δ|τ|)(γR2+Δ2)[(γR-α1+α3)2+Δ2][(γR-α1-α3)2+Δ2]×exp(-γR|τ|)+A sin[-Δk·r+(ω1-ω3)τ+Δ|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR].
I(τ>0)IP2+2|χ(3)|χη exp[-(α1+α2)|τ|]×cos[-Δk·r+(ω1-ω3)τ-θR].
I(τ<0)IP2+A sin[-Δk·r+(ω1-ω3)τ+Δ|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]×cos[-Δk·r+(ω1-ω3)τ-θR].
I(Δ, τ)IP2+χ2γγ+2α1+χRγR(γRχR-2Δχ)γR2+Δ2+4χχRγRΔα1(γ+2α1)[(γR+γ)2+Δ2]+2α1χ22α1+γ-4χχRγRΔα1(2α1+γ)[(γR+γ)2+Δ2]×exp[-(2α1+γ)|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR],
I(Δ, τ)IP2+χ2γγ+2α1+χR2γRγRγR2+Δ2+4χχRγRΔα1(γ+2α1)[(2α1-γR)2+Δ2](γR2+Δ2)[(γR+γ)2+Δ2][5γR4+8γR3γ-2γRγ(γ2-4Δ2)+4α12(2γR+γ)2+Δ4-10γR2Δ2-2α1(4γR3-γ3-2γΔ2+6γγR2)+2γR2(γ2-2Δ2)]+2α1χ22α1+γ-4χχRγRΔα1(2α1+γ)[(2α1-γR)2+Δ2]exp[-(2α1+γ)|τ|]+4χχRγRα1[γR(2α1+γR+γm)sin(Δ|τ|)+Δ(2α1+2γR+γm)cos(Δ|τ|)](γR2+Δ2)[(2α1+γR+γm)2+Δ2] exp(-γR|τ|)+A sin[-Δk·r+(ω1-ω3)τ+Δ|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR].
IP2=(χγη)21γ(γ+2α2)-2α22γ2(4α22-γ2) exp(-2γ|τ|)+6α22-α2γγ2(4α22-γ2) exp[-(2α2+γ)|τ|]
I(τ>0)IP2+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR]
I(τ<0)IP2+A sin[-Δk·r+(ω1-ω3)τ+Δ|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR]
I(Δ, τ)IP2+χ2γγ+2α1+χR2γRγRγR2+Δ2-2χγχRγRΔ(Δ2+2γα1+γ2+4γRα1+2γRγ+γR2)(γ+2α1)(γR2+Δ2)[(γR+γ)2+Δ2]+χ2+2χR2γR2γR-2χχRγRΔ(γR+α1)(γR+α1)(γR2+Δ2)-2χχRγRΔ(γR-2α1)2+Δ2exp(-2α1|τ|)+χ2γ2α1+γ+4χχRγRΔα1(2α1+γ)[(2α1-γR)2+Δ2]exp[-(2α1+γ)|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR],
 I(Δ, τ)IP2+χ2γγ+2α1+χR2γRγRγR2+Δ2-2χγχRγRΔ(Δ2+2γα1+γ2+4γRα1+2γRγ+γR2)(γ+2α1)(γR2+Δ2)[(γR+γ)2+Δ2]+χ2+χR2γR2γR-4χχRγRΔ(γR+α1)(γR+α1)(γR2+Δ2)+2χR2γR2(2α1-γR)(α1-γR)[(2α1+γR)2+Δ2]exp(-2α1|τ|)+(α1+α3-γR)(γR2+Δ2)+γR[(α1+α3-γR)2+Δ2]-2[Δ2+γR2-(α1+α3)2](α1+γR)(α1+γR)(γR2+Δ2)[(α1+α3-γR)2+Δ2]×exp(-2γR|τ|)-2α1χR2γR2exp[-(γR-iΔ)|τ|][α12-(α3+γR-iΔ)2](α1+α3-γR-iΔ)+exp[-(γR+iΔ)|τ|][α12-(α3+γR+iΔ)2](α1+α3-γR+iΔ)+A sin[-Δk·r+(ω1-ω3)τ+Δ|τ|]+2|χ(3)|χη exp[-(α1+α2)|τ|]cos[-Δk·r+(ω1-ω3)τ-θR].
IP2=(χγη)21γ(γ+2α2)+exp(-2α2|τ|)γ2+exp[-(2α2+γ)|τ|]γ(γ+2α2)

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