Freeform optics design for extended sources in paraxial approximation exploiting the expectation maximization algorithm

Abstract

Freeform optics generating specific irradiance distributions have been used in various applications for some time now. While most freeform optics design algorithms assume point sources or perfectly collimated light, the search for algorithms for non-idealized light sources with finite spatial as well as angular extent is still ongoing. In this work, such an approach is presented where the resulting irradiance distribution of a freeform optical surface is calculated as a superposition of pinhole images generated by points on the optical surface. To compute the required arrangement of the pinhole images for a prescribed irradiance pattern, the expectation maximization algorithm from statistics is applied. The result is then combined with a ray-targeting approach for finding the shape of the corresponding freeform optical surface. At its current state, the approach is applicable to Gaussian input irradiances, single-sided freeform optics and for the paraxial case. An example freeform optical surface for laser material processing is shown and discussed demonstrating the performance and the limitations of the presented approach.

Application-adapted beam shaping with cutting-edge optical elements in laser materials processing

“…The design algorithm is based on the solution of the so called Monge-Ampère equation and a following flux optimization step [17]. Additionally, we have developed methods which account for finite divergence angles that are based on phase space considerations [18] or utilize the expectation maximization from statistics [19].…”

Folded light source: a technique for compact and efficient automotive lighting

“…For extended light sources, freeform surfaces have been designed for uniform illumination via edge ray mapping 29 and for specific irradiance distributions via expectation maximization algorithm. 30 Most white LEDs are Lambertian emitters with an intensity distribution that reduces to 50% at 60 deg from the surface normal. To capture all of the rays emitted by a white LED using a reflector, the reflector may need to become undesirably large.…”

“…LEDs are extended sources as compared to ideal point sources and that adds an extra level of challenge for the optical engineers. For extended light sources, freeform surfaces have been designed for uniform illumination via edge ray mapping 29 and for specific irradiance distributions via expectation maximization algorithm 30 . Most white LEDs are Lambertian emitters with an intensity distribution that reduces to 50% at 60 deg from the surface normal.…”

Spherical-freeform lens design for uniform illumination based on differentiable ray tracing with uv polynomials

“…3 For a compact design, the size of the light source cannot be ignored. Some methods have been proposed for tailoring the irradiance distributions of extended sources, including the over compensation methods, [4][5][6][7] the simultaneous multiple surface (SMS)-based methods, 8,9 the deconvolution methods, [10][11][12][13][14] the parametric optimization methods, [15][16][17][18][19] the edge ray mapping method, 20 the expectation maximization algorithm applied for pinhole image arrangement 21 and the energy accumulating optimization method. 22 Among these methods, the parametric optimization methods are easy to implement and have strong universality.…”