content_image/374045.jpg content_image/374046.jpg
neglect the skin depth δ ( ω ) we arrive at the following boundary conditions on the cone: ∂Y$_{ν}$$_{M}$$_{,m}$ ( θ, φ ) /∂ cos θ | S ≃ 0, and Y$_{ν}$$_{E}$$_{,m}$ ( θ, φ ) | S ≃ 0. However, we can make a more realistic approximation and use a finite value for the skin depth in the optical domain, ( δ ≃ 10 nm). In this case, there exist just outside the surface tangential E$_{‖}$ and B$_{‖}$ linked by the relation E$_{‖}$ = √ µ/ϵ B$_{‖}$ × n . In this equation n is inwardly directed normal on the metal surface and √ µ/ϵ can be interpreted as a surface impedance. This relation determines an eigenvalue condition on ν and authorizes an infinite but discrete number of solutions which depend only of the tip angle β . For a value of β on the order of 10 $^{◦}$–15$^{◦}$, we obtain ν$_{E,}$$_{0}$ ≃ 0 . 95 and ν$_{M,}$$_{0}$ ≃ 0 . 96. The ν values and, consequently, the angular profile are not sensitive to variations of the tip angle β unless it exceeds 20 $^{◦}$. 

It is important to note that the boundary conditions cannot be applied inside a sphere of radius a/ (sin β ) because it contains no metal. This so-called aperture zone represents the “terra incognita” near-field zone ( r/λ ≲ 1) of the fiber tip (cf. Fig. 1). The coefficients a$_{ν,m}$ can be approximated by a Taylor series around the tip origin. m = ± 1 are the only authorized terms for linearly polarized light propagating down the fiber. These terms possess the form a E ∼ P$_{eff}$ · E$_{0}$ , a M ∼ M$_{eff}$ · B$_{0}$ which depend on the field E$_{0}$ , B$_{0}$ = ˆ z × E$_{0}$ at the aperture and on the two dipoles: 

<img src='content_image/374045.jpg'>

The normalized angular distribution of the radiated power can then be expressed as 

<img src='content_image/374046.jpg'>

In this formula Re and Im are related to the real and imaginary parts of the generalized harmonic functions. It appears also a characteristic factor 2 between the electric and the magnetic quasi-multipole. It is important to remark that neither the absolute value of the radiated power nor the absolute value of the dipoles are described by our model. This would involve a much more elaborate description of the core/cladding characteristics inside the fiber tip. We can also note that the normalized profile does not depend on the aperture radius. Long wavelength light is blind to subwavelength details, and therefore the aperture size does not constitute a critical parameter of our model. 

Eq. 4 reduces to the Oberm¨ uller-Karrai model when we put ν = 1 (a value close to the real values ν$_{E,}$$_{0}$ and ν$_{M,}$$_{0}$ ). This corresponds to the ideal case of a linear, radiating two-dipole 

