## Abstract

Electromagnetic fields, excited by an electric line source in the presence of an infinite metamaterial wedge, are determined by application of the Kontorovich–Lebedev transform. Uncoupled singular integral equations for the spectral functions are derived and a numerical scheme is devised and implemented to solve them. Numerical results showing the influence of a metamaterial wedge presence on the directivity of a line source are presented and verified through finite-difference frequency-domain simulations.

## 1. Introduction

The concept of left-handed material (LHM) was proposed by Veselago in (1968), where both the dielectric constant and the magnetic permeability are negative. Much attention has been received recently on the study of LHM on theory, experiments and potential applications (IEEE 2003). To mention a few of their applications, LHMs can be used to focus electromagnetic energy; they have the potential to form highly efficient, low reflectance surfaces by cancelling the scattering properties of other materials.

The methods based on the Kontorovich–Lebedev transform (Lebedev 1965) have been successfully applied in the past to a number of problems of diffraction by wedges (Oberhettinger 1954; Forristall & Ingram 1971, 1972; Rawlins 1972, 1999; Osipov 1993; Salem *et al*. 2006). A strategy has been presented by Salem *et al*. (2006) to solve the problem of diffraction by an ordinary material wedge. This paper extends the strategy of Salem *et al*. (2006) to metamaterial wedges of arbitrary negative electric and negative magnetic constants and with arbitrary source-observer configurations. It shows that the singular integral equation formulation is suitable for numerical analysis and develops numerical procedures that provide numerical results.

In §2, the singular integral equations for the Kontorovich–Lebedev spectra are derived by analytically continuing those of an ordinary material wedge; a successive approximation scheme is implemented to solve them numerically; the singularities of the Kontorovich–Lebedev spectrum inside the metamaterial wedge are identified; and the residues are evaluated. In §3, the near fields inside and outside the metamaterial wedge are represented by residue series; the far field inside the metamaterial wedge is represented by direct numerical integration of the Kontorovich–Lebedev integrals; the far field outside the wedge is represented by residue series; and an extension to the plane wave illumination case is also given. In §4, the matched double negative wedge is discussed. In §5, numerical results are presented and compared with the results of finite-difference frequency-domain simulations. Conclusions are given in §6.

## 2. The singular integral equations

### (a) Derivation

The geometry of the problem is shown in figure 1. The parameters *k*_{1}, *ϵ*_{1} and *μ*_{1} are respectively the wavenumber, permittivity and permeability outside the metamaterial wedge. *k*_{2}, *ϵ*_{2} and *μ*_{2} are corresponding quantities inside the metamaterial wedge and *N*=*k*_{2}/*k*_{1}. The line source is located at (*r*_{0}, *φ*_{0}) outside the wedge and the wedge angle is 2*β*. It should be noted that a lossless, dispersionless doubly negative material is not physically realizable. Hence, for the metamaterial wedge, Re (*k*_{2}, *ϵ*_{2}, *μ*_{2})<0 and Im (*k*_{2}, *ϵ*_{2}, *μ*_{2})>0 for the assumed, and omitted throughout, time factor exp (−i*ωt*). It should be noted that the analysis carried out here is for the case |*N*|>1. Modifications for the case |*N*|<1 are straightforward. Since the problem under consideration is one of scattering and diffraction, the wavenumbers *k*_{1,2} are real (complex) for the lossless (lossy) media. However, initially we assume that *k*_{1,2} are such that(2.1)

Under the conditions given by (2.1), the steps leading to the derivation of the singular integral equations are similar to those employed by Salem *et al*. (2006) and will only be summarized here. Fields (*E*_{z}, *H*_{r}, *H*_{φ}) are constructed from symmetric and antisymmetric parts (with respect to the planes *φ*=0, ±*π*) and are represented by Kontorovich–Lebedev spectra. The continuity conditions of the fields *E*_{z}, *H*_{r} on the wedge face at *φ*=*β*, together with the distribution given by Forristall & Ingram (1972), Osipov (1993) and Rawlins (1999) led, for the symmetric part of the field, to(2.2)andwhere v.p. in front of the integral sign denotes that the Cauchy principal value is to be taken.

