High-frequency asymptotic solutions benchmarking skew incidence diffraction by anisotropic impedance half and full planes

Authors


Abstract

[1] The diffraction of plane waves obliquely incident on the edge of anisotropic impedance half and full planes is investigated. Homogeneous anisotropic impedance boundary conditions are defined on both faces of the canonical structures under study, the principal anisotropy axes being parallel and perpendicular to the edge. Rigorous integral representations for the longitudinal field components are derived by applying the Sommerfeld-Maliuzhinets method for a set of specific electrical configurations. Explicit uniform asymptotic expressions for the fields are given in the format of the Uniform Geometrical Theory of Diffraction (UTD). Although obtained for a specific class of geometrical and electrical configurations, these high-frequency solutions provide a contribution for investigating the effects of material anisotropy on edge diffraction, as they represent a new set of reference cases for either developing perturbative solutions or testing numerical and approximate analytical methods of more general validity. Furthermore, these solutions extend the applicability of the Sommerfeld-Maliuzhinets technique and represent a step forward to the solution of more general wedge canonical problems.

1. Introduction

[2] In the framework of high-frequency scattering and radiation phenomena, the diffraction of arbitrarily polarized plane waves obliquely incident on the edge of impedance half and full planes represents an important canonical problem. In the case of isotropic impedance boundary conditions (IBCs), rigorous integral solutions and related uniform high-frequency field expressions are available [Senior and Volakis, 1995]. Conversely, when anisotropic IBCs are introduced the complexity of the scattering problem increases, and the most general problem relevant to oblique incidence and arbitrary surface impedance tensors is still unsolved. Nevertheless, present-day applications (e.g., frequency/polarization selective surfaces, radar targets realized by new absorbing materials) explicitly demand for a characterization of the scattering by electrical and/or geometrical discontinuities in material surfaces which may be modeled by means of anisotropic IBCs.

[3] Solutions to the scattering from discontinuities in anisotropic impedance surfaces available in the literature are based on either numerical approaches or analytical techniques. Analytical solutions have been obtained by alternatively applying two basic techniques, namely, the Wiener-Hopf method [Daniele, 1984] and the Sommerfeld-Maliuzhinets method [Maliuzhinets, 1958; Osipov and Norris, 1999]. Despite of the numerous and continuous variations/extensions of both the above analytical techniques, it is worth noting that exact closed form solutions for the scattering from anisotropic impedance wedges have been derived only for specific electrical and geometrical configurations. For the sake of brevity, an overview of the previous work will be limited to the most recent papers.

[4] The difficulties arising in solving the wedge scattering problem by applying the Sommerfeld-Maliuzhinets technique are due to the fact that the angular spectra for the field components parallel to the edge appear in a system of coupled functional equations. Generally speaking, an exact spectral solution can be derived whenever a couple of suitable field components can be identified, giving rise to decoupled boundary conditions on both wedge faces. These field components are obtained as linear combinations of the electric and magnetic field components parallel to the edge, and their spatial derivatives. By following this approach, rigorous analytical solutions to the scattering by anisotropic impedance wedges or half planes with specific impedance tensors have been presented in [Manara et al., 2000; Nepa et al., 2001; Lyalinov and Zhu, 2003; Manara et al., 2004]. Among alternative approaches, it is worth mentioning the procedure proposed by Senior and Legault [2000], Senior et al. [2001], and more recently applied by Legault and Senior [2002], where the system of coupled functional equations obtained by applying the Somerfeld-Maliuzhinets method is reduced to a second-order functional equation for a linear combination of the angular spectra of the field components parallel to the edge. Lyalinov and Zhu [2005] reduced the solution of the second-order functional equation to the numerical evaluation of an equivalent Fredholm integral equation of the second kind. A simple approach to solve the second-order difference functional equation was presented in [Senior et al., 2001; Senior and Topsakal, 2002], where the original problem is reduced to an inhomogeneous nonsingular integral equation that can be solved by using only a few terms in a Taylor series expansion of the unknown. A similar hybrid analytical-numerical approach has been applied by Osipov and Senior [2005] directly to the first order coupled functional equations to analyze the scattering from an isotropic impedance wedge. Other rigorous analytical solutions valid for specific classes of impedance tensors have been recently derived by resorting to the Wiener-Hopf method [Sendag and Serbest, 2001; Büyükaksoy et al., 1996].

[5] The main objective of this paper is to provide a rigorous analytical solution for the fields scattered by a new class of anisotropic impedance half- and full-plane configurations, illuminated at oblique incidence by an arbitrarily polarized plane wave. Anisotropic IBCs hold on the half- and full-plane faces, the principal anisotropy axes being parallel and perpendicular to the diffracting edge. Although limited to specific geometrical and electric configurations, these analytical solutions do represent important benchmarks in the analysis of the scattering from composite surfaces of finite extent. Indeed, they provide a set of useful reference cases that can be used to check the accuracy of numerical, hybrid or approximate analytical methods, valid for more general configurations. Also, they can be chosen as the starting point for developing perturbative as well as heuristic solutions. Furthermore, they can suggest extensions of the adopted analytical method toward the solution of more general and complex configurations.

