Apertures in shielding enclosures are an important issue for determining shielding
efficiencies. Various mathematical procedures and theories were employed to describe
the coupling between the regions connected via an aperture in a well conducting plane.
Bethe's theory describes the coupling via the equivalent problem of field excited dipole
moments at the location of the aperture. This approach neglects the reaction of the dipole
moments on the exciting field and therefore violates energy conservation.
This work emphasizes an analytical approach for coupling between half-spaces through
small apertures, inspired by the so called

In earlier works,

Geometry of the problem with incoming plane wave and induced magnetic moment

At the aperture the fields must fulfill certain boundary conditions. It is
useful to split the total fields existing in the considered volume such that

The equations Eq. (

The conditions described by Eq. (

The shortcut fields are the superposition of the incoming and reflected
fields in region I only whereas the scattered fields are radiated by the
equivalent sources, so with basic electromagnetic theory it can be written
for region I

The vector potentials

Interpreting

To evaluate the integrals in Eq. (

The basic idea of analytical regularization is the following. The Green's
function representing the interaction between source and field while
fulfilling the respective boundary conditions is split into two parts, as

The singular part

For the half space the Green's function can be found by applying the mirror procedure. The dyadic Green's function for half space is the superposition of the dyadic Green's function for free space of the original and mirrored source.

Mirror procedure for electric and magnetic sources.

The single primed vector

Coordinate systems and aperture.

For the Green's function of free space near a small scatterer regularization
can be done by expanding the exponential term into a Taylor series with

By inserting Eq. (

To calculate the induced renormalized magnetic moment
Eqs. (

The indices

In Eq. (

Power ratio over normalized wave number; on logarithmic scale a ratio above

Here

To demonstrate the improved coupling theory a specific example will be
calculated. The focus lies on power calculation and flow. To investigate
power conservation, the radiated power of circular aperture is considered. In
this case the classical magnetic polarizability is

Here

If the incoming wave is a plane wave too, the ratio of incoming and radiated
power can be calculated as

The relation Eq. (

The difference is also evident in direct analytical calculation of the power
density around the aperture. Figure

A understandable generalization of a simple model for aperture coupling was derived which features a physical interpretation of aperture radiation. As an example the simplest case of aperture coupling, i.e. an aperture in infinite plane, was used. Calculations of the radiated power of the renormalized moments show that in this way power conservation is established for the model. One has to note, that in the case of radiation in half-space, power conservation is violated for high frequencies, where the dipole model isn't valid at all. But for radiation in cavities and waveguides, the radiation resistance, which is represented by the regular part of the Green's function, can be dominant even for lower frequencies. An application of the presented formalism gives the possibility of forward and backward aperture scattering in different environments. This will be attended in future works.Edited by: M. Chandra Reviewed by: two anonymous referees