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Inverse freeform design of two-dimensional reflectors for a finite-source problem using an integro-differential formulation

This paper presents an inverse freeform design method for two-dimensional reflectors that addresses finite-source problems by formulating the task as an optimal transport problem, deriving a linking integral equation, and solving for convex reflector profiles through surface parametrization and loss minimization.

Original authors: Fatéma Goulamaly, Martijn Anthonissen, Wilbert IJzerman, Lisa Kusch, Koondanibha Mitra, Jan ten Thije Boonkkamp

Published 2026-08-10

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CC BY 4.0

Original authors: Fatéma Goulamaly, Martijn Anthonissen, Wilbert IJzerman, Lisa Kusch, Koondanibha Mitra, Jan ten Thije Boonkkamp

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake the perfect cake, but instead of flour and sugar, your ingredients are beams of light. In the world of illumination optics, scientists act like master chefs, trying to shape mirrors and lenses to take a messy, chaotic spray of light from a bulb and turn it into a perfectly smooth, even glow on a table or a street. For a long time, the easiest way to do this was to pretend the light bulb was a single, tiny point with no size at all—a "zero-étendue" source. It's like pretending your cake batter is a single, perfect drop. But real light bulbs aren't drops; they are extended objects with width and depth, emitting light in many directions at once. This makes the math incredibly messy, like trying to bake a cake where the batter is constantly moving and changing shape. The big question for engineers is: how do you design a mirror that takes this messy, real-world light and sculpts it into a specific, desired pattern without the math falling apart?

This paper tackles that exact puzzle by introducing a new mathematical recipe for designing "freeform" reflectors—mirrors that aren't just simple curves like bowls or parabolas, but have complex, custom shapes. The authors, a team from Eindhoven University of Technology and Signify, realized that when you have a real, finite light source, the problem is like trying to map a two-dimensional room (the source) onto a one-dimensional hallway (the target intensity). Because many different points on the source can contribute to the same spot on the target, there isn't just one single mirror shape that works; in fact, without extra rules, there are infinite possibilities. To solve this, the team developed a method that treats the mirror design as an optimization game. They created a "loss function," which is essentially a scorecard that tells them how far off their current mirror shape is from the desired light pattern. By using a clever mix of calculus and computer algorithms, they iteratively tweak the mirror's shape to lower this score, effectively "learning" the perfect curve.

The researchers found that to get a unique, stable answer, they had to impose two specific rules: the mirror must be convex (curving outward like a dome, never dipping inward) and it must sit at a specific average height. Without these rules, the computer might find a mirror that produces the right light but is floating in mid-air or has weird, wavy bumps that don't make physical sense. They tested their method with several scenarios, including flat mirrors, symmetric curved ones, and even tricky, non-symmetric shapes. In every case, their method successfully reconstructed the mirror shape needed to produce the target light. They even managed to approximate a "step function"—a light pattern that suddenly jumps from bright to dark—which is notoriously difficult because real mirrors usually create smooth transitions. Their simulations showed that their approach is highly accurate, with errors as small as 0.000075 in some tests, and they observed that their numerical method converges with fourth-order accuracy, meaning the solution gets incredibly precise very quickly as they add more detail to the calculation.

The paper also compared two ways of solving the math: a traditional method using finite differences (breaking the mirror into a grid of points) and a modern machine learning approach using neural networks. While both worked, the traditional grid method was significantly faster on the computer, though the neural network approach offered a different way to handle the derivatives. Ultimately, the authors demonstrate that by combining an integral equation (a formula that sums up all the light contributions) with a smart optimization strategy, they can design complex mirrors for real-world light sources. They suggest that while their current work focuses on two-dimensional slices, the method is robust enough to potentially handle more complex, three-dimensional systems in the future, offering a powerful new tool for lighting designers who need to tame the chaos of finite light sources.

Technical Summary: Inverse Freeform Design of Two-Dimensional Reflectors for a Finite-Source Problem

Problem Statement
The paper addresses the inverse design problem in illumination optics: determining the shape of a two-dimensional freeform reflector that transforms a prescribed light distribution from a finite source into a specific far-field intensity profile. Unlike traditional approaches that assume a zero-étendue source (idealized point or parallel sources), this work considers a finite source, where light is emitted from multiple spatial positions across a continuum of directions. This introduces a significant mathematical challenge: the problem becomes an optimal transport mapping between domains of different dimensions (a two-dimensional source domain to a one-dimensional target intensity domain). The non-local nature of the problem means that the intensity in a given outgoing direction results from the combined contributions of multiple source points, making standard Monge-Ampère type equations (derived for same-dimensional transport) inapplicable without modification. Furthermore, the authors identify that the inverse problem is inherently non-unique; for instance, the far-field intensity of a horizontal flat reflector is invariant under vertical translation, and solutions are not unique without constraints on convexity and average height.

