Abstract

We show that the backward-stimulated Rayleigh–Bragg scattering (SRBS) can be efficiently generated in a three-photon absorbing medium. Compared with all other known stimulated (such as Brillouin and Raman) scattering effects, the observed effect exhibits the following three features: (i) no frequency shift, (ii) a lower pump threshold, and (iii) no critical requirement for pump spectral linewidths within a range of Δν¯1cm1. The specific nonlinear scattering medium is a three-photon absorbing chromophore solution (PRL-OT04 in chloroform), pumped by 1064nm laser pulses of nanoseconds duration but with three different spectral linewidths. The mechanism for generating backward SRBS in a three-photon active medium is the formation of a stationary Bragg grating enhanced by three-photon-absorption-associated refractive index changes. A superior optical phase-conjugation property of the backward SRBS beam has been experimentally demonstrated by employing two different optical setups. In both cases, a specially introduced aberration influence of 45mrad can be basically removed by the backward SRBS beam that retains a much smaller beam divergence of 0.4mrad.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).
  4. G. S. He and P. N. Prasad, 'Stimulated Kerr scattering and reorientation work of molecules in liquid CS2,' Phys. Rev. A 41, 2687-2697 (1990).
    [CrossRef] [PubMed]
  5. R. M. Herman and M. A. Gray, 'Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,' Phys. Rev. Lett. 19, 824-828 (1967).
    [CrossRef]
  6. D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
    [CrossRef]
  7. D. Pohl, I. Reinhold, and W. Kaiser, 'Experimental observation of stimulated Brillouin scattering,' Phys. Rev. Lett. 20, 1141-1143 (1968).
    [CrossRef]
  8. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  14. G. S. He, J. D. Bhawalkar, P. N. Prasad, and B. A. Reinhardt, 'Three-photon-absorption-induced fluorescence and optical limiting effects in an organic compound,' Opt. Lett. 20, 1524-1526 (1995).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. Q. Zheng, G. S. He, and P. N. Prasad, 'π-conjugated dendritic nanosized chromophore with enhanced two-photon absorption,' Chem. Mater. 17, 6004-6011 (2005).
    [CrossRef]
  20. G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
    [CrossRef] [PubMed]
  21. Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
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  27. D. Liu and G. S. He, 'Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 235-245 (1999).
    [CrossRef]
  28. D. Liu and G. S. He, 'Erratum: Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 1242 (1999).
  29. I. C. Khoo and Y. Liang, 'Stimulated orientational and thermal scattering and self-starting optical phase conjugation with nematic liquid crystals,' Phys. Rev. E 62, 6722-6733 (2000).
    [CrossRef]
  30. G. S. He, Y. Cui, M. Yoshida, and P. N. Prasad, 'Phase-conjugate backward stimulated emission from a two-photon-pumped lasing medium,' Opt. Lett. 22, 10-12 (1997).
    [CrossRef] [PubMed]
  31. G. S. He and P. N. Prasad, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. I. Experimental studies,' J. Opt. Soc. Am. B 15, 1078-1085 (1998).
    [CrossRef]
  32. G. S. He, N. Cheng, P. N. Prasad, D. Liu, and S. H. Liu, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. II. Theoretical studies,' J. Opt. Soc. Am. B 15, 1086-1095 (1998).
    [CrossRef]
  33. G. S. He and P. N. Prasad, 'Phase-conjugation property of one-photon pumped backward simulated emission from a lasing medium,' IEEE J. Quantum Electron. 34, 473-481 (1998).
    [CrossRef]
  34. H. Kogelnik, 'Coupled wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).
  35. A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, 'Light beams wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,' Opt. Commun. 24, 74-76 (1978).
    [CrossRef]
  36. A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
    [CrossRef]

2006 (1)

Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
[CrossRef] [PubMed]

2005 (3)

G. S. He, T.-C. Lin, S.-J. Chung, Q. Zheng, C. Lu, Y. Cui, and P. N. Prasad, 'Two-, three-, and four-photon-pumped stimulated cavityless lasing properties of ten stilbazolium-dyes solutions,' J. Opt. Soc. Am. B 22, 2219-2228 (2005).
[CrossRef]

