Abstract

Linear stability analysis is performed of speckle pattern resulting from multiple, diffuse scattering of coherent light waves in random media with intensity-dependent refractive index (noninstantaneous Kerr nonlinearity). The speckle pattern is shown to become unstable with respect to dynamic perturbations within a certain frequency band, provided that the nonlinearity exceeds some frequency-dependent threshold. Although the absolute instability threshold is independent of the response time of the nonlinearity, the latter significantly affects speckle dynamics (in particular, its spectral content) beyond the threshold. Our results suggest that speckle dynamics becomes chaotic immediately beyond the threshold.

© 2004 Optical Society of America

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    [CrossRef]
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  33. V. M. Agranovich and V. E. Kravtsov, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691–13694 (1991).
    [CrossRef]
  34. V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second-harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931–4942 (1991).
    [CrossRef]
  35. J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
    [CrossRef] [PubMed]
  36. A. J. van Wonderen, “Diagrammatic treatment of light scattering by a collection of Kerr particles,” Phys. Rev. B 50, 2921–2940 (1994).
    [CrossRef]
  37. R. Bressoux and R. Maynard, “On the speckle correlation in nonlinear random media,” Europhys. Lett. 50, 460–465 (2000).
    [CrossRef]
  38. B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of the scattering potential,” Phys. Rev. Lett. 84, 1970–1973 (2000).
    [CrossRef] [PubMed]
  39. S. E. Skipetrov and R. Maynard, “Instabilities of waves in nonlinear disordered media,” Phys. Rev. Lett. 85, 736–739 (2000).
    [CrossRef] [PubMed]
  40. S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
    [CrossRef]
  41. S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
    [CrossRef]
  42. K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
    [CrossRef]
  43. H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
    [CrossRef]
  44. Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982).
    [CrossRef]
  45. Y. Silberberg and I. Bar Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662–670 (1984).
    [CrossRef]
  46. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
    [CrossRef] [PubMed]
  47. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
    [CrossRef] [PubMed]
  48. As regards the effect of the noninstantaneous nature of nonlinearity on speckle dynamics, our first results have been announced in S. E. Skipetrov, “Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity,” Opt. Lett. 28, 646–648 (2003).
    [CrossRef] [PubMed]
  49. S. E. Skipetrov and R. Maynard, “Diffuse waves in nonlinear disordered media,” in Ref. 3.

2003

A. Trombettoni, A. Smerzi, and A. R. Bishop, “Discrete nonlinear Schrödinger equation with defects,” Phys. Rev. E 67, 016607 (2003).
[CrossRef]

S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
[CrossRef]

As regards the effect of the noninstantaneous nature of nonlinearity on speckle dynamics, our first results have been announced in S. E. Skipetrov, “Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity,” Opt. Lett. 28, 646–648 (2003).
[CrossRef] [PubMed]

2002

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297, 2026–2030 (2002).
[CrossRef] [PubMed]

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

2001

P. Sebbah, P. Sixou, C. Vanneste, and H. Guillard, “Nonlinear scattering and optical power limitation in direct and inverse polymer dispersed liquid crystal,” Nonlinear Opt. 27, 377–383 (2001).

D. S. Wiersma and S. Cavalieri, “Light emission: a temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
[CrossRef]

2000

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

R. Bressoux and R. Maynard, “On the speckle correlation in nonlinear random media,” Europhys. Lett. 50, 460–465 (2000).
[CrossRef]

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of the scattering potential,” Phys. Rev. Lett. 84, 1970–1973 (2000).
[CrossRef] [PubMed]

S. E. Skipetrov and R. Maynard, “Instabilities of waves in nonlinear disordered media,” Phys. Rev. Lett. 85, 736–739 (2000).
[CrossRef] [PubMed]

S. E. Skipetrov and R. Maynard, “Nonuniversal correlations in multiple scattering,” Phys. Rev. B 62, 886–891 (2000).
[CrossRef]

1999

B. Shapiro, “New type of intensity correlation in random media,” Phys. Rev. Lett. 83, 4733–4735 (1999).
[CrossRef]

M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

1998

F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

1997

F. Scheffold, W. Härtl, G. Maret, and E. Matijević, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

1996

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

1995

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

1994

A. J. van Wonderen, “Diagrammatic treatment of light scattering by a collection of Kerr particles,” Phys. Rev. B 50, 2921–2940 (1994).
[CrossRef]

