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A Global Method for the Scattering by a Multimode Plane with Arbitrary Primary Sources and Complete Series with Error Functions

1.1. Introduction

In [1], we considered the field scattered by an arbitrary impedance plane in electromagnetism, and 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. In this study, the plane is supposed to be either grounded, that is, a multilayer backed by an impedance plane, or not grounded, that is, a multilayer slab in free space; this will lead us to generalize 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], which presents the particularity to have reflection coefficients that are meromorphic functions. Each one can be then modeled as a rational function with a set of N simple poles {-gj}j=1,..,N, which permits us to assume 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 is lengthy and delicate because of functions in the integral that remain highly oscillating and, most often in literature [3]-[9], analytic expansions are not strictly convergent but asymptotic. Besides, an additional difficulty comes from that, and in multimode case, we have to take into account that the constants gj 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 gj, we are led to transform them to derive a more efficient integral representation, which is able to take account of active modes. Among other specificities, the definition of a parameter e, attached to each pole, is then particularly important to permit complete exact and asymptotic series with error functions. These series allow us to exhibit guiding waves terms near and far from the sources above the multilayer, generalizing [1] and refining [2].

Otherwise, our approach, as in [1], uses a new representation of potentials for the incident field, which possesses the originality to directly 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.

This chapter is organized as follows. In section 1.2, we give a discussion on the representation of the field with potentials, on the boundary conditions and on the positions of gj in the complex plane when metamaterials can be present. Next, we give a global expression of potentials attached to the fields radiated by arbitrary bounded sources in free space in section 1.3, and above the multilayer in section 1.4, which we develop and expand for arbitrarily oriented dipoles in section 1.5. In sections 1.6 and 1.7, we then detail a compact expression of the special function involved in the potentials attached to each mode, intimately depending on a parameter e that is necessary to correctly take account of active modes. The definition of e will be useful for the development of exact (section 1.6) and asymptotic (section 1.7) expansions with error functions for arbitrary cases, allowing in particular a general analysis of guided waves in section 1.8, including backward waves, near and far from the sources.

1.2. Potentials, reflection coefficients and multimode boundary conditions

1.2.1. Fields and potentials

We consider the scattering by an imperfectly reflective plane when it is illuminated by the field radiated by a bounded primary source, which is composed of arbitrary electrical and magnetic currents J and M (see Figure 1.1). In the space of points r with Cartesian coordinates (x, y, z), this plane is defined by z = 0. A harmonic time dependence ei?t, from now on assumed, is suppressed throughout. The constants e0 and µ0 are, respectively, the permittivity and the permeability of the free space above the plane, and k0 = ?(µ0?0)1/2 is its wavenumber. Each component of the scattered field is assumed to be regular in the domain z > 0, and O(e-?|r|) with ? > 0 as |r| 8 when | arg(ik0)| < p/2 (note: no loss is a limit case).

The electric field E and the magnetic field H above the multilayer, following Harrington [23, p. 131] (see also Jones [3, p. 19]), can be written with two scalar potentials E and H, as follows:

where the Helmhotz equations and are verified outside the sources, . Thereafter, we denote (Ei, Hi) and (Es,Hs) the potentials corresponding, respectively, to the incident field (incoming wave) (Ei, Hi) and the scattered field (outgoing wave) (Es, Hs), and we write (1.1) in the compact form:

Figure 1.1 Geometry: sources (J, M) and observation point above the plane z = 0

1.2.2. Multimode boundary conditions for a multilayer backed by an impedance plane

Let us consider a multilayer plane composed of uniform isotropic (or z-axial anisotropic) layers. Any plane wave (Ei, Hi), incident at angle ß with the normal , is then scattered as a reflected plane wave (Es, Hs) that satisfies (i.e. in TM polarization), and (i.e. in TE polarization) [4], [5], [21] (see details in appendix A). If the multilayer is backed by a constant impedance plane, the reflection coefficients Re,e and Rh,h are meromorphic functions of the variable cos ß, which we can model as rational functions [2] with simple poles, following:

for which we have the basic equalities (without superscripts e and h):

where N = 1, and for N = 1. The constants (±1)N refer to limit values when .

The are constants attached to complex modes with Im, passive when or active when , and ordered such that , while, as considered in [26]-[28], we assume that:

(note: this restriction on Ne(,h) will be removed in the more general case of extended boundary conditions). Considering plane waves representation of fields (see appendix A), we can then write a multimode boundary conditions at z = 0+ [2]:

From the symmetry at normal incidence, the condition Rh,h(0) = -Re,e(0) must apply, which leads us to write:

and implies that Rh,h(p) = -Re,e(p). The condition (1.7) has crucial importance to avoid non-physical behaviors of fields derived from potentials, as examined further in this paper. Besides, the reader will notice that (1.7) implies when Ne(,h) = 1, as well known for monomode (impedance) boundaries conditions [1]. The numbers Ne(,h) correspond to truncated infinite products, where the less significant ge(,h) have been neglected, while some ge(,h) have to be modified so that Rh,h(0) = -Re,e(0) remains.

Considering (1.1), we can use:

in (1.6), and we are led to search scattered potentials Es and Hs, satisfying the Helmholtz equation as z > 0, regular and exponentially vanishing as z 08 when | arg(ik0)| < p/2, that verify as z > 0:

where ?i and Hi potentials are attached to radiation of arbitrary primary sources.

1.2.3. Extended multimode boundary conditions

More generally, we can consider an extended form when we want to include the case of a multilayer slab in free space, which is composed of isotropic [4]-[5] (or z-axial anisotropic [21]) layers. The reflection coefficients Re,e and Rh,h remain meromorphic...