Differentiable ray-tracing is an exciting new development in computer graphics to approach all sorts of 3D scene design problems by obtaining gradients of renders produced by ray-tracing with respect to parameters that define the scene. These gradients can then be incorporated in a gradient-descent type optimization pipeline.

One such type of design problem is caustic design with free-form lenses. This is the process of obtaining the geometry of lenses in an optical system such that the light that passes through this system forms a desired illumination distribution on a target screen. A typical application of this is distributing the light from streetlights or car headlights in a pleasing and efficient way over the street.

The non-imaging optics literature offers many methods to design free-form lenses for caustic design, but differentiable ray-tracing is largely unknown in this field. Therefore this thesis proposes a method of optimizing free-form lenses for caustic design with differentiable ray-tracing.

The optimization pipeline starts with a multi-layer perceptron neural network which outputs parameters that define one free-form lens side in the form of a B-spline surface. A specially implemented ray-tracer then produces a caustic render by tracing through this lens from either a plane wave or a (grid of) point source(s). Back-propagation and optimization takes place using automatic differentiation in PyTorch and the Adam optimizer.

The results, which are verified with LightTools, show great promise for this technique, especially for the plane wave optimizations. The point source grid optimizations proved to be more challenging, but also here the optimization was able to achieve improvement. This shows that the proposed technique also has potential in positive etendue optimizations.

This project aims to recreate intensity patterns using Fraunhofer diffraction as a means of simulation. These intensity patterns are created by phase shifting specific parts of an incoming field of light. These phase shifts are determined by a B-spline surface, which is in turn controlled by so-called control points. Only a handful of control points can describe a whole surface. The position of these control points is then determined using machine learning and specifically a technique inspired by ‘physics-informed neural networks’, which were introduced last year by Raissi et al. [1]. With this method, simple experiments which sought to recreate intensity patterns known to be in the solution space were carried out. These experiments showed some success, but suffered from the fact that they used too sensitive parameters in the input of the machine learning model, reducing the sophisticated method to a Monte Carlo search, or they used no input at all, which degraded the machine learning model to simple parameter optimization. Nevertheless, these experiments showed that this method has the potential to be used in more flexible optical setups, where multiple configurations can yield the same intensity pattern or where changing the parameters defining the setup do not induce enormous changes in the resulting intensity pattern. In addition, the proposed method relies upon Fraunhofer diffraction, which, when discretized for numerical computation, introduces aliasing issues when the incoming field changes too rapidly. This phenomenon was especially apparent when using point sources that create spherical wave fronts. A possible solution for this issue is to consider ray tracing techniques in future research.

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.