1993

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
[CrossRef] [PubMed]

1991

V. M. Agranovich and V. E. Kravtsov, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691–13694 (1991).
[CrossRef]

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second-harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931–4942 (1991).
[CrossRef]

R. Berkovits, “Sensitivity of the multiple-scattering speckle pattern to the motion of a single scatterer,” Phys. Rev. B 43, 8638–8640 (1991).
[CrossRef]

1990

M. P. van Albada, J. F. de Boer, and A. Lagendijk, “Observation of long-range intensity correlation in the transport of coherent light through a random medium,” Phys. Rev. Lett. 64, 2787–2790 (1990).
[CrossRef] [PubMed]

A. Z. Genack, N. Garcia, and W. Polkosnik, “Long-range intensity correlation in random media,” Phys. Rev. Lett. 65, 2129–2132 (1990).
[CrossRef] [PubMed]

1989

R. Pnini and B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

1988

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

1987

G. Maret and P.-E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–287 (1987).
[CrossRef] [PubMed]

A. Yu. Zyuzin and B. Z. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–566 (1987).

1986

1984

1983

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

1982

Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982).
[CrossRef]

1980

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Agranovich, V. M.

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second-harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931–4942 (1991).
[CrossRef]

V. M. Agranovich and V. E. Kravtsov, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691–13694 (1991).
[CrossRef]

Akimoto, O.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Al’tshuler, G. B.

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Asaka, S.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Bang, O.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

Bar Joseph, I.

Y. Silberberg and I. Bar Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662–670 (1984).
[CrossRef]

Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982).
[CrossRef]

Berkovits, R.

R. Berkovits, “Sensitivity of the multiple-scattering speckle pattern to the motion of a single scatterer,” Phys. Rev. B 43, 8638–8640 (1991).
[CrossRef]

Bishop, A. R.

A. Trombettoni, A. Smerzi, and A. R. Bishop, “Discrete nonlinear Schrödinger equation with defects,” Phys. Rev. E 67, 016607 (2003).
[CrossRef]

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Bressoux, R.

R. Bressoux and R. Maynard, “On the speckle correlation in nonlinear random media,” Europhys. Lett. 50, 460–465 (2000).
[CrossRef]

Cavalieri, S.

D. S. Wiersma and S. Cavalieri, “Light emission: a temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Chance, B.

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Christiansen, P. L.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

Christodoulides, D. N.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

Coskun, T.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

Cwilich, G.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–287 (1987).
[CrossRef] [PubMed]

Daido, H.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

de Boer, J. F.

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
[CrossRef] [PubMed]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, “Observation of long-range intensity correlation in the transport of coherent light through a random medium,” Phys. Rev. Lett. 64, 2787–2790 (1990).
[CrossRef] [PubMed]

Ermolaev, V. S.

Eugenieva, E.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

Feng, S.

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Freund, I.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Garcia, N.

A. Z. Genack, N. Garcia, and W. Polkosnik, “Long-range intensity correlation in random media,” Phys. Rev. Lett. 65, 2129–2132 (1990).
[CrossRef] [PubMed]

Genack, A. Z.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

A. Z. Genack, N. Garcia, and W. Polkosnik, “Long-range intensity correlation in random media,” Phys. Rev. Lett. 65, 2129–2132 (1990).
[CrossRef] [PubMed]

Gershenfeld, N.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297, 2026–2030 (2002).
[CrossRef] [PubMed]

Grigorishin, K. I.

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second-harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931–4942 (1991).
[CrossRef]

Guillard, H.

P. Sebbah, P. Sixou, C. Vanneste, and H. Guillard, “Nonlinear scattering and optical power limitation in direct and inverse polymer dispersed liquid crystal,” Nonlinear Opt. 27, 377–383 (2001).

Härtl, W.

F. Scheffold, W. Härtl, G. Maret, and E. Matijević, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Hu, B.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

Ikeda, K.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

Itoh, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Kip, D.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

Kravtsov, V. E.