[6] The paper is organized as follows. The formulation of the problem is provided in section 2. Exact integral representations for the electric and magnetic field longitudinal components are derived by the Sommerfeld-Maliuzhinets method in section 3. In section 4, the integral representations are asymptotically evaluated to obtain uniform high-frequency expressions in the UTD format [Kouyoumjian and Pathak, 1974]. Finally, samples of numerical results are presented in section 5.

2. Formulation of the Canonical Problems

[7] A set of canonical plane-wave diffraction problems for half- and full-plane configurations with anisotropic impedance faces are considered in the most general case of oblique incidence. The geometry for both the full- and the half-plane problems is sketched in Figure 1. A standard cylindrical reference system is adopted, with the z-axis aligned with the diffracting edge. The exterior wedge angle is , with n =1 for the full-plane configuration (Figure 1a) and n = 2 for the half-plane configuration (Figure 1b). The observation point is at P ≡ (ρ, ϕ, z). The edge is illuminated by an arbitrarily polarized plane wave impinging from a direction which is identified by the two angles β′ and ϕ′. The longitudinal components of the incident field can be expressed as

display math

where k and ζ denote the free-space wave number and intrinsic impedance, respectively. In (1), kt = ksinβ′ represents the transverse component of the wave vector with respect to the edge. A time harmonic dependence exp(jωt) has been assumed and suppressed.

Figure 1.

Geometry for the canonical problems and corresponding cylindrical coordinate system: (a) full-plane problem (n = 1); (b) half-plane problem (n = 2).

[8] Two different anisotropic IBCs hold on the ϕ = 0 and ϕ = faces. Under the hypothesis that the principal anisotropy directions are parallel and perpendicular to the edge, the tensor impedances normalized with respect to ζ read:

display math

Consequently, the IBCs are expressed as follows:

display math

where ɛ0 = −1 and ɛn = +1.

[9] Since the electric properties of the wedge are independent of z, the scattered field exhibits the same exp(−jkzcosβ′) dependence on z as the incident field, which will be suppressed in the following. Moreover, all field components transverse to the z-axis can be expressed in terms of [Ez, ζHz]. The total field [Ez, ζHz] must satisfy the Helmholtz equation (∇t2 + kt2)[Ez, ζHz] = 0, together with the radiation and the edge conditions. The IBCs in (2) can be expressed in terms of the longitudinal field components [Ez, ζHz] as a set of coupled differential equations:

display math
display math

By adopting the Sommerfeld-Maliuzhinets approach, [Ez, ζHz] are expressed in terms of the following spectral representations [Maliuzhinets, 1958]:

display math

where γ = γ+ + γ is the two-fold Sommerfeld integration path. In order to satisfy the radiation condition, the spectral functions appearing in (4) must be regular in the strip ∣Re{α}∣ ≤ /2, except for a first order pole at α = ϕ′ − /2 accounting for the incident field. In particular, when considering the incident field as in (1), the residue of [se(α), sh(α)] at the above pole must be equal to [ez, hz]. Furthermore, the edge condition requires that [se(α), sh(α)] = O(1) for ∣Im(α)∣ → ∞.

[10] By substituting the spectral representations (4) into (3a) and (3b), the following functional equations are obtained:

display math
display math

The functional equations are coupled, and exact solutions have been derived only for specific electrical configurations. The analytical solution procedure usually starts from identifying a pair of new spectral functions, namely, t1(α) and t2(α), defined as an appropriate combination of se(α) and sh(α). As far as the transformation is concerned, it is required that: (i) the functional equations in (5a) and (5b) must reduce to a set of decoupled first-order functional equations when expressed in terms of t1(α) and t2(α); (ii) the functional equations for t1(α) and t2(α) must exhibit a form similar to that of the functional equations for the normal incidence case [Maliuzhinets, 1958]. The last property guarantees that the spectral solutions only contain the standard Maliuzhinets function, besides simple ratios of trigometric functions.

[11] The aim of this paper is to demonstrate that the functional equations in (5a) and (5b) can be solved for the three specific configurations listed in Table 1. These configurations are characterized by arbitrary values of the elements of the impedance tensor of the ϕ = 0 face, (Z0)ρ and (Z0)z, provided the relationships reported in the third column of Table 1 hold for the impedance tensor of the ϕ = face, (Zn)ρ and (Zn)z.

