《OPTICAL DESIGN OF SINGLE REFLECTOR》.pdfVIP

Journal of Mathematical Sciences, Vol. 117, No. 3, 2003 OPTICAL DESIGN OF SINGLE REFLECTOR SYSTEMS AND THE MONGE–KANTOROVICH MASS TRANSFER PROBLEM T. Glimm and V. Oliker UDC 514; 517.95 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–Ampere 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 ` : 14 titles. corresponding Monge–Ampere equation. Bibliography § 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 Ωa 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 refl