Nonlinearity and Localization in One-Dimensional Random Media
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.
KeywordsCorrelation Length Transmission Coefficient Elliptic Function Nonlinear Medium Random Medium
Unable to display preview. Download preview PDF.
- Abramowitz, M. and Stegun, I. (1965), Handbook of Mathematical Functions Dover Publications, Inc., New YorkGoogle Scholar
- 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
- Carmona, R. (1985), “The Random Schrödinger Equation”, Ecole d’Ete de Saint-Four, ed. P. Hennegrenin, Springer Lecture Notes in Mathematics, 1985Google Scholar
- 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
- Dygas, M. M., Matkowsky, B. J., and Schuss, Z. (1987), “Stochastic Stability of Nonlinear Oscillators,” Preprint Google Scholar
- 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
- 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
- Knapp, R. (1988), “Nonlinearity and Localization in One Dimensional Random Media”, PhD Thesis, New York University, October 1988Google Scholar
- 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
- 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
- Papanicolaou, G. C. (1978), “Asymptotic Analysis of Stochastic Equations,” M.A.A. Studies in Mathematics, 18, ed. M. Rosenblatt, M.A.A.Google Scholar
- Press, W., Flannery, B., Teukolsky S., Vetterling W. (1986), Numerical Recipes Cambridge University PressGoogle Scholar