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2019 | OriginalPaper | Chapter

A Least-Squares Method for a Monge-Ampère Equation with Non-quadratic Cost Function Applied to Optical Design

Authors : N. K. Yadav, J. H. M. ten Thije Boonkkamp, W. L. IJzerman

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2017

Publisher: Springer International Publishing

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Abstract

Freeform optical surfaces can transfer a given light distribution of the source into a desired distribution at the target. Freeform optical design problems can be formulated as a Monge-Ampère type differential equation with transport boundary condition, using properties of geometrical optics, conservation of energy, and the theory of optimal mass transport. We present a least-squares method to compute freeform lens surfaces corresponding to a non-quadratic cost function. The numerical algorithm is capable to compute both convex and concave surfaces.

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previous chapter Derivation of Higher-Order Terms in FFT-Based Numerical Homogenization
next chapter Solving Inverse Illumination Problems with Liouville’s Equation
Literature
1.
C.R. Prins, R. Beltman, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, T.W. Tukker, A least-squares method for optimal transport using the Monge-Ampère equation. SIAM J. Sci. Comput. 37(6), B640–B660 (2015)CrossRef
2.
V. Oliker, Differential equations for design of a freeform single lens with prescribed irradiance properties. Opt. Eng. 53(3), 031302 (2013)
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K. Brix, Y. Hafizogullari, A. Platen, Designing illumination lenses and mirrors by the numerical solution of Monge-Ampère equations. J. Opt. Soc. Am. A 32(11), 2227–2236 (2015)CrossRef
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N.K. Yadav, J.H.M. ten Thije Boonkkamp, W.L. IJzerman. A least-squares method for the design of two-reflector optical system. J. Comput. Appl. Math. (2017, manuscript submitted)
5.
N.K. Yadav, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, A Monge-Ampère problem with non-quadratic cost function to compute freeform lens surfaces. J. Sci. Comput. (2018, manuscript submitted)
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C. Villani, Topics in Optimal Transportation, vol. 58 (American Mathematical Society, Providence, RI, 2003)MATH
7.
V. Oliker, Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport. Arch. Ration. Mech. Anal. 201(3), 1013–1045 (2011)MathSciNetCrossRef
Metadata
Title
A Least-Squares Method for a Monge-Ampère Equation with Non-quadratic Cost Function Applied to Optical Design
Authors
N. K. Yadav
J. H. M. ten Thije Boonkkamp
W. L. IJzerman
Copyright Year
2019
Book
Numerical Mathematics and Advanced Applications ENUMATH 2017

Print ISBN: 978-3-319-96414-0

Electronic ISBN: 978-3-319-96415-7

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-319-96415-7_26

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