Far Field Refractor Problem

Let n1 and n2 be the indexes of refraction of two homogeneous and isotropic media I and II, respectively. Suppose that from a point O inside medium I light emanates with intensity f (x) for x ∈ Ω . We want to construct a refracting surface R parameterized as R = {ρ(x)x : x ∈ Ω}, separating media I and II, and such that all rays refracted by R into medium II have directions in Ω∗ and the prescribed illumination intensity received in the direction m ∈ Ω∗ is f ∗(m). We first introduce the notions of refractor mapping and measure, and weak solution. Let Ω, Ω∗ be two domains on Sn−1, the illumination intensity of the emitting beam is given by nonnegative f (x) ∈ L1(Ω), and the prescribed illumination intensity of the refracted beam is given by a nonnegative Radon measure μ on Ω∗. Throughout the paper, we assume that |∂Ω| = 0 and the physical constraint