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Although wave optics is the standard method of analyzing systems composed of a sequence of lenses separated by arbitrary distances, it is often easier and more intuitive to ascertain the function and properties of such systems by tracing a few rays through them. Determining the location, magnification or scale factor, and field curvature associated with images and Fourier transforms by tracing only two rays is a common skill. In this paper we show how the transform order, scale factor, and field curvature can be determined in a similar manner for the fractional Fourier transform. Our purpose is to develop the understanding and skill necessary to recognize fractional Fourier transforms and their parameters by visually examining ray traces. We also determine the differential equations governing the propagation of the order, scale, and curvature, and show how these parameters are related to the parameters of a Gaussian beam. © 1997 Published by Elsevier Science B.V.

We form optical systems by using only free space portions and cylindrical lenses, and consider these systems as anamorphic fractional Fourier transformers. We dynamically adjust the transform order, scale factor and field curvature of both orthogonal dimensions of anamorphic fractional Fourier transformation by just changing the focal lengths of cylindrical lenses used in the proposed setups. Here, we also consider two approaches for implementing cylindrical lenses with dynamically adjustable focal lengths. There may also be some other methods to obtain cylindrical lenses having adjustable focal lengths which can successfully be used in these proposed setups.