Mode Diffraction: Analytical Justification of Matrix Models and Convergence Problems

The technique of the analytical justification of the matrix models developed via the eigenmode expansion methods (the mode-matching technique, the overlapping regions method, etc.) is proposed. According to this matrix operator technique, the sought-for scattering operator is expressed as a linear-fractional transformation of the given matrix operator, and vice versa. These operator matrices are considered to act in the space with an indefinite metric. Using four fundamental electromagnetic laws, the spectral properties of the scattering operator and, therefore, the given matrix operator are established. On this basis, the correctness of the studied matrix models is proved. The convergence of approximations is analytically investigated, including the relative convergence phenomenon