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Abstract-The paper is devoted to the study of the interaction of the electromagnetic waves with the structure composed of perfectly conducting strip grating, situated on the plane boundary of metamaterial with effective permittivity, depending on the frequency of the wave of excitation. The rigorous solution to the relevant diffraction boundary value problem is developed. The extensive numerical experiments, performed with a help of corresponding algorithm constructed, allowed to establish several regularities in the complicated process of interaction of electromagnetic waves with grating on dispersive metamaterial. The efficient association of analytical and numerical study has provided the understanding of the nature of resonant phenomena appearing in this process.

Abstract-The goal of the present paper is two folded. The first, the methodological one, is the complementation of well established in diffraction theory of gratings C method with certain elements of spectral theory and the development of interactive numerical algorithm that made feed back conjunction between diffraction and spectral problems. As a natural result the second goal appeared: the appearing of such tool for numerical experiments resulted in profound qualitative and quantitative study of rather peculiar phenomena in resonant scattering from periodic surface. Special attention has been paid to the investigation of electromagnetic waves diffraction from periodic boundaries of material with single and double negative parameters.

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The paper is focused on reliable modeling of the effects associated with the resonant transformation of the field of a plane, density modulated electron beam, flying over the periodically uneven boundary of a natural or artificial medium, in the field of volume outgoing waves. Here, the general information (analytical basis) is presented on the peculiarities and principal characteristics of electromagnetic fields arising in the situations under consideration, on the procedures for regularization of model boundary value problems describing these situations, and on possible eigen modes of periodic structures. Without relying on this information, it is impossible to advance considerably effectively in solving numerous urgent physical problems (establishing the conditions providing anomalously high levels of Vavilov-Cherenkov and/or Smith-Purcell radiation; diagnostics of beams of charged particles, artificial materials and media) and in practical implementation of new knowledge about the effects of diffraction radiation and their wave analogues in new devices and instruments of optoelectronics, high-power electronics, antenna, and accelerator technology.