Implementation of arbitrary linear optical transformations with diffractive optics

Abstract

All applications for diffractive optical elements can be seen as some type of linear optical transform implementation. Methods for the implementation of arbitrary linear optical transformations are discussed. Necessary and sufficient conditions for the implementation of linear optical transformations are considered. These are derived from the properties of a linear optical system. A taxonomy of linear optical transformations is provided. Point transforms are one of the groups of linear optical transformations. These are considered in more detail. The necessary and sufficient conditions naturally leads to the discrete phase technique of implementation. This technique comprise the weighted summation of localized diffraction gratings (or Fresnel lenses in the case of a lensless implementation). As an example of an implementation of this technique the Hough transform is considered. This well known transform is used for the processing of two dimensional images. The conventional Hough transform maps lines in an input image to points in a two dimensional output plane. The cartesian coordinates of the points in the output plane denote the orientations and locations of the lines in the input image.