Scattering of an arbitrary plane wave by a dielectric wedge: Integral equations and fields near the edge

Authors


Abstract

[1] The behavior of the field components near the edge has been shown to be that of the static fields, which is derived here without rigor for an infinite wedge. Fields scattered by a finite dielectric wedge illuminated by an arbitrary plane monochromatic wave are computed using either singular or hypersingular integral equations (SIEs or HIEs), derived by the single integral equation method. Field components are then computed near the edge of a finite wedge. Longitudinal components of the fields behave like constants, other components of the electric field behave like those in the transverse magnetic mode, and other components of the magnetic field behave like those in the transverse electric mode. Exceptions occur when approaching the wedge along the bisector. Boundary functions and transverse field components computed with SIEs rise more sharply than predicted approaching the edge after a range in which the agreement with those computed with HIEs is good.

1. Introduction

[2] The behavior of electromagnetic fields near the edge of a dielectric wedge has been the subject of a protracted effort both in terms of theoretical analysis and numerical experiments. Some components of these fields diverge and the rates of divergence are relevant, for instance, for the solution of problems in scattering by general cylinders with sharp edges using singular integral equations (SIEs). Even though the fields at a distance from the edge that is small compared to the wavelength might not be important, the unknown boundary functions, which are jumps in the normal derivatives of longitudinal auxiliary fields, also diverge, and knowledge of their rate of divergence would facilitate numerical solutions. Hypersingular integral equations (HIEs) lead to boundary functions that are constant near the edge, but integrations are much more complicated and time consuming. There is agreement between the results of computations using SIEs and HIEs except for very small distances, where the behavior of the field components computed with SIEs is more divergent than those computed with HIEs. A similar behavior is observed for the boundary functions in SIEs. The latter agree with the behavior of the static field and should be the correct ones. The lack of a rigorous solution of the scattering problem and disagreements between the numerical solution and the expected behavior of the fields make this a subject of continuing interest.

[3] The problem of scattering by a perfectly conducting wedge has been solved rigorously [Sommerfeld, 1896; MacDonald, 1904; Carslaw, 1919; Jones, 1964], and numerical experiments show agreement with the expected behavior [Marx, 1990a]. In Sommerfeld's approach the form of the analytic solution differs depending on whether the wedge angle is a rational multiple of π or not. Meixner [1972] extended his approach from perfectly conducting wedges to dielectric ones, although it was shown [Bach Andersen and Solodukhov, 1978] that some of the coefficients in the power series expansion Meixner derived become infinite for wedge angles that are rational multiples of π. In that case, powers of log ρ, where ρ is the radial distance from the edge, have to be included in the series [Makarov and Osipov, 1986; Marx, 1990b]. Still, the consensus is that the behavior of the fields near the edge of the wedge is that of static fields, which was later proved for oblique incidence by Bergljung and Berntsen [2001]. There are many different approaches to the determination of the fields scattered by a dielectric wedge, such as those by Kim et al. [1991a, 1991b] and by Vasil'ev et al. [1996]. A good general reference of this subject is a book by Van Bladel [1996].

[4] Here we derive a set of two SIEs and a set of two HIEs, either of which can be used to determine the fields scattered by a dielectric or finitely conducting cylinder of arbitrary cross section illuminated by an arbitrary plane monochromatic wave. We then use these equations to determine the fields scattered by a finite dielectric wedge and show the behavior of the fields near the edge of the wedge. By finite wedge we mean an infinitely long wedge of finite cross section. In section 2 we derive an equation for the exponents that gives the behavior of the fields near the edge of an infinite dielectric wedge in the static limit. In section 3 we derive the sets of SIEs and HIEs that are used to find the fields scattered by a cylinder of arbitrary finite cross section using the single integral equation method. In section 4 we discuss the methods used to obtain numerical solutions of the sets of equations, in section 5 we present the results of numerical experiments carried out to find the behavior of the field components near the edge of a finite dielectric wedge, and we conclude with some remarks in section 6.

