Plane wave expansion of cylindrical functions

where V 2 is the two-dimensional Laplace operator and k is the wavenumber of the radiation field. The function Vstands for a typical component of the electric or magnetic field. In cylindrical coordinates, the basic solutions of the Helmholtz equation are of the form Hn (kr ) exp (int~), where Hn is the Hankel function of integer order n while r and ot are a radial and an angular coordinate, respectively. For the sake of brevity, we shall refer to these functions as cylindrical waves of order n (shorthand notation: CW~). We recall that there are two types of Hankel functions, known as Hankel functions of the first and of the second kind. The corresponding CW~ represents outgoing (first kind) or ingning (second kind) fields, when a time factor e x p ( i t o t ) is assumed. Although these functions are so important, their representation in the form of a plane wave expansion is not, to the best of our knowledge, available in the literature, except for the case n = 0. In this paper, we shall derive such an expansion and we shall give some examples of applications. Our study was motivated by a research about quasi-optical techniques for launching electromagnetic power into plasmas [ 1,2 ]. One of the proposed methods [ 2 ] relies on the coupling between the electromagnetic field and the plasma via evanescent waves produced by scattering at a grating. In case the grating is made up by conducting cylinders, the diffracted field can be expressed as a superposition of CWn. The spatial frequency spectrum of such a field is of interest. Although this can be obtained through numerical Fourier transform