### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Solution by Separation of Variables
- 3. Special Wedge Angles
- 4. Determination of the Modal Expansion Coefficients
- 5. Conclusion
- Appendix A:: Function
*F*(*ν*) and the Sommerfeld Functions in the Angular Regions - Acknowledgments
- References

[1] The problem of diffraction by an isorefractive wedge of arbitrary angle is solved with the method of separation of variables in the frequency domain, when the incident field is either a plane wave propagating in a direction perpendicular to the edge of the wedge or a line source parallel to the edge. The dispersion relation is derived, and is solved explicitly for certain wedge angles that are commensurable to *π* radians. The modal expansion coefficients in the eigenfunction series solutions for the electromagnetic field components are determined as residues of suitable meromorphic functions.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Solution by Separation of Variables
- 3. Special Wedge Angles
- 4. Determination of the Modal Expansion Coefficients
- 5. Conclusion
- Appendix A:: Function
*F*(*ν*) and the Sommerfeld Functions in the Angular Regions - Acknowledgments
- References

[2] The problem of diffraction by an isorefractive wedge was solved by *Osipov* [1993], who applied the Kontorovich-Lebedev (KL) transform to the diffraction of a cylindrical wave by a dielectric wedge and reduced that problem to the solution of two coupled integral equations of the second kind, whose integral operators vanish in the isorefractive case and therefore yield an explicit solution to the boundary-value problem. Unaware of Osipov's work, *Knockaert et al.* [1997] rediscovered the same solution a few years later.

[3] The solution obtained by *Osipov* [1993] and by *Knockaert et al.* [1997] is in the spectral domain of KL transforms. In order to express the electromagnetic field in the physical domain, an inverse KL transform resulting in integrals that are difficult to evaluate by saddle point must be performed. On the other hand, the saddle point method is easily applicable to Sommerfeld-type integrals. In order to obtain the wedge diffraction coefficients, one should then proceed from the algebraic solutions in KL domain to Sommerfeld solutions using a method developed by Malyushinetz, then evaluate the Sommerfeld integrals by saddle point. An easier approach in the case of plane-wave incidence is provided by the method of rotating waves developed by *Daniele* [2003], which allows for the direct determination of Sommerfeld integrals.

[4] It should be pointed out that exact results for specific isorefractive wedge structures and incident fields have been obtained by a variety of methods. The special case of four right-angle wedges isorefractive to one another and with a common edge was solved by Wiener-Hopf technique [*Daniele and Uslenghi*, 2002]; in particular, it yielded the exact result obtained previously by *Uslenghi* [2000] via geometrical optics when the intrinsic impedances of the media in the four quadrants satisfy a certain relationship. Some other wedge structures were also solved exactly by geometrical optics [*Uslenghi*, 1997, 2004a, 2004b]. The radiation by a line source located at the edge of an isorefractive wedge was obtained in closed form by separation of variables [*Daniele and Uslenghi*, 1999].

[5] An alternative approach valid for all wedge angles was introduced by *Uslenghi* [2001]. It is based on a straightforward solution of the boundary-value problem by separation of variables in the physical domain, as done by *Macdonald* [1902] for the perfectly conducting wedge. The dispersion relation is identical to that obtained for circular cylindrical waveguides containing an isorefractive sector [*Uslenghi and Roy*, 1997], and may be solved explicitly in some cases when the wedge angle is commensurable to *π* radians, i.e. the ratio of the wedge angle to *π* equals the ratio of two integers. However, the expansion coefficients in the infinite series of eigenfunctions for the electromagnetic field components cannot be found as easily as in the case of a metallic wedge [see, e.g., *Harrington*, 1961] because now the fields occupy two angular regions of space filled with different materials. This difficulty is overcome by the approach illustrated in this paper, wherein the modal expansion coefficients are determined by utilizing properties of meromorphic functions.

[6] In this paper, the solution by separation of variables is given in section 2, and the explicit solution of the dispersion relation is detailed in section 3 for several wedge angles. The determination of the modal expansion coefficients is performed in section 4. The time-dependence factor exp(+*jωt*) is omitted throughout.

