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Numerical Experiments in Scattering by a Dielectric Wedge



Egon Marx


A general discussion of the scattering of plane monochromatic waves by dielectric wedges precedes this paper. Numerical solutions of the problem of scattering by a dielectric wedge of finite cross section, with special emphasis on the divergent behavior of the transverse field components near the edge, are presented here. Integral equations are used because they incorporate the radiation condition and allow for unequal patch distributions close to the edge, so that fields can be computed, for instance, starting at a distance lambda/10,000,000,000, for instance, where lambda is the wavelength. At these small distances from the edge, the SIEs give results that increase rapidly as they approach the edge, in disagreement with the HIEs. Numerical difficulties also occur for small wedge angles. Alternatively, hypersingular equations can be derived for boundary functions that tend to a constant at the edge, but the difficulties with the divergent unknown function are shifted to integration difficulties due to the higher singularity of the kernel. The behavior of the unknown boundary functions near edges affects mainly the fields near the boundary, while the far fields are less sensitive to the details of the approximation, although computed images of a broad dielectric strip on a substrate can present anomalies due to these divergences. The expected behavior of field components near the edge of a wedge in the static limit has also been determined as functions of the angle of incidence of the corresponding scattering problem. The contributions of neighboring patches to the HIEs and to the integrals that give the fields have to be evaluated either by approximating the Hankel functions by their small-argument approximations and doing an analytic integration over the patch or by subdividing the interval further to approximate the kernel better, procedures that are based on the assumption that the unknown function is constant over the patch. The values of the upper limit for the approximation of the Hankel function and the number of subdivisions of an interval have to be chosen carefully. Numerical experiments for TE and TM modes generate fields near the edge of a finite wedge that agree only at times with the expected behavior depending on the direction of approach to the edge. The range of agreement can stretch, for instance, between lambda/1000 and lambda/10,000,000,000. For oblique incidence and arbitrary polarization, the unknown functions for the HIEs decrease with an oscillating behavior near the edge except in the TE and TM cases and diverge as before for the SIEs. The behavior of the transverse field components near the edge remains unpredictable.
Proceedings Title
PIERS Meeting
Conference Dates
January 1, 2003
Conference Location
Unknown, USA
Conference Title
PIERS 2003: Progress in Electromagnetics Research Symposium


dielectric wedge, divergent electromagnetic fields, hypersingluar integral equations, numerical experiments, singular integral equations


Marx, E. (2003), Numerical Experiments in Scattering by a Dielectric Wedge, PIERS Meeting, Unknown, USA (Accessed July 21, 2024)


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Created January 1, 2003, Updated June 2, 2021