Journal of Mathematical Sciences

, Volume 117, Issue 3, pp 4096–4108 | Cite as

Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem

  • T. Glimm
  • V. Oliker


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 by the authors in their work “Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle” [Preprint, 2003], 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 was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.


Design Problem Geometric Optic Optical Design Spherical Wave Elliptic Partial Differential Equation 
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© Plenum Publishing Corporation 2003

Authors and Affiliations

  • T. Glimm
  • V. Oliker

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