### 2. Solution by Separation of Variables

- Top of page
- Abstract
- 1. Introduction
- 2. Solution by Separation of Variables
- 3. Special Wedge Angles
- 4. Determination of the Modal Expansion Coefficients
- 5. Conclusion
- Appendix A:: Function
*F*(*ν*) and the Sommerfeld Functions in the Angular Regions - Acknowledgments
- References

[7] In circular cylindrical coordinates, the faces of the wedge under examination are the half planes ϕ = *α* and ϕ = −*α*, whose common edge coincides with the z-axis. The angular region −*α* ≤ ϕ ≤ *α* is filled with a linear, homogeneous and isotropic material characterized by an electric permittivity ε_{2} and a magnetic permeability *μ*_{2} or, equivalently, by a wavenumber *k* = *ω*(ε_{2}*μ*_{2})^{1/2} and an intrinsic impedance *Z*_{2} = (*μ*_{2}/ε_{2})^{1/2}, where *ω* is the angular frequency. The complementary angular region *α* ≤ ϕ ≤ 2*π* −*α* is filled with a linear, homogeneous and isotropic material characterized by an electric permittivity ε_{1} and a magnetic permeability *μ*_{1} or, equivalently, by a wavenumber *k* = *ω*(ε_{1}*μ*_{1})^{1/2} and an intrinsic impedance *Z*_{1} = (*μ*_{1}/ε_{1})^{1/2}. The isorefractive condition ε_{1}*μ*_{1} = ε_{2}*μ*_{2} is assumed. Without loss of generality, the semi-aperture angle *α* of the wedge may be restricted to the interval 0 ≤ *α* ≤ *π*/2.

[8] The primary field may be either a plane wave propagating in a direction perpendicular to the edge of the wedge and whose Poynting vector forms the angle ϕ_{0} with the negative x-axis, or a line source parallel to the edge and located at position (*ρ*_{0}, ϕ_{0}). In either case, a two-dimensional boundary-value problem must be solved.

[9] For the case of E-polarization, the total field may be written as:

where *l* = 1 for the angular region *α* ≤ ≤ 2*π* − *α*, and *l* = 2 for the angular region −*α* ≤ ≤ *α*; in both regions 0 ≤ *ρ* < ∞. The parameters define the characteristic impedance of the medium = 1, 2. The field components *E*_{lz} and *H*_{lρ}, that are both needed to impose the boundary conditions, may be written in the form:

where *a*_{lν} and *b*_{lν} are constants,

and *R*_{ν} is an appropriate Bessel function of order *ν*. The choices of ϕ_{l} and the introduction of the normalization factor will simplify the calculation of the expansion coefficients in section 4.

[10] The continuity of *E*_{z} and *H*_{ρ} across the faces of the wedge leads to the following dispersion relation:

where

[11] The solution of equation (5) yields the set of allowed values for the separation constant *ν*.

[12] Equation (5) was first obtained in studying the propagation inside a circular waveguide or a coaxial cable sectorally filled with isorefractive material [*Uslenghi and Roy*, 1997]. It has the following interesting properties: (1) it is invariant under the interchange of *Z*_{1} and *Z*_{2}; (2) it is invariant when *α* is replaced by *π* − *α*; (3) *ν* = 0 is a solution; (4) if *ν* is a solution, then −*ν* is also a solution.

[13] The last two properties allow us to limit our considerations to non-negative values of *ν*, and to choose:

where

[14] The choices (7) and (8) with *ν* ≥ 0 imply that both the edge condition and (whenever appropriate) the two-dimensional radiation condition are satisfied.

[15] The situation of a wedge that is a perfect electric conductor (PEC) is obtained as a limit, by letting either *Z*_{1} or *Z*_{2} approach zero in (5), (6). In either case, *η* tends to infinity and (5) yields the two solutions:

where *n* is a positive integer. Case (10) corresponds to a penetrable wedge region of aperture angle 2*α* with PEC faces (concave metal region). Case (11) corresponds to a penetrable wedge region of aperture angle 2(*π* − *α*) with PEC faces (convex metal wedge). Both cases confirm previously known results.

