Speaker
Description
The task in optical design is to find a set of parameters such that an optical system meets certain performance criteria and boundary conditions. Mathematically, this can be described as a least-squares optimization problem, i.e., the objective function is the norm of an objective vector. For many years, the Levenberg-Marquardt algorithm has been the optimization method of choice in the optical design community. However, the efficient evaluation of the Jacobian of the objective vector is a delicate task – also in many other disciplines of applied optimization. We present a workflow based on algorithmic differentiation (AD), where the Jacobian can be evaluated in the same order of computational complexity as the objective vector itself, i.e., linear in the number of parameters and linear in the number of objectives. Sparsity patterns in the specific optimization problem can be exploited to achieve this result in both AD forward mode and AD reverse mode.
Ray tracing as a part of geometrical optics describes the propagation of light through an optical system. Many rays are traced from the object to the image to obtain the data that are required for the evaluation of the objective vector of the optimization. While light travels through homogeneous media in straight lines, its trajectory in inhomogeneous media is curved and described by an ordinary differential equation. Applying AD to a ray tracing routine is in most cases straightforward. However, there are some challenges, e.g., the determination of the intersection of a ray with a surface, especially for inhomogeneous media. We discuss a framework in which this implicit problem can mathematically be described and differentiated.
