## Abstract

The problem of the diffraction of a plane electromagnetic wave incident at an oblique angle on a wedge of arbitrary angle with general tensor impedance boundary conditions is solved using a semi-analytical approach. Application of Maliuzhinets' method transforms the boundary-value problem into coupled functional difference equations (FDEs) for two unknown Sommerfeld integral spectra in a basic strip. By explicitly separating out the singular parts of the spectra in the strip, followed by a partial inversion of the FDEs, we obtain integral representations of the regular parts of the spectra. The regular parts are then expanded in a Taylor series in terms of a new variable that conformally maps the strip on to a disc. This expansion reduces the integral representations to algebraic equations for the series coefficients and these are solved numerically. We examine the convergence of the procedure, compare the numerical solution with an available reference solution and present solutions of new problems.

## 1. Introduction

A wedge with external angle 2*Φ* is illuminated by an arbitrarily polarized plane electromagnetic wave having(1.1)where a time factor exp(−i*ωt*) has been assumed and suppressed. The wave is incident obliquely in the direction(1.2)where is the position vector in a cylindrical coordinate system with the *z*-axis along the edge of the wedge; *k* is the wave number; and , (figure 1). The material properties of the wedge are represented by the tensor impedance boundary conditions(1.3)where *Z*_{0} is the free-space impedance and is the outward unit vector normal to the wedge face . The normalized tensor impedance has the general form(1.4)and may take different values at the upper and lower faces. To ensure uniqueness, the tensor elements are assumed to satisfy the passivity requirements (Senior & Volakis 1995)(1.5)where the asterisk denotes the complex conjugate.

The problem does not have a closed-form solution, but in a number of special cases when the face impedances, incidence direction and/or wedge angle have specific values (Maliuzhinets 1958; Rojas 1988; Bernard 1990, 1998; Senior & Volakis 1995; Pelosi *et al*. 1998*a*; Lyalinov & Zhu 1999, 2003), exact analytical solutions are available. If the incidence is close to normal () or grazing (), or the wedge faces are nearly perfectly conducting, it is possible to construct an approximate solution using a suitable perturbation procedure (Pelosi *et al*. 1998*b*; Buldyrev & Lyalinov 2001; Fei & Guoqiang 2005). The approximate solution presented in Syed & Volakis (1992, 1995; see also Senior & Volakis 1995) applies when the wedge is almost a half plane (). In the general case it is necessary to resort to numerical methods, and some of the procedures that have been used are a moment method solution of surface integral equations (Bilow 1991), a numerical integration of parabolic equations (Pelosi *et al*. 1996; Zhu & Landstorfer 1997), a probabilistic random walk solution of transport equations (Budaev & Bogy 2005, 2006), numerical matrix factorization (Daniele 2003; Daniele & Lombardi 2006) and the numerical solution of integral equations resulting from second-order difference equations for the Sommerfeld spectra (Lyalinov & Zhu 2005, 2006).

Since the wedge is a canonical geometry of practical interest and a knowledge of the diffraction coefficient is an important building block for high-frequency methods, an analytic (or at least semi-analytic) solution is highly desirable. A semi-analytic solution is presented here. The Maliuzhinets method (Maliuzhinets 1958; Senior & Volakis 1995) is used to represent the fields as Sommerfeld integrals that are well suited for a high-frequency analysis. The spectra have singularities responsible for the geometric optics portions of the field and these are explicitly extracted. The remaining unknown parts of the spectra are regular in a strip of the complex plane and they are expanded in a power series in a variable that maps the strip onto a disc. The expansion coefficients are then obtained numerically. The derived solution is semi-analytic because the singular portion is described analytically, whereas the regular portion is defined by a series whose coefficients are found numerically, and only a small number of terms are necessary for most practical applications. In the special case of normal incidence and diagonal impedance tensors, the numerical part of the solution vanishes and the rest reduces to the exact closed-form solution of Maliuzhinets (1958).

