Random self-modulation of radiation in a ring cavity. Case of strong mixing

It is shown that when certain conditions are met, the random dynamics of radiation in a ring cavity containing a nonlinear element can be described by means of a 2-D representation with a rapidly oscillating exponential. Such a representation generates strong mixing in the attracting region of the phase space and is equivalent to a certain random representation when the exciting signal contains a small noise component. Methods of finding the transient and stationary distribution densities in the attracting region are discussed, as is the maximum Lyapunov index characterizing the mixing rate. References 1-3 have shown the fundamental possibility of formation of periodic and random self-oscillations of optical radiation in a ring cavity (RC) containing a nonlinear medium; subsequently, such oscillations have been observed experimentally [4]-[6]. Of interest are a further study and classification of the types of random oscillations (chaos), transformations of chaos, as well as scenarios of the genesis of chaos in RC and other nonlinear optical systems. This article is devoted to a study of developed chaos with strong mixing, which in a certain sense is the simplest type of chaos in RC and permits an analytical description based on the random-phase approximation. Let us assume that a RC is excited by a partially coherent light wave through a beam splitter; after passage through the nonlinear element, part of the light leaves the RC through another beam splitter; no excitation of the counterpropagating wave takes place. Let us also assume that the nonlinear element is a cell with a two-level medium (TM) in which s-quantum transitions are excited (s > 1); the corresponding dynamic equations for a slowly changing amplitude E of the wave electric field, polarization P of the medium, and density n of population differences are: E′ + c−1Ė = iβsgsP ∗(E∗)s−1, (1) Ṗ + P T2 − i(ω0 − sω)P = igs(E∗)sn, (2) ṅ + 1 + n T1 = i(g∗ sE P − c.c.)/2, (3) where T1 is the RC population relaxation time; T2 is the RC phase mismatch time; ω0 is the RC resonance frequency; ω is the frequency of the exciting field; the constants gs and βs are expressed in terms of the ordinary or composite (when s > 1) matrix elements of the transitions; the prime and dot above the letter, respectively, denote derivatives with respect to the coordinate z and time t. Let ∆ = T2(ω0−sω) À 1, so that the RC loss is low in comparison to the loss due to emission through the mirrors. We will also assume that Tc À τk À T1, T2, where Tc is the round-trip time, and τk is the correlation time of the noise component of the