Generalized Analytical Solutions for Planar and Non-planar Focal Curves Reconstruction

Abstract

A generalized solution in the scope of the Fresnel approximation for diffractive optical elements producing focal curves is derived in this paper. The method includes planar and non-planar focal lines parametrized by differentiable functions and different types of holograms (Fourier, Fresnel). The integral describing the complex amplitude distribution generated by such a focal curve is evaluated with the help of the stationary phase method and the points in the hologram plane contributing to a particular point of the focal curve are found to be circles or straight lines. As the result equations for stationary points turn out to be solvable analytically only in some specific cases, an approximate approach is proposed additionally. In this way the class of analytic solutions for non-planar curves can be broadened into cases for which there exists a solution related to their projection on to the hologram plane. The paper is completed by presentation of non-paraxial ray tracing diagrams for elements designed with the help of both methods.