MODELING OF RESONANCE EFFECTS IN ONE-DIMENSIONAL PERIODIC DIFFRACTION GRATINGS CONTAINING GRAPHENE STRIPS Part 1. MATHEMATICAL JUSTIFICATION OF THE SPECTRAL METHOD

Subject and Purpose. This paper presents a theoretical study of the interaction between monochromatic electromagnetic radiation and a one-dimensional periodic strip grating. The grating consists of periodically alternating perfectly conducting and graphene strips located at the boundary of a planar dielectric layer. The aim is to provide a mathematical justification for the spectral method analysis of resonance effects arising during the interaction of electromagnetic radiation with the strip grating. Methods and Methodology. The mathematical justification of the spectral method is based on the theory of non-self-adjoint compact operators in Hilbert spaces and the theory of compact analytic operator functions. In particular, we apply Keldysh’s theorems on the completeness of eigenvectors and associated vectors of non-self-adjoint compact operators, as well as the operator generalization of Rouché’s theorem for analytic operator functions. Results. The spectral approach to solving the diffraction problem of a one-dimensional periodic strip grating, which includes graphene strips, has received a rigorous mathematical treatment. It has been established that the diffraction field can be repre- sented as an expansion in eigenfunctions of the spectral problem, where the spectral parameter (eigenvalue) enters linearly into the boundary condition of conjugation on the graphene strips. The existence of the spectral problem solution has been proved in the case of small widths of the perfectly conducting strips. The completeness of the system of eigenfunctions (eigenvectors) in the corresponding Hilbert space has been demonstrated. As a consequence, in an unbounded region, there is a possibility to expand the diffraction field in resonance terms. An equation for resonance frequencies has been derived, indicating that the imaginary part of the spectral parameter equals the imaginary part of the surface conductivity of the graphene strips in the grating. Conclusions. The developed spectral method enables effective analysis of resonance effects that occur when electromagnetic radiation interacts with a one-dimensional periodic diffraction grating that includes graphene strips. This method can be used in the mathematical modeling of various devices and systems that utilize such gratings.