expansion. Nevertheless, in order to take into account the finite size of the aperture domain in our modelization, we study the field outside a sphere of radius a/ sin β only. This so-called aperture zone represents the “terra incognita” near-field region ( r/λ ≤ 1) of the fiber tip (see Fig. 2). Let ϵ ≃ µ ≃ 1 be the electric and magnetic permittivities of vacuum. The electromagnetic field in the region r/λ ≥ 1 can be described by an expansion using the quasi- multipole formalism. The coefficients a$_{ν,m}$ contain all properties of the electromagnetic field. In order to calculate E and B in the radiation zone, we need (see Eq. 11) to know the field on the spherical section Ω$_{0}$, i. e. in the near-field zone of the aperture. Unfortunately, there is as yet no complete theory of light diffraction by a small aperture in a screen with conical geometry, and we must rely on approximations. The a$_{ν,m}$ coefficients can be evaluated by using a Taylor series written in the vicinity of the tip origin. We obtain, using Eq. 11, the following expressions to first order in r : In Eq. 14, the symmetric terms with prefactors Q λ,α,β and M λ,α,β are neglected in a first approximation. The integrals can be evaluated using the following properties: i) the β value is small and, as a consequence, the fraction (4 π − Ω$_{0}$) / (4 π ) is negligible, ii) the authorized ν values are close to 1, and iii) there are only the two first roots ν$_{E,p}$$_{=1}$ , ν$_{M,p}$$_{=1}$ associated with our boundary problem. These second and third hypothesis will be justified later on but they can be intuitively understood by realizing that in the limit of a very small tip angle β , the theory must reduce to the case of a linear antenna in which the first and dominating radiation mode is the dipole term with l = 1. Hence, using Maxwell’s equations, we have where the average is taken in the spherical domain of radius a/ sin β . It can be noted that in the spherical harmonic expansion, the terms ‖ m ‖ ≥ 1 are prohibited by recursion relations, but this is not the case here. a$_{λ}$ ≃ 2 πik $^{3}${− E (0) · 1 4 π ∫ Ω$_{0}$ r × (ˆ r × E ∗ $_{λ}$) r $^{2}$d Ω + B (0) · 1 2 iπk ∫ Ω$_{0}$ (ˆ r × E ∗ $_{λ}$) r $^{2}$d Ω + ... } + 1 6 ∑ α,β [ Q $_{λ,α,β}$∂$_{β}$ E$_{α}$ (0) − M $_{λ,α,β}$∂$_{β}$ B$_{α}$ (0)] . (14) a E ν$_{E}$,m ≃ − 2 πik $^{3}$〈 E E ∗ ν$_{E}$,m 〉 . 1 3 a 3 (sin β ) $_{3}$E (0) a M ν$_{M}$ ,m ≃ − 2 πik $^{3}$〈 B M ∗ ν$_{M}$ ,m 〉 . 2 3 a 3 (sin β ) $_{3}$B (0) , (15) 8