Helical focus wave modes and helical nondiffracting beams

Abstract

A number of years ago Durnin and colleagues considered the possibility of nondiffracting beams, beams defined to be nonsingular free space exact solutions of the scalar wave equation having the same intensity distribution in every plane normal to the axis of propagation [J. Durnin, J. Opt. Soc. Am. A 4, 651–655 (1987)]. Assuming harmonic time dependence and cylindrical coordinates, i.e., a Fourier decomposition of the time and space coordinates, the Bessel function mode solutions of the scalar Helmholtz equation were pointed out as a class of diffraction-free solutions. Similarly, using a nonorthogonal helical coordinate system, helical mode solutions of the scalar Helmholtz equation that are nondiffracting in the same sense as the Bessel beams can be determined [P. L. Overfelt, Phys. Rev. A 46, 3516–3522 (1992)]. In this paper, when a bidirectional plane wave decomposition [I. M. Besieris et al., J. Math. Phys. 30, 1254–1269 (1989)] is assumed, this same helical coordinate system can be used to obtain helical focus wave mode (FWM) solutions to the scalar wave equation. The fundamental Gaussian FWM can be decomposed into two oppositely polarized helical FWMs. Many relationships between the helical and cylindrical FWMs and their beam solution counterparts will be shown.