Fourier transformation of rotationally invariant two-variable functions: Computer implementation of Hankel transform

A Hankel transform distribution algorithm for paraxial wavefields with an application to free-space optical beam propagation

“…where the z coordinate frequency is written within the paraxial approximation. This Bessel beam expansion from equation (5) is the reason why numerous articles have proposed different methods for evaluating numerically Hankel transforms of arbitrary order [16][17][18][19][20][21][22]. This Hankel transform formalism gives a complete description of radially symmetric wavefields, profiles given by equation (2), and even more importantly can be generalized to profiles that are not radially symmetric.…”

“…The typical approach for solving these problems is first to write the input data via the modal decomposition from equation (7), and to subsequently apply a Hankel transform algorithm to each individual discrete u n n , m x y ¯( ) function, where there exist a variety of algorithms for this task [16][17][18][19][20][21][22]. This can be a lengthy process and by no means constitutes an optimal approach, since it involves two-dimensional projections and a thorough modal decomposition, which in turn is base-set dependent.…”

“…This synthesizes a wave's spatial distribution into a series of propagation invariant Bessel beam spatial modes [13][14][15] and gives a complete OAM description of the analyzed profile. Even though numerous works have provided different means of evaluating the Hankel transform [16][17][18][19][20][21][22], most of these focus on analyzing a circularly symmetric profile whose only angular dependence is a helical wave-front. Scarce effort has been devoted to obtaining, fast and accurately, the set of Hankel transforms encompassed by a more general profile, one that is not circularly symmetric.…”