Table 1. Half-Plane and Full-Plane Electrical Configurations Solved in This Paper
CaseWedge ConfigurationTensor Impedance Relationships
ahalf plane (n = 2)(Zn)z = −(Z0)z, (Zn)ρ = −(Z0)ρ
bfull plane (n = 1)(Zn)z = −1/(Z0)ρ, (Zn)ρ = −1/(Z0)z
chalf plane (n = 2)(Zn)z = 1/(Z0)ρ, (Zn)ρ = 1/(Z0)z

[12] Indeed, for the three cases listed, a transformation that provides the decoupling of the functional equations in (5a) and (5b) is found. Thus, auxiliary spectral functions t1(α) and t2(α) are defined by applying the following transformation to the spectra se(α) and sh(α):

display math
display math

Once the spectral functions t1(α) and t2(α) are known, the spectra for the longitudinal field components are recovered by simply inverting the previous transformation:

display math
display math

where

display math

with

display math

We note that 0 < Re{ϑ0±} < π/2 for passive surfaces. By substituting (7) into (5) a set of decoupled functional equations is obtained for the configurations listed in Table 1. Explicit solutions for the spectral functions will be given in the next section, and the asymptotic evaluation of the corresponding rigorous integral representations for the fields will be provided in section 4; there, it will be shown that the complex angles ϑ0± defined in (9) denote the generalized Brewster's angles associated with the ϕ = 0 face of the wedge. The generalized Brewster's angles associated with the ϕ = face, ϑn±, are defined by analogous expressions:

display math

It is worth noting that (9) and (10) are valid for the most general wedge configuration, since they define the generalized Brewster's angles for a wedge with an arbitrary exterior angle , illuminated at oblique incidence (β′ ≠ π/2), for any value of the surface impedance tensor [Nepa et al., 2007].

[13] It is worth noting that Senior [1978] emphasized that for the cases listed in Table 1 the scattering canonical problem can be scalarized, in the sense that the IBCs can be decoupled. Also, Hurd and Luneburg [1985] noted that for the wedge configuration corresponding to case (c) in Table 1 the scattering problem becomes equivalent to that of a half plane with two different but isotropic IBCs on both faces, whose solution was known at that time. Nevertheless, to the best of our knowledge, the above authors have not presented explicit solutions of the canonical problems that are listed in Table 1.

[14] Finally, it is worth noting that the passivity condition on the impedance surfaces, namely, Re{(Z0,n)ρ,z} ≥ 0, requires that the impedance faces must be purely reactive when considering cases (a) and (b) in Table 1. Nevertheless, the corresponding analytical solutions are still useful as they can be used to test numerical or heuristic solutions valid for more general electrical configurations. On the other hand, the above constraint can be removed when considering the solution relevant to case (c) in Table 1, in the sense that arbitrary passive surfaces can be accounted for.

3. Rigorous Spectral Solutions

[15] The functional equations for the auxiliary spectral functions t1(α) and t2(α) are determined in the following. It is done for the three specific canonical problems listed in Table 1, noting that cases (b) and (c) can be treated at the same time.

display math

[16] The passivity of the half-plane faces directly implies that Re{(Z0,n)ρ,z} = 0, i.e., they must be purely reactive surfaces. It follows that the complex angles ϑ0± and ϑn± defined in (9) and (10) are purely imaginary and ϑn± = −ϑ0. Moreover, the IBCs in (3a) and (3b) become identical.

[17] The transformation defined in (6) produces a set of decoupled functional equations for [t1(α), t2(α)], namely:

display math

By accounting for the edge condition, the solution can be written as:

display math
display math

where

display math

with n = 2. In (12)A1,2, c1,2, c1,2′, c1,2″ denote constant coefficients that have to be determined. In particular, the coefficients A1,2 accounting for the incident field can be determined from (6) by considering that the residue of σ(α) at the pole associated with the incident field, α = ϕ′ − /2, is equal to unity:

display math
display math

[18] We note that we are allowed to introduce the terms containing the unknown constants c1,2, c1,2′, c1,2″, as the edge condition requiring that [se(α), sh(α)] = O(1) for ∣Im(α)∣ → ∞ is met. The constants are determined by solving a system of linear equations, which is obtained by imposing the cancellation of some nonphysical poles at αp = −ϑ0±, ϑ0± + π, ϑ0±π introduced in se,h(α) by the function Δ(α + π) and lying in the region ∣Re{α}∣ ≤ π (see also Appendix A). In particular, from (7a) the following equations are obtained:

display math

It can be shown that (15) also guarantees the cancellation of the same poles in sh(α) (Appendix A). Finally, the function Δ(α + π) also introduces the poles α0sw = −2π − ϑ0± and αnsw = 2π − ϑ0± = 2π + ϑn that are associated with the surface waves potentially propagating along the ϕ = 0 and ϕ = 2π face, respectively.

display math

and

display math

When considering case (b), the angles ϑ0± and ϑn± result to be purely imaginary (purely reactive surfaces), with ϑn± = −ϑ0. Conversely, in case (c) the condition Re{(Z0)ρ,z} ≥ 0 is sufficient to guarantee that both faces are passive. Consequently, the complex angles ϑ0± and ϑn± can exhibit a non vanishing real part. Moreover, from (9) and (10) it results ϑn± = ϑ0±.

