Use of Nonlinear Operators for Solving Geometric Optics Problems

Abstract

The aim of this work is to develop and apply a mathematical apparatus based on nonlinear operators for solving problems of geometric optics, namely the construction of images of objects in systems of thin lenses. The problem of constructing the image of a point in a thin lens was considered, on the basis of which the concept of the lensing operator was defined. The mathematical properties of the operator were investigated. The model problem of constructing an image in thin lenses folded together was investigated, on the basis of which it became possible to establish a physical interpretation of the previously determined properties. The problem of a system of lenses located at a distance was also considered, which resulted in the introduction of the concept of shift operator. The properties of the shift operator were studied, which together with the properties of the lens operator made it possible to determine the rules for using the created operators for solving the problems. In addition to solving the model problems, the following problems were considered: the speed of the moving point image, the magnification factor and the construction of the curve image. As an example, images of a segment and an arc of the circle were constructed. The segment was transformed into the segment, and the arc of the circle into the arc of the curve of the second order. The presented mathematical apparatus is very convenient for implementation in computer programs, as well as for the study of images of different curves.