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Computers in Physics

Computational method for intrinsic optical bistability: Steady-state solutions and their stability

Joseph W. Haus, Tuyen K. Tran

Published Mar 1, 1988• 0 citations
Publisher

Abstract

An efficient numerical algorithm is described for solving problems with coupled nonlinear differential equations with boundary conditions applied at opposite ends of the medium. It is applied to a problem of optical bistability with coupled Maxwell equations and heat-flow equations, and associated boundary conditions. It is shown that the numerical procedure is well suited to perform a stability analysis of the solution.

DOI
10.1063/1.168297

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An efficient numerical algorithm is described for solving problems with coupled nonlinear differential equations with boundary conditions applied at opposite ends of the medium. It is applied to a problem of optical bistability with coupled Maxwell equations and heat-flow equations, and associated boundary conditions. It is shown that the numerical procedure is well suited to perform a stability analysis of the solution.
The theory of optical bistability can be developed at various levels of sophistication.(1) In this overview I have chosen an approach which I hope will make the simplicity and generality of the phenomenon clear, and be suited to the emphasis of this meeting. I will describe a deterministic, semiclassical theory of optical bistability, based on coupled Maxwell-Bloch equations. Statistical and quantum statistical theories will not be included. I also limit my attention to questions concerning steady states and their stability, leaving the theory of higher bifurcations to other speakers.
A single-mode dynamic equation is obtained from the Maxwell equation with the slowly-varying amplitude approximation, and the boundary conditions of the cavity. Here the mean-field limit and the good-quality cavity case have been used. This equation is useful in describing the dynamic behavior of intrinsic optical bistable systems and/or lasers with injected signals for any kinds of materials, including two-level atoms. Furthermore, according to the linear stability analysis, a generalized instability criterion is derived.
We consider the problem of the wave propagation through a nonlinear medium. We derive a dynamic system that governs the behavior of standing (or solitary) waves. The form of this system alone suffices to understand the qualitative dependence of solutions of the original equation on the intensity of the incident wave. We solve this dynamical system in the leading order in the nonlinearity strength. We find multiple solutions of the original problem for a given incoming wave and turning points of these solutions as a function of the wave intensity. We briefly investigate stability of different branches. Our results yield an analytic description of the optical bistability phenomenon.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.