[19] Again, by applying the transformation given in (6), the functional equations in (5) reduce to the following set of decoupled functional equations for [t1(α), t2(α)]:

display math
display math

where Δ(α) = sin2β′ (sin α − sin ϑ0±)(sin α − ϑ0). Let us define

display math

where ψ/2(α) is the Maliuzhinets special function defined in [Maliuzhinets, 1958]. By considering that ψ(α, ϑ) = O (exp[Im{α}/(2n)]) when ∣Im{α}∣ → ∞ and accounting for the edge condition, the following solution is obtained:

display math
display math

where

display math
display math

with c1′ = c2′ = 0 if n = 1 (case (b)). The values for A1,2 are still given by (14), and again we are allowed to introduce the terms containing the unknown constants c1,2, c1,2′, c1,2″, as the edge condition is met. As before, a system of linear equations must be solved to obtain the cancellation of some nonphysical poles introduced in se,h(α) by the function Δ(α + /2) and lying in the strip ∣Re{α}∣ ≤ /2 (see Appendix A for a detailed analysis). These equations read:

display math

In case (c), equation (20) must be solved for the four nonphysical poles at αp = −ϑ0±, ϑ0±π, so determining the constants c1,2, c1,2′. Moreover, the nonphysical poles at αp = π + ϑ0±, 2π − ϑ0± are cancelled out by the zeros of the function ψ(α + /2, ϑn±) = ψ(α + /2, ϑ0±). In case (b), the two unknown constants c1,2 are determined by solving (20) for the nonphysical poles at αp = ϑ0±π/2. Again, the nonphysical poles at αp = −ϑ0± + π/2, ϑ0± + 3π/2 are cancelled out by the zeros of the function ψ(α + /2, ϑn±) = ψ(α + /2, −ϑ0±). Finally, the poles introduced by the function Δ(α + /2) that are located at α0sw = −π/2 − ϑ0± are associated with the surface waves that can propagate on the ϕ = 0 face. Conversely, the poles associated with the surface wave that can propagate along the ϕ = face are those located at αnsw = π + /2 + ϑn±, and they are introduced by the function ψ(α + /2, ϑn±).

4. High-Frequency Asymptotic Solutions

[20] Once the longitudinal components of the total field are expressed in terms of Sommerfeld integrals, uniform asymptotic expressions in the UTD format can be derived by applying the Cauchy residue theorem. In particular, the original integral representations for the total fields in (4) are expressed as the contribution of the residues of the poles which can be captured in the contour deformation process, namely, the Geometrical Optics (GO) poles and the surface wave poles, and that of two integrals defined along the Steepest Descent Paths (SDPs) through the saddle points at ±π (the latter integrals account for the diffraction phenomenon originating at the edge of the wedge):

display math

[21] An asymptotic evaluation procedure of the diffraction integrals in (21) is described in Kouyoumjian et al. [1996], providing the proper discontinuity compensation also when the poles cross the SDPs away from saddle points, as for instance in the case of surface wave poles. The procedure is applied to derive uniform high-frequency expressions for the diffracted fields. As far as the GO fields are concerned, the residues of the geometrical poles account for the incident field (αi = ϕ′ − /2), the field reflected from the ϕ = 0 face (αi = −ϕ′ − /2), and the field reflected from the ϕ = face (αi = −ϕ′ + 3/2). Thus, the complete expression of the GO fields can be written as:

display math

where U(·) is the Heaviside unit step function and the reflection matrix entries R0,nee, R0,neh, R0,nhe, R0,nhh are given in Appendix B.

[22] The amplitude of the surface wave contributions can be easily determined by evaluating the residues of the poles that are introduced by Δ(α + /2) (cases (a), (b) and (c)) and ψ(α + /2, ϑn±) (cases (b) and (c)), as explained in section 3 (see also Appendix A). Surface wave propagation along the ϕ = 0 face can take place when the following existence condition is met: {gd[Im(ϑ0±)] − Re(ϑ0±)} > 0, where gd(·) denotes the Gudermann function [Kouyoumjian et al., 1996]. Likewise, the surface wave existence condition for the ϕ = face is: {gd[Im(ϑn±)] − Re(ϑn±)} > 0. We note that for each face just one out of the two surface waves is allowed to propagate at the most, as the existence condition cannot be satisfied for both the Brewster's angles at the same time. Hence, the surface wave field contributions can be expressed as:

display math

where ϕ0sw = gd[Im(ϑ0±)] − Re(ϑ0±), ϕnsw = −{gd[Im(ϑn±)] − Re(ϑn±)}, and the complete expressions of the surface wave amplitude A0,ne, A0,nh are given in Appendix B.

