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From Notices of the AMS

A blue oval with red rays radiating from point O to the oval's edge and purple lines from where those lines intersect the edge of the oval refracting out to point P
Refraction property of oval k = 0.7, P = (10,0), b=8.

Freeform Optics: Optimal Transport, Minkowski Method and Monge-Ampère-Type Equations

by Henok Mawi
Communicated by Reza Malek-Madani

1. Description and Background

A freeform optical surface, simply stated, refers to a surface whose shape lacks rotational symmetry. The use of such surfaces allows generation of complex, compact, and highly efficient imaging systems. Ever since lenses without symmetries were used in World War I in periscopes, the engineering and design of freeform optical surfaces have gone through remarkable evolution, with applications in a wide range of areas, including medical devices, clean energy technology, military surveillance equipments, mobile displays, remote sensing, and several other areas of imaging and nonimaging optics that can benefit from distributing light from a source to a target in a controlled fashion using spatially and energy-efficient systems. See [17] and the references therein.

Mathematically, the design of freeform optical surfaces is an inverse problem related to optimal transportation theory and leads to a class of nonlinear partial differential equations (PDE) called generated Jacobian equations for which the Monge-Ampère equation is a prototype.

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