## Abstract

Electromagnetic fields excited by a line source in the presence of an infinite dielectric wedge with refractive index *N* are determined by application of the Kontorovich–Lebedev transform. Singular integral equations for spectral functions are solved by perturbation procedure, and the solution is obtained in the form of a Neumann series in powers of . The devised numerical scheme permits evaluation of the higher-order terms and, thus, extends the perturbation solution to values of *N* not necessarily close to unity. Asymptotic approximations for the near and far fields inside and outside the dielectric wedge are derived. The combination of the Neumann-type expansion of the transform functions with the representation of the field as a Bessel function series extends solutions derived with the Kontorovich–Lebedev method to the case of real-valued wavenumbers and arbitrarily positioned source and observer. Numerical results showing the influence of wedges with various values of dielectric and magnetic constants on the directivity of a line source are presented and verified through finite-difference frequency-domain simulations.

## 1. Introduction

Electromagnetic wave propagation from one medium into a different infinite wedge-shaped medium is a difficult and unsolved problem. No explicit closed-form solution for a wedge with an arbitrary refractive index *N* has been obtained so far. A formulation based on the Kontorovich–Lebedev transform (Kontorovich & Lebedev 1938; Oberhettinger 1954; Rawlins 1972) is one of the most promising approaches to the solution of the dielectric wedge problem. Osipov (1993) applied the method to the case of a dielectric wedge excited by a line source and formulated singular and regular integral equations for the transform functions. He described the pole structure of the exact solution and presented a closed-form solution for the special case of the diaphanous wedge (*N*=1). For the almost diaphanous and for the strongly conducting wedge, an iterative solution of the regular integral equations was suggested. The case of the diaphanous wedge was analysed later in more details in Knockaert *et al*. (1997).

Rawlins (1999) studied diffraction of a plane wave by a dielectric wedge. He developed a perturbation scheme for the solution of the singular integral equations and constructed the transform functions in the form of Neumann-type series in powers of . The first-order correction terms were analytically computed, which permitted analytic continuation of the solution represented by Kontorovich–Lebedev integrals from the diffusion case (imaginary wavenumbers) to the diffraction case (real-valued wavenumbers). Numerical results for the electric field diffracted by a perfect dielectric wedge with were given.

This paper extends Rawlins' perturbation approach to a more general form of excitation (line source) of a dielectric wedge. Since the closed-form analytical evaluation of the correction terms is no longer possible, a numerical scheme is developed. We show that the correction terms can be efficiently evaluated at every level of the perturbation procedure, which permits solutions for values of *N* not necessarily close to unity.

Expansion of the solution in the series in powers of solves the problem of the analytic continuation of the solution, since the series coefficients are given by integrals that converge for real-valued and complex-valued wavenumbers. The problem of the anomalous convergence of the Kontorovich–Lebedev field representations, when the wavenumbers are not purely imaginary numbers and the source is placed far from the edge (Felsen & Marcuvitz 1973), is avoided by reducing the integrals into series of Bessel functions. An alternative case, in which the observer is located far from the edge, is treated by invoking the reciprocity property of the Green's functions.

In §2, the problem is formulated. The fields are represented by Kontorovich–Lebedev spectral integrals with unknown spectra. Singular integral equations for the spectra are deduced from meeting the continuity relations on the wedge faces. The derivations of this section follow the analysis of Osipov (1993) and Rawlins (1999). In §3, we apply Rawlins' perturbation scheme to the solution of the singular integral equations and extend the approach to the case of a line source. In §§4 and 5, we describe the reduction of the Kontorovich–Lebedev spectral integrals into residue series of Bessel functions and give useful representations of the fields inside and outside the wedge. In §6, numerical results are presented and compared with the results of finite-difference frequency-domain (FDFD) simulations. Conclusions are given in §7. Appendix A, making use of the analytic continuation process, describes the pole singularities of the Kontorovich–Lebedev spectra of the dielectric wedge and shows expressions for the computation of their respective residues.