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second-harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931–4942 (1991).
[CrossRef]

V. M. Agranovich and V. E. Kravtsov, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691–13694 (1991).
[CrossRef]

Krylov, K. I.

Lagendijk, A.

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
[CrossRef] [PubMed]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, “Observation of long-range intensity correlation in the transport of coherent light through a random medium,” Phys. Rev. Lett. 64, 2787–2790 (1990).
[CrossRef] [PubMed]

Manenkov, A. A.

Maret, G.

F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

F. Scheffold, W. Härtl, G. Maret, and E. Matijević, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

G. Maret and P.-E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Matijevic, E.

F. Scheffold, W. Härtl, G. Maret, and E. Matijević, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Matsuoka, M.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Maynard, R.

S. E. Skipetrov and R. Maynard, “Nonuniversal correlations in multiple scattering,” Phys. Rev. B 62, 886–891 (2000).
[CrossRef]

S. E. Skipetrov and R. Maynard, “Instabilities of waves in nonlinear disordered media,” Phys. Rev. Lett. 85, 736–739 (2000).
[CrossRef] [PubMed]

R. Bressoux and R. Maynard, “On the speckle correlation in nonlinear random media,” Europhys. Lett. 50, 460–465 (2000).
[CrossRef]

Nakatsuka, H.

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Nieuwenhuizen, Th. M.

M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Pappu, R.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297, 2026–2030 (2002).
[CrossRef] [PubMed]

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Pnini, R.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

R. Pnini and B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

Polkosnik, W.

A. Z. Genack, N. Garcia, and W. Polkosnik, “Long-range intensity correlation in random media,” Phys. Rev. Lett. 65, 2129–2132 (1990).
[CrossRef] [PubMed]

Prokhorov, A. M.

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Rasmussen, J. J.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

Recht, B.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297, 2026–2030 (2002).
[CrossRef] [PubMed]

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Scheffold, F.

F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

F. Scheffold, W. Härtl, G. Maret, and E. Matijević, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

Sebbah, P.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

P. Sebbah, P. Sixou, C. Vanneste, and H. Guillard, “Nonlinear scattering and optical power limitation in direct and inverse polymer dispersed liquid crystal,” Nonlinear Opt. 27, 377–383 (2001).

Segev, M.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

Shapiro, B.

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

B. Shapiro, “New type of intensity correlation in random media,” Phys. Rev. Lett. 83, 4733–4735 (1999).
[CrossRef]

R. Pnini and B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57, 2168–2171 (1986).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Silberberg and I. Bar Joseph, “Optical instabilities in a nonlinear Kerr medium,” J. Opt. Soc. Am. B 1, 662–670 (1984).
[CrossRef]

Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982).
[CrossRef]

Sixou, P.

P. Sebbah, P. Sixou, C. Vanneste, and H. Guillard, “Nonlinear scattering and optical power limitation in direct and inverse polymer dispersed liquid crystal,” Nonlinear Opt. 27, 377–383 (2001).

Skipetrov, S. E.

S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
[CrossRef]

As regards the effect of the noninstantaneous nature of nonlinearity on speckle dynamics, our first results have been announced in S. E. Skipetrov, “Instability of speckle patterns in random media with noninstantaneous Kerr nonlinearity,” Opt. Lett. 28, 646–648 (2003).
[CrossRef] [PubMed]

S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
[CrossRef]

S. E. Skipetrov and R. Maynard, “Nonuniversal correlations in multiple scattering,” Phys. Rev. B 62, 886–891 (2000).
[CrossRef]

S. E. Skipetrov and R. Maynard, “Instabilities of waves in nonlinear disordered media,” Phys. Rev. Lett. 85, 736–739 (2000).
[CrossRef] [PubMed]

Smerzi, A.

A. Trombettoni, A. Smerzi, and A. R. Bishop, “Discrete nonlinear Schrödinger equation with defects,” Phys. Rev. E 67, 016607 (2003).
[CrossRef]

Soljacic, M.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

Spivak, B.

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of the scattering potential,” Phys. Rev. Lett. 84, 1970–1973 (2000).
[CrossRef] [PubMed]

Spivak, B. Z.

A. Yu. Zyuzin and B. Z. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–566 (1987).

Sprik, R.

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
[CrossRef] [PubMed]

Stephen, M. J.

