Dielectric corrected plane reflector: an unusual solution

The design of the dielectric coating necessary to convert a conducting plane into a focusing reflector is shown to be identical to that of designing an astigmatic lens with a focus at infinity, which has two surfaces with reflection symmetry about a central plane. The design of such a symmetrical lens with asymmetrical foci is a longstanding problem in optics. The solution proposed is to extend a geometrical interpretation of the completely symmetrical lens to accommodate by inference the asymmetrical. This states that the four points, the two foci, the centre of the lens and the point at which a bisected lens would focus, form an homographic range on the axis. This is true for the symmetrical lens, and gives an extremely good approximation to the asymmetrical one. The approximation improves as the system becomes paraxial, and can be used as the first step in an iterative procedure to obtain results to any specified degree of accuracy. The designed ´half-lens´ with a conducting layer over the plane face, then constitutes the corrected plane reflector.