Freeform lens for irradiance tailoring on 3D surface


Traditional optics has been dominated by spherical, aspherical and cylindrical surface shapes for a long time. These surfaces are limited to their rotational and translational symmetries. Freeform optics has at least one freeform optical surface which breaks the rotational and translational symmetries. Freeform optics has much more design freedom which can complete very complicated tasks that are previously unimaginable. One of the most interesting topics for freeform optics design is the conversion of a light source emission into a desired illumination pattern on a given target (i.e., the prescribed irradiance problem). This technology has various applications including general lighting, automotive lighting and laser material processing, etc. However, freeform optics design for the prescribed irradiance problem is a very difficult inverse problem and there is only a rigorous mathematical model under the point source approximation. This inverse problem for a point source can be simplified to  fully nonlinear partial differential equations of Monge-Ampère type by mainly merging two type equations. The first type is the energy conservation between the source and target. The second type is the ray tracing equations that describe the coordinate relationships from source to target. In addition, the freeform optical surface is constrained to be smooth to facilitate fabrication. Although such a direct determination has the advantage that the design problem is reduced to only one equation, but the formula derivation process is extremely complicated and tedious. The developments of simplified methods have attracted a lot of researchers not only from the optical community but also from other fields including computer graphics and applied mathematics. Ray mapping methods are also widely used, and they simplify the design with two separate steps: an approximate ray mapping computation and surface construction from the ray mapping. However, the traditional ray mapping methods are only accurate for paraxial or small angle approximations but not suitable for off-axis and non-paraxial cases because the approximate ray mappings are no longer integrable for surface integration. There have been several advanced ray mapping methods developed for obtaining integrable ray mapping. However, most of the current methods focus on producing prescribed irradiance distributions on planar targets. Very limited work has been done for curved targets.

Fig.1 Irradiance tailoring on a curved target using a freeform lens

The research group of Professor Yongtian Wang at School of Optics and Photonics of Beijing Institute of Technology proposed a new method applicable for curved targets based on iterative wavefront tailoring (IWT). The IWT method has been applied successfully for planar targets. Instead of directly determining the required freeform lens, the IWT method simplify the design problem through the intermediate construction of a sequence of outgoing wavefronts. A Monge–Ampère equation of a parameterized outgoing wavefront was first established by connecting the outgoing wavefront properties to energy conservation. A ray mapping can be acquired from the solution of the wavefront equation, which is used to obtain an approximate freeform lens. Using the freeform lens data, the wavefront equation is updated to generate a more accurate ray map. This process is iterated to reduce the surface errors and improve the target irradiance accuracies. For a planar target, the z-coordinates are constant and irrelevant with the ray map. For a curved target, the z-coordinate are no longer constant and intertwined with the ray map, which add much more design complexities. The new IWT method can artfully dissolve the mathematical difficulties by involving the varying z-coordinates of the curved target in the IWT procedure. The ray map generated at the i-th iteration is obtained from the solution of the outgoing wavefront equation that imbeds the target’s z-coordinates at the (i-1)-th iteration, and then the target’s z-coordinates is immediately updated following the i-th ray map. The new method is developed with the stereographic coordinate system, which is suitable for light sources with large emission angles such as LED sources. Although a series of Monge-Ampère equations of the wavefront need to be solved, a multi-scale strategy can be used to speed up the computation. The proposed method is verified by designing two spherical-freeform lenses for generating a rectangular uniform illumination pattern and a circular image pattern on an undulating surface, respectively. The method is also applicable for designing plano-freeform, aspherical-freeform or double freeform lenses. The result of this research can be beneficial to those applications where the targets cannot be considered as perfect 2D planes but 3D surfaces, for examples, road surfaces on mountain areas, sand tables, surfaces of sculptures and cultural relics.

Fig. 2. Two examples of generating a rectangular uniform illumination pattern and a circular image pattern on an undulating surface from a Lambertian light source. The simulation are implemented in LightTools 8.6


About The Group

Beijing Engineering Research Center of Mixed reality and Advanced Display, led by Professor Yongtian Wang, was established in 2012 under the recognition of Beijing Municipal Commission of Science and Technology and was rated as excellent in performance in early 2017. The center’s main research areas include freeform optics, micro-nano optics, novel 3D display, virtual reality and augmented reality, medical image processing and surgical navigation. There are more than 200 research members including faculty members, post-doctors and graduate students. The center members have undertaken many important research projects including 973 Program, 863 Program and National Key Research and Development Project funded by the Ministry of Science and Technology of China, and Major Scientific Research Instruments Development Project and Key Projects funded by the National Natural Science Foundation of China (NSFC). In recent years, the center members have published more than 200 SCI papers, many of which have been published in high-level journals such as Nature Communications, Light: Science & Applications, Advanced Materials, Nano Energy. The center members have been authorized more than 100 invention patents including 5 US and European invention patents. Recently, the scientific and technological awards won by the center include the second prize of National Technological Invention Award in 2017, first prize of Innovation Technology Award of the Chinese Society for Optical Engineering, first prize of Science and Technology Award of China Society of Image and Graphics, first prize of Technological Invention Award of Beijing in 2019, and first prize of Innovation Achievement Award of China Industry-University-Research Institute Collaboration Association. The center has made outstanding achievements in the application and conversion of the research results which have been widely used in aerospace, aviation, military, medical, industry, education, culture, sports, etc.


Feng Z X, Cheng D W, Wang Y T. Iterative freeform lens design for prescribed irradiance on curved target. Opto-Electron Adv 3, 200010 (2020).