### 2007-03

by Kunik, M., Skrzypacz, P..

**Series:** 2007-03, Preprints

- MSC:
- 78A45 Diffraction, scattering
- 42A50 Conjugate functions, conjugate series, singular integrals
- 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and ~Wiener-Hopf type)

**Abstract:**

The diffraction of light is considered for a plane screen with an open

infinite slit by solving the Maxwell-Helmholtz system

in the upper half space with the Fourier method.

The corresponding solution

is given explicitely in terms of the Fourier-transformed

distributional boundary fields. The method deals with all components

of the electromagnetic field and leads to a

modification of Sommerfeld's scalar diffraction

theory. Using this approach we can represent each vectorial solution

satisfying an appropriate energy condition

by its boundary fields in the Sobolev spaces $H^{\pm 1/2}$.

This representation includes also solutions with

smooth boundary fields, which are not covered by Sommerfeld's solutions

of boundary integral equations (or integro-differential equations)

with Hankel kernels. On the other hand we show that

Sommerfeld's theory using a boundary integral equation for the so called

B-polarisation leads in general to vectorial solutions

which violate a necessary energy condition.

For the physically admissible regular solutions

in the upper half space we derive the necessary and sufficient

energy conditions in terms of the Fourier transformed

distributional boundary fields.

**Keywords:**

Maxwell-Helmholtz equations, Fourier analysis, Sobolev spaces, energy conditions, singular boundary fields, Hankel functions

**This paper was published in:**

Mathematical Methods in the Applied Sciences