On the electromagnetic field singularities near the vertex of a dielectric wedge



[1] The problem of electromagnetic scattering by a dielectric wedge has attracted attention recently because of reported disagreement between the theory and numerical experiments concerning a certain behavior of the fields near the wedge vertex. The goal of this paper is to show that what is reported has been not a disagreement but a misinterpretation of the existing theory, which predicts all of the numerically observed effects.

1. Introduction

[2] The problem of electromagnetic scattering by a dielectric wedge has attracted considerable attention because of numerous reports [Marx, 1993; Anderson and Solodukhov, 1978; Beker, 1991] of disagreement between numerical experiments and the theory [Meixner, 1972; Makarov and Osipov, 1986; Budaev, 1991]. The theory implies that the asymptote of the electromagnetic field near the vertex of a dielectric wedge is described by a generalized Meixner series [Makarov and Osipov, 1986; Budaev, 1991] which starts from the same leading term as the asymptote of the electrostatic field in a similar configuration. However, numerical results obtained by Marx [1993], Anderson and Solodukhov [1978] and Beker [1991] show only partial agreement with the prediction that an electromagnetic field near the edge of a dielectric wedge behaves like a static field, ‘and numerical experiments indicate that the discrepancy is not due to imprecision of the calculations [Marx, 1993]. Such a situation is not acceptable because it undermines the credibility of both the numerical and analytical approaches to the equations that model electromagnetic fields in dielectric bodies, and more generally, because it undermines the credibility of the mathematical models themselves.

[3] To clarify the issue, we analyzed the theoretical asymptotes of electromagnetic fields near the apex of dielectric wedges, and this analysis revealed that the asymptotes have ‘hidden’ second terms which are normally dominated by the leading terms, but, in certain circumstances, they become dominant. Here we revisit the problem considered by Marx [1993] and demonstrate that the discrepancies between the numerical simulations and the theory reported by Marx [1993] do not contradict the theory but, on the contrary, confirm the theory which, if applied correctly, predicts all of the observed effects.

2. Analysis

[4] Let (ρ, ϕ, z) be the standard cylindrical polar coordinates and let V1 and V2 be the wedges defined as

equation image

where β is an angle of the wedge V2. The wedge V1 is occupied by a homogeneous dielectric with permittivity ε1 and permeability μ1. Similarly, the wedge V2 has permittivity ε2 and permeability μ2. This notation coincides with that by Marx [1993], but for our purposes it is convenient to introduce an additional coordinate system (ρ, θ, z) related with the system (ρ, ϕ, z) by the formula

equation image

which makes it possible to define the wedges V1 and V2 by

equation image

which imply that −π < θ < π. The described configuration is illustrated in Figure 1 in which both of the coordinate axes ϕ = 0 and θ = 0 are shown.

Figure 1.

Geometry of the problem.

[5] To simplify the analysis, we restrict ourselves to the transverse magnetic mode (TM) where the disagreement reported by Marx [1993] between the theory and numerical simulation is especially noticeable. In this mode the electromagnetic field can be defined in terms of a scalar solution Hz of the Helmholtz equation

equation image

where λ is an arbitrary parameter determining the scaling of the polar radius ρ. The field Hz and its derivative (1/equation image)(∂Hz/∂ϕ) have to be continuous everywhere except possibly at the vertex ρ = 0. Other components of the electromagnetic fields equation image and equation image are defined by the formulas

equation image

[6] It is convenient to set the scaling parameter λ by λk1 = 1 which means that distances are measured in wavelength units. Then the structure of the field Hz near the vertex ρ = 0 is described in the coordinates (ρ, θ) by the generalized Meixner series [Budaev, 1991]

equation image

determined by the roots 0 ≡ t0±t1±… of the transcendental equations

equation image


equation image

[7] It is worth observing that since (7) has a fixed root t00± = 0, the series always contains the “subseries” A00± + A01ρ + A02ρ2 + …. In certain particular circumstances the generic series (6) has to be modified. Namely, if for some n > 0 a root tn± can be represented as tn± = tm± + q, where m, n and q are integers satisfying inequalities m < n and q > 0, then, instead of terms with ρt+ν and ρt+q+ν, the series should involve independent terms with ρt+ν(lnρ)l, l = 0, 1, …, L, where L is the number of all possible different representations tn± = tm± + q of the root tn±.

[8] The last modification, however, never affects the first few terms of (6) which are given by the expression

equation image

from which it follows that the fields Eϕ and Eρ have the local structure

equation image

with constants B10± and B01± proportional to A10± and A01±, respectively.

[9] It is instructive to observe that the middle terms in (10), which do not depend on the radius ρ, appear in electrodynamics but do not appear in electrostatics. Electrostatic fields near the vertex of the dielectric wedge are described by the Meixner series [Meixner, 1972] which may be considered as a particular case of the series (10) with Anv± = 0 for all ν ≥ 1. In this case the asymptotes corresponding to those in (10) degenerate to the expressions

equation image

which are, in general, similar to (10) but may diverge from (6) as cos(t1±θ) → 0 or sin(t1±θ) → 0.

[10] From elementary graphics shown in Figures 2 and 3it is easy to see that

equation image

and that

equation image

as either (ε1 − ε2) → 0 or as β → 0.

Figure 2.

Graphical estimate of the root t1 in the case ε2 > ε1.

Figure 3.

Graphical estimate of the root t1+ in the case ε2 > ε1.