[23] Finally, uniform high-frequency expressions for the longitudinal components of the diffracted fields are obtained by asymptotically evaluating the integrals along the SDP±π in (21):

display math

In (24), F(·) is the standard UTD transition function [Kouyoumjian and Pathak, 1974], extended to complex argument as in Kouyoumjian et al. [1996], and the summation includes all poles lying in the strip ∣Re(α)∣ < 2π, namely, the GO poles and the surface wave poles. Note that the residues appearing in (24) are the same as those in (21), and they coincide with the reflection coefficients and the surface wave amplitudes given in Appendix B. We also note that in the far-field region ( sin β′ ≫ 1), the following relationships hold: Eβd = −Ezd/sinβ′, Eϕd = ζHzd/sinβ′. Moreover, the longitudinal components of the incident field [ez, hz] can be expressed in terms of Eβi and Eϕi as follows [Kouyoumjian and Pathak, 1974]: ez = Eβisinβ′ and hz = −Eϕ′isinβ′. Hence, once Ezd and ζHzd are calculated from (24), the diffracted field components in the standard UTD reference frame can be easily derived through the above relationships.

5. Numerical Results

[24] Samples of numerical results, which are obtained through the evaluation of the asymptotic high-frequency expressions given in section 4, are presented here in order to demonstrate the correctness of the implemented analytical solutions. Note that the diffracted field is discontinuous at both the GO shadow boundaries and the surface wave shadow boundaries, so that the total field results to be smooth and continuous. Figure 2 refers to case (a), where the half plane (n = 2) surface impedance tensors satisfy the relationships (Zn)z = −(Z0)z, (Zn)ρ = −(Z0)ρ. A TM polarized plane wave (Eβi = 1, Eϕ′ = 0) impinges on the edge from the direction ϕ′ = π/4, β′ = π/6. The observation point is at a normalized distance ktρ = 5 from the edge. The normalized surface impedance values on the ϕ = 0 face are (Z0)z = , (Z0)ρ = j(0.8 −δ), with δ = 0, 0.2, 0.4, 0.6, 0.8. This particular choice allows comparisons with two limit cases that have been solved in Manara et al. [2000], namely, the cases of a half-plane with faces exhibiting a vanishing surface impedance in the direction either parallel ((Z0)z = 0) or perpendicular ((Z0)ρ = 0) to the edge, which are obtained for δ = 0 and δ = 0.8, respectively. The limit case of isotropic impedance faces [Senior and Volakis, 1995] is also obtained for δ = 0.4 (solid curve in the plots), with Z0 = (Z0)z = (Z0)ρ = j0.4 and Zn = (Zn)z = (Zn)ρ = −j0.4. The amplitude of the copolar and cross-polar components of the longitudinal diffracted field (Ezd and ζHzd) is plotted for different values of δ in Figures 2a and 2b, respectively. The numerical results provided by the present solution coincide with those obtained by the known solution of all the limit cases above mentioned [Manara et al., 2000; Senior and Volakis, 1995], as expected.

Figure 2.

Amplitude of the (a) copolar and (b) cross-polar component of the longitudinal diffracted field in the presence of an anisotropic impedance half plane (n = 2) illuminated by a TM polarized (Eβi = 1, Eϕ′i = 0) plane wave impinging from ϕ = π/4′, β′ = π/6. Normalized surface impedance tensors: (Z0)z = , (Z0)ρ = j(0.8 − δ), (Zn)z = −(Z0)z, (Zn)ρ, with δ = 0, 0.2, 0.4, 0.6, 0.8. Continuous line: isotropic impedance half plane (δ = 0.4). The field is evaluated at a normalized distance ktρ = 5 from the edge.

