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The dipole model of Oberm¨ uller and Karrai, which corresponds to ν = 1 (a value close to the real values ν$_{E,}$$_{0}$ and ν$_{M,}$$_{0}$ ), gives good results for the S polarization but cannot reproduce the backward emission in the P plane. The fact that the two values of ν$_{E,M}$ approach unity explains the efficiency of the dipole model for small azimuthal angles ( in this regime Y$_{ν,m}$ ≃ Y$_{1}$$_{,m}$ ). In addition we can formally justify the dipole model considering Eq. 15 as the generalization for conical geometry of the fundamental definition of multipole moments which enter into the spherical harmonic multipole expansion, and which are used for localized source distributions. Using this analogy, we deduce the two effective dipoles associated with the aperture 

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Once again, the characteristic factor of 2 between the magnetic and electric moments appears in this formula. Figs. 4 and 5 are two equivalent representations of the spatial dependence of radiated power. In particular, the “orbital” representation of Fig. 4 shows the important backward emission in the plane containing the effective electric dipole P and the tip axis (P plane). This backward effect is less important in the S plane containing the tip axis and the magnetic dipole due to the strong near-field effect on the metal surface. The factor of two in Eq. 26 (as in Eq. 17) stresses the difference between electric and magnetic dipoles as discussed in the introduction (namely an oscillating charge in the case of an electric dipole and a rotating charge for a magnetic dipole). This factor of two and the dependence of the dipoles on a and on E$_{0}$ are similar to the Bethe model and the Rayleigh diffusion theory by subwavelength particles and holes, but the dipole orientations are different, a result unexplained by the Bethe-Bouwkamp solution. 

We can note in addition that the far fields radiated by a small distribution of electric and magnetic dipoles located in the apex zone which satisfy a relation of the type given by Eq. 26, are equivalent to those deduced from Eq. 15 (and consequently Eq. 17) in the limit of subwavelength distributions. This fact, which can be obtained by means of scalar Green’s functions discussed above, gives further support to the analogy between dipoles and aperture at the apex of a cone. 

It is worthwhile to mention that A. Roberts[43] has used the method of quasi-multipoles for the interior problem of “small hole coupling of radiation into a near field probe”. In 

