Controlling the path of light is fundamental to many technologies, yet designing optical systems that precisely shape light beams remains a significant challenge. Pieter Braam, Jan ten Thije Boonkkamp, and Martijn Anthonissen, all from Eindhoven University of Technology, alongside colleagues including Koondanibha Mitra and Wilbert IJzerman from both Eindhoven University of Technology and Signify Research, now present a new method for designing optical elements that transform light from multiple sources into multiple desired patterns. This inverse design approach calculates the precise shape of reflectors or lenses needed to achieve this transformation, effectively solving for the surface geometry given the desired input and output light distributions. The team’s work represents a substantial advance in freeform optics, offering greater flexibility and control over light manipulation than traditional methods and opening possibilities for innovative illumination and imaging systems.
Freeform Optics via Optimal Transport Theory
This research details a new approach to designing freeform optical surfaces using advanced mathematical techniques, particularly optimal transport theory and least-squares methods based on the Monge-Ampère equation. The goal is to create optical elements, such as lenses and reflectors, that shape light in a desired way, extending beyond traditional imaging optics to achieve complex beam shaping and non-imaging applications. This work addresses the limitations of conventional optics by allowing for arbitrary surface shapes and greater control over light manipulation. The research centers on optimal transport theory, a mathematical framework used to map a desired light distribution to a source distribution, effectively defining how light travels through the system.
This involves solving the Monge-Ampère equation using efficient least-squares methods. The team developed a unified mathematical framework for designing a wide range of freeform optical systems, accommodating various design constraints like surface slope and material properties. This method is applicable to diverse optical systems, including beam shaping for illumination or laser processing, non-imaging optics for solar concentration, point-to-point light transport, and designs incorporating Fresnel reflections and scattering surfaces. Key advancements include local grid refinement, which improves the accuracy of numerical solutions, and methods for handling boundary conditions on the optical surfaces. This work builds upon existing research in freeform optics, non-imaging optics, and optimal transport, offering a more general and robust approach to inverse design, ultimately providing powerful mathematical tools and numerical techniques for designing freeform optical systems with unprecedented control over light manipulation.
Freeform Surface Design via Inverse Least Squares
Scientists developed a novel inverse method to compute freeform optical surfaces capable of transforming light distributions from two source planes into two separate target distributions. This technique allows for the design of both reflector and lens surfaces, precisely controlling the spatial and directional coordinates of light rays. The research builds upon established principles of geometrical optics and energy conservation, employing a three-stage least-squares algorithm to numerically solve the resulting equations and determine surface shapes. This approach differs from forward methods and offers a direct computation of surface geometry from source and target distributions.
Researchers adapted and refined a generating least-squares method to solve the standard Monge-Ampère problem with Dirichlet boundary conditions, incorporating a transport boundary condition to ensure all light rays originating from the source reach the intended target. Further advancements involved extending the method to solve generalized Jacobian equations, enabling the modeling of complex optical systems. The team applied this refined technique to design parallel-to-parallel optical systems, building upon previous work that utilized supporting quadric methods and ray-mapping approaches, specifically computing distances from the source and target planes to the optical surfaces.
Freeform Optics Design via Energy Conservation Principles
Scientists developed a novel method for designing freeform optical surfaces, specifically reflectors and lenses, capable of transforming light distributions. The work centers on mapping light from one arrangement of source planes to another, precisely controlling both the position and direction of light rays. This is achieved through a mathematical framework that links the geometry of the optical surfaces to the conservation of energy, deriving equations that govern how light propagates through the system. The team established a mathematical model based on generating functions and optical path length, allowing them to define the shapes of the reflectors and lenses.
By applying energy conservation principles, they derived Jacobian equations that describe the mapping of light rays, and these equations, combined with the generating functions, determine the precise form of the optical surfaces. The method accounts for both reflection and refraction, with the lens design incorporating a refractive index to accurately model light behavior within the material. Experiments. Measurements confirm that the derived equations are valid, with the arguments of the square roots within the generating functions remaining strictly positive, indicating no internal reflections. This ensures that the light rays propagate efficiently through the system without being lost. The team’s approach provides a robust and accurate method for designing complex optical systems, offering precise control over light manipulation and paving the way for advanced optical technologies.
Freeform Optics Design via Energy Conservation
This research presents a novel inverse method for designing freeform optical surfaces, specifically reflectors and lenses, that transform light distributions between two source and two target planes. The team successfully developed a mathematical model based on energy conservation and optical path length, deriving equations that relate the geometry of the surfaces to the desired light transformations. This approach allows for precise control over both the spatial position and directional properties of light rays. By defining relationships between distances and angles within the optical system, the researchers created a framework for designing complex optical elements with tailored light manipulation capabilities. Demonstrations with complex source and target distributions confirm the method’s versatility and potential for advanced optical design. Future work will focus on refining the algorithm to improve computational efficiency and exploring applications of this method to more complex optical systems, including those with far-field targets.