Diffraction by perfectly conducting plane screens

Abstract

A previously developed Hilbert-space formulation of electromagnetic diffraction is employed to solve the general problem of diffraction by perfectly conducting plane screens. There appears to be a trade-off between employing such a constrained mathematical context and the more loosely defined contexts previously employed to solve these problems. Whereas the latter type of context admits more of the idealized physical elements, such as infinite linearly polarized plane waves, etc., the Hilbert-space context possesses more algebraic and geometrical structure, giving rise to fundamental insights concerning the diffractive process. For screens of bounded edge (i.e. screens for which either the aperture or screen is spatially bounded), the diffracted fields are shown to be unique. For this latter type of screen, the diffracted field isexplicitly given in terms of the incident field by a single linear transformation which is both closed and idempotent. The use of orthogonal projections to express the boundary conditions enables us to obviate the usual dual integral equations, and hence to characterize the diffractive process by this single linear transformation. In case zero is not in the spectrum of a certain positive self-adjoint operator, the diffracted field can be described by a von Neumann series. In such a series, the zeroth term has the appearance of an electromagnetic « black screen » term, with the higher-order terms of the series representing increasing degrees of scattered contributions from the induced currents in the screen.