An inverse design method for generalized zero-étendue sources and two targets

TL;DR

This work addresses the inverse design of freeform optical surfaces that map a general light source described on two planes to two target irradiances using two freeform surfaces (reflectors or lenses). It introduces a generating least-squares framework grounded in energy conservation and optical-path-length-based generating functions $G$ and $H$, solved via a three-stage least-squares algorithm to compute the mappings, $V$, and surface shapes. The approach handles zero-étendue sources and complex target distributions, and is validated through reflector and lens examples with rigorous ray-tracing verification. The method advances freeform optics by enabling simultaneous control of ray position and direction at both the source and target, with potential extensions to non-planar targets and system concatenation.

Abstract

We present an inverse method to compute freeform optical surfaces that transform a light distribution, parameterized by two source planes, into two separate target distributions. The surfaces can be reflectors or lenses, and control both the spatial and directional source and target coordinates of light rays. From energy conservation we derive Jacobian equations for optical mappings, and the optical path length provides generating functions for the optical surfaces. A three-stage least-squares algorithm numerically solves the resulting equations. We present examples with complex source and target distributions.

Figures (6)

• Figure 1: Sketch of a reflector system with two freeform reflectors $\mathcal{R}_1$ and $\mathcal{R}_2$, and a lens system with two freeform lens surfaces $\mathcal{L}_1$ and $\mathcal{L}_2$, where light rays originate from sources $\mathcal{S}_1$ with distribution $f_1$ and $\mathcal{S}_2$ with distribution $f_2$ and are directed to targets $\mathcal{T}_1$ with distribution $g_1$ and $\mathcal{T}_2$ with distribution $g_2$.
• Figure 2: Sketch of the reflector system.
• Figure 3: Sketch of the lens system.
• Figure 4: Flowchart of the least-squares algorithm.
• Figure 5: Results of the reflector system with $L_0=-10$, $L_1=15$, $L_2=25$, $V_0=55$, $u_{1,0}=12$, computed after $50$ iterations on a $1000\times1000$ grid and verified by ray-tracing using $10^6$ rays and $100\times100$ bins.