On the use of the supporting quadric method in the problem of designing double freeform surfaces for collimated beam shaping

Abstract

We propose a version of the supporting quadric method for calculating a refractive optical element with two working surfaces for collimated beam shaping. Using optimal mass transportation theory and generalized Voronoi cells, we show that the proposed method can be regarded as a gradient method of maximizing a concave function, which is a discrete analogue of the Lagrange functional in the corresponding mass transportation problem. It is demonstrated that any maximum of this function provides a solution to the problem of collimated beam shaping. Therefore, the proposed method does not suffer from “trapping” at a local extremum, which is typical for gradient methods. We present design examples of refractive optical elements illustrating high performance of the method.

Support quadric method in non-imaging optics problems that can be reformulated as a mass transfer problem

“…Как и для случая двух отражающих поверхностей, мы будем рассматривать первую поверхность R 1 как огибающую семейства поверхностей, фокусирующих в точки второй поверхности R 2 [11,19]. В этом случае имеют место уравнения огибающей (4) с функцией…”

“…где Q -оптическая длина пути лучей между плоскостями z = 0 и z = l. Данная задача также эквивалента ЗПМ Монжа-Канторовича с функцией стоимости (11) [11,15,18]. Формулировка ЗПМ приведена в следующем параграфе.…”

“…Формулировка ЗПМ приведена в следующем параграфе. Подробное описание данной задачи приведено в [11,15,18,19].…”

“…Для всех этих задач мы единообразно формулируем метод опорных квадрик (МОК), доказывая его совпадение с градиентным методом для нахождения максимума некоторой вогнутой функции. Это переносит результаты статьи [11] на широкий класс задач.…”

A New Conformal Cooling System for Plastic Collimators Based on the Use of Complex Geometries and Optimization of Temperature Profiles

“…Collimating lenses are found in the group of secondary optics, and are in charge of shaping the light, expanding the light beam, or concentrating it on a specific point. The low operating temperatures in LED head lamps allow the use of transparent polymers to replace materials such as glass in the manufacturing of collimating lenses [ 5 ]. This fact encourages more and more automobile manufacturers to use injection molded thermoplastics to produce collimating lenses with complex geometries for the design of new automotive headlights.…”