Nonlinearity and Localization in One-Dimensional Random Media

  • R. Knapp
  • G. Papanicolaou
  • B. White
Part of the Springer Proceedings in Physics book series (SPPHY, volume 39)


A nonlinear Fabry-Perot etalon with random inhomogeneities is modeled by a one-dimensional stochastic Helmholtz equation. An asymptotic estimate of the threshold intensity needed for optical bistability is found for homogeneous media as a function of length. In the random case localization is affected and estimates of the energy growth are derived for large lengths with fixed output. Comparisons of the theory and numerical simulations are presented.


Correlation Length Transmission Coefficient Elliptic Function Nonlinear Medium Random Medium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Abramowitz, M. and Stegun, I. (1965), Handbook of Mathematical Functions Dover Publications, Inc., New YorkGoogle Scholar
  2. [2]
    Al’tschuler, G.B., Ermolaev, V.S. (1982), “Enhanced Transmission Due to Nonlinear Light Scattering by Static Optical Nonuniformities”, Sov. Phys. Dokl. 28 2 p 146, American Institute of PhysicsADSGoogle Scholar
  3. [3]
    Al’tschuler, G.B., Ermolaev, V.S., Krylov, K.I., Manenkov, A.A. (1983), “Effects of Light Scattering in Nonuniform Media with Kerr Nonlinearity”, Sov. Phys. Dokl. 28 11 p 951, American Institute of PhysicsADSGoogle Scholar
  4. [4]
    Arnold, L., Papanicolaou, G., and Wihstutz, V. (1986), “Asymptotic Analysis of the Lyapunov Exponent and Rotation Number of the Random Oscillator and Applications.” Siam Journal of Applied Math, 28 3Google Scholar
  5. [5]
    Blankenship, G. and Papanicolaou G. C (1978), “Stability and Control of Stochastic Systems with Wide-Band Noise Disturbances,” Siam Journal of Applied Math, 34 3 p 347–476MathSciNetGoogle Scholar
  6. [6]
    Burridge R., Papanicolaou, G., and White, B. (1987), “Statistics for Pulse Reflection From a Randomly Layered Medium”, Siam J. Appl. Math., 47 p 146–168MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Carmona, R. (1985), “The Random Schrödinger Equation”, Ecole d’Ete de Saint-Four, ed. P. Hennegrenin, Springer Lecture Notes in Mathematics, 1985Google Scholar
  8. [8]
    Chen, W. and D.L. Mills (1987), “Optical Response of a Nonlinear Dielectric Film”, Physical Review B 35 2 p 524ADSCrossRefGoogle Scholar
  9. [9]
    Danileiko Y.K., Manenov A.A., Nechitailo V.S., Khaimov-Mal’kov V.Y. (1971), “Nonlinear Scattering of Light in Inhomogeneous Media”, Sov. Phys. JETP 33 4 p 674ADSGoogle Scholar
  10. [10]
    Devillard, P. and Souillard, B. (1986), “Polynomially Decaying Transmission for the Nonlinear Schrödinger Equation in a Random Medium” Journal of Statistical Physics 43 3–4Google Scholar
  11. [11]
    Dygas, M. M., Matkowsky, B. J., and Schuss, Z. (1987), “Stochastic Stability of Nonlinear Oscillators,” Preprint Google Scholar
  12. [12]
    Flytzanis, C., (1984), “Bistability, Instability, and Chaos in Passive Nonlinear Optical Systems”, Lecture notes from the Proceedings of the Third International School on Condensed Matter Physics, Nonlinear Phenomena in Solids-Modem Topics, ed. M. Borissov, World ScientificGoogle Scholar
  13. [13]
    Fürstenberg H. (1963), “Noncommuting Random Products”, Trans. Am Soc., 108 p 377–428CrossRefGoogle Scholar
  14. [14]
    Gibbs, H.M., McCall, S.M., Venkatesan, T.N.C., Gossard, A.C., Passner, A., Wiegmann, W. (1979), “Optical Bistability in Semiconductors”, Appl. Phys. Lett. 35 6 p 451ADSCrossRefGoogle Scholar