## 2. Formulation

We consider the problem of scattering of harmonic electromagnetic waves by an infinite dielectric wedge embedded in an infinite medium (see figure 1). In what follows denote the usual cylindrical coordinates, such that on the surface of the wedge one has with . The region will be referred to as the *medium* I, while the region (i.e. the dielectric wedge) will be called the *medium* II. A time factor is assumed and omitted throughout. , and are, respectively, the wavenumber, permittivity and permeability of *medium* I. , and are corresponding quantities of *medium* II and . Since the problem under consideration is one of scattering and diffraction, the wavenumbers are real (complex) for the lossless (lossy) media. However, for a while we shall assume that are such that(2.1)Condition (2.1) transforms the Hankel functions in the subsequent Kontorovich–Lebedev integral representations into the modified Bessel functions *I* and *K*, and thus into the standard form of the Kontorovich–Lebedev transformation, for which the results of Kontorovich & Lebedev (1938) guarantee the convergence. This is required to be able to use the boundary conditions conveniently, leading to the derivation of integral equations on the Kontorovich–Lebedev spectra. If the wavenumbers are not purely imaginary, then the convergence of the Kontorovich–Lebedev representations is not guaranteed (Felsen & Marcuvitz 1973; Jones 1980; Osipov 1993). Once integral equations are derived and a numerical scheme is proposed to solve them, we examine the restrictions under which could be extended to real or complex values (to tackle the original scattering and diffraction problem) while maintaining the validity of the numerical scheme. It should be noted that the analysis carried out here is for the case . Modifications for the case are straightforward.

In what follows, the field excitation is provided by an impressed *z*-directed unit electric line current located at in and normalized with respect to . Extension to excitation by a normally (with respect to *z*) incident plane wave is also provided.

With and assumed constant, the longitudinal electric field component in , , satisfies the scalar inhomogeneous Helmholtz equation(2.2)where . stands for the transverse (with respect to *z*) Laplacian. The longitudinal electric field component in , , satisfies the scalar homogeneous Helmholtz equation(2.3)where .

The magnetic fields are derived from as(2.4)(2.5)where and .

By using the symmetry of the problem structure with respect to the planes , we split the problem into two independent sub-problems. The boundary conditions on the symmetry planes correspond to either an electric wall ( is zero on the wall) or a magnetic wall ( is zero on the wall). In what follows, the detailed analysis of the electric wall case in the angular domain will be reported. Only the difference with the electric wall case will be reported for the magnetic wall case. Numerical results will be presented for the original wedge geometry.

Tangential field components are continuous across the surface of the wedge, namely(2.6a)(2.6b)The fields are required to decay exponentially as . This replaces the Sommerfeld radiation condition for the case.

The field behaviour near the edge of the wedge is given by (Meixner 1972)(2.7)We propose to solve the problem by means of a Kontorovich–Lebedev transform (Kontorovich & Lebedev 1938)(2.8)(2.9)(2.10)where and are the standard Bessel and Hankel functions, respectively, and *ν* is purely imaginary.

Applying the Kontorovich–Lebedev transform to (2.2) and (2.3), we get the ordinary differential equations(2.11)(2.12)for spectra .

We represent the field in as the sum of an unperturbed field and a scattered field due to the presence of the wedge. Hence,(2.13a)

(2.13b)

### (a) The unperturbed field

Function satisfies the source conditions of (2.11), namely

is continuous across ,(2.14a)

is discontinuous across ,(2.14b)

From (2.14*a*), (2.14*b*) and the electric wall requirement at , we obtain(2.15)where is the greater|lesser of and .

It follows from (2.4), (2.10) and (2.15) that is given by(2.16a)

(2.16b)

(2.16c)

### (b) The scattered field

Since must satisfy the electric wall boundary condition at *ϕ*=0, we represent it by(2.17)Since must satisfy the electric wall boundary condition at we represent it by(2.18) and are Kontorovich–Lebedev spectra to be determined from the boundary conditions. The equation (Gradshteyn & Ryzhik 1980)(2.19)implies(2.20)(2.21)Additionally, the convergence of the Kontorovich–Lebedev integrals at *ϕ*=*β*, which is guaranteed by the condition in (2.1), implies that the spectral functions must vanish aswhen .

Substituting the fields in (2.6*a*) and (2.6*b*) with their Kontorovich–Lebedev representations, we obtain(2.22a)(2.22b)and(2.23a)

(2.23b)

### (c) Derivation of singular integral equations

While the boundary conditions in (2.22*a*) and (2.23*a*) are integral equations, more useful forms are derived by multiplying them by and integrating with respect to *r* from 0 to . To that end we make use of (Forristall & Ingram 1972; Rawlins 1999)(2.24a)(2.24b)(2.24c)where and are imaginary. is the Gauss hypergeometric function. For the particular case which is considered here (, ), the in (2.24*b*) can be removed.

We reach, after some algebraic manipulations, the singular integral equation for ,(2.25)(2.26a)(2.26b)(2.26c)(2.27a)(2.27b)(2.27c)(2.27d)and in front of the integral sign denotes that the Cauchy principal value is to be taken.

