Structure of solutions of the nonlinear equations in ring resonator

V. A. Lukyanenko   ; Yu. A. Khazova

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2023), pp. 49-66

Résumé

A mathematical model of the process of formation of phase spatial structures in the cross section of a coherent light beam in a non-linear optical system with spatially distributed feedback as a nonlinear ring resonator is considered. The simultaneous consideration of diffraction and non-linearity results in a variety of spatial structures. Nonlinear wave dynamics of spatially distributed optical systems is represented by the following phenomena: spatial bistability and multi-stability, formation of regular spatial diffraction structures, formation of optical vortices, solitons, dissipative structures, etc. The model of a ring resonator containing a layer of a nonlinear medium with cubic nonlinearity is based on the study of two related equations: the equation which are describing the time dynamics of phase modulation of a light wave in a nonlinear medium, and equations which are describing the time dynamics of the complex amplitude of the light field within the resonator, taking into account the diffraction. This model takes into account the local transverse interactions of the light wave with the nonlinear medium caused by both the diffusion of the particles of the nonlinear medium and the diffraction of the light wave. Due to simultaneous action of two physical processes in the system there are spatio-temporal phase structures. The nonlinear functional-differential equations of the parabolic type with feedback and transformation of spatial variables (which is given by involution operator) were previously considered. The involution operator property (rotation, reflection) allows to reduce the original equation to a system of equations without transformation of spatial variables. The set of solutions of such equations is determined by two parameters: low diffusion coefficient, and high flow rate. Also considered are the initial boundary problems for a circle, circle, ring with involution operator. This article considers a mathematical model of a ring resonator, consisting of a system of linear and nonlinear equations in partial derivatives. Stationary solutions are analyzed and stability zones of the corresponding linearized problem solutions are studied. For the case of a thin ring (circle), the problem is represented as a nonlinear integral equation.

Détail

Keywords: nonlinear ring resonator, phase pattern.
Mots-clés : spatial mode