Computation of double freeform optical surfaces using a Monge–Ampère solver: Application to beam shaping

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

Research output: Contribution to journalArticleAcademicpeer-review

18 Citations (Scopus)
114 Downloads (Pure)


In this article, we present a formulation for the design of double freeform lens surfaces to control the intensity distribution of a laser beam with plane wavefronts. Double freefrom surfaces are utilized to shape collimated beams. Two different layouts of the freeform lens optical system are introduced, i.e., a single lens with double freeform surfaces, and two separate lenses with two flat and two freeform surfaces. 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 and the constraint imposed on the optical path length between source and target planes. Numerical solutions are computed using a generalized least-squares algorithm which is presented by Yadav et al. (2018). The algorithm is capable to compute two solutions of the Monge–Ampère boundary value problem, corresponding to either c-convex or c-concave freeform surfaces for both layouts. The freeform surfaces are validated for several numerical examples using a ray-tracer based on Quasi-Monte Carlo simulation.

Original languageEnglish
Pages (from-to)251-259
Number of pages9
JournalOptics Communications
Publication statusPublished - 15 May 2019


  • Freeform optical surfaces
  • Laser beam shaping
  • Least-squares method
  • Monge–Ampère equation
  • Monge-Ampere equation


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