# A Monge–Ampère problem with non-quadratic cost function to compute freeform lens surfaces

N.K. Yadav (Corresponding author), J.H.M. ten Thije Boonkkamp, W.L. IJzerman

### Abstract

In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.

Language English 475-499 25 Journal of Scientific Computing 80 1 27 Mar 2019 10.1007/s10915-019-00948-9 Published - Jul 2019

### Fingerprint

Cost functions
Lens
Cost Function
Lenses
Lens Design
Free-form Surface
Least Square Algorithm
Refraction
Least Square Method
Conservation
Half line
Differential equation
Boundary conditions
Target
Differential equations
Energy

### Keywords

• Freeform lens surfaces
• Inverse problem
• Least-squares method
• Monge–Ampère equation
• Optical design
• Transport boundary conditions

### Cite this

@article{69ebac8716f1435299e237a841e21535,
title = "A Monge–Amp{\`e}re problem with non-quadratic cost function to compute freeform lens surfaces",
abstract = "In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Amp{\`e}re type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.",
keywords = "Freeform lens surfaces, Inverse problem, Least-squares method, Monge–Amp{\`e}re equation, Non-quadratic cost function, Optical design, Transport boundary conditions",
author = "N.K. Yadav and {ten Thije Boonkkamp}, J.H.M. and W.L. IJzerman",
year = "2019",
month = "7",
doi = "10.1007/s10915-019-00948-9",
language = "English",
volume = "80",
pages = "475--499",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer",
number = "1",

}

A Monge–Ampère problem with non-quadratic cost function to compute freeform lens surfaces. / Yadav, N.K. (Corresponding author); ten Thije Boonkkamp, J.H.M.; IJzerman, W.L.

In: Journal of Scientific Computing, Vol. 80, No. 1, 07.2019, p. 475-499.

TY - JOUR

T1 - A Monge–Ampère problem with non-quadratic cost function to compute freeform lens surfaces

AU - ten Thije Boonkkamp,J.H.M.

AU - IJzerman,W.L.

PY - 2019/7

Y1 - 2019/7

N2 - In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.

AB - In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.

KW - Freeform lens surfaces

KW - Inverse problem

KW - Least-squares method

KW - Monge–Ampère equation

KW - Optical design

KW - Transport boundary conditions

UR - http://www.scopus.com/inward/record.url?scp=85064340947&partnerID=8YFLogxK

U2 - 10.1007/s10915-019-00948-9

DO - 10.1007/s10915-019-00948-9

M3 - Article

VL - 80

SP - 475

EP - 499

JO - Journal of Scientific Computing

T2 - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 1

ER -