Řešení difrakčních úloh metodou hraničních integrálních rovnic

Abstract

Optical diffraction is one of the less usual applications of the boundary integral equations. The main intent of the dissertation thesis is to describe and realize the solution of boundary problem for Helmholtz equation with transition conditions on the common periodical boundary. The formulation of the boundary integral equations is usually based on variation methods. To derive the boundary integral equations, we use the single- and double-layer potentials which enable appropriate implementation of boundary conditions. Special attention is payed to properties of fundamental solution of Helmholtz equation represented by the Hankel function in two-dimensional case. Due to periodicity of the problem, the fundamental solution is expressed by quasi-harmonic series. Its properties determine characterics of singularities arising in integral kernels. The key theorems enabling us the algorithmization of the problem are formulated and proved. The collocation method is chosen for the numerical implementation of obtained algorithm. The boundary is represented by equidistant points and trigonometric polynomials are introduced as the basis functions. As the first application, we suggest the sine boundary between air and glass. Obtained results are compared with classical Rigorous Coupled Waves Method (RCWM). The smoothness of the boundary is one of the neccessary conditions of designed algorithm. For non-smooth boundaries, we propose original approach of the binary profile smoothing by using suitable goniometrical smoothing function. Obtained algorithm can be generalized for multilayer optical systems. Method of the algorithm as well as the discretization model are introduced in the concluding part of the thesis. For the application, we analyze optical structure glass-gold-air with the same period of boundary between the metal and the air that produce plasmon waves.