## 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 **k**_{i} that has a *z* component *k*_{z} and angular spherical coordinates θ_{i} and ϕ_{i}. Then the *z* dependence of the fields is a factor exp(*ik*_{z}*z*). We designate by _{x}, _{y}, and _{z} 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, *E*_{z}(** ξ**) and

*H*

_{z}(

**), where**

*ξ***=**

*ξ**x*

_{x}+

*y*

_{y}, and we decompose vectors into longitudinal and transverse parts, which are parallel and perpendicular to the generator of the cylinder, respectively. We set

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

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

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

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*)

_{x}+

*Y*(

*s*)

_{y}in terms of the arc length

*s*. We assume that

*C*starts at the edge and goes through the third quadrant first.