[5] For a scatterer that is a cylinder of an arbitrary cross section we choose the z axis parallel to the generator of the cylinder. We assume that the incident fields are those of a plane monochromatic wave of frequency ω, with a time dependence exp(–iωt), and a propagation vector ki that has a z component kz and angular spherical coordinates θi and ϕi. Then the z dependence of the fields is a factor exp(ikzz). We designate by equation imagex, equation imagey, and equation imagez the unit vectors along the corresponding axes. A field that satisfies Maxwell's equations can be expressed in terms of two scalar fields defined in the xy plane, Ez(ξ) and Hz(ξ), where ξ = xequation imagex + yequation imagey, and we decompose vectors into longitudinal and transverse parts, which are parallel and perpendicular to the generator of the cylinder, respectively. We set

display math

where k is the wave number of the plane wave, λ is the wavelength, and ɛ is the permittivity and μ the permeability of the medium. The fields are then decomposed by setting

display math
display math

where the perpendicular parts are expressed in terms of the longitudinal components by

display math

The longitudinal components of the fields satisfy the Helmholtz equation in two dimensions,

display math

The time dependence is suppressed in what follows. In addition to satisfying the Helmholtz equation, the field components have to satisfy the boundary conditions at the interface. In Figure 1 the interface is the curve C, given by the parametric equations ξ(s) = X(s) equation imagex + Y(s)equation imagey in terms of the arc length s. We assume that C starts at the edge and goes through the third quadrant first.

Figure 1.

Finite cross section of a wedge.

2. Static Limit of Fields

[6] We consider the electromagnetic scattering of a plane monochromatic wave by an infinite wedge of angle β, permittivity ɛ2 and permeability μ2, surrounded by a medium characterized by ɛ1 and μ1. We assume that 0 < β < π. A static field does not define a direction of incidence, but here we consider the static limit of monochromatic fields with an incident plane wave. The z components of the fields satisfy the two-dimensional Helmholtz equation, and the solutions in polar coordinates have the form R(ρ)Φ(ϕ), where R and Φ satisfy

display math
display math

where τ is a separation constant. The tangential components of the perpendicular fields in terms of the normal and tangential derivatives of the longitudinal components, from equations (4), are

display math
display math
display math
display math

where the subindices + and − correspond to the regions V1 outside and V2 inside the wedge, respectively, and

display math

[7] The fields Ez in the transverse electric (TE) mode and Hz in the transverse magnetic (TM) mode behave like a constant plus a term proportional to ρτ, where the separation constant τ obeys the equation

display math

and where r = ∣μ1μ2∣/(μ1 + μ2) (TE) or r = ∣ɛ1 − ɛ2∣/(ɛ1 + ɛ2) (TM). Clearly, τ = 0 is always a solution of equation (13), and the next larger solution satisfies 0 < τ ≤ 1 [Meixner, 1972].

[8] We have to match the fields along the sides of the wedge for all ρ, thus the fields have to depend on ρ by means of the same solution, R(ρ), of equation (6). We solve equation (7) and set

display math
display math
display math
display math

where in this section we define V1 as the region between ϕ = 0 and ϕ = β to simplify the notation. In the static limit the solution of equation (6) for small ρ is R(ρ) ≈ ρτ, the lowest order term of the Meixner series. The continuity of Ez, equation image(s), ·E, Hz, and equation image(s), ·H, where equation image(s) is the unit tangent vector dξ(s)/ds, gives

display math
display math
display math
display math
display math
display math
display math
display math

where we set

display math

The solution of this set of homogeneous linear equations for A, B, A′, B′, A″, B″, A′″, and B′″ is the trivial solution unless the determinant of the coefficient, D(τ), vanishes. The vanishing of this 8 × 8 determinant gives a transcendental equation for τ that allows us to find the behavior of the components of the fields near the edge. This equation reduces to

display math

[9] We note that τ = 0 is a solution of equation (27). We define a new variable q by setting

display math

and equation (27) reduces to

display math

Then, once we determine q, equation (28) gives τ, which also depends on the wedge angle β.

[10] From equation (26) we obtain

display math

and we use equations (26) and (30) to derive

display math

[11] Although κ is a function of θi, when we substitute from equations (31) into equation (29) we obtain

display math

The coefficients of qn, n = 2, 1, 0, do not depend on θi, whence the solutions for q and τ are independent of the direction of incidence. Solving the quadratic equations, we find that the two nonnegative solutions,

display math

are precisely the values of r for the TE mode and for the TM mode shown after equation (13).

3. Integral Equations

[12] We first recall the solution of the homogeneous Helmholtz equation for a function that has a given jump across a closed curve C and whose normal derivative also has a given jump. We then derive SIEs and HIEs that allow us to find the fields scattered by a dielectric or finitely conducting cylinder when an oblique monochromatic wave is incident on that cylinder. Similar SIEs and HIEs were derived by Marx [1990c, 1993], and for acoustics by Kleinman and Martin [1988]. As stated in the latter, the contour should be smooth for derivatives and a unique normal to exist. Nevertheless, we use these equations for contours with edges such as that of the finite wedge and stay away from the edge itself, where the boundary function diverges for SIEs.

