### 2007-03

#### Diffraction of light revisited

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