The singular integral equation satisfied by the Kontorovich–Lebedev spectrum in the magnetic wall case is derived from (2.25) by replacing by , and the remaining , by , , respectively, in (2.27*c*). Similar comments apply for the subsequent sections and will not be repeated.

## 3. Numerical scheme to solve the singular integral equation

It can be shown (Kamel & Osipov 2004) that a numerical solution of the singular integral equation with switched back from purely imaginary to real (complex for lossy media) values is only possible if(3.1)otherwise the integral equation includes terms growing as . The latter condition imposes severe restrictions on the position of the source with respect to the wedge face.

Hence, we devise a scheme, inspired by the Neumann series expansion approach introduced in Rawlins (1999), which is found to be valid for arbitrary source–boundary separations. We remark that Rawlins implements the Neumann series only to the first term, whereas the present approach goes to arbitrary order. The main steps of the procedure are as follows.

Multiply both sides of (2.25) by .

Expand all functions of

*N*in Neumann series in powers of , that is(3.2)(3.3)(3.4)(3.5)(3.6)(3.7)(3.8)Equate equal powers of from both sides of the equation.

Hence,(3.9)(3.10)(3.11)(3.12)The above defines an iterative scheme in which one starts with the known and generates the rest of the Neumann series coefficients by integrations. All integrands of such integrals decay exponentially as . Hence, the integrals and the subsequent coefficients , *n*>0 exist unconditionally. The evaluation of the principal value integrals above is carried out numerically by evaluating the integral from *ν*=0 to *σ*−*ζ*, then from to , where *ζ* is an arbitrary imaginary constant that is made as small as possible.

## 4. Field representations in the wedge region

### (a) Near field in the wedge region

As is shown in appendix A, is a meromorphic function, whose only singularities in the complex *σ*-plane are poles. It is worth noting that (see appendix A) is the Meixner's roots equation (Meixner 1972; Osipov 1993). Additionally, when higher-order poles exist, the electric field near the edge of the wedge may contain the logarithm of the distance in addition to its power (Makarov & Osipov 1986), namely , with and *m*, *n* being non-negative integers.

For , making use of (2.9) and (2.18), we obtain(4.1)Closing the integration contours in (4.1) in the right-hand side of the complex *ν*-plane and collecting residue contributions, we express(4.2)where the residues are given by (A 13*a*) and (A 14) in appendix A. Equation (4.2) is valid for first-order poles only. Modifications are required for higher-order poles if they exist. Since the poles accumulate at infinity (see (A 11)), the Kontorovich–Lebedev spectra may have an essential singularity at infinity. This could result in the divergence of the residue sum in (4.2). The truncated residue sum should, therefore, be understood as an asymptotic approximation in terms of first identified poles. The same comments apply to residue sums over in the following sections and will not be repeated.

Next we analyse the behaviour of the summand as . From Jones (2001),(4.3)(4.4)For *l*=0, , the series summand is dominated by(4.5)From (Felsen & Marcuvitz 1973)(4.6)(4.7)equation (4.5), characterizing the dominant behaviour of the series summand, reduces to(4.8)For , the dominant behaviour of the summand in (4.2) is given by(4.9)One notes that if the main interest is to compute the singular behaviour of the field near the edge of the wedge, then one could use(4.10)where is the singular field near the edge of the wedge, together with the corresponding sums from (2.4) and (2.5) for the magnetic field with the sums running to . One then needs to compute only the residues of for .

The near field for the magnetic wall case is recovered from (4.2) on replacing by and making use of the appropriate Kontorovich–Lebedev spectrum and pole singularities. A contribution of half a residue from the pole at *ν*=0 exists and quantifies the field at the edge of the wedge. Similar comments apply to the subsequent sections and will not be repeated.

### (b) Far field in the wedge region

It is known (Felsen & Marcuvitz 1973) that the Kontorovich–Lebedev representation diverges when the rate of exponential decay of with *ν* is less than the rate of exponential increase of , and one needs to continue the integrand analytically to get the far field. With the lack of an analytic formula for , such a continuation needs to be done numerically, which is very cumbersome for the dielectric wedge problem. We propose an alternative method to calculate the far field. We invoke the reciprocity principle stating that at the observation point , due to an impressed *z*-directed unit electric line current located at , is equal to at the observation point , due to an impressed *z*-directed unit electric line current located at . Hence, to calculate the fields when , we employ(4.11)where the sum is over the residues of , with the source located at in and the observer located at in . It is worth mentioning that the above residue sum cannot be used to derive far field results for plane wave illumination, since then the convergence is an asymptotic result for source near the edge of the wedge.