In order to address the original scattering and diffraction problem, we continue (2.2) analytically with respect to *k*_{1} and *k*_{2} as we switch them back to real (complex) for the lossless (lossy) media. For the metamaterial wedge under consideration (|*N*|>1), one can note that the argument of the Gauss hypergeometric function, *Z*=(1−(1/*N*^{2})), will attain values in the range |1−*Z*|<1, possibly with |*Z*|>1 for which the series representation of the hypergeometric function in (2.2) is invalid. Therefore, analytic continuation of the hypergeometric function in (2.2) is required in order to obtain a singular integral equation (SIE) valid for all possible values of *Z*≶1 in the range |1−*Z*|<1. To that end, we make use of Whittaker & Watson (1990) to obtain(2.3)where |arg (1−*Z*)|<2*π* and |1−*Z*|<1. In what follows, we will term the r.h.s. of (2.3) *F*_{c}(σ, *ν*, *N*). The equation (2.2) with the hypergeometric function in the kernel replaced by *F*_{c}(*σ*, *ν*, *N*), and with [*k*_{2}/*k*_{1}]^{±1} continued to [*N*]^{±1}, is the one satisfied by the spectral function *A*_{2}(*σ*) of the electric field inside the metamaterial wedge.

The spectral function *A*_{1}(*σ*) of the scattered electric field outside the metamaterial wedge is found from *A*_{2}(*σ*) (see Salem *et al*. 2006) with the same analytic continuation procedure as given above.

The singular integral equations satisfied by the Kontorovich–Lebedev spectra of the antisymmetric part is derived by replacing sin(*σφ*_{0}) by −cos(*σφ*_{0}) and the remaining sin(.), cos(.) by cos(.), −sin(.), respectively.

### (b) Numerical scheme to solve the singular integral equation

We apply the scheme devised, detailed by Salem *et al*. (2006) and inspired by the Neumann series expansion approach introduced by Rawlins (1999), to solve (2.2) numerically; now with *k*_{1,2} switched back to real (complex) for the lossless (lossy) diffraction and scattering problem. A summary of the scheme is as follows:

Multiply both sides of (2.2) by .

Expand all functions of

*N*in the Neumann series in powers of (1−(1/*N*^{2})), i.e.(2.4)(2.5)(2.6)(2.7)(2.8)(2.9)(2.10)and(2.11)Equate equal powers of (1−(1/

*N*^{2})) from both sides of the equation.Hence,(2.12)(2.13)(2.14)and(2.15)

The above equation defines an iterative scheme in which one starts with the known and generates the rest of the Neumann series coefficients from (2.14).

### (c) The pole singularities of the spectra

The processes of identifying the pole singularities, their order and the quantification of their residues have been detailed by Salem *et al*. (2006); a summary is given here.

Making use of the analytic continuation process (see Salem *et al*. 2006), one identifies, for *A*_{2}(*σ*), the set of poles *σ*_{pl} where(2.16)and(2.17)

The order of the *σ*_{pl} poles is analysed next.

With the observation that the *σ*_{p0} poles are simple (first-order poles), one can note that the wedge angle, 2*β*, being a rational multiple of *π* (i.e. 2*β*=*r*_{1}*π*/*r*, where *r*_{1} and *r* are positive integers, *r*_{1}<2*r*) is the condition for higher order poles to take place.

With denoting the *q*th strip, one can note that the poles in the strips *S*_{q}, 2*r*>*q*≥0, are of first order; the poles in the strips *S*_{q}, 4*r*>*q*≥2*r*, are of second order and the order increases by one every band of 2*r* strips. When the wedge angle is not a rational multiple of *π*, the *σ*_{pl} poles are of first order.

#### (i) Residues computation

Poles of the type

*σ*_{p0}in the*S*_{q}strip(2.18)(2.19)(2.20)(2.21)(2.22)and(2.23)where (2.14) is used to find*A*_{2}(*ν*) under the integral sign.Poles of the type

*σ*_{pl},*l*=1,2, …, ∞(2.24)

The residues of higher order poles, if available, are not detailed here but are straightforward and require the use of the higher order residue formula instead of the first-order formula used in this analysis.