[25] Figure 3 refers to case (b) of Table 1. An anisotropic impedance full plane (n = 1) is considered, exhibiting purely imaginary values of the surface impedance on the ϕ = 0 face, and (Zn)ρ = −1/(Z0)Z, (Zn)Z = −1/(Z0)ρ. In particular, (Z0)ρ = j(2 + δ) and (Z0)Z = j(2 − δ), with δ = −0.5, 0, 0.5, 1. The full plane is illuminated by a TM polarized plane wave impinging from ϕ′ = π/4, β′ = π/4. The amplitude of the copolar and the cross-polar components of the uniform scattered (i.e., reflected plus diffracted) field is evaluated at a normalized distance ktρ = 10 from the edge and plotted in Figures 3a and 3b, respectively. Note that when δ = 0 the presented solution exactly recovers that for the limit case of an isotropic impedance full plane [Senior and Volakis, 1995], with Z0 = (Z0)z = (Z0)ρ = j2, Zn = (Zn)z = (Zn)ρ = j0.5 (solid line in the plots). Analogous results are presented in Figure 4 for the canonical problem defined in case (c); an anisotropic impedance half plane (n = 2) is considered, with the following values for the normalized surface impedances: (Z0)ρ = 0.5 −j(0.5 + δ), (Z0)ρ = 0.5 −j(0.5 − δ), with δ = −0.5, 0, 0.5, 1 on the ϕ = 0 face, and (Zn)ρ = 1/(Z0)Z, (Zn)Z = 1/(Z0)ρ on the ϕ = π face. The incident plane wave is TE polarized (Eβi = 0, Eϕ′i = 1) and impinges from the direction ϕ′ = π/4, β′ = π/4. The amplitude of the copolar and cross-polar components of the total field, calculated at a normalized distance ktρ = 5 from the edge, is plotted in Figures 4a and 4b, respectively. Again, in both figures the solid curve refers to the limit case of an isotropic impedance half plane, with Z0 = (Z0)z = (Z0)ρ = 0.5 − j0.5, Zn = (Zn)ρ = 1 + j, and this curve is exactly recovered by our solution when δ = 0.

Figure 3.

Amplitude of the (a) copolar and (b) cross-polar component of the longitudinal scattered field (diffracted plus reflected) in the presence of an anisotropic impedance full plane (n = 1) illuminated by a TM polarized (Eβi = 1, Eϕ′i = 0) plane wave impinging from ϕ′ = π/4, β′ = π/4. The normalized surface impedance values are as follows: (Z0)ρ = j(2 + δ), (Z0)Z = j(2 − δ), (Zn)ρ = −1/(Z0)Z, (Zn)Z = −1/(Z0)ρ, with δ = −5.5, 0.5, 1. Continuous line: isotropic impedance half plane (δ = 0). The field is evaluated at a normalized distance ktρ = 10 from the edge.

Figure 4.

Amplitude of the (a) copolar and (b) cross-polar component of the longitudinal total field in the presence of an anisotropic impedance half plane (n = 2) illuminated by a TE polarized (Eβi = 0, Eϕ′i = 1) plane wave impinging from ϕ′ = π/4, β′ = π/4. The normalized surface impedance values are as follows: (Z0)ρ = 0.5 − j(0.5 + δ), (Z0)ρ = 0.5 − j(0.5 − δ), (Zn)ρ = 1/(Z0)Z, (Zn)Z = 1/(Z0)ρ, with δ = −0.5, 0.5, 1. Continuous line: isotropic impedance half plane (δ = 0). The field is evaluated at a normalized distance ktρ = 5 from the edge.

6. Conclusions

[26] Plane wave scattering by a specific set of anisotropic impedance half and full planes with principal anisotropy axes parallel and perpendicular to the edge has been analyzed and exact integral representations for the total field derived. The analysis has been performed for an arbitrary polarization of the incident field and in the most general case of oblique incidence. The integral representations are defined along the Sommerfeld integration path, so that they are suitable for a uniform asymptotic evaluation through standard techniques. The diffracted field is expressed in the UTD format and can be efficiently computed since it contains simple rational trigonometric functions together with the Maliuzhinets special function, whose calculation can be performed by means of efficient algorithms available in the literature [see, e.g., Osipov, 2005]. Besides the edge diffracted contribution, the asymptotic analysis has revealed surface wave fields that are excited by the diffraction at the edge and can propagate along the impedance faces of the half or full plane. It is worth noting that the proposed asymptotic solution is uniform also at the surface wave shadow boundaries.

[27] The uniform asymptotic solutions here derived constitute a new reference set for checking the accuracy of novel numerical or approximate analytical procedures, which may be applicable to a larger class of wedge scattering problems. Finally, the results of this paper represent a step ahead towards the analytical solution of the anisotropic impedance wedge scattering problem.

Appendix A:: Nonphysical Pole Cancellation

[28] This appendix presents a detailed analysis of the nonphysical poles that need to be cancelled out from the analytical solution of the functional equations for the angular spectra. In particular, it is shown that the number of equations needed to cancel the above poles exactly matches the number of unknown constants appearing in the spectra se,h(α). Besides standard GO contributions, other first-order pole singularities that can be captured during the contour deformation process are those related to surface waves that can be excited on the impedance faces, in agreement with the results in [Nepa et al., 2007].

