Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation

Abstract

Our purpose is to bring to the attention of the optical community our recent work on the numerical evaluation of zero-order Hankel transforms; such techniques have direct application in optical diffraction theory and in optical beam propagation. The two algorithms we discuss (Filon–Simpson and Filon-trapezoidal) are reasonably fast and very accurate; furthermore, the errors incurred are essentially independent of the magnitude of the independent variable. Both algorithms are then compared with the recent (fast-Fourier-transform-based Hankel transform algorithm developed by Magni, Cerullo, and Silvestri (MCS algorithm) [J. Opt. Soc. Am. A9, 2031 (1992)] and are shown to be superior. The basic assumption of these algorithms is that the term in the integrand multiplying the Bessel function is relatively smooth compared with the oscillations of the Bessel function. This condition is violated when the inverse Hankel transform has to be computed, and the Filon scheme requires a very large number of quadrature points to achieve even moderate accuracy. To overcome this deficiency, we employ the sampling expansion (Whittaker’s cardinal function) to evaluate numerically the inverse Hankel transform.