The spectral amplitude *A*_{1}(*σ*) has an extra set of poles *σ*_{s} with(2.25)

There are simple poles with(2.26)

A more detailed presentation of the residue computation is given by Salem *et al*. (2006).

## 3. Field representations inside and outside the wedge

### (a) Near field inside and outside the metamaterial wedge

For *r*<*r*_{0}, we represent the electric field inside the wedge, , and the electric field scattered outside the wedge, , by the Kontorovich–Lebedev integrals(3.1)and(3.2)

From Gradshteyn & Ryzhik (1980)(3.3)and(3.4)we obtain(3.5)(3.6)and(3.7)

Next, the behaviour of *A*_{2}(*σ*), as *σ*→i∞ is estimated.

Write (2.2) in the form(3.8)

Express as the sum of three integrals, with

*δ*very small,(3.9)Let and, making use of the asymptotic formulae by Jones (2001), we obtain(3.10)to reach(3.11)

Make use of(3.12)to obtain(3.13)

From Jones (2001), we obtain(3.14)to reach

Hence,(3.15)i.e. the behaviour of *A*_{2}(*σ*) as *σ*→i∞ is bounded by that of *Q*(*σ*). Similar analysis led to the same conclusion for *σ*→−i∞, as well as for the behaviour of *A*_{1}(*σ*) as *σ*→±i∞, (see Salem *et al*. (2006) for the SIE corresponding to *A*_{1}(*σ*)) with the inhomogeneous term of that SIE given by

The above conclusion is confirmed by the numerical results of §2*b*.

Hence, one concludes that the integral in (3.1) converges and that in (3.2) diverges, which was also confirmed numerically.

Following Salem *et al*. (2006), closing contours in the r.h.s. of the complex *ν*-plane and collecting residue contributions, we express(3.16)and(3.17)

The behaviour of the *ν*_{pl} and *ν*_{s} series summands are as reported by Salem *et al*. (2006).

One can note that, since (3.1) converges, an alternative representation for the near field in the metamaterial wedge exists in terms of the direct numerical integration of (3.1).

### (b) Far field inside and outside the metamaterial wedge

For *r*>*r*_{0}, we represent the electric field inside the wedge, , and the electric field scattered outside the wedge, , by the Kontorovich–Lebedev integrals(3.18)and(3.19)

From Gradshteyn & Ryzhik (1980)(3.20)(3.21)and(3.22)we obtain(3.23)and(3.24)which, together with the corresponding behaviour of *A*_{1,2}(*ν*), show that the integral in (3.18) converges while that in (3.19) diverges. Hence, we represent the far field inside the metamaterial wedge by direct numerical integration of (3.18). The far field outside the wedge is found, similar to Salem *et al*. (2006), by invoking reciprocity to obtain(3.25)where the sums are on the residues of *A*_{1}(*ν*) with the source located at (*r*, *φ*) and the observer located at (*r*_{0}, *φ*_{0}), both outside the wedge.

### (c) The plane wave illumination case

The fields due to a normally (with respect to *Z*) incident plane wave are recovered by replacing by in the line source results; hence, the expressions are similar to (3.17) and (3.16) with the residues replaced by those for plane wave illumination. One can note that the field inside the metamaterial wedge could, alternatively, be obtained by direct numerical integration of (3.18) with the Kontorovich–Lebedev spectrum *A*_{2}(*ν*) replaced with that of the plane wave illumination.

**Remark.** Similar to Salem *et al*. (2006), the truncated residue sums on *ν*_{pl} should be understood as giving some asymptotic approximation in terms of first identified poles; the antisymmetric part of the field, the field structure when higher order poles exist and the suitability of the Bessel series for field computations in the plane wave illumination case is as reported by Salem *et al*. (2006).