Q. Zheng, G. S. He, and P. N. Prasad, 'π-conjugated dendritic nanosized chromophore with enhanced two-photon absorption,' Chem. Mater. 17, 6004-6011 (2005).
[CrossRef]

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

2004 (2)

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

G. S. He, T.-C. Lin, and P. N. Prasad, 'Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation,' Opt. Express 12, 5952-5961 (2004).
[CrossRef] [PubMed]

2003 (1)

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, 'Observation of three-photon enhanced four-photon absorption,' Phys. Rev. Lett. 91, 063902 (2003).
[CrossRef] [PubMed]

2002 (2)

G. S. He, 'Optical phase conjugation: principles, techniques, and applications,' Prog. Quantum Electron. 26, 131-191 (2002).
[CrossRef]

G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, 'Observation of stimulated emission by direct three-photon excitation,' Nature 415, 767-770 (2002).
[CrossRef] [PubMed]

2000 (1)

I. C. Khoo and Y. Liang, 'Stimulated orientational and thermal scattering and self-starting optical phase conjugation with nematic liquid crystals,' Phys. Rev. E 62, 6722-6733 (2000).
[CrossRef]

1999 (2)

D. Liu and G. S. He, 'Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 235-245 (1999).
[CrossRef]

D. Liu and G. S. He, 'Erratum: Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 1242 (1999).

1998 (3)

1997 (2)

1995 (1)

1990 (1)

G. S. He and P. N. Prasad, 'Stimulated Kerr scattering and reorientation work of molecules in liquid CS2,' Phys. Rev. A 41, 2687-2697 (1990).
[CrossRef] [PubMed]

1978 (3)

A. Yariv, 'Phase conjugate optics and real-time holography,' IEEE J. Quantum Electron. QE-14, 650-660 (1978).
[CrossRef]

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, 'Light beams wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,' Opt. Commun. 24, 74-76 (1978).
[CrossRef]

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

1977 (1)

1970 (1)

P. M. Rentzepis, C. J. Mitschele, and A. C. Saxman, 'Measurement of ultrashort laser pulses by three-photon fluorescence,' Appl. Phys. Lett. 17, 122-125 (1970).
[CrossRef]

1969 (1)

H. Kogelnik, 'Coupled wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).

1968 (1)

D. Pohl, I. Reinhold, and W. Kaiser, 'Experimental observation of stimulated Brillouin scattering,' Phys. Rev. Lett. 20, 1141-1143 (1968).
[CrossRef]

1967 (2)

R. M. Herman and M. A. Gray, 'Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,' Phys. Rev. Lett. 19, 824-828 (1967).
[CrossRef]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
[CrossRef]

1965 (1)

D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).

1964 (1)

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, 'Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,' Phys. Rev. Lett. 12, 592-595 (1964).
[CrossRef]

1962 (1)

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Baev, A.

Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
[CrossRef] [PubMed]

Bhawalkar, J. D.

Boyd, R. W.

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2002).

Brekhovskikh, G. L.

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, 'Light beams wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,' Opt. Commun. 24, 74-76 (1978).
[CrossRef]

Cheng, N.

Chiao, R. Y.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, 'Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,' Phys. Rev. Lett. 12, 592-595 (1964).
[CrossRef]

Cho, C. W.

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
[CrossRef]

Chung, S.-J.

Cui, Y.

Dai, J.

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

Dombroskie, A. G.

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

Eckhardt, G.

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Fabelinskii, I. L.

D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).

Foltz, N. D.

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
[CrossRef]

Gazengel, J.

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

Gray, M. A.

R. M. Herman and M. A. Gray, 'Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,' Phys. Rev. Lett. 19, 824-828 (1967).
[CrossRef]

Hasselbeck, M. P.

He, G. S.

Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
[CrossRef] [PubMed]

G. S. He, T.-C. Lin, S.-J. Chung, Q. Zheng, C. Lu, Y. Cui, and P. N. Prasad, 'Two-, three-, and four-photon-pumped stimulated cavityless lasing properties of ten stilbazolium-dyes solutions,' J. Opt. Soc. Am. B 22, 2219-2228 (2005).
[CrossRef]

Q. Zheng, G. S. He, and P. N. Prasad, 'π-conjugated dendritic nanosized chromophore with enhanced two-photon absorption,' Chem. Mater. 17, 6004-6011 (2005).
[CrossRef]

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

G. S. He, T.-C. Lin, and P. N. Prasad, 'Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation,' Opt. Express 12, 5952-5961 (2004).
[CrossRef] [PubMed]

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, 'Observation of stimulated emission by direct three-photon excitation,' Nature 415, 767-770 (2002).
[CrossRef] [PubMed]

G. S. He, 'Optical phase conjugation: principles, techniques, and applications,' Prog. Quantum Electron. 26, 131-191 (2002).
[CrossRef]

D. Liu and G. S. He, 'Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 235-245 (1999).
[CrossRef]

D. Liu and G. S. He, 'Erratum: Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 1242 (1999).

G. S. He and P. N. Prasad, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. I. Experimental studies,' J. Opt. Soc. Am. B 15, 1078-1085 (1998).
[CrossRef]

G. S. He, N. Cheng, P. N. Prasad, D. Liu, and S. H. Liu, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. II. Theoretical studies,' J. Opt. Soc. Am. B 15, 1086-1095 (1998).
[CrossRef]

G. S. He and P. N. Prasad, 'Phase-conjugation property of one-photon pumped backward simulated emission from a lasing medium,' IEEE J. Quantum Electron. 34, 473-481 (1998).
[CrossRef]

G. S. He, Y. Cui, M. Yoshida, and P. N. Prasad, 'Phase-conjugate backward stimulated emission from a two-photon-pumped lasing medium,' Opt. Lett. 22, 10-12 (1997).
[CrossRef] [PubMed]

G. S. He, J. D. Bhawalkar, P. N. Prasad, and B. A. Reinhardt, 'Three-photon-absorption-induced fluorescence and optical limiting effects in an organic compound,' Opt. Lett. 20, 1524-1526 (1995).
[CrossRef] [PubMed]

G. S. He and P. N. Prasad, 'Stimulated Kerr scattering and reorientation work of molecules in liquid CS2,' Phys. Rev. A 41, 2687-2697 (1990).
[CrossRef] [PubMed]

G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 2000).

Hellwarth, R. W.

R. W. Hellwarth, 'Generation of time-reversed wave fronts by nonlinear refraction,' J. Opt. Soc. Am. 67, 1-3 (1977).
[CrossRef]

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Herman, R. M.

R. M. Herman and M. A. Gray, 'Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,' Phys. Rev. Lett. 19, 824-828 (1967).
[CrossRef]

Kaiser, W.

D. Pohl, I. Reinhold, and W. Kaiser, 'Experimental observation of stimulated Brillouin scattering,' Phys. Rev. Lett. 20, 1141-1143 (1968).
[CrossRef]

Kannan, R.

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

Khoo, I. C.

I. C. Khoo and Y. Liang, 'Stimulated orientational and thermal scattering and self-starting optical phase conjugation with nematic liquid crystals,' Phys. Rev. E 62, 6722-6733 (2000).
[CrossRef]

Kogelnik, H.

H. Kogelnik, 'Coupled wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).

Kudriavtseva, A. D.

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

Kudryavtseva, A. D.

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, 'Light beams wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,' Opt. Commun. 24, 74-76 (1978).
[CrossRef]

Liang, Y.

I. C. Khoo and Y. Liang, 'Stimulated orientational and thermal scattering and self-starting optical phase conjugation with nematic liquid crystals,' Phys. Rev. E 62, 6722-6733 (2000).
[CrossRef]

Lin, T.-C.

