Resonance Properties of Scattering and Generation of Waves on Cubically Polarisable Dielectric Layers

Abstract

In this paper we investigate the problem of scattering and generation of waves on an isotropic, non-magnetic, linearly polarised (E-polarisation), non-linear, layered, cubically polarisable, dielectric structure, which is excited by a packet of plane waves, in the range of resonant frequencies. We consider wave packets consisting of both strong electromagnetic fields at the excitation frequency, leading to the generation of waves, and of weak fields at the multiple frequencies, which do not lead to the generation of harmonics but influence on the process of scattering and generation of waves. The analysis of the quasi-homogeneous electromagnetic fields of the non-linear dielectric layered structure made it possible to derive a condition of phase synchronism of waves. If the classical formulation of the mathematical model is supplemented by this condition of phase synchronism, we arrive at a rigorous formulation of a system of boundary-value problems with respect to the components of the scattered and generated fields (see Angermann & Yatsyk (2011)). This system is transformed to equivalent systems of non-linear problems, namely a system of one-dimensional non-linear Fredholm integral equations of the second kind and a system of non-linear boundary-value problems of Sturm-Liouville type. The numerical algorithms of the solution of the non-linear problems are based on iterative procedures which require the solution of a linearised system in each step. In this way the approximate solution of the non-linear problems is described by means of solutions of linearised problems with an induced dielectric permeability. The analytical continuation of these problems into the region of complex values of the frequency parameter allows us to switch to the analysis of spectral problems. The corresponding eigen-frequencies form a discrete, countable set of points, with the only possible accumulation point at infinity, and lie on a complex two-sheeted Riemann surface. In the frequency domain, the resonant scattering and generation properties of non-linear structures are determined by the proximity of the excitation frequencies of the non-linear structures to the complex eigen-frequencies of the corresponding homogeneous linear spectral problems with the induced non-linear dielectric permeability of the medium. 15