[13] The Helmholtz equations (5) can be written in the generic form (equation image2 + k2) U(ξ) = 0, where we have dropped the subindex ⊥. We assume that the function U satisfies this equation in the whole plane, satisfies the radiation condition at infinity, has a jump ϕ on C, and its normal derivative, ∂U/∂n, has a jump η on C. The distribution [Schwartz, 1966; Petit, 1980] that corresponds to the function U satisfies the equation

display math

where equation image(s) is the unit normal to C. The source contains distributions of the form f(s)δ(C), which is defined by its action on a test function F(ξ),

display math

where f(s) is a given function of the arc length. We use the Green function or distribution, which is essentially the fundamental solution of the Helmholtz equation, g(ξ) = (i/4)H0(1)(), to find the solution

display math

where the functionals G{η} and N{ϕ} evaluated at an arbitrary point ξ are defined by

display math

where R = ξξ′, R = ∣R∣, equation image = R/R, ξ′ = ξ(s′), equation image′ is the normal to C at the point ξ′ pointing into V1, and Hn(1) stands for the Hankel function of the first kind of order n that, in terms of the Bessel function Jn and the Neumann function Yn, is Hn(1) = Jn + iYn. We use Hankel functions of the first kind because they satisfy the radiation condition when the time dependence is exp(−iωt), and we interpret the improper integrals in equation (37) in the sense of principal values. The functionals G and N are defined for all ξ. The normal and tangential derivatives of G when ϕ = 0 are

display math

The functionals N′ and N″ are defined only if ξ is on a curve that provides normal and tangential unit vectors, equation image and equation image, at the point ξ(s).

[14] The values of U on the boundary C are given by

display math

and the normal derivatives on the boundary C, when ϕ = 0, are given by

display math

When ϕ does not vanish, the normal derivative of U is expressed in terms of M′, the normal derivative of N, which is

display math

The limit is evaluated as the point ξ approaches C along equation image. If χ → 0 and R → 0, then equation image′ ≈ equation image and H1 behaves like 1/R, whence the integrand behaves like 1/R2. This makes the integral hypersingular. If the jump in the derivative vanishes, the limit should be the same from both sides. There are complications due to the δ distribution behavior on C; this is discussed by Marx [1993]. It is clear that the integral is symmetric under an interchange of the field point and the source point. A related functional is the tangential derivative of N,

display math

When χ → 0, equation image·equation image′ → 0 and equation image·equation image → 0, whence this integral is not hypersingular.

[15] As shown in section 1, the transverse components of the fields in a problem with symmetry under translations along the z axis can be expressed in terms of Ez and Hz. For a perfectly conducting cylinder the TE and TM modes remain separate, and the problem decomposes into two independent problems. This is not the case for a dielectric cylinder, in which the polarization of the incident field is not maintained for the scattered field.

[16] The total fields in V1 are separated into incident and scattered fields; that is,

display math

The scattered fields satisfy the radiation condition.

[17] If we want to obtain a set of SIEs, we define auxiliary fields U1 and U1 that are equal to the longitudinal components of the scattered electric and magnetic fields, Ezsc and Hzsc, respectively, in V1, obey the same Helmholtz equations in V1 and V2, are continuous on the curve C, and the jumps in their normal derivatives are the unknown boundary functions η and η′, respectively. The auxiliary fields U2 and U2′ are equal to the total fields in V2 and vanish in V1. We impose the physical continuity conditions on the tangential components across C, i.e., on Ez, equation image·E, Hz, and equation image·H. Since the longitudinal components of the fields, Ez and Hz, are continuous across the boundary, so are the tangential derivatives ∂Ez/∂s and ∂Hz/∂s. The continuity of the tangential components Ht and Et obtained from equations (8) to (11) gives

display math

We use these relations to express the auxiliary fields in terms of the unknown boundary functions.