## 4. The matched double negative wedge

To recover the case of the lossless double negative wedge,(4.1)one can start with (2.2), with *k*_{2}/*k*_{1} replaced by *N*, which is valid (i.e. the series representation of the hypergeometric function converges) in the vicinity of (and at) *N*∼−1, continue the SIE into the right half *σ*-plane, identify the poles and quantify the residues of *A*_{2}(*σ*) as detailed by Salem *et al*. (2006). This is followed by taking the limit of the field as *N*→−1 and *μ*_{2}→−*μ*_{1}. We obtain, for the total field (the symmetric and antisymmetric parts) of a line source located at (*r*_{0}, *φ*_{0}),(4.2)and(4.3)where , *n*=0, 1, 2, …, ∞, *ϵ*_{n}=1 for *n*=0 and *ϵ*_{n}=2 for *n*≠0 and *r*_{<}|*r*_{>} is the lesser|greater of *r* and *r*_{0}.

The above is in agreement with Monzon *et al*. (2005).

Equation (4.2) reveals, in agreement with the predictions of geometrical optics, that when(4.4)where Rem (*a*, *b*) is the remainder of the division of *a* by *b*, an image of the field at the source is formed at the location (*r*_{0}, 2*β*−*φ*_{0}) inside the metamaterial wedge (when the inequality holds) or on the boundary (when the equality holds).

For an odd geometry (metamaterial wedge of angle *β* on top of a ground plane), the symmetric part of (4.2) reveals that when(4.5)an image of the field at the source is formed at the boundary (*r*_{0}, *β*). The symmetric part of (4.3) shows a zero electric field along the plane *φ*=(*φ*_{0}+*β*)/2 outside the wedge.

One should note that, contrary to an infinite planar interface, a metamaterial wedge geometry, which is formed of truncated planar interfaces, is incapable of forming a faithful image of the source. This can clearly be seen from (4.2) where, even though the field at the image location is infinite and has a logarithmic nature as the observation point approaches the image location, the expression in (4.2) is not equal to the field of a line source, namely .

## 5. Numerical results

Using the developed numerical scheme, the electric field modulus |*E*_{z}| is calculated for line source excitation and plane wave illumination in the near- and far-field regions, with *β*=2.521, *φ*_{0}=*π*/4 and (for the line source case). Figure 2 shows the plots |*E*_{z}| due to a unit strength line source when and , along with |*E*_{z}| due to a unit strength line source when and (ordinary material wedge).

For the results in figure 2, eight terms from the Neumann series of *A*_{2}(*ν*) and subsequently residues with *ν*_{pl}≤5.02 for the symmetric part and the antisymmetric parts were required to produce fields within 1.1% accuracy in comparison with the finite-difference frequency-domain simulation.

Figure 3 shows the same plots when *ϵ*_{2}/*ϵ*_{1} and *μ*_{2}/*μ*_{1} are changed to and . Six terms from the Neumann series of *A*_{2}(*ν*) and subsequently residues with *ν*_{pl}≤5.02 for the symmetric part and the antisymmetric parts were sufficient to produce results within 1.4% accuracy in comparison with the finite-difference frequency-domain simulation.

The magnitude of *E*_{z} due to a normally incident unit strength plane wave illumination, when , and along with |*E*_{z}| due to the same excitation when and (ordinary material wedge) are shown in figure 4.

Eight terms from the Neumann series of *A*_{2}(*ν*) and subsequently residues with *ν*_{pl}≤6.23 for the symmetric part and *ν*_{pl}≤7.40 for the antisymmetric part were sufficient to produce results within 2.3% accuracy in comparison with the finite-difference frequency-domain simulation.

The developed finite-difference frequency-domain algorithm was detailed by Salem *et al*. (2006) and will not be repeated here.

## 6. Conclusions

The application presented here, together with that by Salem *et al*. (2006), establishes the Kontorovich–Lebedev formulation as a viable solution strategy for diffraction problems in wedges with field continuity-type boundary conditions on their faces. The proposed problem solving strategy is applicable for two- and three-dimensional problems of thermal conductivity, acoustics and elastodynamics wedges and cones. The work presented here could be extended to the problems of monochromatic as well as transient diffraction by moving wedges and cones and by wedges and cones composed of single-negative materials.

## Footnotes

- Received January 25, 2008.
- Accepted March 14, 2008.

- © 2008 The Royal Society