M.Sc Student Shelenkova Ekatarina Minimal Optical Surfaces for Perfect Imaging of Two Point Objects Department of Applied Mathematics Professor Jacob Rubinstein

Abstract

One of the fundamental problems in optical design is perfect imaging of a given set of objects or wave fronts by an optical system. An optical system is defined as a finite number of refractive and reflective surfaces and is considered to be perfect if all the light rays from the object on one side of the system arrive to a single image on the other side of the system. Descartes was the first who used the law of refraction and solved the problem of perfect imaging of one point object by a single refractive surface that is now called a Cartesian oval. However, in the more general case, where we need to find a set of optical surfaces that map a given set of n objects into n respective images, the problem remains open.

In the current work we study the problem of perfect imaging of two given point objects into two corresponding images by an optical system consisting of two refractive or reflective surfaces. We limit our study to two-dimensional geometry. At the beginning we consider the case where the objects and the images are given as single points. We study the simultaneous multiple surface method in 2-D geometry that generates two freeform C 1curves and provides a solution, or more precisely a large family of solutions.

We propose a modification to the algorithm, which yields a solution that is C2 smooth. Our method includes computations of wave fronts refraction at the anchor points of generated curves and therefore allows us to calculate the second derivatives at these points.  The main advantage of the method we propose is that it generates two smooth optical curves with minimal degrees of freedom. In order to demonstrate the implementation of the algorithm we present two illustrative examples of constructing the optical element, which is composed of two refractive surfaces, and the optical element consisting of two reflective surfaces.

Further, we consider the case of coupling two point objects and two general outgoing wave fronts, and show that the proposed algorithm can be applied to construct an optical system consisting of two C2 differentiable curves as well. One application of this algorithm is developing a multi-surface customized model of human eye.  We show that it is possible to calculate a two-surface individual eye model using the data which involves two wave fronts    refracted from two point sources on the retina.  In order to validate our approach we examine the algorithm on the Gullstrand eye model. The two refractive surfaces that were constructed turned out to be a good approximation of the surfaces of the lenses in the Gullstrand model. In addition to the Gullstrand model we consider the situation where the surfaces of the eye lens are not spherical.

At the end of our work we talk about possible directions of its extension and discuss the problem of the dependence of the proposed algorithm on the choice of initialization.