[16] The dispersion equation (5) can be factorized in the form:

Consequently, besides the solution *ν* = *ν*_{0} = 0, the zeros are also given by the solutions of the two equations:

The solutions of (5) form a countable set: *ν* = *ν*_{n} where *n* is an integer and

This property will be shown in the next section, for the case of rational values of *α* (*α* = *π* with *h*, *m* integers with *h* ≤ *m* rational). However it holds even if *α* is irrational. In fact by putting: *α* = *π* + and arbitrarily small, the Taylor expansion yields:

It will be shown:

consequently equation (16) provides the condition (15) even when *α* is irrational. With the property (15) and if and are bounded as *n* ∞, (2) and (3) are convergent series. This follows from the D'Alembert ratio test for the convergence of the series, taking into account the asymptotic expansion for large values of *ν* of the function *R*_{ν} (*kρ*).

### 3. Special Wedge Angles

- Top of page
- Abstract
- 1. Introduction
- 2. Solution by Separation of Variables
- 3. Special Wedge Angles
- 4. Determination of the Modal Expansion Coefficients
- 5. Conclusion
- Appendix A:: Function
*F*(*ν*) and the Sommerfeld Functions in the Angular Regions - Acknowledgments
- References

[17] Let us consider the cases when the ratio of the wedge aperture angle 2*α* to *π* radians is equal to the ratio of two integers:

Set

then (5) becomes:

Since cos *mξ* is a polynomial of degree *m* in *u* = cos *ξ*, then the dispersion relation (19) is an algebraic equation of degree *m* in *u*. This equation is exactly solvable by radicals for *m* ≤ 4 (and may also be solvable exactly in some cases for *m* > 4).

[18] The case *m* = 2, *h* = 1 corresponds to a 90° wedge (*α* = *π*/4). The solution of (19) yields:

where *n* is an integer. For a PEC right-angle wedge (*η* = ∞), solution (20) yields *ν* = 2*n*/3, which is a well-known result.

[19] The case *m* = 3, *h* = 1 corresponds to a 60° wedge (*α* = *π*/6), and (19) becomes:

[20] The case *m* = 3, *h* = 2 corresponds to a 120° wedge (*α* = *π*/3), and (19) becomes:

The solution of (22) yields:

where *l* = 0, 1, 2 and *n* is an integer. For a PEC wedge (*η* = ∞), solution (23) gives *ν* = *n*, a well-known result.

[21] The case *m* = 4, *h* = 1 corresponds to a 45° wedge (*α* = *π*/8), and (19) becomes:

[22] Finally, the case *m* = 4, *h* = 3 corresponds to a 135° wedge (*α* = 3*π*/8), and (19) becomes:

[23] The algebraic equations (21), (24) and (25) also can be solvable by radicals. However, the expressions become quite cumbersome. In general the algebraic equation of order m admits m solutions of the quantity *u*_{i} = cos , *i* = 1, 2,.*m*. For every m, we get:

that is in accord with condition (15).

### 4. Determination of the Modal Expansion Coefficients

- Top of page
- Abstract
- 1. Introduction
- 2. Solution by Separation of Variables
- 3. Special Wedge Angles
- 4. Determination of the Modal Expansion Coefficients
- 5. Conclusion
- Appendix A:: Function
*F*(*ν*) and the Sommerfeld Functions in the Angular Regions - Acknowledgments
- References

[25] The modal expansion coefficients satisfy the equations:

where:

To obtain explicit expressions for *a*_{lν} and *b*_{lν}, we interprete these coefficients as residues of a suitable meromorphic function. To familiarize ourselves with this approach, let us first consider the case in which the two media *l* = 1 and *l* = 2 are identical. In this case, the total field coincides with the primary field and we have:

where:

In relation to *a*_{ln}^{i} and *b*_{ln}^{i}, we introduce the meromorphic function *F*^{i}(*ν*) defined by:

where the series have been evaluated in closed form by using the formula:

The function *F*^{i}(*ν*) has poles at *ν* =±*n* (*n* = 0, 1, 2,…) with residues *a*_{ln}^{i} and *b*_{ln}^{i}, and excluding these poles, it vanishes as *ν* ∞.

[26] When two different media are present, the function *F*^{i}(*ν*) can be replaced by the unknown function:

This function has its poles (with residues *a*_{lν} and *b*_{lν}) at the zeros of the dispersion relation (5) and it is interesting to observe that it is related by a suitable Fourier transforms to the Sommerfeld functions of the two regions 1 and 2 (see Appendix A). Let us indicate with *F*^{s}(*ν*) the scattered field function:

Excluding the poles ±*n* and ±*ν*_{n} (*n* = 0, 1, 2.), both functions *F*^{s}(*ν*) and *g*(*ν*) · *F*^{s}(*ν*) vanish as *ν* ∞. The proof of this statement is cumbersome and is not reported here. However, we can verify these properties a posteriori, after we obtain the explicit evaluation of *F*^{s}(*ν*) (see equations (38)).