The idea of expanding the spectra in a Taylor series was proposed (Senior *et al*. 2001; Senior & Topsakal 2005) to solve the second-order difference equations that arise in diffraction by a half-plane and related configurations. A modification to this approach is to apply the Taylor series to the coupled first-order functional difference equations (FDEs) satisfied by the spectra; this was used by Osipov & Senior (2005) to solve the problem of an arbitrary-angled wedge with isotropic impedances. The present paper extends this to the case of general tensor impedances, and an advantage is that the solution is expressed in terms of the Maliuzhinets function (1958) for which there are computational algorithms available (e.g. Senior & Volakis 1995; Osipov 2005).

The layout of this paper is as follows. In §2 the fields are cast in the form of Sommerfeld integrals and FDEs for the spectra are obtained. In §3 the singular parts of the spectra are extracted and, by partial inversion of the FDEs, integral representations for the regular parts are derived. The regular parts are then expressed (§4) as a Taylor series, the coefficients of which are given by a system of algebraic equations. Section 5 describes the high-frequency components of the solution, including the geometric optics, surface and edge-diffracted waves. The numerical results in §6 demonstrate the convergence of the procedure and illustrate the behaviour of the diffraction coefficient as a function of *ϕ*, *ϕ*_{0} and *β*.

## 2. Functional difference equations

At the two faces of the wedge, the components of the boundary condition (1.3) are(2.1)and(2.2)By using (2.1) we can eliminate *H*_{ρ} from (2.2) and rewrite it as(2.3)where(2.4)

Owing to the translational invariance of the wedge in the *z*-direction, the scattered (and therefore total) field must have the same *z*-dependence as the incident field, and all field components can be expressed in terms of *E*_{z} and *Z*_{0}*H*_{z}. In particular,enabling us to rewrite (2.1) and (2.3) as(2.5)and(2.6)

Following Maliuzhinets (1958), the total field components *E*_{z} and *Z*_{0}*H*_{z} are written as the Sommerfeld integrals(2.7)and(2.8)where *γ* is the Sommerfeld double loop contour. The spectra must be such that(2.9)to ensure that |*E*_{z}|, |*H*_{z}|=*O*(1) as *ρ*→0 (edge condition), and apart from a pole singularity at *α*=*ϕ*_{0} with(2.10)which is necessary to recover the incident field (1.1), must be free of poles in the strip(2.11)

When (2.7) and (2.8) are inserted into (2.5) and (2.6), can be replaced by and, using integration by parts, by , and the boundary conditions then demand(2.12)and(2.13)where(2.14)

(2.15)

(2.16)and(2.17)

From (2.12) and (2.13), by accounting for the behaviour of the spectra as , we obtain the functional difference equations(2.18)with(2.19)and(2.20)

Although (2.18) is valid over the entire complex *α* plane, it is convenient to impose it only on the imaginary axis, , thereby confining the analysis to the strip . Once the spectra are determined here, they can be extended to the rest of the complex plane by using (2.18) as a functional relation (e.g. equation (4.8) below).

## 3. Integral representation of the spectra

Since an explicit solution of (2.18) is not possible for general values of *Φ*, *β* and , we now transform the equations to facilitate a numerical solution. The transform involves the separation of the singular parts of the spectra, followed by a partial inversion of the FDEs and the derivation of integral representations for the regular parts.

We start by writing(3.1)where are auxiliary functions that satisfy the functional equations(3.2)with(3.3)Owing to the uniqueness conditions (1.5), the real parts of the terms on the r.h.s. of (3.3) are non-negative, which implies that can be chosen such that .

The auxiliary functions satisfying (3.2) are known from the work of Maliuzhinets (1958) and can be expressed as(3.4)with(3.5)where is the Maliuzhinets function. The functions are free of poles and zeros in the strip and have the asymptotic behaviour(3.6)as with . Formulae (3.1) and (3.6) imply that(3.7)as in accordance with the edge condition (2.9).

Substituting (3.1) into (2.18) leads to the system of equations(3.8)and(3.9)where(3.10)and(3.11)with(3.12)and(3.13)At within the strip , the functions must have pole singularities to comply with (2.10); these are conveniently described by(3.14)where(3.15)The functions are solutions of (3.8) and (3.9) with .

