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

折射和反射光学元件的表面与传统透镜和反射器（轴对称和圆柱形球体和非球面）相比提供了更高的自由度，这些元件被归入术语“自由光学”。对于它们的计算机辅助设计，使用从给定的源和目标光密度以及其他边界条件计算光学表面的算法。该项目解决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展，以便使它们可用于复杂光学系统的工程应用。最先进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，因为它们也不能满足生产要求，特别是，关于表面的规则性，它们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。在初步工作中，申请人已经开发了一种方法，该方法通过严格地根据Monge-Ampère方程对基本问题进行建模来为给定的点光源生成光滑表面。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，逆反射器问题可以表示为Monge-Ampère方程的第二边值问题，Monge-Ampère方程是一个受附加边界条件约束的强非线性二阶偏微分方程。为了解决这个问题的数值，B样条配点法和有限差分法用于相互验证。采用嵌套迭代法保证了求解器的收敛性，加快了计算速度。特别具有挑战性的边界条件是通过皮卡德型迭代实现的。作为提交项目的一部分，申请人开发的算法被推广到对于现实生活应用至关重要的空间扩展光源。此外，开发了处理具有多个表面的光学系统的设计的方法。通过组合这两个步骤，将构建同时处理任意有界光学表面、多个光学表面和空间扩展光源的方法。除了正确性之外，该项目的其他基本目标是计算解决方案的鲁棒性，数值效率和可制造性。

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.