[27] Since *g*(*ν*) is an entire function, the poles of *g*(*ν*) · *F*^{s}(*ν*) are the poles *ν* = ±*ν*_{n} (*n* = 0, 1, 2,…) pertaining to the total field plus the poles *ν* = ±*n* (*n* = 0, 1, 2,…) pertaining to the primary field. The Mittag-Leffler expansion of *g*(*ν*) · *F*^{s}(*ν*) yields:

[28] Because of the boundary conditions (27), the first term in the right-hand side of (35) vanishes, thus yielding an explicit expression for the scattered meromorphic function *F*^{s}(*ν*):

Considering the expressions of *g*(*n*), *a*_{ln}^{i} and *b*_{ln}^{i}, the following series are involved in the solution (36):

The above series have been evaluated in closed form by using formula (32).

[29] Taking into account these results, a tedious calculation yields the following explicit expressions for the four components *F*_{i}^{s}(*ν*), (*i* = 1,.4) of *F*^{s}(*ν*):

where:

It is immediate to verify that *F*^{s}(*ν*) vanishes when the two media are the same (*Z*_{2} = *Z*_{1}), yielding the expected result:

Also, it is easily verified that *F*^{s}(*ν*) and *g*(*ν*) *F*^{s}(*ν*) vanish as *v* ∞.

[30] In order to better understand the physical meaning of *F*^{s}(*ν*), let us consider its poles. They arise from the zeros of sin (*πν*): *ν* = ±*n* (*n* = 0, 1, 2,…), and from the zeros *ν* = ±*ν*_{n} of the characteristic equation:

The first set defines the poles of the primary field *F*^{i}(*ν*), while the second set defines the poles of the total field *F*(*ν*). Since *F*(*ν*) = *F*^{i}(*ν*) + *F*^{s}(*ν*) does not have the poles *ν* = ±*n* of the primary field, the residues of *F*^{s}(*ν*) must exactly compensate those of *F*^{i}(*ν*). With the exception of some particular cases (for example, the PEC wedge), the proof of this fact is cumbersome. However, we checked the absence of offending poles in *F*(*ν*) by evaluating numerically several power series of *F*^{s}(*ν*) about the points *ν* = ±*n*, (*n* = 1, 2,…).

[31] In the PEC case, we observe that when *Z*_{2} = 0 equations (38) reduce to:

By indicating with *F*_{3}^{i}(*ν*) and *F*_{4}^{i}(*ν*) the components of *F*^{i}(*ν*) relevant to the interior of the wedge, trigonometric manipulations yield:

thus correctly yielding zero field in the interior of the PEC wedge.

[32] By indicating with *F*_{1}(*ν*), *F*_{1}^{i}(*ν*) and *F*_{2}(*ν*), *F*_{2}^{i}(*ν*) the first two components of *F*(*ν*) and *F*^{i}(*ν*), trigonometric manipulations yield:

Equations (45) and (46) show that there are no offending poles at *ν* = ±*n*, (*n* = 1, 2,…) (the poles of the primary field). The poles of *F*_{1}(*ν*) and *F*_{2}(*ν*) are only those of the PEC wedge:

The evaluation of the residues at these poles yields:

which is the well known result of the MacDonald expansion coefficients for the PEC wedge, since:

[33] In the general case, for *n* ≠ 0 the coefficients in the series (30) are expressed by the residues of *F*(*ν*) = *F*^{s}(*ν*) + *F*^{i}(*ν*) at the poles *ν*_{n}. Since *F*_{1}^{i}(*ν*) does not present these poles, we evaluate the residues of *F*_{1}^{s}(*ν*):

where:

Since *F*_{1}(*ν*) and *F*_{3}(*ν*) are regular at *ν* = 0, we obtain the expected result:

For *F*_{2}(*ν*) and *F*_{4}(*ν*), the pole located at the origin yields:

thus providing the value of *E*_{z} at the edge:

which vanishes for a PEC wedge (*Z*_{2} = 0) and yields *E*_{z} (0) = *E*_{o} when *Z*_{2} = *Z*_{1}.