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–287 (1987).
[CrossRef] [PubMed]

Sukhorukov, A. A.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

Taylor, J.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297, 2026–2030 (2002).
[CrossRef] [PubMed]

Trombettoni, A.

A. Trombettoni, A. Smerzi, and A. R. Bishop, “Discrete nonlinear Schrödinger equation with defects,” Phys. Rev. E 67, 016607 (2003).
[CrossRef]

van Albada, M. P.

M. P. van Albada, J. F. de Boer, and A. Lagendijk, “Observation of long-range intensity correlation in the transport of coherent light through a random medium,” Phys. Rev. Lett. 64, 2787–2790 (1990).
[CrossRef] [PubMed]

van Rossum, M. C. W.

M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

van Wonderen, A. J.

A. J. van Wonderen, “Diagrammatic treatment of light scattering by a collection of Kerr particles,” Phys. Rev. B 50, 2921–2940 (1994).
[CrossRef]

Vanneste, C.

P. Sebbah, P. Sixou, C. Vanneste, and H. Guillard, “Nonlinear scattering and optical power limitation in direct and inverse polymer dispersed liquid crystal,” Nonlinear Opt. 27, 377–383 (2001).

Vishwanath, A.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Wiersma, D. S.

D. S. Wiersma and S. Cavalieri, “Light emission: a temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

Wolf, P.-E.

G. Maret and P.-E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Yodh, A.

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Zyuzin, A.

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of the scattering potential,” Phys. Rev. Lett. 84, 1970–1973 (2000).
[CrossRef] [PubMed]

Zyuzin, A. Yu.

A. Yu. Zyuzin and B. Z. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–566 (1987).

Europhys. Lett.

R. Bressoux and R. Maynard, “On the speckle correlation in nonlinear random media,” Europhys. Lett. 50, 460–465 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Nature

D. S. Wiersma and S. Cavalieri, “Light emission: a temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

Nonlinear Opt.

P. Sebbah, P. Sixou, C. Vanneste, and H. Guillard, “Nonlinear scattering and optical power limitation in direct and inverse polymer dispersed liquid crystal,” Nonlinear Opt. 27, 377–383 (2001).

Opt. Lett.

Phys. Rep.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Phys. Rev. B

V. M. Agranovich and V. E. Kravtsov, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691–13694 (1991).
[CrossRef]

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second-harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931–4942 (1991).
[CrossRef]

A. J. van Wonderen, “Diagrammatic treatment of light scattering by a collection of Kerr particles,” Phys. Rev. B 50, 2921–2940 (1994).
[CrossRef]

R. Pnini and B. Shapiro, “Fluctuations in transmission of waves through disordered slabs,” Phys. Rev. B 39, 6986–6994 (1989).
[CrossRef]

S. E. Skipetrov and R. Maynard, “Nonuniversal correlations in multiple scattering,” Phys. Rev. B 62, 886–891 (2000).
[CrossRef]

F. Scheffold, W. Härtl, G. Maret, and E. Matijević, “Observation of long-range correlations in temporal intensity fluctuations of light,” Phys. Rev. B 56, 10942–10952 (1997).
[CrossRef]

R. Berkovits, “Sensitivity of the multiple-scattering speckle pattern to the motion of a single scatterer,” Phys. Rev. B 43, 8638–8640 (1991).
[CrossRef]

Phys. Rev. E

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Nonlinearity and disorder: classification and stability of nonlinear impurity modes,” Phys. Rev. E 63, 036601 (2001).
[CrossRef]

A. Trombettoni, A. Smerzi, and A. R. Bishop, “Discrete nonlinear Schrödinger equation with defects,” Phys. Rev. E 67, 016607 (2003).
[CrossRef]

S. E. Skipetrov, “Temporal fluctuations of waves in weakly nonlinear disordered media,” Phys. Rev. E 63, 056614 (2001).
[CrossRef]

S. E. Skipetrov, “Langevin description of speckle dynamics in nonlinear disordered media,” Phys. Rev. E 67, 016601 (2003).
[CrossRef]

Phys. Rev. Lett.

