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A Minimization Approach for the Design of TIR Collimators with Freeform Surfaces

This paper presents an inverse optical design method that utilizes a gradient-descent algorithm to minimize the Wasserstein distance between simulated and target light distributions, enabling the efficient construction of freeform total internal reflection (TIR) collimators with rotational symmetry.

Original authors: Sarah Nysen, Martijn Anthonissen, Jan ten Thije Boonkkamp, Wilbert IJzerman, Lotte Romijn

Published 2026-09-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sarah Nysen, Martijn Anthonissen, Jan ten Thije Boonkkamp, Wilbert IJzerman, Lotte Romijn

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

Light is everywhere, yet controlling exactly where it goes is a profound challenge for engineers. Most light sources, like the tiny diodes that now illuminate our homes and streets, emit light in a broad, soft spread, much like a candle flame. However, many applications require a tight, focused beam, such as a spotlight on a stage or a headlight on a car. To achieve this, designers use lenses and mirrors to gather that scattered light and redirect it. For decades, the most reliable way to design these complex optical shapes has been a process of trial and error. Engineers build a computer model, simulate how light rays travel through it, see if the result is too wide or too narrow, tweak the shape, and repeat. This forward-looking approach works, but it is slow and often relies on guesswork to find the perfect curve.

A new approach flips this process on its head. Instead of guessing a shape and seeing what light comes out, researchers start with the exact light pattern they want and work backward to find the shape that creates it. This inverse design method is not new for simple lenses or mirrors, but it has been incredibly difficult to apply to systems that mix both refraction (bending light as it passes through glass) and reflection (bouncing light off a surface). These mixed systems, known as catadioptric systems, are common in compact lighting devices but have resisted a unified mathematical solution. A team of researchers from the Eindhoven University of Technology and Signify Research has now developed a method to solve this specific puzzle for a type of device called a TIR collimator, which uses total internal reflection—a phenomenon where light bounces inside a material like a mirror without needing a silver coating—to create sharp beams.

The researchers focused on a specific device used in LED spotlights. Imagine a small, clear plastic piece with a point of light at its center. As light rays shoot out from this center, some pass straight through the front of the plastic, while others hit the sides and bounce off the inner walls before exiting. The goal is to shape the front and the inner walls so that every single ray, regardless of where it started, exits the device pointing in the exact direction needed to form a perfect beam. The difficulty lies in the fact that the rays taking the straight path and the rays taking the bouncing path are governed by different physical rules, yet they must work together to create a single, smooth light distribution.

To solve this, the team treated the design problem as a mathematical minimization task. They imagined the light coming from the source as a pile of energy and the desired light pattern as a target pile. The challenge was to move the energy from the source to the target with the least amount of "effort" or distortion. They used a mathematical tool called the Wasserstein distance, which essentially measures the cost of rearranging one distribution of mass into another. In this context, it measured how far the actual light pattern was from the desired one. The researchers then set up a computer algorithm that could adjust the shape of the optical surfaces step by step. With every small adjustment, the algorithm calculated the new light pattern, measured the distance to the target, and moved the surfaces in the direction that reduced that distance.

Because the two types of light rays in this device follow separate paths, the researchers could treat them independently during the calculation. This separation allowed them to use a gradient-descent method, a technique that efficiently finds the bottom of a valley by always moving downhill. The algorithm started with a simple, rough guess for the shape of the surfaces. It then iteratively refined this guess, updating the curves thousands of times. The system was designed to ensure that the final shape was not just mathematically correct but also physically possible to manufacture, avoiding sharp edges or impossible angles. The process continued until the light produced by the virtual device matched the target pattern almost perfectly.

The team tested their method with two distinct scenarios. In the first, they created a known, ideal light pattern and asked the computer to rediscover the shape that produced it. The algorithm successfully recovered the original design, proving that the method works in theory. In the second, more realistic test, they used the actual light profile of a commercial LED bulb and asked the system to create a specific, focused beam. The result was a precise set of freeform surfaces—shapes that do not follow simple circles or spheres—that could transform the messy light of the LED into a clean, controlled beam. The simulations showed that the light distribution produced by the new design matched the target with extremely high accuracy, with errors so small they were barely detectable.

What makes this work significant is that it provides a unified framework for designing these complex mixed systems. Previously, combining refraction and reflection in a single inverse design was a major hurdle. By breaking the problem down and using the Wasserstein distance as a guide, the researchers demonstrated that it is possible to calculate the exact shape needed for a TIR collimator without relying on guesswork. The study confirms that for these specific devices, the optical paths can be decoupled, allowing for an efficient and reliable solution. While the current work focuses on rotational symmetry and point sources, the success of this approach suggests a path forward for designing even more complex lens and reflector systems in the future, potentially leading to more efficient and precise lighting for a wide range of applications.

Technical Summary: A Minimization Approach for the Design of TIR Collimators with Freeform Surfaces

Problem Statement
Inverse optical design methods have been extensively developed for systems comprising solely refractive or reflective elements. However, catadioptric systems, which combine both, remain less explored from a mathematical perspective, particularly in the context of non-imaging optics. While industrial practice often relies on forward simulation techniques like Monte Carlo ray tracing followed by iterative trial-and-error adjustments, inverse design methods aim to compute optical surfaces directly from prescribed source and target distributions.