Extracting the pole singularities by writing(3.16)leads to new unknown functions that are free of poles in the strip and satisfy the equations(3.17)and(3.18)The functions on the r.h.s. of (3.17) and (3.18) can be expressed as(3.19)and(3.20)where and are given by (3.10) and (3.11) with replaced by and , respectively. The functions are, therefore, known and serve as source terms in the inhomogeneous equations (3.17) and (3.18).

When and , the source terms vanish and the system of equations (3.17) and (3.18) becomes homogeneous, with trivial solutions . We then recover the closed-form Maliuzhinets solution .

If we assume for the moment that the r.h.s. of (3.17) and (3.18) are known, the equations form a system of first-order difference equations with constant coefficients. For pure imaginary *α*, the arguments of the unknown functions lie on the boundaries of the designated strip of analyticity of , whereas the source functions decay as at infinity. Such equations can be solved by applying a Fourier transform (modified by mapping on to the imaginary axis on the complex *α*-plane), leading to the representations(3.21)that are valid in the strip where they are analytic functions of *α*. By noting that are odd functions of *τ*, we can rewrite this as(3.22)where(3.23)

As the integrand of (3.22) includes the unknown boundary values , we cannot use (3.22) to determine directly, but by analytically continuing it to the boundaries of the strip , we can transform (3.22) into a set of linear equations for the boundary values. The result is a system of eight singular integral equations with Cauchy kernels defined over the semi-axis for the eight unknown functions . However, there is an alternative approach based on (3.22) as it is, and this is described in §4.

## 4. The Taylor series solution

Equation (3.22) suggests the possibility of expanding the solution in a series of powers of(4.1)To this end, we introduce(4.2)and, rewriting the kernel functions as(4.3)we conclude that (4.3) can be expanded in powers of *w*(*α*) if and . For pure imaginary *τ*, these are equivalent to since . The resulting expansions of the kernel functions are(4.4)where and(4.5)

Substituting (4.4) into (3.22) leads to the series representation of the solutions as(4.6)where the expansion coefficients are as yet unknown. Since (4.1) defines a conformal mapping of the strip in the complex *α*-plane on to the unit disc |*w*|≤1 in the complex *w*-plane, (4.6) is a Taylor series expansion about *w*=0 in the complex *w*-plane. Convergence inside the disc follows from the analyticity of inside the strip , and when |*w*|=1, the convergence depends on whether has singularities on the boundaries of . The singularities of the spectra can be located by using (2.18) as functional relations to extend the spectra from to the neighbouring strips(4.7)The extension relations are(4.8)where(4.9)Thus, is analytic inside except for pole singularities at (images of the geometric optics singularity of from ) and at the poles of the elements of the matrices . It then follows that are free of poles in a strip where *δ*>0 if . This last is simply the condition for passive impedances and is always true for any physical surface.

Under the assumption of the passivity of the wedge faces, the series (4.6) now converges inside and on its boundaries, and can be used to represent the unknown functions in the integrand of (3.22). By inserting these Taylor series expansions into (3.22), expanding the kernel functions using (4.4) and (4.5), and then equating the coefficients of like powers of *w*, we obtain the coupled system of linear algebraic equations(4.10)and(4.11)where The source terms and matrix elements are(4.12)(4.13)(4.14)and(4.15)where(4.16)

The problem is therefore reduced to the solution of the algebraic equations (4.10) and (4.11). Once the coefficients are determined, the Taylor series (4.6) defines the functions within the regularity strip , and the spectra can be assembled from (3.1) and (3.16). Extension of the spectra from into the rest of the complex *α*-plane is achieved by using the FDEs (2.18). For example, (4.8) extends the spectra from into .

