Mathematics

A global method for the scattering by a multimode plane with arbitrary primary sources and complete series using error functions

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Authors: J.M.L. Bernard

In a recent paper [1], we considered the field scattered by an arbitrary impedance plane in electromagnetism. We here exploit this formalism to analyze the scattering by a structure composed of several homogeneous planar layers, with isotropy or uniaxial anisotropy, illuminated by arbitrary bounded sources above the multilayer, when it is grounded (multilayer on an impedance plane) or not (multilayer slab in free space), which generalizes our previous approach for a multilayer given in [2]. The field scattered of such structures is usually given by its plane wave expansion (Fourier representation) [3]-[6] with reflection coefficients that are meromorphic functions. Each one, when modeled as a rational function with a set of N simple poles -g_j leads to a multimode boundary condition of order N [2]. The Fourier expansion is well-adapted in far field or for plane wave illuminations, but is not suitable for an analysis at any distance or for complex incident waves. Even when double Fourier integrals are reduced to single Fourier-Bessel integrals, calculation remains lengthy and delicate because of functions in the integral that remains highly oscillating and, most often in literature [3]-[7], analytic expansions are not strictly convergent but asymptotic. Besides, an additional difficulty comes from that, in multimode case, we have to take account that the constants g_j can have real parts of any sign, which signifies that passive but also active modes are present, even if the complete system is strictly passive. In this frame, after expanding potentials into a combination of Fourier-Bessel integrals depending on each g_j, we are led to transform them and to derive an original integral representation, able to take account of active modes from the definition of a parameter % , which permits novel exact and asymptotic series with error functions. These series allow to exhibit guiding waves terms near and far from the sources, generalizing and refining [1]. Otherwise, our approach, as in [1], uses a new representation of potentials for the incident field, which possesses the originality to consider arbitrarily oriented electric and magnetic primary currents sources. Thus, we have no more to solve separately the problem for vertical or horizontal dipolar source as commonly done in the literature for passive impedance planes [7]-[14], isotropic or uniaxial slabs [15]-[17], or multilayers [3]-[6],[18]-[22]. In practice, the analytic method so developed can be applied in whole generality to various problems, in particular for the determination of coupling between antennas above an imperfectly reflective plane, or for the calculus of Green's functions for planar lines printed on a multilayer.