For such a reciprocal situation, satisfies the singular integral equation(4.12)

(4.13a)

(4.13b)

(4.13c)

(4.13d)

(4.13e)

(4.13f)

### (c) The plane wave illumination case

The field due to a normally (with respect to *z*) incident plane wave, , is recovered by replacing by in the line source results. Hence,(4.14)where is the Kontorovich–Lebedev spectrum for the plane wave case.

From (4.3), (4.4) and (4.6), the series summand in (4.14) is dominated by(4.15)(4.16)The above indicates that as increases, the number of residues required to produce a good approximation to the field increases as well. Hence, while (4.14) is suitable for field computations when is small or moderate, it is not suitable for large .

## 5. Fields outside the wedge region

The spectral amplitude is obtained from on making use of(5.1)(5.2a)(5.2b)(5.2c)Equation (5.1) reveals that is meromorphic with pole singularities at and, when , (with , ).

For , the field is given by(5.3)(5.4a)(5.4b)(5.4c)Similar to the results in §4*a*, the above series converge exponentially. The singular field behaviour near the edge of the wedge is expressed similar to (4.10).

For , the field is again found by invoking the reciprocity. The field expressions result from (5.3)–(5.4*c*) upon interchanging and . Comments, given in §4*b*, regarding the convergence of the series representations apply.

The field due to a normally (with respect to *z*) incident plane wave is recovered in a manner similar to that described in §4*c*. Comments given there regarding the suitability of the representation as well as the convergence of the series apply.

## 6. Numerical results

Using the numerical schemes developed in §3, the electric field modulus is calculated for line source excitation and plane wave illumination in the near and far field regions, with , and . For the line source excitation, was used.

Figure 2 shows the plots of the electric field modulus due to a unit strength line source outside the source–boundary separation region , as discussed in §3. Two terms from the Neumann series of and subsequently residues with for the electric wall and the magnetic wall cases were sufficient to produce results within 0.2% accuracy in comparison with the FDFD simulation. Figure 3 shows the same plots when the source is inside the source–boundary separation region . Four terms from the Neumann series of and subsequently residues with for the electric wall and the magnetic wall cases were sufficient to produce results within 0.3% accuracy in comparison with the FDFD simulation. Fields due to a unit strength plane wave illumination, when the incident plane wave angle is outside the source–boundary separation region , are shown in figure 4.

Four terms from the Neumann series were sufficient to represent . For , residues with for the electric wall case and for the magnetic wall case were sufficient to produce results within 0.4% accuracy in comparison with the FDFD simulation. For , residues with for the electric wall case and for the magnetic wall case were sufficient to produce results within 0.3% accuracy in comparison with the FDFD simulation.

The electric field was also computed for line source excitation when *N* is increased to , 4.95 and 11. The computations were made with , , , and .

Nine, 12 and 28 terms from the Neumann series of and subsequently residues with for the electric wall and the magnetic wall cases were sufficient to produce results within 0.2, 0.19 and 0.66% accuracy in comparison with the FDFD simulations, respectively, for , 4.95 and 11.

Figure 5 shows the increase of the deviation of the computed field from the FDFD simulation, as the number of Neumann series terms used to represent is reduced.

The above computations, performed with increasing values of *N*, show that the proposed numerical scheme is suitable for field computations when the deviation of *N* from unity is small or moderate, whereas it is not suitable for large values of *N*.

The developed FDFD algorithm is based on a Cartesian staggered grid to discretize the space, with , where and are the step size in either direction and is the wavelength in . An absorbing boundary condition is used to truncate the computational domain, which requires(6.1)where * n* is the outward unit vector normal to the computational domain boundary.

A Gaussian function of the form was used to simulate the line source excitation, with *α* being an arbitrary constant that is chosen to depress the Gaussian function to 10% of its maximum value after two grid points. The plane wave excitation was simulated by imposing the condition(6.2)on a phase front line traversing the computational domain and starting from the lower (upper) right-hand corner of the domain for , and where * n* is the unit vector in the direction of propagation of the incident plane wave.

The FDFD has been tested on configurations of known analytic results (a line source in an infinite space and a line source in a parallel plane waveguide).

## 7. Conclusion

The approach of this paper is applicable to two- and three-dimensional problems of thermal conductivity, acoustics and elastodynamics in a wedge and a cone with boundary conditions, both of the continuity and impedance types, on the radial direction. Further extensions may include problems of monochromatic and transient diffraction by moving wedges and cones and by wedges and cones composed of left-handed materials.

## Footnotes

- Received July 29, 2005.
- Accepted February 10, 2006.

- © 2006 The Royal Society