  15. [15]
    Gibbs, H.M., Tarng, J.L., Weinberger, D.A., Tai, K. and Gossard, A.C., McCall, S.L., Passner, A., Wiegman, W. (1982), “Room Temperature Excitonic Optical Bistability in a GaAs-GaAlAs Superlattice Etalon”, Appl. Phys. Lett. 1 Aug, 1982Google Scholar
  16. [16]
    Gihman, I.I., Skorohod, A.V. (1972), Stochastic Differential Equations Springer VerlagzbMATHGoogle Scholar
  17. [17]
    Knapp, R. (1988), “Nonlinearity and Localization in One Dimensional Random Media”, PhD Thesis, New York University, October 1988Google Scholar
  18. [19]
    Kohler, W. and Papanicolaou, G. (1973), “Power Statistics for Wave Propagation in one Dimension and Comparison With Transport Theory”, J. Math. Phys. 14, p 1733–1745 and 15 p 2186–2197 (1974)MathSciNetADSCrossRefGoogle Scholar
  19. [18]
    Klyatskin, V.I. (1984), “Solution of a Nonlinear Problem on Self-action of a Plane Wave in a Layered Medium Using the Immersion Method”, PreprintGoogle Scholar
  20. [20]
    Klyatskin, V.I., Kozlov, F.F., and Yaroshchuk, E.V. (1982), “Reflection Coefficient in the One-dimensional Problem of Self-action of a Wave”, Sov. Phys. JETP 55 2 p 220 American Institute of PhysicsGoogle Scholar
  21. [21]
    Marburger, J. H. and Felber, F. S. (1978), “Theory of a Lossless nonlinear Fabry-Perot Interferometer,” Physical Review A 17 p 335–342ADSGoogle Scholar
  22. [22]
    McCall, S.L., Gibbs, H.M., Venkatesan, T.N.C. (1975), J. Opt. Soc. Am. 65 p 1184ADSGoogle Scholar
  23. [23]
    Miller, D.A.B., Smith, S.D., and Johnston, A. (1979), “Optical Bistability and Signal Amplification in a Semiconductor Crystal: Applications of New Low-power Nonlinear Effects in InSb”, Appl. Phys. Lett. 35 9 p 658ADSCrossRefGoogle Scholar
  24. [24]
    Papanicolaou, G. C. (1978), “Asymptotic Analysis of Stochastic Equations,” M.A.A. Studies in Mathematics, 18, ed. M. Rosenblatt, M.A.A.Google Scholar
  25. [25]
    Papanicolaou, G. C. (1977), “Introduction to the Asymptotic Analysis of Stochastic Equations,” Lectures in Applied Mathematics 16 p 109–147MathSciNetGoogle Scholar
  26. [26]
    Press, W., Flannery, B., Teukolsky S., Vetterling W. (1986), Numerical Recipes Cambridge University PressGoogle Scholar
  27. [28]
    Sheng, P., Zhang, B., White, B., Papanicolaou, G. (1986), “Multiple Scattering Through Localization Length Scales”, Phys. Rev. Letters 578 p 1000–1003ADSCrossRefGoogle Scholar
  28. [27]
    Sheng, P., Zhang, B., White, B., Papanicolaou, G. (1986), “Minimum Wave Localization Length in a one Dimensional Random Medium”, Phys. Rev. B 34 7 p 4757–4761MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    Spigler, Renato (1986), “Mean Power Reflection from a One-dimensional Nonlinear Random Medium”, J. Math. Phys. 27 7 p 1760–1771MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [30]
    Spigler, Renato (1985), “Nonlinear Parametric Oscillations in Certain Stochastic Systems: A Random van der Pol Oscillator”, J. Stat. Phys. 41 p 175MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. [31]
    Warren, M.E., Koch, S.W., Gibbs, H.M. (1987), “Optical Bistability, Logic Gating, and Waveguide Operation in Semiconductor Etalons”, IEEE Computer 20 12 p 68–81ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • R. Knapp
    • 1
  • G. Papanicolaou
    • 2
  • B. White
    • 3
  1. 1.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Exxon Research and Engineering CompanyAnnandaleUSA

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