G. S. He, T.-C. Lin, S.-J. Chung, Q. Zheng, C. Lu, Y. Cui, and P. N. Prasad, 'Two-, three-, and four-photon-pumped stimulated cavityless lasing properties of ten stilbazolium-dyes solutions,' J. Opt. Soc. Am. B 22, 2219-2228 (2005).
[CrossRef]

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

G. S. He, T.-C. Lin, and P. N. Prasad, 'Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation,' Opt. Express 12, 5952-5961 (2004).
[CrossRef] [PubMed]

G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, 'Observation of stimulated emission by direct three-photon excitation,' Nature 415, 767-770 (2002).
[CrossRef] [PubMed]

Liu, D.

D. Liu and G. S. He, 'Erratum: Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 1242 (1999).

D. Liu and G. S. He, 'Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 235-245 (1999).
[CrossRef]

G. S. He, N. Cheng, P. N. Prasad, D. Liu, and S. H. Liu, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. II. Theoretical studies,' J. Opt. Soc. Am. B 15, 1086-1095 (1998).
[CrossRef]

Liu, M.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, 'Observation of three-photon enhanced four-photon absorption,' Phys. Rev. Lett. 91, 063902 (2003).
[CrossRef] [PubMed]

Liu, S. H.

Lu, C.

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

G. S. He, T.-C. Lin, S.-J. Chung, Q. Zheng, C. Lu, Y. Cui, and P. N. Prasad, 'Two-, three-, and four-photon-pumped stimulated cavityless lasing properties of ten stilbazolium-dyes solutions,' J. Opt. Soc. Am. B 22, 2219-2228 (2005).
[CrossRef]

Markowicz, P. P.

G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, 'Observation of stimulated emission by direct three-photon excitation,' Nature 415, 767-770 (2002).
[CrossRef] [PubMed]

Mash, D. I.

D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).

McClung, F. J.

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Mitschele, C. J.

P. M. Rentzepis, C. J. Mitschele, and A. C. Saxman, 'Measurement of ultrashort laser pulses by three-photon fluorescence,' Appl. Phys. Lett. 17, 122-125 (1970).
[CrossRef]

Morozov, V. V.

D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).

Nguyen, P. X.

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

Pilipetsky, N. F.

B. Ya. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Pohl, D.

D. Pohl, I. Reinhold, and W. Kaiser, 'Experimental observation of stimulated Brillouin scattering,' Phys. Rev. Lett. 20, 1141-1143 (1968).
[CrossRef]

Polyakov, S.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, 'Observation of three-photon enhanced four-photon absorption,' Phys. Rev. Lett. 91, 063902 (2003).
[CrossRef] [PubMed]

Prasad, P. N.

Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
[CrossRef] [PubMed]

Q. Zheng, G. S. He, and P. N. Prasad, 'π-conjugated dendritic nanosized chromophore with enhanced two-photon absorption,' Chem. Mater. 17, 6004-6011 (2005).
[CrossRef]

G. S. He, T.-C. Lin, S.-J. Chung, Q. Zheng, C. Lu, Y. Cui, and P. N. Prasad, 'Two-, three-, and four-photon-pumped stimulated cavityless lasing properties of ten stilbazolium-dyes solutions,' J. Opt. Soc. Am. B 22, 2219-2228 (2005).
[CrossRef]

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

G. S. He, T.-C. Lin, and P. N. Prasad, 'Stimulated Rayleigh-Bragg scattering enhanced by two-photon excitation,' Opt. Express 12, 5952-5961 (2004).
[CrossRef] [PubMed]

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, 'Observation of stimulated emission by direct three-photon excitation,' Nature 415, 767-770 (2002).
[CrossRef] [PubMed]

G. S. He, N. Cheng, P. N. Prasad, D. Liu, and S. H. Liu, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. II. Theoretical studies,' J. Opt. Soc. Am. B 15, 1086-1095 (1998).
[CrossRef]

G. S. He and P. N. Prasad, 'Phase-conjugation properties of two-photon-pumped backward-stimulated emission. I. Experimental studies,' J. Opt. Soc. Am. B 15, 1078-1085 (1998).
[CrossRef]

G. S. He and P. N. Prasad, 'Phase-conjugation property of one-photon pumped backward simulated emission from a lasing medium,' IEEE J. Quantum Electron. 34, 473-481 (1998).
[CrossRef]