[11] From (9) and (10) it follows that near the vertex Hz approaches a bounded constant and that the fields Eρ, Eϕ have singularities of the type ρt−1 or ρt−1 depending on which of the inequalities t1 < 1 or t1+ < 1 is valid.

[12] To analyze the fields Eρ, Eϕ in detail, we assume for definiteness and for compatibility with the examples from Marx [1993] that ε2 > ε1 which means, in view of (12), that t1 < 1 < t1+.

[13] Consider first the field Eϕ. From the inequality t1 < 1 < t1+ we see that, in general, this field becomes unbounded as ρ → 0, but if the vertex is approached along the bisector θ = 0, (ϕ = 90°, in the original coordinates) then B10sin(t1θ)ρt−1 = 0 and Eϕ remains bounded approaching the finite value EϕB01+, which explains the results in Figure 4.

Figure 4.

Comparison of Eϕ in (TM) mode (reproduction of Figure 4 from Marx [1993]).

[14] Figure 4 reproduces Figure 4 from Marx [1993], which compares the electromagnetic fields near the vertex of the dielectric wedge as predicted by the Meixner asymptotes [Meixner, 1972] with the numerical results obtained by Marx [1993] using a hypersingular equation (HIE) and with the results obtained by Marx [1990] using a singular integral equation (SIE). All of the reported results correspond to the configuration in Figure 1 with β = 90° and ε2 = 10ε1. Both of the axes in Figure 4 have logarithmic scales, and they correspond to the distance from the vertex and to the magnitude of the electric field Eϕ. The dotted line corresponds to the straight line log(Eϕ) = (t1 − 1)log(ρ) + const, which represents the Meixner asymptote. The solid and dashed lines correspond to the results obtained by the methods using hypersingular and singular integral equations, respectively. Since the first method is more accurate, we do not discuss any dashed lines below.

[15] The solid line labeled “1” shows the electric field Eϕ computed along the bisectrix θ = 0, and this is essentially a horizontal line, which agrees well with the prediction that EϕB01+ = const as ρ ≪ 1. Similarly, the solid line labeled “2” shows that Eϕ(ρ, θ) along the perpendicular direction θ = −90° approaches an inclined line, which agrees well with the prediction

equation image

based on the asymptote (10).

[16] Next we observe that (10) implies that for any θ ≠ 0, the length of the interval 0 < ρ < ρ*(θ, δ) where the relative error of the asymptotic approximation of Eϕ(ρ, θ) by the first term of (10) does not exceed δ > 0 can be estimated as

equation image

[17] Then, applying this formula to the configuration in Figure 1 with the permittivities related by ε2 = 5ε1, we get the estimates

equation image

which explains why some of the curves in Figure 5 fit the theoretical predictions while others do not. Indeed, if β > 30°, then the power (t1 − 1) varies from −0.15 to −0.2, and therefore the principal terms of the asymptotes in (10) should dominate starting from distances of 10−7 to 10−10, included in Figure 5. However, if β = 10°, then (t1 − 1) ≈ −0.1 and the principal term of (10) is expected to dominate at a distances smaller than 10−14 which are far beyond the range studied by Marx [1993].

Figure 5.

Dependence of Eϕ on the wedge angle in the TM mode (Figure 5 from Marx [1993]).

[18] Finally, we consider the field Eρ. It is clear from (10) that in general, Eρ(ρ, θ) grows as O(ρt−1) when ρ → 0, but if the vertex is approached along the rays θ = ±π/2t1 where the coefficient cos(t1θ) of the singular term in (10) vanishes, then Eρ does not grow and approaches a bounded constant. Unlike the case for the field Eϕ these directions are not fixed but depend on the particular configurations. However, if the root t1 is close to unity (t1 ≈ 1), then the rays θ = ±π/2t1 are close to the rays θ = ±90° referred to by Marx [1993] as the perpendicular directions. Therefore, if t1 ≈ 1 the field Eρ is expected to approach a constant along the perpendicular direction θ = ±90° which agrees with the results in Figure 6. Indeed, the curve of Figure 6 labeled “2” and “ε = 2ε0” corresponds to t1 ≈ 0.9 and it is close to a horizontal line as predicted by the theory.

Figure 6.

Dependence of Eρ on the permittivity in the TM mode (Figure 6 from Marx [1993]).

[19] As for the behaviors of the field Eρ along other directions, they are similar to the behavior of the field Eϕ, and (16) estimates the region where the field may be approximated by the first term of (10) with the relative error of 5%. These estimates obviously agree with the numerical results in Figure 6.

3. Conclusion

[20] Thus we see that the numerical results presented by Marx [1993] do not contradict the theory but, instead, confirm the theory: theoretical curves fit the numerical data in areas where they are expected to fit; they fail to fit in the areas where they should not fit; and the differences between the asymptotic and numerical results have the trends predicted by the analysis. This validates the computations by Marx [1993] and, at the same time, reconfirms that the generalized Meixner series (6) correctly describes electromagnetic fields near the vertex of the dielectric wedge. However, our analysis reveals that the leading term of the generalized Meixner series, which describes electrodynamic fields, may not always coincide with the leading term of the standard Meixner series [Meixner, 1972], which describes static fields. The discrepancy may occur in narrow domains where the leading term of the generalized Meixner series either completely annihilates or becomes comparable with lower-order terms, as happens in the examples considered above. Therefore the observations [Marx, 1993; Anderson and Solodukhov, 1978; Beker, 1991] that the electrodynamic fields near the vertex of the wedge do not behave like static fields are correct, but the theory gives no reason to expect overwise.


[21] This research was supported by NSF grant CMS-0408381.