of a quasi spherical eigenfunction expansion of the scalar Green’s functions for a cone[31–35] with Dirichlet (or Neumann type) boundary conditions depending on ν$_{E}$ (respectively ν$_{M}$ ) and m . The scalar field · E (respectively r · B ), and consequently the coefficients a E ν$_{E}$,m (respectively a M ν$_{M}$,m ) given by Eq. 15 can be easily deduced from a surface integral on Ω$_{0}$. Conical Green’s functions (which have already been used in the context of near-field optics for aperturless microscopes[36, 37]) connect surface integrals, given by Eq. 11 and Eq. 15, to the Huygens-Fresnel principle directly. In order to complete this section we can note than the ratio 1 P dP d Ω represents the physical quantity in measurements of the angular power emitted by actual fiber tips as explained later on. The total radiated power of the fiber tip can be expressed as a function of the field E (0) This formula can be compared with the Bethe[18] result P = 5 ck $^{6}$a6 108 π $^{2}$k$_{2}$E 2 $_{0}$, where E$_{0}$ is the field at the center of the aperture. The same behavior appears in the two formulas: a dependence in a 6 and in E 2 $_{0}$. The difference arises from the (sin β ) 6 which represents the second geometrical parameter of the fiber tip. It is important to note that this formula contains an unknown variable: the field E$_{0}$ . This field is a function of the transmission of the tip which depends on the over-exponential decay in the conical wave guide due to the cut-off of the propagating mode in the fiber[38]. This decay is not taken into account in our model which prevents us from computing the tip transmission. In order to complete the solution of Maxwell’s equations, we will describe the boundary conditions on the cone and justify the above assumption ν ≃ 1. A simple analogy with wave guide theory permits to establish the boundary conditions in a conical geometry (this corresponds to the case of a “wave guide” of variable cross section). For a perfect metal , we have These equations impose a condition on ν$_{E}$ and ν$_{M}$ which restricts the allowed values to infinite, growing sequences ν$_{E,p}$ and ν$_{M,p}$ that depend on β and m (p integer). Because P total ≃ 5 c 864 k 2 ( ka sin β ) 6 E (0) 2 . (18) Y$_{ν}$$_{E}$$_{,m}$ ( θ, φ ) | S = 0 , ∂Y$_{ν}$$_{M}$$_{,m}$ ( θ,φ ) ∂ cos θ ∣ S = 0 (19) V. BOUNDARY CONDITIONS AND PERTURBATION METHOD 10