G. S. He, Y. Cui, M. Yoshida, and P. N. Prasad, 'Phase-conjugate backward stimulated emission from a two-photon-pumped lasing medium,' Opt. Lett. 22, 10-12 (1997).
[CrossRef] [PubMed]

G. S. He, J. D. Bhawalkar, P. N. Prasad, and B. A. Reinhardt, 'Three-photon-absorption-induced fluorescence and optical limiting effects in an organic compound,' Opt. Lett. 20, 1524-1526 (1995).
[CrossRef] [PubMed]

G. S. He and P. N. Prasad, 'Stimulated Kerr scattering and reorientation work of molecules in liquid CS2,' Phys. Rev. A 41, 2687-2697 (1990).
[CrossRef] [PubMed]

Rank, D. H.

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
[CrossRef]

Reinhardt, B. A.

Reinhold, I.

D. Pohl, I. Reinhold, and W. Kaiser, 'Experimental observation of stimulated Brillouin scattering,' Phys. Rev. Lett. 20, 1141-1143 (1968).
[CrossRef]

Rentzepis, P. M.

P. M. Rentzepis, C. J. Mitschele, and A. C. Saxman, 'Measurement of ultrashort laser pulses by three-photon fluorescence,' Appl. Phys. Lett. 17, 122-125 (1970).
[CrossRef]

Rivoire, G.

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

Samoc, M.

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

Saxman, A. C.

P. M. Rentzepis, C. J. Mitschele, and A. C. Saxman, 'Measurement of ultrashort laser pulses by three-photon fluorescence,' Appl. Phys. Lett. 17, 122-125 (1970).
[CrossRef]

Schwarz, S. E.

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Sheik-Behae, M.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Shkunov, V. V.

B. Ya. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Sokolovskaya, A. I.

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, 'Light beams wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,' Opt. Commun. 24, 74-76 (1978).
[CrossRef]

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

Starunov, V. S.

D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).

Stegeman, G.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, 'Observation of three-photon enhanced four-photon absorption,' Phys. Rev. Lett. 91, 063902 (2003).
[CrossRef] [PubMed]

Stoicheff, B. P.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, 'Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,' Phys. Rev. Lett. 12, 592-595 (1964).
[CrossRef]

Tan, L.-S.

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

Townes, C. H.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, 'Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,' Phys. Rev. Lett. 12, 592-595 (1964).
[CrossRef]

Vaia, R. A.

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

Van Stryland, E. W.

Weiner, D.

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Wiggings, T. A.

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
[CrossRef]

Woodbury, E. J.

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

Yariv, A.

A. Yariv, 'Phase conjugate optics and real-time holography,' IEEE J. Quantum Electron. QE-14, 650-660 (1978).
[CrossRef]

Yoshida, M.

Yoshino, F.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, 'Observation of three-photon enhanced four-photon absorption,' Phys. Rev. Lett. 91, 063902 (2003).
[CrossRef] [PubMed]

Zel'dovich, B. Ya.

B. Ya. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Zerom, P.

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

Zheng, Q.

Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
[CrossRef] [PubMed]

Q. Zheng, G. S. He, and P. N. Prasad, 'π-conjugated dendritic nanosized chromophore with enhanced two-photon absorption,' Chem. Mater. 17, 6004-6011 (2005).
[CrossRef]

G. S. He, T.-C. Lin, S.-J. Chung, Q. Zheng, C. Lu, Y. Cui, and P. N. Prasad, 'Two-, three-, and four-photon-pumped stimulated cavityless lasing properties of ten stilbazolium-dyes solutions,' J. Opt. Soc. Am. B 22, 2219-2228 (2005).
[CrossRef]

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

P. M. Rentzepis, C. J. Mitschele, and A. C. Saxman, 'Measurement of ultrashort laser pulses by three-photon fluorescence,' Appl. Phys. Lett. 17, 122-125 (1970).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, 'Coupled wave theory for thick hologram gratings,' Bell Syst. Tech. J. 48, 2909-2947 (1969).

