Solving the Monge-Ampère equation on triangle-meshes for use in optical freeform design

Rolf Wester; Annika Völl; Michael Berens; Jochen Stollenwerk; Peter Loosen

doi:10.1117/12.2313454

28 May 2018 Solving the Monge-Ampère equation on triangle-meshes for use in optical freeform design

Rolf Wester, Annika Völl, Michael Berens, Jochen Stollenwerk, Peter Loosen

Proceedings Volume 10693, Illumination Optics V; 1069307 (2018) https://doi.org/10.1117/12.2313454
Event: SPIE Optical Systems Design, 2018, Frankfurt, Germany

Abstract

Designing freeform optical surfaces with a large number of degrees of freedom has been a field of extensive research and development. Several design methods have been proposed. Starting point in the design process often is an idealized light source that has zero étendue (e.g. point source or collimated light). With this assumption the solution is unique and corresponds to the solution of an equation of Monge-Ampère type. We propose a method to solve the Monge-Ampère equation on convex bounded domains by using triangle meshes and by minimizing the difference between prescribed and actual target light distribution which is computed by tracing rays through the optical surface. The mathematical solution has to comply with two conditions: the boundary of the source domain has to be mapped onto the boundary of the target domain and the solution has to be convex. The boundary condition problem is solved using a signed distance function that is computed in advance by a fast marching algorithm. The actual light distribution is computed by tracing rays along the triangle nodes and computing the light irradiance on the target by dividing the light flux through a triangle by its mapped area on the target. Under certain conditions this is also an approximate solution to the Optimal Transportation Problem with quadratic cost.

Citation Download Citation

Rolf Wester, Annika Völl, Michael Berens, Jochen Stollenwerk, and Peter Loosen "Solving the Monge-Ampère equation on triangle-meshes for use in optical freeform design", Proc. SPIE 10693, Illumination Optics V, 1069307 (28 May 2018); https://doi.org/10.1117/12.2313454

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available