K. Ikeda, H. Daido, and O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45, 709–712 (1980).
[CrossRef]

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett. 50, 109–112 (1983).
[CrossRef]

Y. Silberberg and I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear optical interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef] [PubMed]

B. Spivak and A. Zyuzin, “Mesoscopic sensitivity of speckles in disordered nonlinear media to changes of the scattering potential,” Phys. Rev. Lett. 84, 1970–1973 (2000).
[CrossRef] [PubMed]

S. E. Skipetrov and R. Maynard, “Instabilities of waves in nonlinear disordered media,” Phys. Rev. Lett. 85, 736–739 (2000).
[CrossRef] [PubMed]

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947–3950 (1993).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

B. Shapiro, “New type of intensity correlation in random media,” Phys. Rev. Lett. 83, 4733–4735 (1999).
[CrossRef]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800–5803 (1998).
[CrossRef]

M. P. van Albada, J. F. de Boer, and A. Lagendijk, “Observation of long-range intensity correlation in the transport of coherent light through a random medium,” Phys. Rev. Lett. 64, 2787–2790 (1990).
[CrossRef] [PubMed]

A. Z. Genack, N. Garcia, and W. Polkosnik, “Long-range intensity correlation in random media,” Phys. Rev. Lett. 65, 2129–2132 (1990).
[CrossRef] [PubMed]

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial-field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
[CrossRef] [PubMed]

B. Shapiro, “Large intensity fluctuations for wave propagation in random media,” Phys. Rev. Lett. 57, 2168–2171 (1986).
[CrossRef] [PubMed]

M. J. Stephen and G. Cwilich, “Intensity correlation functions and fluctuations in light scattered from a random medium,” Phys. Rev. Lett. 59, 285–287 (1987).
[CrossRef] [PubMed]

Phys. Today

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

Rev. Mod. Phys.

M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Science

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297, 2026–2030 (2002).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef] [PubMed]

Sov. Phys. JETP

A. Yu. Zyuzin and B. Z. Spivak, “Langevin description of mesoscopic fluctuations in disordered media,” Sov. Phys. JETP 66, 560–566 (1987).

Z. Phys. B

G. Maret and P.-E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B 65, 409–413 (1987).
[CrossRef]

Other

F. Scheffold, S. Romer, F. Cardinaux, H. Bissig, A. Stradner, V. Trappe, C. Urban, S. Skipetrov, L. Cipelleti, and P. Schurtenberger, “New trends in optical microrheology of complex fluids and gels,” Prog. Colloid Polym. Sci., to be published.

Waves and Imaging through Complex Media, P. Sebbah, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001).

Wave Scattering in Complex Media: From Theory to Applications, B. A. van Tiggelen and S. E. Skipetrov, eds. (Kluwer Academic, Dordrecht, The Netherlands, 2003).

N. Bloembergen, Nonlinear Optics (World Scientific, Singapore, 1996).

R. W. Boyd, Nonlinear Optics (Academic, New York, 2002).

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer-Verlag, Berlin, 1999).

S. E. Skipetrov and R. Maynard, “Diffuse waves in nonlinear disordered media,” in Ref. 3.

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Figures (6)

Fig. 1
Fig. 1

(a) Diagrammatic representation of the correlation function jext(i)(r, t)jext(j)(r1, t1) of Langevin currents jext(r, t). (b)–(d) Diagrams contributing to the correlation function q(i)×(r, r, Δt)q(j)(r1, r1, Δt1) of random response functions q(r, r, Δt). (e), (f) Examples of diagrams neglected in our analysis. Solid traces with right (left) arrows denote retarded (advanced) Green’s functions of the linear disordered wave equation. Vertical dashed lines denote scattering of the two connected wave fields on the same heterogeneity. Wavy lines are k02 vertices. The diagrams (b)–(f) are obtained by functional differentiation of diagram (a) with respect to the dielectric constant of the random medium.

Fig. 2
Fig. 2

Stability diagram of speckle pattern in a nonlinear disordered medium. Physically realizable combinations of excitation frequency Ω, Lyapunov exponent Λ, and bifurcation parameter p belong to the surface shown in the figure. Heavy curve corresponds to Λ=0 and splits the surface into two parts. Instability (Λ>0) can develop only in the upper part of the surface, hence requires p>1. In the lower part of the surface, p<1 and Λ<0 for all Ω.

