Multistability resulting from random rearrangements in a ring resonator with a nonlinear element

Certain types of random rearrangements of attracting sets (attractors) arising in the phase space of a ring resonator containing a nonlinear medium are studied numerically. It is shown that for certain values of the parameters we can reduce the 2-D mapping describing the evolution of the field in the resonator to 1-D and give a qualitative explanation of the threshold phenomena observed. A method is considered for expansion in terms of a small parameter that allows us to calculate the fine structure of a random attractor analytically. As it has been shown in [1] [5], self-oscillations transformed into random motion (optical turbulence) can arise in a ring resonator with a nonlinear element (henceforth to be called the nonlinear ring resonator – NRR). This paper is devoted to the study of random rearrangements of attractors in the phase space of the NRR (including random-regular motion transitions), leading to behavior similar to hysteresis. Such behavior was observed in the numerical modeling of NRR dynamics in the adiabatic regime (the Ikeda mode [1]); it was assumed that nonlinear conversion of light takes place due to the ν-photon transitions in the two-level medium. It is important that at the time of rearrangements of an attractor, an exchange (jump) of the type of NRR dynamics takes place; such a transition can result both in the smooth variation of some parameter of the system and in impulse perturbation of the exciting field. This indicates, in principle, feasibility of the design of new types of multistable (having a series of stable states) optical devices based on NRR (we call multistable dynamic systems or states, to which correspond before and after the jump isolated attracting sets, but stationary points are optional). We consider a theoretical NRR model. Let a two-level medium placed in the ring resonator be excited by an external light source (stationary and coherent) through a beam splitter; after passing through the active medium part of the light leaves the resonator through another beam splitter; the intensity reflectance of each of the mirrors is equal to κ. In the case of ν-photon transitions, the slowly varying electric field amplitude E of the wave, the polarization P of the medium, and the density W of population inversion satisfy the equations E ′ + c−1Ė = iβgP ∗(E∗)ν−1, (1) Ṗ = − P T2 + i(ω0 − νω)P + ig(E∗)νW, (2) Ẇ = − + W T1 + i(g∗EνP − c.c.)/2, (3) where c is the speed of light; T1 is the energy relaxation time; T2 is the dephasing time; ω0 is the resonant frequency of the medium; ω is the frequency of the exciting field; g and β are constants