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Geodesic focussing in parallel-plate systems Mosevich, Jack Walter

Abstract

This thesis is concerned with the mathematics of the design of parallel-plate equivalents of optical systems, in particular with the parallel-plate equivalent of the parabolic mirror. A parallel-plate microwave system consists of a pair of metal plates, not necessarily plane, which are parallel in the sense that they share common normals at every point, and the normal separation is constant throughout. Consider the mean surface M , which is the locus of midpoints of the double normals, and suppose that microwave radiation is fed into the region between the plates. If M is not excessively curved the rays travel along its geodesies and a natural problem arises of how to shape M so that all rays from a point source between the plates emerge in a parallel beam (thus the system is equivalent to a parabolic mirror). It is assumed that M consists of two flats connected by a focusing bend B , where B is part of a tubular surface with directrix Δ . The problem is to determine the curve Δ so that the geodesies on the tube generated by Δ satisfy the focusing condition. The exact mathematical formulation of this problem yields an extremely involved functional differential equation in terms of the polar equation of Δ, r = r(θ) , which proves unsuitable for solving for r(θ) . Methods are developed by which an approximate solution is given in terms of an implicit non-linear integro-differential equation in r(θ) . This equation also proves too involved to solve exactly, but numerical approximations are calculated by two different schemes. One scheme is an analog of Euler's method, and the other is based on Galerkin's method of undetermined coefficients. The problem is so involved that analytical error analysis appears too difficult to handle. The best that can be achieved at this time is to calculate numerically the deviation of a beam from true parallelism. The results prove to be encouraging, the maximum deviation from true parallell of a geodesic was 6/10 of one degree, at the periphery of the system. The necessary modifications of these methods for solving other optical problems are also taken up.

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