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 Geodesic focussing in parallelplate systems
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Geodesic focussing in parallelplate systems Mosevich, Jack Walter
Abstract
This thesis is concerned with the mathematics of the design of parallelplate equivalents of optical systems, in particular with the parallelplate equivalent of the parabolic mirror. A parallelplate 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 nonlinear integrodifferential 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.
Item Metadata
Title  Geodesic focussing in parallelplate systems 
Creator  Mosevich, Jack Walter 
Publisher  University of British Columbia 
Date Issued  1972 
Description 
This thesis is concerned with the mathematics of the design of parallelplate equivalents of optical systems, in particular with the parallelplate equivalent of the parabolic mirror.
A parallelplate 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 nonlinear integrodifferential 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.

Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110323 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080442 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.