427	RWTH Publication No: 479655        2015        IGPM427.pdf
TITLE	Designing Illumination Lenses and Mirrors by the Numerical Solution of Monge-Ampère Equations
AUTHORS	Kolja Brix, Yasemin Hafizogullari, Andreas Platen
ABSTRACT	We consider the inverse refractor and the inverse reflector problem. The task is to design a free-form lens or a free-form mirror that, when illuminated by a point light source, produces a given illumination pattern on a target. Both problems can be modeled by strongly nonlinear second-order partial differential equations of Monge-Ampère type. In [Math. Models Methods Appl. Sci. 25 (2015), pp. 803–837, DOI: 10.1142/S0218202515500190] the authors have proposed a B-spline collocation method which has been applied to the inverse reflector problem. Now this approach is extended to the inverse refractor problem. We explain in depth the collocation method and how to handle boundary conditions and constraints. The paper concludes with numerical results of refracting and reflecting optical surfaces and their verification via ray tracing.
KEYWORDS	inverse refractor problem, inverse reflector problem, elliptic Monge-Ampère equation, B-spline collocation method, Picard-type iteration
DOI	10.1364/JOSAA.32.002227
PUBLICATION	Journal of the Optical Society of America A Volume 32, Issue 11, pp. 2227-2236, 2015 Peer reviewed article Copyright 2015 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 prohibited.