Chem. Mater. (1)

Q. Zheng, G. S. He, and P. N. Prasad, 'π-conjugated dendritic nanosized chromophore with enhanced two-photon absorption,' Chem. Mater. 17, 6004-6011 (2005).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Yariv, 'Phase conjugate optics and real-time holography,' IEEE J. Quantum Electron. QE-14, 650-660 (1978).
[CrossRef]

G. S. He and P. N. Prasad, 'Phase-conjugation property of one-photon pumped backward simulated emission from a lasing medium,' IEEE J. Quantum Electron. 34, 473-481 (1998).
[CrossRef]

J. Chem. Phys. (1)

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, 'Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores,' J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

J. Phys. Chem. B (1)

Q. Zheng, G. S. He, A. Baev, and P. N. Prasad, 'Experimental and quantum chemical studies of cooperative enhancement of three-photon absorption, optical limiting, and stabilization behaviors in multibranched and dendritic structures,' J. Phys. Chem. B 110, 14604-14610 (2006).
[CrossRef] [PubMed]

JETP Lett. (1)

D. I. Mash, V. V. Morozov, V. S. Starunov, and I. L. Fabelinskii, 'Stimulated scattering of light of the Rayleigh-line wing,' JETP Lett. 2, 25-27 (1965).

Nature (1)

G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, 'Observation of stimulated emission by direct three-photon excitation,' Nature 415, 767-770 (2002).
[CrossRef] [PubMed]

Opt. Commun. (2)

A. I. Sokolovskaya, G. L. Brekhovskikh, and A. D. Kudryavtseva, 'Light beams wavefront reconstruction and real volume image reconstruction of the object at the stimulated Raman scattering,' Opt. Commun. 24, 74-76 (1978).
[CrossRef]

A. D. Kudriavtseva, A. I. Sokolovskaya, J. Gazengel, P. X. Nguyen, and G. Rivoire, 'Reconstruction of the laser wave-front by stimulated scatterings in the pico-second range,' Opt. Commun. 26, 446-448 (1978).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (2)

G. S. He, C. Lu, Q. Zheng, P. N. Prasad, P. Zerom, R. W. Boyd, and M. Samoc, 'Stimulated Rayleigh-Bragg scattering in two-photon absorbing media,' Phys. Rev. A 71, 063810 (2005).
[CrossRef]

G. S. He and P. N. Prasad, 'Stimulated Kerr scattering and reorientation work of molecules in liquid CS2,' Phys. Rev. A 41, 2687-2697 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (1)

I. C. Khoo and Y. Liang, 'Stimulated orientational and thermal scattering and self-starting optical phase conjugation with nematic liquid crystals,' Phys. Rev. E 62, 6722-6733 (2000).
[CrossRef]

Phys. Rev. Lett. (6)

R. M. Herman and M. A. Gray, 'Theoretical prediction of the stimulated thermal Rayleigh scattering in liquids,' Phys. Rev. Lett. 19, 824-828 (1967).
[CrossRef]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggings, 'Stimulated thermal Rayleigh scattering,' Phys. Rev. Lett. 19, 828-830 (1967).
[CrossRef]

D. Pohl, I. Reinhold, and W. Kaiser, 'Experimental observation of stimulated Brillouin scattering,' Phys. Rev. Lett. 20, 1141-1143 (1968).
[CrossRef]

G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, D. Weiner, and E. J. Woodbury, 'Stimulated Raman scattering from organic liquids,' Phys. Rev. Lett. 9, 455-457 (1962).
[CrossRef]

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, 'Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,' Phys. Rev. Lett. 12, 592-595 (1964).
[CrossRef]

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, 'Observation of three-photon enhanced four-photon absorption,' Phys. Rev. Lett. 91, 063902 (2003).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

G. S. He, 'Optical phase conjugation: principles, techniques, and applications,' Prog. Quantum Electron. 26, 131-191 (2002).
[CrossRef]

Sov. Phys. JETP (2)

D. Liu and G. S. He, 'Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 235-245 (1999).
[CrossRef]

D. Liu and G. S. He, 'Erratum: Quasi-collinear and partially degenerate four-wave mixing: an alternative explanation of the phase-conjugation property of backward stimulated scattering,' Sov. Phys. JETP 88, 1242 (1999).

