A least-squares method for the design of two-reflector optical systems

Abstract

The purpose of this paper is to present a method for the design of two-reflector optical systems that transfer a given energy density of the source to a desired energy density at the target. It is known that the two-reflector design problem gives rise to a Monge-Ampère (MA) equation with transport boundary condition. We solve this boundary value problem using a recently developed least-squares algorithm (Prins et al 2015 J. Sci. Comput. 37 B937-61). It is one of the few numerical algorithms capable to solve these type of problems efficiently. The least-squares algorithm can provide two solutions of the MA problem, one is concave and the other one is convex. The reflectors are validated for several numerical examples by a ray-tracer based on Monte-Carlo simulation.

Computation of double freeform optical surfaces using a Monge–Ampère solver: Application to beam shaping

“…The residuals J I and J B stall after around 1000 iterations at a value of approximately 10 −7 , and the algorithm shown similar convergence for the both layouts, although for the second layout the algorithm converges a little slower than for the single lens problem. The freeform surfaces are validated using a ray tracing algorithm [8]. We run the algorithm for 10 million uniformly distributed quasi-random points on the source to verify the illumination pattern at the target.…”

“…Furthermore, combined with energy conservation, this relation gives rise to a non-standard Monge-Ampère type equation with transport boundary condition. There are several numerical methods available which can be employed to compute freeform surfaces of optical systems governed by the standard Monge-Ampère equation corresponding to a quadratic cost function [7,8,9,10]. However, numerical methods for problems with a non-quadratic cost function are rare [1,11].…”

A Monge–Ampère Problem with Non-quadratic Cost Function to Compute Freeform Lens Surfaces

“…The optical design problem involving freeform surfaces is a challenging problem, even for a single mirror/lens surface which transfers a given intensity/emittance distribution of the source into a desired intensity/illuminance distribution at the target [1][2][3]. More specifically, the freeform design problem is an inverse problem: "Find an optical system containing freeform refractive/reflective surfaces that provides the desired target light distribution for a given source distribution".…”

“…To convert a given emittance profile with parallel light rays into a desired illuminance profile with parallel light rays, one requires at least two freeform lens/mirror surfaces [2,5]. This freeform problem can be formulated as a second order partial differential equation of Monge-Ampère (MA) type, with transport boundary conditions, applying the laws of geometrical optics and energy conservation [2,3,6,7]. For the two-reflector problem [2,8], one can obtain the following mathematical formulation using properties of geometrical optics, i.e., u 1 (x) + u 2 (y) = c(x, y) :…”

“…A least-squares (LS) method [2,7] has been presented to solve the standard MA-equation to compute single reflector/lens or double reflector freeform surfaces optical systems. The method provides the optical mapping which transfers the given emittance of the source into the desired illuminance at the target, and the freeform surfaces are obtained via this mapping.…”