Relationships among ray optical, Gaussian beam, and fractional Fourier transform descriptions of first-order optical systems

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.