aperture in a metallic screen is introduced in general terms. In section III, a method to solve electromagnetic problems with conical geometry is developed. The radiation pattern of a conical optical probe is described in section IV for the case of a tip coated with a perfect metal, which is extended to real metals in section V. Section VI compares the theoretical results with experimental emission patters of metal coated, optical fiber tips. A summary is given in section VII. 

## II. INTRODUCTION TO THE PROBLEM OF DIFFRACTION BY SMALL APERTURES : THE BETHE-BOUWKAMP SOLUTION 

In classical optics, the diffraction of light by an aperture in a screen is studied within the Kirchhoff approximation [7, 8]. The use of this scalar method is justified in numerous cases where the polarization of light can be ignored and where the aperture size is large compared to the wavelength of the incident radiation. In order to apply the Huygens-Fresnel principle to light diffraction by small apertures, Smythe[9, 10] has developed a vectorial Kirchhoff integral formula based on the Green function method. The formalism is very general but - like Kirchhoff’s scalar theory - its practical application is restricted to long wavelengths compared to the geometrical parameters of the aperture. The first-order solution consists in neglecting the influence of boundary effects on the light in the aperture zone. This method, which may be viewed as the electromagnetic counterpart of the Born[11, 12] approximation in quantum mechanics, is adapted to the quasi geometrical regime of Maxwell’s equations. However, it is not adapted to long wavelength radiation. 

Bethe [13] (and after him Bouwkamp [14]) have developed a rigorous solution for the case of a small circular aperture in a perfect metallic plane by solving an electrostatic and a magne- tostatic eigenvalue problem for the static near-field existing in the vicinity of the aperture. This solution, which generalizes previous work by Rayleigh [15], can be easily recovered by using oblate spheroidal coordinates and harmonical wavefunctions. The Helmoltz equation [ ∇ 2 + k $^{2}$] ψ = 0, which reduces to the Laplace equation for small hole sizes ∇ $^{2}$ψ = 0, has the solution ψ ( ξ, η, φ ) = P m l ( ξ ) · P m l ( iη ) .e ± $^{imφ}$, where P m l denote Legendre wavefunctions indexed by two integers l, m . ξ, η, φ are the oblate spheroidal coordinates. The Bethe so- lution exhibits a complicated behavior in the vicinity of the aperture. Nevertheless, the Bethe solution reduces to a multipole expansion in the near-field domain limited by r/λ ∼ 1 

