Efficient and robust algorithms for the design of manufacturable optical freeform surfaces and their experimental validation

用于可制造光学自由曲面设计及其实验验证的高效且稳健的算法

基本信息

批准号：
259180742
负责人：
Professor Dr. Wolfgang Dahmen
金额：
--
依托单位：
Institut für Geometrie und Praktische Mathematik (IGPM)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/259180742?language=en
关键词：
Efficient robust algorithms design manufacturable

项目摘要

Refractive and reflective optical components, whose surfaces offer a much higher number of degrees of freedom in comparison with conventional lenses and reflectors (axially symmetric and cylindric spheres and aspheres), are subsumed under the term "freeform optics". For their computer-aided design, algorithms are used that compute the optical surfaces from the given source and target light densities and other boundary conditions. This project addresses the first interdisciplinary advancement of these algorithms on the mathematical as well as on the technical and optical side in order to make them available for engineering applications in complex optical systems.State of the art design algorithms do not offer satisfactory solutions to this problem even for a point light source, because neither do they meet production requirements, in particular, concerning the regularity of the surfaces, nor do they offer the possibility for the design of multiple surfaces (two lens surfaces, lens and reflector, etc.). Given the source intensity distribution none of the known methods can take a nonzero extent of the light source into account without employing an iterative correction.In preliminary work, the applicants have developed a method that generates for given point light sources smooth surfaces by rigorously modeling the underlying problem in terms of a Monge-Ampère equation. Furthermore, the method is capable of generating surfaces that produce highly complex target illuminations. Similar to optimal transport problems, the inverse reflector problem can be formulated as a second boundary value problem for the Monge-Ampère equation which is a strongly nonlinear second order partial differential equation subject to additional boundary conditions. In order to solve this problem numerically, a B-spline collocation method and a finite-difference method are used for mutual validation. Nested iteration methods are applied to ensure the convergence of the solver and to accelerate the calculations. The particularly challenging boundary condition is realized through a Picard-type iteration. Until now, no other solution algorithms are known for this boundary value problem.As part of the submitted project, the algorithms developed by the applicants are generalized to spatially extended light sources which are essential for real-life applications. Moreover, methods are developed that handle the design of optical systems with multiple surfaces. By combining both of these steps, a method shall be constructed that simultaneously handles arbitrarily bounded optical surfaces, multiple optical surfaces and spatially extended light sources. Besides correctness, further essential goals of the project are robustness, numerical efficiency and manufacturability of the calculated solutions.

与传统的镜头和反射器相比（轴向对称和圆柱形球体和沥青）相比，其表面的折光和反射性光学组件均在“ freeform optics”术语中归为。对于其计算机辅助设计，使用了从给定源和目标光密度和其他边界条件的光学表面计算光学表面的算法。该项目介绍了这些算法在数学以及技术和光学方面的第一个跨学科进步，以便使它们适用于复杂的光学系统的工程应用程序。艺术设计算法的态度。表面（两个镜头表面，镜头和反射器等）。鉴于源强度分布没有任何已知方法可以在不使用迭代校正的情况下考虑到非零的光源。此外，该方法能够产生产生高度复杂目标照明的表面。与最佳传输问题相似，可以将反反射器问题作为Monge-ampère方程的第二个边界值问题提出，这是一个强烈的非线性二阶偏差方程，但要受其他边界条件的影响。为了以数值为单位解决此问题，使用B-Spline搭配方法和有限差分方法进行相互验证。应用嵌套的迭代方法以确保求解器的收敛性并加速计算。特定的挑战边界条件是通过PICARD型迭代实现的。到目前为止，尚无其他解决方案算法因此边界值问题而闻名。作为提交项目的一部分，应用程序开发的算法被推广到经常扩展的光源，这对于现实生活中必不可少。此外，开发了处理具有多个表面的光学系统设计的方法。通过结合这两个步骤，应构建一种方法，该方法简单地处理任意界限的光学表面，多个光学表面和空间扩展的光源。除正确性外，该项目的进一步基本目标是鲁棒性，数值效率和计算解决方案的制造。