Ray mappings and the weighted least action principle

Two basic problems in optics are presented. The solutions to both problems are formulated in terms of the associated ray mappings. An alternative formulation based on a weighted sum of the actions along the rays is derived. Existence of solutions is established via the Weighted Least Action Principle. Numerical methods for computing the ray mappings are discussed. Finally, we demonstrate the theoretical considerations by presenting complete solutions to a phase retrieval problem and to a specific beam shaping lens design.

[1]  Fred M. Dickey,et al.  Laser Beam Shaping , 2003 .

[2]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[3]  Genaro Saavedra,et al.  Photonic Structures: Fractal Zone PlatesProduce Axial IrradianceWith Fractal Profile , 2003 .

[4]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[5]  Jacob Rubinstein,et al.  Intensity control with a free-form lens. , 2007, Journal of the Optical Society of America. A, Optics, image science, and vision.

[6]  J. H. M. ten Thije Boonkkamp,et al.  A Monge-Ampère-Solver for Free-Form Reflector Design , 2014, SIAM J. Sci. Comput..

[7]  Jacob Rubinstein,et al.  A weighted least action principle for dispersive waves , 2005 .

[8]  Benjamin B. Wells.  Weak compactness of measures , 1969 .

[9]  M. Teague.  Deterministic phase retrieval: a Green’s function solution , 1983 .

[10]  Jacob Rubinstein,et al.  A variational principle in optics. , 2004, Journal of the Optical Society of America. A, Optics, image science, and vision.

[11]  Jacob Rubinstein,et al.  Geometrical optics and optimal transport. , 2017, Journal of the Optical Society of America. A, Optics, image science, and vision.

[12]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[13]  Vladimir Oliker,et al.  Designing Freeform Lenses for Intensity and Phase Control of Coherent Light with Help from Geometry and Mass Transport , 2011 .

[14]  Jacob Rubinstein,et al.  Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light , 2015, Adv. Appl. Math..

[15]  Allen R. Tannenbaum,et al.  An Efficient Numerical Method for the Solution of the L2 Optimal Mass Transfer Problem , 2010, SIAM J. Sci. Comput..

[16]  Jayoung Nam,et al.  Wavelength adjustment using an eye model from aberrometry data. , 2010, Journal of the Optical Society of America. A, Optics, image science, and vision.

[17]  Daniel G. Smith.  Field Guide to Physical Optics , 2013 .

[18]  Steven Haker,et al.  Minimizing Flows for the Monge-Kantorovich Problem , 2003, SIAM J. Math. Anal..

[19]  Jacob Rubinstein,et al.  Ray mapping and illumination control , 2013 .

[20]  C. Villani.  Topics in Optimal Transportation , 2003 .

[21]  L. Mandel,et al.  Optical Coherence and Quantum Optics , 1995 .

[22]  Vladimir Oliker,et al.  Optical Design of Two-reflector Systems, the Monge-Kantorovich Mass Transfer Problem and Fermat's Principle , 2003 .

[23]  Herbert Gross,et al.  Single freeform surface design for prescribed input wavefront and target irradiance. , 2017, Journal of the Optical Society of America. A, Optics, image science, and vision.

[24]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[25]  Emil Wolf,et al.  Principles of Optics: Contents , 1999 .

[26]  Vladimir Oliker,et al.  On design of free-form refractive beam shapers, sensitivity to figure error, and convexity of lenses. , 2008, Journal of the Optical Society of America. A, Optics, image science, and vision.

[27]  F Roddier,et al.  Curvature sensing and compensation: a new concept in adaptive optics. , 1988, Applied optics.