Abstract
In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.
Highlights
The optical design problem involving freeform surfaces is a challenging problem, even for a single mirror/lens surface which transfers a given intensity/emittance distribution of the source into a desired intensity/illuminance distribution at the target [1,2,3]
We show that a similar mathematical expression can be obtained for the freeform surfaces of a lens with parallel ingoing and outgoing light rays applying the laws of geometrical optics: u1(x) + u2(y) = c(x, y) := b1 − b2 + b3|x − y|2, (3)
We apply the algorithm to four test problems to compute c-convex lens surfaces: first, we map a square with uniform emittance into a circle with uniform illuminance, second, we map an ellipse with uniform emittance into another ellipse with uniform illuminance, third, we map a square with uniform emittance into a non-convex target distribution, and we challenge our algorithm to map the same distribution into a light pattern given by a picture on the target screen
Summary
The optical design problem involving freeform surfaces is a challenging problem, even for a single mirror/lens surface which transfers a given intensity/emittance distribution of the source into a desired intensity/illuminance distribution at the target [1,2,3]. The freeform design problem is an inverse problem: “Find an optical system containing freeform refractive/reflective surfaces that provides the desired target light distribution for Journal of Scientific Computing (2019) 80:475–499 a given source distribution”. To convert a given emittance profile with parallel light rays into a desired illuminance profile with parallel light rays, one requires at least two freeform lens/mirror surfaces [2,5]. This freeform problem can be formulated as a second order partial differential equation of Monge–Ampère (MA) type, with transport boundary conditions, applying the laws of geometrical optics and energy conservation [2,3,6,7].
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