[18] From equation (36) and the definitions of the auxiliary functions we have

display math
display math

where G and N are the functionals in equations (37) using k instead of k. The subindex 1 or 2 refers to the constants of the medium in V1 or V2. The boundary conditions determine the jumps in U2, U2′, and their normal derivatives. We find

display math

and, using equations (44),

display math

We impose the conditions that the auxiliary fields U2 and U2′ vanish just outside V2, which by equation (39) become

display math
display math

We substitute from equations (47) and (48) into these equations and obtain two SIEs that reduce to

display math
display math

where the meaning of a composite functional such as N2G1 is

display math

The incident fields are given and the boundary functions η and η′ are determined from equations (51) and (52), which can be solved, for instance, by the point-matching method.

[19] The incident fields can be specified by giving the real and imaginary parts of the x and z components of the amplitude of the electric field. Then the y component is found from ki·E0, the magnitude of the electric field can be normalized so that E0·E0* = 1, and we compute the amplitude of the incident magnetic field from H0 = ki \times E0/μ1ω. Once η and η′ are known, the z components of the scattered electric and magnetic fields outside the wedge are computed by integration using equations (45). The x and y components, as well as the radial and azimuthal components, of the scattered fields are computed using equation (4).

[20] To obtain HIEs we define the auxiliary fields U1 and U1 to have jumps ϕ and ϕ′, respectively, across C and normal derivatives that have zero jumps. Instead of equations (45) we have

display math

and we retain equations (46). Now equation (39) gives

display math

on C, whence

display math

Instead of equations (47) and (48) we have

display math
display math

Note that ∂Ez/∂s and ∂Hz/∂s are the same on both sides of C, but they are expressed in terms of the incident and scattered fields only in V1. Furthermore equation imageN is not defined on C because the function U has a discontinuity across C, but the limits from either side of C are the same.

[21] From equations (49) and (50) we obtain the integral equations

display math
display math

The terms with the derivatives of ϕ and ϕ′ actually transform these equations into integrodifferential equations.

[22] The z components of the scattered electric and magnetic fields outside the wedge are now computed by integration using equation (54), and the x and y components are still computed from equation (4). We assume that the points where the fields are evaluated are not on the curve C, so that there is no jump that complicates the taking of the derivative.

[23] Thus we have to solve the singular integral equations (51) and (52) to obtain the boundary functions η and η′ or the hypersingular integral equations (59) and (60) to obtain ϕ and ϕ′. The field components are then computed by integration.

[24] We have used the same integral equations for complex permittivities, although we have not analyzed what happens in the static case. We have used similar SIEs in scattering by strips on substrates including conducting media such as Si and Cr, compared the results to those of other simulation programs and to measured results, and the agreement is reasonably good [Silver et al., 2002].

4. Numerical Solution of Integral Equations

[25] In this section we discuss the computation of the fields near the edge of a finite dielectric wedge, which is obtained by closing the two flat surfaces of the wedge with a circular cylindrical surface (see Figure 1). We match the position and the tangent between the flat and the cylindrical surfaces at both sides.

[26] To solve for the boundary functions in one of the sets of integral equations obtained in section 3, we transform them into linear algebraic equations by the point-matching method. We divide the curve C into a number of patches of length Δsm centered about s = sm and call the coordinates ξm. We assume that the unknown ηm or ϕm is approximately constant over each patch and set, for instance, G{η}(ξl) ≈ ∑mGlmηm, where

display math

where Rlm = ∣ξlξm∣. We have to take special care in the evaluation of the self-patch contributions to the integrals [Marx, 1985a], where the kernel diverges, and especially for the HIEs, the neighboring-patch contributions [Marx, 1985b], where the kernel varies rapidly. The self-patch coefficients are

display math

where κl is the curvature of C at the point ξl. When Δsl is small, the value of Mll′ above is large and inaccurate. We prefer to follow the regularization procedure described by Meyer et al. [1978], adapted to electromagnetic fields. We set ξ″ = ξ + χequation image and let χ → 0 to obtain the value of M′ at a point on C and rewrite equation (41) as

display math

where ∇ operates on ξ or ξ″ and equation image′ operates on ξ. We substitute

display math

in equation (63) and obtain

display math

The first integral vanishes because equation imageg = − equation imageg, and the curl of a gradient is identically zero. The third integral can be changed because g(ξ) satisfies the homogeneous Helmholtz equation when ξ ≠ 0, whence

display math

The first term is still hypersingular. We use the identity

display math

and integrate by parts to obtain, assuming that χ ≠ 0,

display math

[27] This relation is not particularly useful as it stands because we do not want to introduce dϕ/ds′ as another unknown. We substitute a constant for ϕ and find that, since in that case dϕ/ds′ = 0,

