r and the quasi-harmonic functions Y$_{ν,m}$ ∝ P m ν (cos θ ) · e imφ defined in the literature[23–25]. The latter represent the generalization to a conical geometry[23] of the well-known spherical harmonics Y$_{l,m}$ used in spherical eigenvalue problems. However, an important modification with respect to spherical harmonics is the presence of a non-integer parameter ν which replaces the integer l and which depends on the boundary conditions on the cone. The possible values of ν in Eq. 3 follow an infinite sequence which is a function of β : ν$_{p}$ = ν$_{p}$ ( β ) ( p integer). The exact relation depends on the boundary conditions on the cone. In analogy with the spherical case and with wave guide theory, we now decompose the electromagnetic field obeying a specific boundary condition on the cone into two parts: a magnetic quasi-multipole part (M) of the transverse electric field (TE), and an electric quasi- multipole part (E) of the transverse magnetic field (TM). These two parts are characterized by components of order ( ν, m ) satisfying In this article we consider only outgoing radiation propagating in vacuum . For reasons of generality, we must use two types of indices ν$_{E}$ ( β ) and ν$_{M}$ ( β ) because of a difference between the boundary conditions for TE and TM waves. For example, if the conductivity of the metallic cone is infinite (the perfect conductor), we must have the boundary conditions: These expressions fix the authorized ν values as a function of β . Hence, an electromagnetic field satisfying specific boundary conditions can be written as an expansion in TE and TM waves, and we have and r .E E, ± ν,m = − ( µ ϵ ) 1 4 √ ν ( ν +1) k .h ( ± ) ν ( kr √ ϵµ ) .Y$_{ν,m}$ ( θ, φ ) r .H M, ± ν,m = ( ϵ µ ) 1 4 √ ν ( ν +1) k .h ( ± ) ν ( kr √ ϵµ ) .Y$_{ν,m}$ ( θ, φ ) (4) r .H E, ± ν,m = 0 , r .E M, ± ν,m = 0 (5) Y$_{ν}$$_{E}$$_{,m}$ ( θ, φ ) | θ = π − β = 0 ∂Y$_{ν}$$_{M}$$_{,m}$ ( θ, φ ) ∂θ ∣ θ = π − β = 0 , (6) H = ∑ ν$_{E}$,m a E ν$_{E}$$_{,m}$.h (+) ν$_{E}$ ( kr ) . L Y$_{ν}$$_{E}$$_{,m}$ ( θ, φ ) √ ν$_{E}$ ( ν$_{E}$ +1) − i k ∑ ν$_{M}$ ,m a M ν$_{M}$ $_{,m}$. ∇ × ( h (+) ν$_{M}$ ( kr ) . L Y$_{ν}$$_{M}$$_{,m}$ ( θ, φ ) √ ν$_{M}$ ( ν$_{M}$ +1) ) (7) 6