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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.

折射和反射光学部件，其表面提供比传统透镜和反射器(轴对称和柱面球面和非球面)更多的自由度，被归入术语“自由形式光学”。对于它们的计算机辅助设计，使用了根据给定源和目标光密度以及其他边界条件来计算光学表面的算法。本项目致力于这些算法在数学以及技术和光学方面的第一次跨学科进展，以使它们可用于复杂光学系统的工程应用。最新的设计算法即使对于点光源也不能提供令人满意的解决方案，因为它们既不满足生产要求，特别是关于表面的正则性，也不提供设计多个表面(两个透镜表面、透镜和反射器等)的可能性。在给定源强度分布的情况下，任何已知方法都不能在不采用迭代校正的情况下考虑光源的非零范围。在初步工作中，申请人已经开发了一种方法，该方法通过根据Monge-Ampère方程对潜在问题进行严格建模来为给定点光源生成平滑表面。此外，该方法能够生成产生高度复杂的目标照明的表面。与最优输运问题类似，反射面反问题可以表示为Monge-Ampère方程的第二边值问题，Monge-Ampère方程是一个带有附加边界条件的强非线性二阶偏微分方程组。为了对这一问题进行数值求解，采用了B-样条配点法和有限差分法进行了相互验证。采用嵌套迭代法保证了求解器的收敛，加快了计算速度。特别具有挑战性的边界条件是通过Picard类迭代实现的。作为提交的项目的一部分，申请者开发的算法被推广到对现实生活应用至关重要的空间扩展光源。此外，还发展了处理多表面光学系统设计的方法。通过将这两个步骤结合起来，将构建一种同时处理任意边界光学表面、多个光学表面和空间扩展光源的方法。除了正确性之外，该项目的另一个基本目标是计算解的稳健性、数值效率和可制造性。