## 5. High-frequency analysis

Once the spectra are determined, the Sommerfeld integrals (2.7) and (2.8) give the exact solution to the problem in the entire space outside the wedge. In the high-frequency limit when , the integrals can be evaluated asymptotically using the steepest descent method (e.g. Maliuzhinets 1958; Senior & Volakis 1995). The steepest descent paths (SDPs) ,where is the Gudermannian function , pass through the saddle points and enclose a region of the complex *α*-plane. Deformation of the Sommerfeld contour *γ* into the SDPs leads to the representation(5.1)where(5.2)In (5.1), denotes the integrals over the SDPs and and are the residues of the poles captured by the contour deformation. is the geometrical optics field produced by the pole in and its images resulting from the extension of the spectra into the rest of the *α*-plane. represents the surface waves that are associated with the upper and lower wedge faces and result from the poles of in and their images in the further strips. If, for simplicity, we restrict attention to acute-angle wedges such that , at most five poles of may lie in the strip and these are at , and . The quantities are the zeros of(5.3)and, since the elements of the matrices (2.20) are entire functions of *α*, the poles of (4.9) are the zeros of .

The geometrical optics field is produced by the poles and and consists of the incident plane wave and the waves specularly reflected off the upper and lower faces of the wedge. We have(5.4)where(5.5)and *u*(*x*)=1 if *x*>0 and 0 otherwise. In (5.4),(5.6)with(5.7)

(5.8)

(5.9)and(5.10)

The residues at are(5.11)with(5.12)and(5.13)where the prime denotes the derivative. The two terms in (5.11) describe the surface waves on the upper and lower faces of the wedge that are excited by the plane wave incident on the edge. For example, in the case of isotropic impedances with and , the zeros of (5.3) that lie in are either if or if . The exponent in (5.12) then describes a wave propagating away from the edge and concentrated near to the appropriate face.

The third term on the r.h.s. of (5.1) is the pair of SDP integrals, and their approximate evaluation using the steepest descent method gives the edge-diffracted field(5.14)The geometrical theory of diffraction describes the edge diffraction in a ray-fixed coordinate system based on the unit vectors , and according to the relations , , , , and . The incident and edge-diffracted fields are then and , and their connection to the *z*-components of the fields is as follows: , , and . From (5.14) the edge-diffracted field(5.15)and the incident field(5.16)are related by the formula(5.17)where the diffraction coefficient is the tensor(5.18)whose elements are expressed in terms of the spectra as(5.19)where and and(5.20)where and .

If the impedance tensors are either symmetric () or antisymmetric (), then the elements of the diffraction coefficient tensor satisfy the symmetry relations(5.21)or(5.22)respectively. It can be shown that properties (5.21) and (5.22) result from three- and two-dimensional versions of the Lorentz reciprocity theorem. For diagonal impedance tensors , both (5.21) and (5.22) are valid simultaneously.

Further symmetry relations hold when the impedance tensors are polarization independent , in which case(5.23)If the impedance tensors are both polarization independent and symmetric or antisymmetric, the corresponding symmetry relations apply simultaneously.

## 6. Numerical results

A practical realization of the Taylor series method described in the previous sections assumes truncation of the series (4.6) and the corresponding systems of algebraic equations (4.10) and (4.11). As we increase the number *N* of terms retained in the Taylor expansions of the spectra, the accuracy improves and the solution becomes exact in the limit *N*→∞. In this section, we present a numerical analysis which shows that the solution does satisfy the FDEs (2.18) and is convergent. Having proved that, we present numerical results illustrating the behaviour of the tensor diffraction coefficient (5.18) as a function of the various problem parameters. The choice of the impedance tensors(6.1)allows a comparison with previously published data (Daniele & Lombardi 2006).

The source terms and matrix elements (4.12)–(4.15) in the systems (4.10) and (4.11) are given by integrals that are exponentially convergent. These have been computed by using adaptive quadratures and exact representations of the Maliuzhinets function.

We begin with an analysis of the error in satisfying the FDEs (2.18). The error is measured by inserting the spectra into the FDEs (2.18) and finding the maximum difference between the l.h.s. and r.h.s. over the imaginary axis for both polarizations and for both upper and lower face boundary conditions. A formal definition of the error is as follows. Let be solutions for *E*- (, ) and *H*- (, ) polarization cases, respectively, and *N* be the truncation number. Then the matrixis a solution of the equationswhere are the error terms that are expected to vanish as *N*→∞. These terms are 2×2 matrices and their matrix norm (which is defined in this paper as the maximum singular value of the respective matrix)is a scalar non-negative even function of *α*. Since the coefficients of the FDEs grow as when , the division of the error functions by cos *α* eliminates their growth at infinity. It is then possible to find the maximum value over the imaginary axisFinally, choosing the greatest value of the twogives a single numerical measure of the error with which a solution with a finite number of the retained terms satisfies the FDEs (2.18).

