Chaotic dynamics of electromagnetic field in plane-parallel resonator with semitransparent nonlinear layer

A plane-parallel resonator, containing a thin translucent layer of active or passive material with a nonlinear dependence of the current density (magnetic moment) on the electric (magnetic) field, can be considered as the simplest model of many electronic devices in the microwave, terahertz, and optical frequency ranges. Very often the nonlinear layer has its size along the wave propagation direction much smaller of the oscillation’s wavelength. It may be placed either at the resonator wall, or at the arbitrary point inside the resonator. In this case the nonlinear transformation of the resonator partial waves by the active element takes place, and the spatial-temporal dynamics of the field may be either a simple standing wave or become a very complicate nonlinear, but still deterministic dynamics. Finally, the field dynamics may be a stochastic one. These processes are very convenient to describe within the frame of the approach proposed in [1, 2] and developed in later publications, e.g. [3]. The main idea of that approach consists in accounting the nonlinear field-current interaction in the boundary conditions, but not in the wave equations. Introducing Reimann invariants for electromagnetic field components of plane waves in the resonator, and using their invariance on the wave equation characteristics one may reduce the primary complicated initial-boundary value problem for partial differential equations with nonlinear boundary conditions to the much simpler mathematical object - nonlinear difference or difference-differential equation, or to the set of such equations for the one of the field components (or on the several ones). This simplification enables to carry on an efficient analysis of spatial-temporal dynamics of electromagnetic field in case of self-oscillatory active devices or in resonant systems with passive nonlinear thin inserts, etc. In the paper, the common statement of the problems is formulated for the case of the surface with the set of lamped active elements and also for passive semitransparent layer, such as semiconductor or ferromagnetic metasurfaces [4]. Solutions for the obtained equations of self-oscillations in the resonator with the active layer placed at 1/3 of the resonator length are shown in Figure 1: we observe relaxation oscillations for a small parameter of nonlinearity and for strong nonlinearity we observe chaos