display math

[29] The angular spectra obtained by substituting (12) into (7) exhibit first-order pole singularities at αp = −ϑ0±, ϑ0± ± π, −ϑ0± ± 2π, ϑ0± ± 3π… introduced by the function Δ(α + π). Some of them can be captured by the contour deformation process if the observation point is located in the angular sector −π + gd[Im(ap)] ≤ Re(ap) − ϕ + /2 ≤ + π + gd[Im(ap)]. By considering that ϑ0± are purely imaginary, it follows that, in the above angular regions and independently of the sign of Im(ϑ0±), the residues of the poles at αp = −ϑ0±, ϑ0± ± π correspond to inhomogeneous plane waves approaching the edge at grazing incidence (incoming waves), which are not physical contributions since the only incoming wave contribution is the incident field in (1). Therefore, the unknown coefficients c1,2, c1,2′, c1,2″ can be determined by imposing the condition of cancelling the above six nonphysical poles (equation (15)). Furthermore, the poles at αp = α0sw = −2π − ϑ0± are associated to physical poles accounting for surface waves that can be excited on the ϕ = 0 face, when the excitation condition gd[Im(ϑ0±)] − Re(ϑ0±) > 0 is met. Also, the poles at αp = αnsw = +2π − ϑ0± = 2π + ϑn are associated with physical poles accounting for surface waves that can be excited on the ϕ = 2π face, if the excitation condition gd[Im(ϑn±)] − Re(ϑn±) > 0 is satisfied. The above surface waves are characterized by a propagation vector with a real component parallel to the corresponding half-plane face, while its imaginary component is strictly perpendicular to the surface. Indeed, due to the purely reactive characteristic of the impedance faces, the surface wave is not losing energy while it propagates along the surface. All the other poles at αp = −ϑ0± ± 3π, −ϑ0± ± 4π,. are far from the SDPs and can never be captured by the contour deformation. Equation (15) is obtained by requiring that the numerator of equation (7a) vanishes at αp = −ϑ0±, ϑ0± ± π. Moreover, upon multiplying equation (15) for the term [−sinαp −sinβ′/(Z0)z] and considering that Δ(αp + π) = 0, it follows that equation (15) also implies that

display math

which shows that even sh(α) is vanishing at the nonphysical poles αp = −ϑ0±, ϑ0± ± π. This demonstration is also valid for cases (b) and (c).

display math

and

display math

[30] Let us remember that, when considering case (b), the angles ϑ0± and ϑn± must be purely imaginary (purely reactive surfaces), with ϑn± = −ϑ0±. In this case, the angular spectra obtained by substituting (18) into (7) still exhibit the first-order pole singularities introduced by the function Δ(α + /2). They are located at αp = ϑ0±π/2, −ϑ0± ± 2ππ/2,… and αp = −ϑ0± ± ππ/2, −ϑ0± ± 3ππ/2,… Between them, only the poles located at αp = ϑ0±π/2, −ϑ0± + π/2, −ϑ0± ± 3π/2, −ϑ0± ± 3π/2 can be captured by the contour deformation process, in the sense that the condition −π + gd[Im(ap)] ≤ Re(ap) − ϕ + π/2 ≤ π + gd [Im(ap)] could be satisfied by some values of the observation angle ϕ. By considering that ϑ0± are purely imaginary, it follows that, in the above angular regions and independently of the sign of Im(ϑ0±), the residues of the poles at αp = ϑ0±π/2, −ϑ0± + π/2, −ϑ0± − 3π/2 correspond to inhomogeneous plane waves approaching the edge (incoming waves), which are not physical contributions since the only incoming wave contribution is given by the incident field in (1). The two unknown constants c1,2 appearing in (19) are then determined by solving (20) for the nonphysical poles at αp = ϑ0±π/2. In particular, equation (20) can be rewritten as follows:

display math

[31] Since the function ψ(α + /2, ϑn±) has zeros at α = π/2 + /2 ± ϑn±π/2 = π/2 + /2 ∓ ϑ0π/2 and α = −π/2 − /2 ±ϑn±π/2 = −π/2 − /2 ∓ ϑnπ/2, the remaining nonphysical poles at αp = π/2 − ϑ0±, 3π/2 + ϑ0± are cancelled out by the above zeros, so that equation (A2) is satisfied when αp = π/2 − ϑ0±, 3π/2 + ϑ0±. Finally, the poles located at αp = −3π/2 − ϑ0± do not need to be cancelled out since they account for surface waves that can be excited on the ϕ = 0 face. On the contrary, surface waves that can be excited on the ϕ = π face correspond to the residues of the poles at αp = 3π/2 + ϑn±, which are introduced by the function ψ(α + /2, ϑn±) appearing in (18).

