Initial design with L(2) Monge-Kantorovich theory for the Monge-Ampère equation method in freeform surface illumination design.

The Monge-Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L(2)-Kantorovich (LMK) theory. An efficient approach is proposed to find the optimal mapping of the LMK problem. The characteristics of the new approach are introduced and the limitations of the LMK theory in illumination design are presented. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.

[1]  Roland Glowinski,et al.  Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type , 2006 .

[2]  Harald Ries,et al.  Tailored freeform optical surfaces. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[3]  Jochen Stollenwerk,et al.  High resolution irradiance tailoring using multiple freeform surfaces. , 2013, Optics express.

[4]  Mali Gong,et al.  Designing double freeform optical surfaces for controlling both irradiance and wavefront. , 2013, Optics express.

[5]  Jochen Stollenwerk,et al.  Limitations of the ray mapping approach in freeform optics design. , 2013, Optics letters.

[6]  Julio Chaves,et al.  Simultaneous multiple surface optical design method in three dimensions , 2004 .

[7]  V. Oliker,et al.  Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem , 2003 .

[8]  Zhenrong Zheng,et al.  A mathematical model of the single freeform surface design for collimated beam shaping. , 2013, Optics express.

[9]  Yi Luo,et al.  Discontinuous free-form lens design for prescribed irradiance. , 2007, Applied optics.

[10]  N. Trudinger,et al.  The Monge-Ampµere equation and its geometric applications , 2008 .

[11]  John C. Bortz,et al.  Iterative generalized functional method of nonimaging optical design , 2007, SPIE Optical Engineering + Applications.

[12]  Lei Zhu,et al.  Optimal Mass Transport for Registration and Warping , 2004, International Journal of Computer Vision.

[13]  V. Oliker.  Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics , 2003 .

[14]  Zhenrong Zheng,et al.  Conceptual design of dedicated road lighting for city park and housing estate. , 2013, Applied optics.

[15]  Gian Luca Delzanno,et al.  An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization , 2008, J. Comput. Phys..

[16]  Brittany D. Froese.  A Numerical Method for the Elliptic Monge-Ampère Equation with Transport Boundary Conditions , 2011, SIAM J. Sci. Comput..

[17]  Jannick Rolland,et al.  Optimization of freeform lightpipes for light-emitting-diode projectors. , 2008, Applied optics.

[18]  Thomas L. R. Davenport,et al.  Optimization for illumination systems: the next level of design , 2004, SPIE Photonics Europe.

[19]  Xu Liu,et al.  Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation. , 2013, Optics letters.

[20]  J. F. Williams,et al.  An efficient approach for the numerical solution of the Monge-Ampère equation , 2011 .

[21]  Jannick P Rolland,et al.  Fast freeform reflector generation usingsource-target maps. , 2010, Optics express.

[22]  Xu Liu,et al.  Freeform LED lens for uniform illumination. , 2008, Optics express.