Fig. 3
Fig. 3

Frequency dependence of instability threshold in a nonlinear disordered medium with instantaneous Kerr nonlinearity (τNL/TD=0) for kl/(L/l)0 (upper solid curve) and kl/(L/l)=0.1, 1, 10 (other three solid curves from top to bottom). Inclined dashed lines show scaling laws Ω2 and Ω1/2. Horizontal dashed lines show asymptotic values of p for large Ω.

Fig. 4
Fig. 4

Lyapunov exponent Λ versus excitation frequency Ω (both in units of TD-1=D2/L) in a nonlinear disordered medium with instantaneous Kerr nonlinearity (τNL/TD=0) slightly above the absolute instability threshold (p=1.1) for kl/(L/l)0 (lower solid curve) and kl/(L/l)=0.1, 1, 10 (other three solid curves from bottom to top). Horizontal dashed line shows Λ=0.

Fig. 5
Fig. 5

Frequency dependence of instability threshold in a nonlinear disordered medium with noninstantaneous Kerr nonlinearity (τNL/TD=1, 10-2, 10-3, top three solid curves from top to bottom) compared with the case of instantaneous nonlinearity (τNL/TD=0, lower solid curve). kl/(L/l)=1 for all curves. Inclined dashed line shows scaling law Ω2, horizontal dashed line shows asymptotic value of p for large Ω and τNL/TD=0.

Fig. 6
Fig. 6

Lyapunov exponent Λ versus excitation frequency Ω (both in units of TD-1=D2/L) in a nonlinear disordered medium with noninstantaneous Kerr nonlinearity (τNL/TD=1, 10-1, 10-2, bottom three solid curves from bottom to top), slightly above the absolute instability threshold (p=1.1), compared to the case of instantaneous nonlinearity (τNL/TD=0, upper solid curve). kl/(L/l)=1 for all curves. Horizontal dashed line shows Λ=0.

Equations (20)

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

2-1c2 2t2[1+δ(r)+ΔNL(r, t)]E(r, t)=J(r, t),
δI(r, t)δI(r1, t1)short=|E(r, t)E(r, t1)|2f(r-r1)2,
tδI(r, t)-D2δI(r, t)=-·jext(r, t),
δI(r, t)δI(r1, t)longI02k2lΔr.
Δjext(r, t)=Vd3r-tdtq(r, r, t-t)×Δ[ΔNL(r, t)],
q(i)(r, r, Δt)q(j)(r1, r1, Δt1)
=3πD2(c2/l)δijδ(r-r1){I(r)G(r, r1;Δt-Δt1)×G(r1, r;Δt1)I(r)+I(r1)G(r1, r;Δt1-Δt)×G(r, r;Δt)I(r)-I(r)G(r, r;Δt)×I(r1)G(r1, r;Δt1)+k2l/(3πD)×δ(Δt-Δt1)f2(r-r1)I(r)×[I(r)G(r, r;Δt)+I(r1)G(r1, r;Δt)]}.
t jext(r, t)=Vd3r0dΔtq(r, r, Δt)×tΔNL(r, t-Δt).
τNL tΔNL(r, t)=-ΔNL(r, t)+2n2δI(r, t),
iνδI(r, ν)-D2δI(r, ν)=-·jext(r, ν),
iνjext(r, ν)=iνVd3r0dΔt×q(r, r, Δt)×ΔNL(r, ν)×exp(-iνΔt),
iντNLΔNL(r, ν)=-ΔNL(r, ν)+2n2δI(r, ν).
p1+kl/(L/l)h(ΩTD, ΛTD)+kl/(L/l)g(ΛTD)H(ΩτNL, ΛτNL),
g(y)=2[2-3 exp(-1/2)] 12y+1×1-1+12 2y+1exp-12 2y+1.
p=Δn2Ll2kl+Ll.
ΔϕNL=kn20sds1ΔI(r),
ΔϕNL2=ΔϕL2k2n220sds10sds2δI(r1)δI(r2),
ΔϕNL2pΔϕL2,
ΔϕNL2Δn2Ll2kl+Ll(ΩTD)-1/2ΔϕL2.
p1+kl/(L/l)(ΩTD)-1/2+kl/(L/l).

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