Other (5)

R. A. Fisher, ed., Optical Phase Conjugation (Academic, 1983).

B. Ya. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2002).

G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 2000).

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Figures (13)

Fig. 1
Fig. 1

(a) Linear absorption spectral curves of a 1 mm thick PRL-OT04 in CHCl 3 (chloroform) and of a 1 mm pure solvent sample; the chemical structure of the solute is shown in the top-right corner. (b) One- and three-photon-induced fluorescence spectra of the PRL-OT04 solution samples with two different concentration values.

Fig. 2
Fig. 2

Measured 2PA spectrum of PRL-OT04 solution in chloroform by using the femtosecond continuum-generation technique.

Fig. 3
Fig. 3

Measured waveforms of 1064 nm and Q-switched laser pulses working at three different operating modes: (a) BDN-sheet switching, (b) Pockels cell switching, and (c) seeded and Pockels cell switching.

Fig. 4
Fig. 4

(a) Measured decay curve of three-photon-excited fluorescence. (b) Measured relative intensity of three-photon-excited fluorescence versus the input excitation energy (both in logarithmic scales).

Fig. 5
Fig. 5

F–P interferograms measured for (a) the input pump beam of 1064 nm (half-ring), (b) the backward SRBS beam from the dye solution sample (full ring), (c) both of the above two beams, and (d) both the pump beam (half-ring) and the backward SBS beam (full ring) from a pure solvent (chloroform) sample at a higher pump level. The spectral linewidth of the pump beam was 0.005 cm 1 .

Fig. 6
Fig. 6

F–P interferograms measured for (a) the input pump beam of 1064 nm (half-ring), (b) the backward SRBS beam from the dye solution sample (full ring), and (c) both of the above two beams. The spectral linewidth of the pump beam was 0.08 cm 1 .

Fig. 7
Fig. 7

Measured waveforms of the input pump pulses (solid curves) and the backward SRBS pulses (dashed curves) at three different pump levels.

Fig. 8
Fig. 8

(a) Measured output backward-stimulated scattering energy as a function of the input pump energy. (b) Measured transmitted pump energy after passing through the dye solution sample as a function of the input pump energy.

Fig. 9
Fig. 9

Two optical layouts for the phase-conjugation experiment: (a) placing an aberration plate in the optical path of the input pump beam, and (b) filling a glass vial of poor optical quality with the dye solution.

Fig. 10
Fig. 10

Measured far-field patterns for (a) the backward SRBS beam without use of an aberrator, (b) the pump beam that passed an aberrator twice, and (c) the stimulated scattering beam that backward passed through the aberrator. The optical layout is shown in Fig. 9a.

Fig. 11
Fig. 11

Measured far-field patterns for (a) the input pump beam of 0.4 mrad divergence, (b) the transmitted pump beam after passing through a dye-solution-filled glass vial producing 4 5 mrad aberrations, and (c) the backward-stimulated scattering beam from the same dye-solution-filled vial sample. The optical layout is shown in Fig. 9b.

Fig. 12
Fig. 12

Schematic illustration for the formation of an induced stationary Bragg grating inside the scattering medium, as well as the energy transfer from the strong pump beam to the weak backward-scattering beam through the grating’s reflection.

Fig. 13
Fig. 13

Schematic illustration for the formation of the phase-conjugation property of backward SRBS in a multiphoton absorbing medium through a degenerate FWM process or equivalent holographic reconstruction process.

