Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle

Abstract

It is shown that the problem of designing a two-reflector system transforming a plane wave front with given intensity into an output plane front with prescribed output intensity can be formulated and solved as the Monge-Kantorovich mass transfer problem 1 .where P d is the map ofΩ onT d and J is the Jacobian, is the expansion ratio and it measures the expansion of a tube of rays due to the two reflections [5]. It is assumed that both R 1 and R 2 are perfect reflectors and no energy is lost in the transformation process. Consequently, the corresponding relation between the input intensity I on Ω and output intensity L on T d is given by(1)The "two-reflector" problem that needs to be solved by designers of optical systems consists in determining the reflectors R 1 and R 2 so that all of the properties of the two-reflector system above hold for prescribed in advance domains Ω, T and positive integrable functions I(x), x ∈ Ω, and L(p), p ∈ T ;

Design of freeform optics

“…Thus, the surface R defines a map γ: m→y. This freeform design problem is solved using the variational method [25,26]. The geometrical optics is combined with calculus of variations, which could solve the beam-shaping problem of the known illumination distribution of target surface and complete the one-reflector or two-reflector design.…”

“…Variational method is about solving the extremum problem and makes the boundary to be discrete under geometrical optics to obtain a linear solution. Oliker found a set of optimal solutions according to the MongeKantorovich quality problems [25,26]. Figure 3 is a design example of two reflectors using this method.…”

“…The rays in B 1 are reflected by igure 2 Sketch of the setting and irradiance distribution [13]. reflector R 1 and then intercepted by reflector R 2 and at last reach the target suface (z = d) [25]. The approach is based on a rigorous mathematical theory and does not assume rotational or other symmetry of the data.…”

Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations

“…One can, for example, consider problems of refraction instead of reflection with a point light source, as considered in works by Gutiérrez and Huang [GH14]; and Oliker, Rubinstein, and Wolansky [ORW15]. In another direction, one can change the light source to be a parallel beam instead of a point source (see [Kar14]), or consider multiple optical instruments instead of just one (see work of Glimm and Oliker [GO04], and Oliker [Oli11]). Another interesting family of problems are models with nonperfect energy transmission, as studied by Gutiérrez and Mawi [GM13] and Gutiérrez and Sabra [GS14].…”

Síntese De Antenas Refletoras Utilizando Seções Cônicas E Superfícies Quádricas Confocais

“…Para a solução do mapeamento,é desenvolvido um algoritmo iterativo para solução do problema de redistribuição da energia na forma integral, obtendo-se a energia referente a cada uma das elipses que compõem a superfície em vez de avaliar a equação diferencial de Monge-Ampère numericamente. Outra abordagem alternativa para a síntese geométrica de sistemas duplo-refletoresé explorada em [53] e [54] onde a redistribuição da energiaé formulada e resolvida como um problema de transferência de massa de Monge-Kantorovich.…”