Methodology
The authors propose a mathematical framework and an optimization-based solution strategy:

Forward Model Derivation:
Starting from the conservation of luminous flux and the law of reflection, the authors derive an exact integro-differential equation linking the source luminance $L_s(q_s, p_s)$ to the target intensity $I_t(\hat{t})$ .
$I_t = -\int_{\Omega} L_s(q_s(\sigma, \hat{t}), p_s(\sigma, \hat{t})) \frac{\hat{t} \cdot n(\sigma)}{\hat{t} \cdot \hat{e}_z} d\sigma$
Here, the reflector is parameterized by $r(\sigma) = (x(\sigma), z(\sigma))^T$ . The source coordinates $(q_s, p_s)$ are determined by tracing rays backward from the reflector point $P$ in the direction $\hat{t}$ to the source plane.
Surface Parameterization:
To handle the unknown spatial extent of the reflector, a specific parameterization is introduced where the parameter $\sigma$ corresponds to a fixed domain $[x_{min}, x_{max}]$ . The surface is defined by a distance function $u(\sigma)$ along a direction vector $\hat{v}(\sigma)$ that varies linearly with $\sigma$ . This ensures all emitted rays intersect the reflector.
Inverse Problem Formulation:
The inverse problem is cast as a least-squares minimization problem. The goal is to find the function $u(\sigma)$ that minimizes the difference between the computed intensity and a desired target intensity. The total loss functional $\mathcal{L}(u)$ comprises three terms:
- Intensity Loss ( $\mathcal{L}_I$ ): The squared $L_2$ difference between the target and computed intensities.
- Convexity Penalty ( $\mathcal{L}_C$ ): A term penalizing non-convex regions ( $d^2z/dx^2 < 0$ ) to ensure a unique, physically realizable convex solution.
- Height Penalty ( $\mathcal{L}_H$ ): A term enforcing a specific average height $H$ to resolve the translational invariance of the solution.
Numerical Implementation:
The optimization is solved using a quasi-Newton BFGS algorithm. The authors compare two discretization strategies:
- Finite Differences: Using compact fourth-order finite-difference schemes to approximate derivatives on a uniform grid.
- Neural Networks: Representing the surface profile $u(\sigma)$ as a neural network $u_\theta$ , with derivatives computed via automatic differentiation.

Key Contributions

Integral Equation Formulation: The derivation of an exact integral equation (Eq. 1) that relates the target intensity to the source luminance for a general finite source and arbitrary reflector shape.
Identification of Non-Uniqueness: A rigorous analysis demonstrating that unique reflector solutions require additional constraints on convexity and average height, illustrated through special cases (horizontal flat reflectors and parallel sources).
Optimization Framework: The development of a robust least-squares optimization method that successfully reconstructs reflector profiles by minimizing the derived loss functional, effectively handling the non-local coupling of the finite source.
Convergence Analysis: Demonstration of fourth-order convergence for the numerical method when using compact finite-difference schemes combined with Simpson's rule integration.

Results
The proposed method was validated through four numerical experiments:

Horizontal Flat Reflector: The method successfully reconstructed a flat reflector profile. Without height constraints, the solution converged to a flat surface at a different height than the original, confirming the theoretical non-uniqueness. Adding the height penalty recovered the exact original profile.
Symmetric Curved Reflector: The algorithm accurately reconstructed a symmetric curved reflector. The ray-tracing validation showed a low error ( $4.4 \times 10^{-5}$ ), and the convergence study confirmed the expected fourth-order accuracy with respect to grid resolution.
Non-Symmetric Curved Reflector: The method handled asymmetric configurations effectively. A comparison between the finite-difference approach and the neural network approach showed that while both converged to similar accuracy, the finite-difference approach required significantly less computational time.
Unknown Reflector (Step Function): The method attempted to approximate a reflector producing a step-function intensity profile. While smooth reflectors cannot produce discontinuous intensities, the algorithm successfully approximated the profile for smoothed transitions ( $\epsilon > 0$ ). For a sharp step ( $\epsilon = 0$ ), the solution was limited by the convexity constraint, highlighting the trade-off between intensity matching and geometric constraints.

Significance and Claims
The paper claims to provide a novel mathematical model and a robust numerical method for the inverse design of freeform reflectors for finite sources. The authors state that their method is optimal up to discretization error because the forward model (Eq. 1) is exact. They emphasize that the proposed approach successfully constructs convex reflector profiles for various target intensities, overcoming the challenges posed by the non-local nature of finite-source problems. The work establishes a foundation for solving these inverse problems without relying on iterative ray-tracing corrections or deconvolution techniques, instead utilizing a direct optimization of an integro-differential formulation. The authors note that the method naturally extends to higher dimensions and more complex optical systems, which they plan to investigate in future research.

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