display math

[28] We use this identity to show that

display math

since ϕ(s) is not a function of s′. This subtraction regularizes the integral, since the first factor vanishes like R and the second becomes singular like 1/R2 in the limit χ → 0. The product can again be handled like a principal value if necessary. We rewrite equation (66) as

display math

Further vector manipulations allow us to rewrite this expression as

display math

and

display math

We evaluate the function M′{ϕ} at the center of the patch l and obtain

display math

The terms that contain Rlm when l = m need special treatment, since the Hankel functions diverge for Rll = 0. In the first sum the term with l = m was eliminated because it can be shown to vanish by expanding ϕ(s) about s′, while in the second sum this term reduces to the usual self-patch term proportional to H0(1)(kR). Reintroducing k, the self-patch coefficient is

display math

which has been found to be more accurate than that in equation (62). Since this expression for Mll′ contains a sum of matrix elements Mlm′ at all other patches, the Mlm′ have to be computed accurately to avoid large errors in the self-patch contribution.

[29] To compute the contributions from neighboring patches, we need to improve on the simple approximation of a constant integrand. Near the edge we concentrate the patches by choosing the size of the first patch and the number of patches on the straight part of the boundary and putting the sizes of the patches in a geometric progression. We expand the Hankel functions for small arguments up to a limit τH for kR, such as 10−3 or 10−4, that is specified as input. After the Hankel functions are expanded we can do the integrations analytically. We subdivide the patch if the approximation of a constant integrand is inaccurate. To estimate the effect of the variation of the integrand over the patch, we evaluate a comparable integral over the patch of length Δs centered about s0 such as

display math

and we compare this value to that obtained by assuming the integrand has a constant value, J′. If the ratio of the integrals J and J′ differs from 1 by more than a given small number τS such as 10−6 or 10−7, i.e., if ∣J/J′ − 1∣ > τS, we subdivide the patch to do the integration. We apply these methods to solve the equations for the boundary functions and again to obtain the field components by integration.

[30] To express the terms dϕ/ds and dϕ′/ds in terms of ϕ and ϕ′, we use a second-order expansion. The patches are not the same size, and we use a patch on each side except at the ends of the interval.

[31] Using these methods, we obtain the scattered fields for the finite wedge. We plot the magnitudes of the field components near the edge of the wedge approaching the wedge from different directions given by ϕ, and we mostly get the expected behavior except for some of the perpendicular components along the bisector of the wedge, ϕ = 90°. These exceptions have been explained by Budaev and Bogy [2007] for the examples presented by Marx [1993]. There is good agreement between results obtained via SIEs and HIEs except at distances smaller than about 10−5λ, where the field amplitudes obtained via the SIEs increase much faster than the predicted rate and than results obtained via HIEs. We show results for a particular example in section 5.

5. Results From Numerical Experiments

[32] We compute the field components for a plane wave of unit wavelength incident on a dielectric wedge with unit side and an angle of 60° from a direction of incidence given by the polar coordinates θi = 60° and ϕi = 60°. We show results for three polarizations described by Ex = 0, Ez = Ex, and Ez = 0. We assume that V1 is free space and choose the permittivity ɛ2 = 5ɛ0 and the permeability μ2 = 3μ0, although dielectrics normally have unit relative permeability. We determine the behavior of the field components as we approach the wedge along the directions ϕ = 0°, ϕ = 90°, and ϕ = 45°.

[33] The resulting magnitudes of the azimuthal components of the electric field, Eϕ, shown in Figure 2a, follow the power law ρτ − 1 as they approach edge of the wedge, where τ is obtained from equation (13) with the value of r for the TM mode, with the exception of those computed along the bisector of the wedge for all three polarizations. (In Figures 2, 3, and 4we show the relative values of the constants ɛ and μ of the medium.) The azimuthal components of the magnetic field, Hϕ, shown in Figure 2b, have the same behavior but for a value of τ obtained from equation (13) for r from the TE mode. The radial components, Eρ, all follow the corresponding power law, and the longitudinal components, Ez and Hz, behave like a constant near the edge of the wedge. The expected asymptotic behavior sets in somewhere between 10−2λ and 10−4λ. In Figure 3 we compare the curves for Hϕ computed using the HIEs with those computed with the SIEs, and we find that the agreement is quite good at distances greater than 10−5λ, especially if one considers that logarithmic scales tend to emphasize differences in small values. At smaller distances the fields computed using the SIEs grow at a much faster rate, a behavior that we have seen before. Since the fields computed with the HIEs generally follow one of the power laws derived in section 2, we conclude that the behavior of the fields computed with the SIEs very close to the edge of the wedge is incorrect. The computed values of the longitudinal components do tend to a constant when calculated with SIEs. Considering the difference between SIEs and HIEs and in the integrals used to compute the field components, we conclude that the computations are essentially correct except for boundary functions and transverse field components very near the edge for the SIEs.