The specific problem addressed in this work is the inverse design of a Total Internal Reflection (TIR) collimator, a rotationally symmetric device commonly used in LED spotlights. This system combines refraction and total internal reflection to produce a narrow beam. The challenge lies in determining two freeform surfaces—one for refraction and one for TIR—such that the resulting light distribution matches a prescribed target intensity. Unlike simpler systems, the interaction between refractive and reflective components in general catadioptric systems creates coupled optical paths, complicating inverse design. The authors focus on the TIR collimator as a foundational step, noting that its rotational symmetry and point source assumption allow for a two-dimensional formulation that yields a three-dimensional design, where refractive and reflective ray paths can be treated independently.

Methodology
The authors propose an inverse design framework that formulates the problem as a minimization of the Wasserstein distance between the computed light distribution and a prescribed target distribution.

Mathematical Model:
- The TIR collimator is modeled with four surfaces: two fixed planar surfaces and two freeform surfaces (A and C) to be determined.
- The angular domain of the source is divided at a transition angle $\vartheta_c$ into two sub-ranges. Rays in the first range follow a "lens route" (refraction only), while rays in the second follow a "TIR route" (refraction, reflection, and refraction).
- Two optical mappings, $m_1$ and $m_2$ , relate source angles to exit angles for each route. These mappings must satisfy the conservation of luminous flux.
- Given the mappings, the freeform surfaces are constructed using functional methods derived by Bortz and Shatz, which yield differential equations describing the surface geometry.
Optimization Formulation:
- The inverse problem is posed as finding mappings $m_1$ and $m_2$ that minimize the squared Wasserstein-2 distance ( $W_2^2$ ) between the intermediate target intensity $H$ (generated by the collimator) and the desired target intensity $G$ .
- The objective functional is transformed from the target domain to the source domain to explicitly depend on the mappings.
- Constraints include energy conservation (flux balance) and geometric requirements for manufacturability, such as ensuring the surfaces intersect the optical axis perpendicularly and that the target domains of the two routes overlap to avoid discontinuities.
Numerical Solution:
- A gradient-descent algorithm is employed to solve the minimization problem. The optical mappings are parameterized using Legendre polynomials to ensure smoothness and avoid oscillatory behavior.
- The transport map $T$ , which relates the intermediate intensity to the target intensity, is not an independent optimization variable; it is uniquely determined by the current mappings via energy conservation constraints at each iteration.
- The algorithm utilizes adaptive learning rates for the two independent mappings ( $m_1$ and $m_2$ ) and includes a projection step to enforce linear constraints on the polynomial coefficients (e.g., boundary conditions and domain alignment).
- An adaptive strategy is used for the number of basis functions: the algorithm starts with a low-order basis and increases complexity if convergence stalls, using the previous solution as an initial guess for the expanded basis.

Key Contributions

Unified Inverse Framework: The paper presents a first unified inverse design framework for catadioptric systems that specifically addresses the TIR collimator. In this specific setting, the refractive and reflective ray paths are treated independently, allowing the problem to be solved efficiently. This serves as a starting point for addressing more complex coupled systems where such independence does not hold.
Wasserstein Distance Metric: The authors utilize the Wasserstein distance as a metric to compare light distributions, which remains meaningful even when the computed and target distributions are defined on different or overlapping domains.
Decoupled Optimization: By exploiting the rotational symmetry and the specific geometry of the TIR collimator, the authors demonstrate that the refractive and reflective ray paths can be treated independently. This allows the resulting minimization problem to be solved efficiently using a gradient-descent method, avoiding the complex variational structures found in fully coupled lens-reflector systems where optical paths are coupled.
Numerical Implementation: The work details a robust numerical implementation involving the discretization of optical mappings, the computation of transport maps via integral balance (avoiding unstable ODEs near zero intensity), and a projection-based constraint handling mechanism.

Numerical Results
The proposed approach was validated through two numerical examples:

Validation Case: A uniform source intensity with prescribed shifted-reciprocal optical mappings was used to generate a known target distribution. The algorithm successfully recovered the target distribution and the associated freeform surfaces from a simple linear initial guess, demonstrating the method's ability to reconstruct a known solution.
Realistic Scenario: A more complex configuration using a measured LED source profile (LED16) and a Gaussian target intensity (representing a 20° spotlight) was tested. The algorithm converged to a solution where the computed intensity distribution closely matched the desired Gaussian profile. The resulting freeform surfaces were physically realizable, and the optical mappings satisfied the necessary continuity and boundary conditions.

In both cases, the algorithm successfully reconstructed the desired target intensity distribution and the associated freeform optical surfaces. The error between the computed and target distributions was minimal, with $L_2$ -norms of the difference on the order of $10^{-4}$ to $10^{-6}$ .

Significance and Claims
The authors position this work as a "first step" toward the design of more general lens–reflector systems. They explicitly state that while the current method relies on the independence of ray paths in the TIR collimator, more general systems involve coupled optical paths, leading to more complex variational derivatives where gradient-based approaches are not directly applicable.

The paper claims that the presented methodology provides a mathematically grounded inverse design approach for catadioptric systems, moving beyond forward simulation and trial-and-error. By demonstrating that the problem can be formulated as a minimization of the Wasserstein distance and solved efficiently via gradient descent for the TIR collimator, the authors establish a foundation for extending these techniques to more complex, coupled optical systems in future research. The work does not claim to solve the general coupled case but rather to provide a viable framework for a specific, practically relevant class of catadioptric devices.

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