Double freeform illumination design for prescribed wavefronts and irradiances

A mathematical model in terms of partial differential equations (PDE) for the calculation of double freeform surfaces for irradiance and phase control with predefined input and output wavefronts is presented. It extends the results of B\"osel and Gross [J. Opt. Soc. Am. A 34, 1490 (2017)] for the illumination design of single freeform surfaces for zero-\'etendue light sources to double freeform lenses and mirrors. The PDE model thereby overcomes the restriction to paraxiality or the requirement of at least one planar wavefront of the current design models in the literature. In contrast with the single freeform illumination design, the PDE system does not reduce to a Monge-Amp\`ere type equation for the unknown freeform surfaces, if nonplanar input and output wavefronts are assumed. Additionally, a numerical solving strategy for the PDE model is presented. To show its efficiency, the algorithm is applied to the design of a double freeform mirror system and double freeform lens system.Comment: Copyright 2018 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibite

Single freeform surface design for prescribed input wavefront and target irradiance

In beam shaping applications, the minimization of the number of necessary optical elements for the beam shaping process can benefit the compactness of the optical system and reduce its cost. The single freeform surface design for input wavefronts, which are neither planar nor spherical, is therefore of interest. In this work, the design of single freeform surfaces for a given zero-\'etendue source and complex target irradiances is investigated. Hence, not only collimated input beams or point sources are assumed. Instead, a predefined input ray direction vector field and irradiance distribution on a source plane, which has to be redistributed by a single freeform surface to give the predefined target irradiance, is considered. To solve this design problem, a partial differential equation (PDE) or PDE system, respectively, for the unknown surface and its corresponding ray mapping is derived from energy conservation and the ray-tracing equations. In contrast to former PDE formulations of the single freeform design problem, the derived PDE of Monge-Amp\`ere type is formulated for general zero-\'etendue sources in cartesian coordinates. The PDE system is discretized with finite differences and the resulting nonlinear equation system solved by a root-finding algorithm. The basis of the efficient solution of the PDE system builds the introduction of an initial iterate constuction approach for a given input direction vector field, which uses optimal mass transport with a quadratic cost function. After a detailed description of the numerical algorithm, the efficiency of the design method is demonstrated by applying it to several design examples. This includes the redistribution of a collimated input beam beyond the paraxial approximation, the shaping of point source radiation and the shaping of an astigmatic input wavefront into a complex target irradiance distribution.Comment: 11 pages, 10 figures version 2: Equation (7) was corrected; additional minor changes/improvement

Freeform illumination design in optical systems with partial differential equations

Zahlreiche Anwendungen in der Beleuchtung und Messtechnik erfordern das Design kompakter, energieeffizienter, nicht-abbildender optischer Systeme zur Generierung nichttrivialer Zielintensitätsverteilungen. Eine moderne Möglichkeit dieses Anforderungsprofil zu erfüllen bieten refraktive oder reflektive optische Flächen ohne jegliche Symmetrien, sogenannte Freiformflächen. Im Gegensatz zu klassischen Projektionsmethoden wie zum Beispiel der Durchlichtprojektion bieten Freiformen durch die geeignete Wahl ihrer lokalen Oberflächenkrümmung theoretisch die Möglichkeit beliebige Beleuchtungsmuster nahezu verlustfrei zu erzeugen. Für eine gegebene Lichtquelle und ein gewünschtes Muster besteht die Hauptschwierigkeit hierbei in der Berechnung der entsprechenden Freiformflächen, welche die Energieumverteilung realisieren. Dieses sogenannte inverse Problem der nicht-abbildenden Optik erfordert zum einen dessen mathematische Modellierung und zum anderen die numerische Lösung des Models. Das Ziel dieser Arbeit ist demzufolge die Entwicklung einer allgemeinen mathematischen Beschreibung des inversen Problems und dessen numerischer Lösung, sowie die Entwicklung anwendungsorientierter Freiformbeleuchtungskonzepte.Numerous applications in illumination and metrology require the design of compact, energy-efficient nonimaging optical systems for nontrivial irradiance or intensity distributions. A modern way to fulfill the profile of requirements are freeform surfaces, meaning refractive or reflective surfaces without any symmetries.In contrast to classical projection methods, for instance transmitted-light illuminators, freeform surfaces offer the possiblity to generate nearly arbitrary target distributions by an appropriate choice of the local surface curvature. For a given light source and a desired target distribution the main difficulty is thereby the computation of the freeform surfaces, which realize the required energy redistribution. This so-called inverse problem of nonimaging optics necessitates on the one hand a mathematical description and on the other hand the numerical solving of the corresponding model. Therefore, the goal of this thesis is to develop a general mathematical description of the inverse problem, the numerical solving of the corresponding model as well as the development of application oriented freeform illumination design concepts