07-03

Diffraction of light revisited

by Kunik, M.; Skrzypacz, P.

 

Preprint series: 07-03, Preprints

The paper is published: Mathematical Methods in the Applied Sciences

MSC:
78A45 Diffraction, scattering, See also {34E20 for WKB methods}
42A50 Conjugate functions, conjugate series, singular integrals
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type), See also {47B35}

 

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

Notes: Dedicated to Professor Wendland on the occasion of his 70th birthday


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Letzte Änderung: 01.03.2018 - Ansprechpartner:

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