An iterative least-squares method for generated jacobian equations in freeform optical design

The design of freeform optical surfaces is an inverse problem in illumination optics. Combining the laws of geometrical optics and energy conservation gives rise to a generalized Monge-Ampère equation. The underlying mathematical structure of some optical systems allows for an optimal-transport formulation of the problem with an associated cost function. This motivates the design of optimal-transport-based numerical algorithms. However, not all optical systems can be cast in the framework of optimal transport. In this paper, we derive a formulation in terms of generating functions where the generalized Monge-Amp\'ere equation becomes a generated Jacobian equation. We present an iterative least-squares algorithm that can be used to solve generated Jacobian equations. We consider two example systems: System 1 is a single freeform lens with a point source and far-field target, and System 2 is a single freeform reflector with a parallel source beam and near-field target. We introduce a novel derivation of the generating functions via Hamilton's characteristics. We can associate a cost function to System 1, and we compare the performance of the numerical algorithm to a previous optimal-transport-based version. System 2 cannot be formulated as an optimal-transport problem, which demonstrates the wider applicability of the new version of the algorithm to any optical system that can be described by a smooth generating function.