Diffraction by a Grating Consisting of Absorbing Screens of Different Height. New Equations

Abstract

UDC 534.26 The problem of diffraction of a plane wave by a grating consisting of absolutely absorbing screens of different height is studied. It is assumed that the angle of incidence is small. The problem is considered in the parabolic approximation. Edge Green functions are introduced. An embedding formula and a spectral equation for the edge Green functions are derived. An OE-equation for the coefficients of the spectral equation is constructed. The latter is solved numerically. The evolution equation describing the dependence of the edge Green functions on a geometric parameter (the height of the screens) is proved. Using this equation, the asymptotics of the reflection coefficient is calculated for the principal mode. Bibliography: 7 titles. The problem of scattering of a waveguide mode by an open end of a waveguide with acoustically hard thin walls was studied by L. A. Weinstein [1]. The problem was solved by applying the Wiener–Hopf method [2]. The coefficients of scattering of the incident mode (the mode moving toward the open end of the waveguide) into reflected modes (the modes moving back from the open end) were obtained. This problem is of significant practical importance. It helps to describe oscillations in a Fabry–Perot resonator, which is considered as part of a waveguide in this case. The wave process in such a resonator is represented as successive reflections by open ends of the waveguide. The key result obtained by L. A. Weinstein is as follows. A high-frequency mode (i.e., a mode with wavelength much smaller than the width of the waveguide) close to the cut-off frequency (i.e. composed of partial waves traveling almost perpendicularly to the axis of the waveguide) has reflection coefficient close to −1. This result is quite surprising, because the open end of the waveguide has no reflective structures. The coefficient close to 0 would be rather expected because of the radiation of the wave energy into the open space. The fact that the reflection coefficient is close to −1 explains a high Q-factor of Fabry–Perot resonators in the absence of focusing elements. The problem of high-frequency scattering close to the cut-off frequency is quite complicated since the problem cannot be solved in terms of the geometrical theory of diffraction: the field contains multiple penumbra components. Here we mean that a primary penumbral field being scattered by an edge generates a secondary penumbra, etc. L. A. Weinstein noticed that, using the reflection principle, one can reformulate the problem of diffraction by a waveguide outlet as a problem of scattering on a branched surface by a periodic diffraction grating formed by branch points. This formulation seems to be more convenient because of the possibility of analyzing this problem with the help of the formalism of the edge Green functions. One can use a parabolic approximation in the high-frequency case. In the parabolic approximation, one can study a grating composed of perfectly absorbing screens instead of a branched surface.