∂Y$_{ν}$$_{M}$$_{,m}$ ∂θ ∣ S = i √ 1 ϵ$_{c}$ ν$_{M}$ ( ν$_{M}$ + 1) kr Y$_{ν}$$_{M}$$_{,m}$ | S . (24) ν (0) E ( ν (0) E + 1 ) − ν$_{E}$ ( ν$_{E}$ + 1) ≃ (1 . 4 + i 0 . 23) kr ν (0) M ( ν (0) M + 1 ) − ν$_{M}$ ( ν$_{M}$ + 1) ≃ (1 . 4 + i 0 . $_{23) 10}$− 5 1 kr . (25) As expected, this reduces to the unperturbed condition for σ = ∞ . By using the substitution ν → ν (0) on the right-hand side of Eq. 24 we have to lowest order ψ$_{E}$ | S ≃ f$_{E}$ ∂ψ (0) E ∂n | S and ∂ψ$_{M}$ ∂n | S ≃ f$_{M}$ ψ (0) M | S where the factors ‖ f$_{E,M}$ ‖ are small. In order to calculate the perturbed ν , we use the two-dimensional Green’s theorem in its integral form[16] and we obtain: ν$_{E,}$$_{0}$ and ν$_{M,}$$_{0}$ are functions of two radii r$_{E}$ ≃ 18 nm and r$_{M}$ = λ = 633 nm that are adjusted to reproduce the experimental results discussed below. The two values of ν are complex and the real parts are close to unity as expected: ν$_{E}$ ≃ 0 . 95 − 0 . 01 i , ν$_{M}$ ≃ 0 . 96. These fits reveal that the adapted boundary conditions depend on a typical length r < λ that is probably linked to the aperture radius a ≃ 20 nm (this value is in agreement with the aperture radius obtained by scanning electron microscopy[5]). With these values of ν we can calculate the radiated power distribution of the tip[40]. Measurements of the far-field radiated power of tapered fiber tips in S and P plane have been carried out extensively by Oberm¨ uller and Karrai [5, 6] . As seen in Fig. 6, there is an important backscattering effect in the P plane. The experimental data and the results obtained with the quasi-dipole model (using the values of ν$_{E,M}$ obtained above) are also shown in Fig. 6 for comparison[40]. The agreement is good for most angles with the exception of extreme θ values corresponding to grazing observation angles on the metallic cone. The disagreement between theory and experiment is probably due to higher terms in the expansion that we have neglected. This is confirmed by the recent independent results obtained by Shin et al.[41] which have fitted the experiment to an expansion in classical spherical multipoles. Their fit contains dipolar terms (where the factor 2 plays the same role as in our model) as well as quadrupolar ( l = 2) and octopolar ( l = 3) terms. The behavior of the field in the extreme regions probably also depends on a modified diffusion due to surface roughness[42]. VI. COMPARISON WITH EXPERIMENTAL DATA 12