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

In a recent paper [1], we considered the field scattered by an arbitrary impedance plane inelectromagnetism. We here exploit this formalism to analyze the scattering by a structurecomposed 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), whichgeneralizes 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 meromorphicfunctions. 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 isnot suitable for an analysis at any distance or for complex incident waves. Even whendouble Fourier integrals are reduced to single Fourier-Bessel integrals, calculationremains lengthy and delicate because of functions in the integral that remains highlyoscillating and, most often in literature [3]-[7], analytic expansions are not strictlyconvergent but asymptotic. Besides, an additional difficulty comes from that, inmultimode case, we have to take account that the constants g_j can have real parts of anysign, which signifies that passive but also active modes are present, even if the completesystem is strictly passive.In this frame, after expanding potentials into a combination of Fourier-Bessel integralsdepending on each g_j, we are led to transform them and to derive an original integralrepresentation, 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 allowto 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 ...