A Matrix Riccati Equation for Fast Modeling of Wave Propagation and Absorption in Complex Inhomogeneous Media with Spatial Dispersion and Anisotropy

We describe a simple general method for addressing problems of electromagnetic wave propagation and linear mode-coupling in complex stationary media characterized by anisotropic and gyrotropic dielectric or magnetic response and spatial dispersion. A specific feature of such media is the possibility of propagation of several coupled normal waves with different polarizations. We develop a physically unambiguous interpretation of the invariant embedding technique applied to corresponding Maxwell equations. A set of normal waves is divided into two groups of counter-propagating modes, then a matrix impedance operator that characterizes the coupling of different counter-propagating modes is defined (R-operator of an inhomogeneous medium). This operator may be determined by solving a nonlinear evolution equation of Riccati type with the universal boundary conditions unrelated to the characteristics of incident radiation, accordingly, R-operator is also independent of them. Once R-operator is known, the field for any given structure of incident radiation can be reconstructed over the entire space by a trivial linear operation. The method provides a fast and convenient tool for studying the propagation of electromagnetic waves in effectively one-dimensional geometries, such as plane stratified or spherically symmetric media, in situations where the standard fast approximate methods like geometric optics or quasi-optics are not inapplicable due to presence either a strong spatial inhomogeneity or evanescent and mode-coupling regions or essential spatial dispersion (non-locality).