The Maxwell Equations
Up until now, we have considered only the direct and inverse obstacle scattering problem for time-harmonic acoustic waves. In the following two chapters, we want to extend these results to obstacle scattering for time-harmonic electromagnetic waves. As in
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The Maxwell Equations
Up until now, we have considered only the direct and inverse obstacle scattering problem for time-harmonic acoustic waves. In the following two chapters, we want to extend these results to obstacle scattering for time-harmonic electromagnetic waves. As in our analysis on acoustic scattering, we begin with an outline of the solution of the direct problem. After a brief discussion of the physical background of electromagnetic wave propagation, we derive the Stratton–Chu representation theorems for solutions to the Maxwell equations in a homogeneous medium. We then introduce the Silver– M¨uller radiation condition, show its connection with the Sommerfeld radiation condition and introduce the electric and magnetic far field patterns. The next section then extends the jump relations and regularity properties of surface potentials from the acoustic to the electromagnetic case both for H¨older spaces and Sobolev spaces. For their appropriate presentation, we find it useful to introduce a weak formulation of the notion of a surface divergence and a surface curl of tangential vector fields. We then proceed to solve the electromagnetic scattering problem for the perfect conductor boundary condition. Our approach differs from the treatment of the Dirichlet problem in acoustic scattering since we start with a formulation requiring H¨older continuous boundary regularity for both the electric and the magnetic field. We then obtain a solution under the weaker regularity assumption of continuity of the electric field up to the boundary and also in Sobolev spaces. For orthonormal expansions of radiating electromagnetic fields and their far field patterns, we need to introduce vector spherical harmonics and vector spherical wave functions as the analogues of the spherical harmonics and spherical wave functions. Here again, we deviate from the route taken for acoustic waves. In particular, in order to avoid lengthy manipulations with special functions, we use the results on the well-posedness of the direct obstacle scattering problem to justify the convergence of the expansions with respect to vector spherical wave functions. The last section of this chapter presents reciprocity relations for electromagnetic waves and completeness results for the far field patterns corresponding to the scattering of electromagnetic plane waves with different incident directions and
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, DOI 10.1007/978-1-4614-4942-3 6, © Springer Science+Business Media New York 2013
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6 The Maxwell Equations
polarizations. For this, and for later use in the analysis of the inverse problem, we need to examine Herglotz wave functions for electromagnetic waves and also the electromagnetic far field operator. For the Maxwell equations, we only need to be concerned with the study of three-dimensional problems since the two-dimensional case can be reduced to the two-dimensional Helmholtz equation. In order to numerically solve the bo
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