content_image/330876.jpg content_image/330877.jpg content_image/330878.jpg
optical axis when the curvature radius decreases, i. e. when boundary effects invade space. We will see below that these general trends are confirmed by theory. 

## III. THE QUASI-MULTIPOLE METHOD 

The method of quasi-multipoles developed here is the result of a convergence of three domains in electromagnetism. It lies at the intersection of i) the metallic wave guide theory for guides with variable cross sections [19–21], ii) the spherical multipole expansion method developed by Bouwkamp[16, 22] for electromagnetic fields applicable to localized source distributions, and iii) the Hall[23] equilibrium solutions for conductors with conical geometry. For instance, the electromagnetic field radiated by a conical antenna or a conical hole can be described using this formalism. In order to simplify the discussion, we will use the same notations for spherical multipoles as those presented by Jackson[16]. Let us consider a source-free region of space with permittivities ϵ and µ . A conical surface with half-angle β separates this space into an “oustide” region and an “inside” region. If β exceeds π/ 2, the external and the internal regions are mutually exchanged. For simplicity, these zones are considered as decoupled and totally independent. This is possible in the case of a perfectly conducting metallic surface in which tunneling of the field through the surface is forbidden. From now on, we will only consider the external problem. Maxwell’s equations can be written (assuming an e − iωt time dependence): 

<img src='content_image/330876.jpg'>

Making use of the identity ∇ $^{2}$( r .A ) = r · ( ∇ $^{2}$A ) + 2 ∇ · A , these equations reduce to 

<img src='content_image/330877.jpg'>

By writing the Laplace operator in spherical coordinates, and separating the angular and radial variables, a typical solution of Eq. 2 can be expressed as follows: 

<img src='content_image/330878.jpg'>

Both Hankel functions h ± ν appear in this expansion ( the ± signs refer to outgoing and incom- ing waves, respectively). They satisfy the spherical Bessel equations for the radial variable 