Tables (1)

Tables Icon

Table 1 Measured Pump Threshold Values for Stimulated Scattering in a 2 cm Long PRL OT04 CHCl 3 Solution of 0.01 M Concentration and in a 2 cm Long Pure CHCl 3 Sample a

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

I ( z ) = ( I p + I s ) + Δ I ( z ) = ( I p + I s ) + 2 I p I s cos ( 4 π n 0 z λ 0 ) .
I ( z ) = I ( z ) ( I p + I s ) = 1 + 2 I p I s ( I p + I s ) cos ( 4 π n 0 z λ 0 ) = 1 + δ I cos ( 4 π n 0 z λ 0 ) .
Δ n ( z ) = n 2 Δ I ( z ) = 2 n 2 I l I s cos ( 4 π n 0 z λ 0 ) = δ n cos ( 4 π n 0 z λ 0 ) .
n ( ω ) = 1 + χ s l v ( 1 ) ( ω ) + Re [ χ d y e ( 1 ) ( ω ) ] = [ 1 + χ s l v ( 1 ) ( ω ) ] { 1 + Re [ χ d y e ( 1 ) ( ω ) ] [ 1 + χ s l v ( 1 ) ( ω ) ] } = n s l v ( ω ) 1 + Re [ χ d y e ( 1 ) ( ω ) ] n s l v 2 ( ω ) n s l v ( ω ) + Re [ χ d y e ( 1 ) ( ω ) ] 2 n s l v ( ω ) .
χ d y e ( 1 ) ( ω ) = N g ( p 0 ) t o 2 ε 0 1 ω t o ω i Γ ,
n ( ω ) = n s l v ( ω ) + N g ( p 0 ) t o 2 2 n s l v ( ω ) ε 0 ω t o ω ( ω t o ω ) 2 + Γ 2 .
N g = N 0 ( N 0 N g ) = N 0 Δ N ,
Δ N ( z ) N 0 σ 2 I 2 ( z ) ( 2 PE ) ,
Δ N ( z ) N 0 σ 3 I 3 ( z ) ( 3 PE ) ,
Δ n ( z ) Δ N ( z ) N 0 σ 2 I 2 ( z ) N 0 σ 2 ( I p + I s ) 2 I 2 ( z ) .
I 2 ( z ) = [ 1 + δ l cos ( 4 π n 0 z λ 0 ) ] 2 = [ 1 + 2 δ I cos ( 4 π n 0 z λ 0 ) + ( δ I ) 2 cos 2 ( 4 π n 0 z λ 0 ) ] .
Δ n ( z ) N 0 σ 2 ( I p + I s ) 2 [ 1 + 2 δ I cos ( 4 π n 0 z λ 0 ) ] ( 2 PE ) .
Δ n ( z ) Δ N ( z ) N 0 σ 3 I 3 ( z ) N 0 σ 3 ( I p + I s ) 3 I 3 ( z ) ,
I 3 ( z ) = [ 1 + δ I cos ( 4 π n 0 z λ 0 ) ] 3 [ 1 + 3 δ I cos ( 4 π n 0 z λ 0 ) ] .
Δ n ( z ) N 0 σ 3 ( I p + I s ) 3 [ 1 + 3 δ I cos ( 4 π n 0 z λ 0 ) ] ( 3 PE ) .
Δ n ( z ) = X 2 P E ( ω ) 4 N 0 σ 2 ( I p + I s ) I p I s cos ( 4 π n 0 z λ 0 ) = δ n cos ( 4 π n 0 z λ 0 ) ,
δ n = X 2 P E ( ω ) 4 N 0 σ 2 I p I p I s ( 2 PE ) .
Δ n ( z ) = X 3 P E ( ω ) 6 N 0 σ 3 ( I p + I s ) 2 I p I s cos ( 4 π n 0 z λ 0 ) = δ n cos ( 4 π n 0 z λ 0 ) ,
δ n = X 3 P E ( ω 0 ) 6 N 0 σ 3 I p 2 I p I s ( 3 PE ) .
R = tanh 2 ( π δ n L λ 0 ) .
R I p I s { 1 exp [ α ( λ 0 ) L ] } ,
( π δ n λ 0 ) 2 I p L I s α ( λ 0 ) .
( 4 π X 2 P E N 0 σ 2 λ 0 ) 2 I p 4 L α ( λ 0 ) ( 2 PE ) .
( 6 π X 3 P E N 0 σ 3 λ 0 ) 2 I p 6 L α ( λ 0 ) ( 3 PE ) .

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