OPTICAL DESIGN OF SINGLE REFLECTOR SYSTEMS AND THE MONGE–KANTOROVICH MASS TRANSFER PROBLEM

Abstract

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge-AmpèreAmp`Ampère type. We show that this problem can be solved as a variational problem within the framework of Monge-Kantorovich mass transfer problem. We develop the techniques used in [1], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem [2] was based on the construction of very special approximate solutions to the corresponding Monge-Ampère equation. Bibliography: 14 titles. § 1. Introduction In this paper, we study the following inverse problem of geometric optics. We consider a reflector system consisting of a point source of light O illuminating through an aperture Ω a perfect reflector R. Let I(m) be the intensity of the source in the direction m. If the reflector R is known, we can determine a region T on the far-sphere covered by the reflected rays and calculate the intensity L(y) in the reflected direction y. In practice (in particular, in design of reflector antennas; cf., for example, [3]) it is often required to solve the inverse problem, i.e., to determine the reflector R from a given position of the source, the aperture Ω, far-field T , and the input and output intensities I and L. In contrast to problems with more than one reflector, this problem will be referred to as the single reflector problem, or, simply, the reflector problem. In the geometric optics approximation, the analytic form of the reflector problem entails solving a second order nonlinear partial differential equation involving the Jacobian of the ray tracing map associated with the unknown reflector R. In a weak formulation (in the sense of partial differential equations) existence of solutions in n (2) dimensions was established in [4] (cf. also [2, 5]). The existence and uniqueness of smooth solutions was proved in [6] (cf. also [7]). In [2] a scheme for solving the reflector problem numerically was introduced and shown to converge. In this paper, we give a variational formulation of the reflector problem, applying a new technique in geometric optics introduced in [1] for solving a different version of the reflector problem involving two reflecting surfaces. This technique is based on relatively recent developments in the study of Monge-Kantorovich mass transfer (or transportation) problems (cf. [8-10]). The Monge-Kantorovich mass transfer problem, stated in general form, is to transfer a mass (or energy) distribution I given on some measurable set Ω in an Euclidean space to a given distribution L on