Figure 2.

Components of the fields near the edge of the wedge.

Figure 3.

Comparison between field components computed with the SIEs and with the HIEs.

Figure 4.

Boundary functions for (a) HIE and (b) SIE.

[34] In Figure 4a we show the magnitudes of the boundary functions for the HIEs, which are essentially constant near the edge of the wedge. In Figure 4b we show the boundary functions for the SIEs, which first increase at a moderate rate for small distances from the edge of the wedge, and then increase at a much faster rate as one approaches still closer to the edge. We have not derived the rate of divergence of the boundary functions, which would be useful to take into account the behavior of boundary functions near edges without the need to have a large number of patches to represent these functions accurately.

[35] If μ2 = μ0, q′ vanishes, and equation (13) implies that the values of the perpendicular components of the magnetic field, Hρ and Hϕ, tend to constants. The computed fields show this behavior for all angles ϕ, including the bisector of the wedge. The computed Eϕ shows the expected behavior derived from q″, but not along the bisector. The unexpectedly high rate of increase of the perpendicular field components and boundary functions very close to the edge, computed using SIEs, is still present.

6. Concluding Remarks

[36] We have derived systems of two SIEs or HIEs that can be used to compute boundary functions on the interface between a dielectric or finitely conducting cylinder of arbitrary cross section and the surrounding medium. The fields scattered by the cylinder are then computed by integration. We computed the fields scattered by finite wedges, especially near the edge of the wedge. The agreement between the results obtained with SIEs and those obtained with HIEs, as well as with the asymptotic behavior derived in section 2, indicates that the equations and the programs that we used are essentially correct. Similar equations can be derived for configurations such as a cylinder on a substrate.

[37] The unknown boundary functions and the perpendicular field components computed via SIEs diverge as we approach the edge of the wedge as expected, but they unexpectedly diverge at a faster rate for distances less than about 10−5λ from the edge. The rapidly varying behavior of the kernels for contributions of patches near the self-patch in the HIEs causes difficulties that require special approximations for these neighboring patches. The regularization procedure for HIEs potentially can lead to sizeable errors because the self-patch contribution is computed by summing over those of all other patches. The precise behavior of the boundary functions of SIEs near an edge appears to have only a minor influence on the fields computed far from this edge.

[38] For oblique incidence the longitudinal components of the fields behave like constants near the edge of the wedge, and the transverse components mostly diverge like ρτ − 1, where τ is obtained from equation (13) with a value of r that corresponds to the TM mode for the electric fields and to the TE mode for the magnetic fields. Boundary functions and perpendicular components of the fields computed via the SIEs increase more sharply than expected at distances less than 10−5λ in the example in section 5. This incorrect behavior is also seen in most other cases we have analyzed. The agreement between the results of computations obtained from SIEs and HIEs at distances larger than that indicates that the equations are correct and that we probably need to overcome numerical problems with the SIEs at the shorter distances from the edge of the wedge.

[39] One stills needs to determine the behavior of the boundary function for SIEs or to find better ways of representing the hypersingular integrals for HIEs. Also of interest are the distances at which the asymptotic behavior sets in and any exceptions to the expected behavior such as that found along the bisector of the wedge. We have not solved the dynamical problem for infinite dielectric wedges, which should explain how the fields propagating at different speeds along both sides of the interfaces match. The behavior of fields near the edge of a finitely conducting infinite wedge, as well as the behavior of the boundary function for SIEs, remains to be determined. For a complex dielectric constant, τ is complex, and the behavior of the divergent field components near the edge is determined by the real part of τ.

Acknowledgments

[40] I wish to thank Andrey Osipov of the German Aerospace Center, Bruce Miller of NIST, Piergiorgio L. E. Uslenghi of the University of Illinois, Bair Budaev of the University of California, and Svend Berntsen of Aalborg University for helpful comments and suggestions.

Ancillary