Scattering and generation properties of cubically polarisable layers with negative and positive cubic susceptibility of the medium

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 plane wave, is investigated in the domain of resonance frequencies. The resulting mathematical model can be represented equivalently by a system of non-linear boundary-value problems of Sturm - Liouville type or by a system of one-dimensional non-linear Fredholm integral equations. The solutions of these problems are approximated numerically by the help of quadrature methods and iterative procedures which require the solution of a linear system in each step. In this way the approximate solution of the non-linear problems is described by means of solutions of linear problems with an induced non-linear dielectric permeability. The analytical continuation of these linear problems into the region of complex values of the frequency parameter allows us to switch to the analysis of spectral problems. Their eigenfrequencies form a discrete, countable set of points, with the only possible accumulation point at infinity, and lie on a complex two-sheeted Riemann surface. Some results of calculations of characteristics of the scattered field of a plane wave are presented, taking into account the third harmonic generated by nonlinear cubically polarisable layers with both negative as well as positive values of the cubic susceptibility of the medium. It is shown that layers with negative and positive values of the coefficient of cubic susceptibility of the non-linear medium have fundamentally different scattering and generation properties. For instance, in the case of negative values of the susceptibility, a decanalisation of the electromagnetic field can be detected.