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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 

<img src='content_image/337783.jpg'>

and 

<img src='content_image/337784.jpg'>

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: 

<img src='content_image/337785.jpg'>

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 

<img src='content_image/337786.jpg'>

