L. Angermann, V.V. Yatsyk

Mathematical models for the analysis of processes of resonance scattering and generation of the third harmonic in the case of the diffraction of a plane electromagnetic wave through an isotropic, cubically polarisable, non-magnetic, linearly polarised (E polarisation) medium with a non-linear, layered dielectric structure are considered. The electromagnetic waves in a non-linear medium with a cubic polarisability are described by an infinite system of non-linear equations. In the study of particular non-linear effects it proves to be possible to restrict the examination to a finite number of equations, and also to leave certain terms in the representation of the polarisation coefficients, which characterize the physical problem under investigation. Thus, in the analysis of the non-linear effects of the processes of resonance scattering of the field (at the frequency of excitation) and generation of the third harmonic (at the triple frequency) it is possible to restrict the investigation to a system of two equations, where only the non-trivial terms in the expansion of the polarisation coefficients are taken into account. This leads to the strong formulation of a boundary-value problem, which in turn can be reduced to a system of one-dimensional non-linear integral equations (defined along the height of the structure) with respect to the complex Fourier amplitudes of the scattered fields in the non-linear layer at the basic and triple frequencies. The system of non-linear integral equations and also the boundary-value problem are reduced to a system of non-linear boundary-value problems of Sturm-Liouville type, which indicates the equivalence of both problems. In the paper both the variational approach to the solution of the system of non-linear boundary-value problems of Sturm-Liouville type (based on the application of a finite element method) and one of the possible iterative schemes of the solution of the system of non-linear integral equations (based on the application of a quadrature rule to each of the non-linear integral equations) are considered. The algorithm of the solution of the system of non-linear integral equations is given. The analysis of the scattering problem and the generation of the third harmonic by excitation by a plane wave passing a non-linear three-layered structure is carried out. Results of the numerical investigation of both the values of the non-linear dielectric constants corresponding to a given amplitude of the incident field and of the scattered and generated fields are presented. The dependence characterizing the portions of generated energy in the third harmonic on the value of the amplitude of the excitation field of the non-linear structure is given. Within the framework of the conservative system under consideration it is shown that the imaginary part of the dielectric constant, determined by the value of the non-linear part of the polarization at the excitation frequency, characterises the loss of energy in the non-linear medium (at the frequency of the incident field) due to the generation of the electromagnetic field of the third harmonic (at the triple frequency).