Figure 2 illustrates the accuracy of meeting FDEs for the impedance tensors (6.1), the incident angle *ϕ*_{0}=0 and various values of the wedge angle (, , , and *π*) and the illumination aspect (, and ). The convergence of the solution process is clearly seen from the monotonic decrease in error as *N* increases. As evident from the first five curves of the plot, the value of the wedge angle has little impact on the convergence. In contrast to that, the value of the aspect angle *β* considerably influences the convergence such that more terms in the Taylor series are needed to achieve a prescribed accuracy when the incidence direction is grazing to the edge () than when it is perpendicular to the edge (). This is illustrated by the last two curves in figure 2.

Figure 3 shows the diffraction coefficients (5.19) and (5.20) as a function of *ϕ* for , and (bistatic case). The infinities at and correspond to the shadow boundaries for the direct wave () and the wave reflected from the upper face of the wedge (), where the GTD diffraction coefficient (5.18) is singular. For *E*-polarization the problem has been studied by Daniele & Lombardi (2006). Comparison with the results in figure 3 shows agreement for co-polarization but not for cross-polarization. In the latter case, the curves are similar in shape but differ in level by the value of , which is due to a different normalization of the diffraction coefficient for the magnetic field.

Figure 4 shows the monostatic diffraction coefficient as a function of *ϕ*_{0}=*ϕ* for and *β*=*π*/3. In the monostatic case the total field consists of the edge-diffracted field only, and the curves for , , and also represent those for the field components in the edge-fixed coordinate system, since, according to (5.17) and (5.18), the latter can be obtained from the former by multiplying by or and dividing by . The curves are not symmetric with respect to the bisecting plane *ϕ*_{0}=0 because the upper and lower face impedance tensors are different. The maxima in the curves correspond to the specular reflection directions at . Owing to the diagonal structure of the tensor , the cross-polar components of the diffraction coefficient are bounded at .

The number of terms *N* in the Taylor expansion (4.6) needed to achieve a prescribed accuracy depends on the value of *α*. The convergence rate is the slowest at the boundary of the strip when Re as this corresponds to the boundary |*w*|=1 of the convergence disc on the complex *w*-plane. For the diffraction coefficient that is defined by equations (5.19) and (5.20), this implies the slowest convergence when . Figure 5 illustrates this for the monostatic configuration analysed in figure 4. Parameter *δ* is the required accuracy, which is defined as the matrix norm of the difference between the exact (*N*=∞) and approximate (*N* finite) tensor diffraction coefficients. As expected, the higher the required accuracy, the more the terms are needed, and the number of terms depends on *ϕ* with maxima at . The dotted line shows that for a lower precision setting *δ*=0.1, a solution with *N*=0 is sufficient over a considerable range of values of the aspect angle.

Dependence of the diffraction coefficient on angle *β* is shown in figure 6 for and *ϕ*=*ϕ*_{0}=0. The symmetry with respect to the plane perpendicular to the edge (*β*=*π*/2) is not present owing to the general form of the impedance tensor in (6.1), which is neither symmetric, nor antisymmetric, nor polarization independent.

## 7. Conclusion

The semi-analytical method presented here is applicable to the most general wedge problem for all wedge angles, directions of incidence and (first-order) tensor impedance boundary conditions. The Taylor series approach to the coupled integral equations (3.21) uses the known analytical properties of the Sommerfeld spectra and leads to the exact solution of the diffraction problem. The convergence of the procedure has been demonstrated and results presented for a variety of aspect angles and wedge parameters. An important advantage of the approach is that a relatively small number of terms in the Taylor series provides good accuracy.

The method is also applicable to other wedge-shaped configurations, including those with resistive sheets dividing the region outside the wedge.

## Footnotes

- Received August 8, 2007.
- Accepted September 25, 2007.

- © 2007 The Royal Society