Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A second fourth-order ordinary differential equation for the general Fourier coefficent is derived from an integral representation of the coefficient, and both differential equations are shown to be equivalent. Series solutions for the various Fourier coefficients are also given, mostly in terms of Legendre functions and Bessel/Hankel functions. These are derived from the closed form hypergeometric solutions or an integral representation, or both. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented

Annular nanoantenna on Fibre micro-axicon

International audienceIn this paper, we propose to extend the concept of loop antenna to the optical domain. The aim is to develop a new generation of optical nanocollectors that are sensitive to specific electric or magnetic vectorial field components. For validating our approach, a preliminary one-micron-diameter gold nanoring is micromachined on the apex of a cone lens obtained from a tapered optical fibre. It is shown that such a nano-object behaves as a nano-antenna able to detect the longitudinal electric field from a Bessel beam in radial polarization and the longitudinal magnetic component from a Bessel beam in azimuthal polarization. In the latter case, the annular nano-antenna exhibits the properties of an optical inductance