Freeform lens predictions by a Neural Network and B-splines

Abstract

There exist a lot of methods to find the electromagnetic field behind a lens, so called the forward lens problem. In contrast very few methods can do the inverse, namely finding a lens that would produce a given image or light distribution for a known source. Here a solution to the forward problem, the Fraunhofer approximation, is used to find an approximate solution of the inverse problem using a neural network. Given an input image the network would predict the lens that can produce this image. In general this lens is not the classic convex/concave shape but is a freeform lens. The predicted image produced by the predicted lens can be computed by the Fraunhofer approximation. To train the network this predicted image is compared to the
desired image. This unsupervised training method is similar to that of Physics Informed Neural Networks (PINN), which is a recent approach to solve PDE’s.
The phase contour of the lens is represented by a B-spline curve. The control points of this spline are the output of the network. In this way very few output variables can be specified by the network to achieve a smooth detailed lens shape.
To give a proof of concept a one dimensional version of the Fraunhofer approximation is used. For this case the training already depends on many parameters. These are optimised for the lowest resulting loss.
The Fraunhoffer approximation limits the images that can be created. If the network, however, is trained on images that can definitely be created, it is able to almost exactly predict a lens that delivers the desired image.
The potential of the unsupervised training method in combination with a spline approximation should therefore be explored with other solutions of the forward problem. Namely a ray-tracing algorithm could be used, since this can create a wider variety of images. This would be computationally heavy compared to the
Fraunhofer approximation.