[32] As far as case (c) is concerned, it is worth noting that the surfaces can be lossy, namely, the surface impedances can exhibit a nonvanishing real part. As in case (a), the angular spectra obtained by substituting (18) into (7), with n = 2, exhibit some poles introduced in se,h(α) by the function Δ(α + /2). Some of these poles can give rise to incoming waves approaching the edge: αp = −ϑ0±, ϑ0± ± π, −ϑ0± ± 2π. However, by imposing the condition of vanishing of the numerator of se(α) at αp = −ϑ0±, ϑ0±π, the following system of four linear equations, corresponding to equation (20), is obtained:

display math

As in case (b), it turns out that the remaining nonphysical poles at αp = ϑ0± + π, −ϑ0± + 2π are cancelled out by some of the zeros introduced by the function ψ(α + /2, ϑn±). Indeed, the closest zeros of ψ(α + /2, ϑn±) to the origin of the complex plane α are located at α = 3π/2 ± ϑn±π/2 = 3π/2 ∓ ϑ0π/2 and α = −3π/2 ± ϑn±π/2 = −3π/2 ∓ ϑ0π/2. Finally, the poles located at αp = −2π − ϑ0± do not need to be cancelled out as they account for surface waves that can be excited on the ϕ = 0 face. On the contrary, surface waves that can be excited on the ϕ = π face correspond to the residues of the poles at αp = 2π + ϑn± introduced by the function ψ(α + /2, ϑn±) appearing in (18).

[33] In case (c), the correctness of the proposed solution when the surfaces are purely reactive can be verified by considering the limit case of surfaces with small losses. Under this hypothesis, Re{ϑ0,n±} > 0 and it becomes apparent that the nonphysical poles, namely, those lying in the strip ∣Re{α}∣ < π, are located at αp = −ϑ0±, ϑ0±π, which are exactly those considered to define the system of linear equations for determining the unknown constants appearing in the analytical solution. A similar simple check has not been done in cases (a) and (b). Nevertheless, the detailed analysis of the nonphysical poles given in this appendix and the numerical convergence of the proposed solution to known analytical solutions valid for some limit cases (see section 5, Numerical Results) prove the correctness of the proposed analytical solutions for all the cases listed in Table 1. In particular, it is worth mentioning that the location of the surface wave poles exhibited by the proposed solutions always coincides with that given in other papers dealing with the scattering from anisotropic impedance wedges [Nepa et al., 2007]. Moreover, all the nonphysical poles appearing in the solution have been cancelled. Indeed, some of them are cancelled through a proper evaluation of a set of unknown constants, while others are automatically cancelled from some zeros of the Maliuzhinets special function contained in our analytical solution.

Appendix B:: Reflection Coefficients and Surface Wave Amplitude Coefficients

[34] The reflection coefficients appearing in (22) are obtained by evaluating the residues of the geometrical poles at αi = −ϕ′ − /2 and αi = −ϕ′ + 3/2. They coincide with the expressions that could be derived by evaluating plane wave reflection at an infinite impedance surface characterized by the IBCs holding on the corresponding wedge face.

[35] The reflection coefficients related to the ϕ = 0 face read:

display math
display math
display math
display math

Conversely, the reflection coefficients related to the ϕ = face read:

display math
display math
display math
display math

In (B1) and (B2), the functions R1(γ, Zρ, Zz) and R2(γ, Zρ, Zz) are defined as follows:

display math
display math

We note that the above reflection coefficients are given for the most general case and they can be applied to all of the three cases examined, provided the proper surface impedance tensor is adopted.

[36] To calculate the amplitude of the surface waves appearing in (23), we need to distinguish between the pole singularities introduced by the function Δ(α + /2) (cases (a), (b) and (c)) and those introduced by the function ψ(α + /2, ϑn±) (cases (b) and (c)), when evaluating their residues.

[37] For all of the three canonical problems under study the surface waves that can propagate along the ϕ = 0 face are associated with the poles at α0sw = −π/2 − ϑ0±, which are related to the function Δ(α + /2). Their amplitudes read:

display math
display math

with

display math

Note that only one out of the two surface waves can propagate at most, as it will always occur that gd[Im(ϑ0±)] − Re(ϑ0±) < 0 for at least one of the two Brewster's angles.

[38] In case (a), the surface wave propagating along the ϕ = face is associated with one out of the two poles at αnsw = 2π − ϑ0± = 2π + ϑn, which are again introduced by the function Δ(α + π). We note that, as a consequence of the periodicity (with period 4π) of the function sin(α/2), it results t1,2(αnsw) = t1,2(α0sw + 4π) = t1,2(α0sw), and consequently:

display math

where A0e and A0h are calculated through equations (B5a) and (B5b) with the parameters of case (a) (α0sw = −2π − ϑ0±).

[39] Conversely, in cases (b) and (c), the surface wave on the ϕ = face is associated with one out of the two poles introduced by the functions ψ(α + /2, ϑn±), namely, the poles αnsw = π + /2 + ϑn±, and its amplitude reads:

display math
display math

with

display math

The functions σ1(α) and σ2(α) appearing in (B8a) and (B8b) are defined in (19a) and (19b). Finally, it is worth observing that even on the ϕ = face, only one out of the two possible surface waves can propagate at most, and the choice between ϑn+ and ϑn in (B8a), (B8b) and (B9) depends on which of the surface waves satisfies the excitation condition.

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