Radio Science

Isorefractive wedge

Authors


Abstract

[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

[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

[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 ε2 and a magnetic permeability μ2 or, equivalently, by a wavenumber k = ω2μ2)1/2 and an intrinsic impedance Z2 = (μ22)1/2, where ω is the angular frequency. The complementary angular region α ≤ ϕ ≤ 2πα is filled with a linear, homogeneous and isotropic material characterized by an electric permittivity ε1 and a magnetic permeability μ1 or, equivalently, by a wavenumber k = ω1μ1)1/2 and an intrinsic impedance Z1 = (μ11)1/2. The isorefractive condition ε1μ1 = ε2μ2 is assumed. Without loss of generality, the semi-aperture angle α of the wedge may be restricted to the interval 0 ≤ απ/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 ϕ0 with the negative x-axis, or a line source parallel to the edge and located at position (ρ0, ϕ0). 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:

equation image

where l = 1 for the angular region αequation image ≤ 2πα, and l = 2 for the angular region −αequation imageα; in both regions 0 ≤ ρ < ∞. The parameters equation image define the characteristic impedance of the medium equation image = 1, 2. The field components Elz and H, that are both needed to impose the boundary conditions, may be written in the form:

equation image
equation image

where a and b are constants,

equation image

and Rν is an appropriate Bessel function of order ν. The choices of ϕl and the introduction of the normalization factor equation image will simplify the calculation of the expansion coefficients in section 4.

[10] The continuity of Ez and Hρ across the faces of the wedge leads to the following dispersion relation:

equation image

where

equation image

[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 Z1 and Z2; (2) it is invariant when α is replaced by πα; (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:

equation image
equation image

where

equation image

[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 Z1 or Z2 approach zero in (5), (6). In either case, η tends to infinity and (5) yields the two solutions:

equation image
equation image

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

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

equation image

Consequently, besides the solution ν = ν0 = 0, the zeros are also given by the solutions of the two equations:

equation image
equation image

The solutions of (5) form a countable set: ν = νn where n is an integer and

equation image

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

equation image

It will be shown:

equation image

consequently equation (16) provides the condition (15) even when α is irrational. With the property (15) and if equation image and equation image 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 Rν ().

3. Special Wedge Angles

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

equation image

Set

equation image

then (5) becomes:

equation image

Since cos 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:

equation image

where n is an integer. For a PEC right-angle wedge (η = ∞), solution (20) yields ν = 2n/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:

equation image

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

equation image

The solution of (22) yields:

equation image

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

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

equation image

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

equation image

[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 ui = cos equation image, i = 1, 2,.m. For every m, we get:

equation image

that is in accord with condition (15).

4. Determination of the Modal Expansion Coefficients

[24] The expansion coefficients a, b for an electric line source of intensity Io are identical to those for a plane wave of electric field amplitude Eo, if we replace Eo with equation imageIo. Thus, it is not restrictive to consider only the case in which the primary electric field is the plane wave Eoequation image. By indicating with ν = νn (n = 0, ±1, ±2,…) the zeros of the dispersion equation, in the following we introduce the coefficients equation image and equation image.

[25] The modal expansion coefficients satisfy the equations:

equation image

where:

equation image

To obtain explicit expressions for a and b, 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:

equation image

where:

equation image

In relation to alni and blni, we introduce the meromorphic function Fi(ν) defined by:

equation image

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

equation image

The function Fi(ν) has poles at νn (n = 0, 1, 2,…) with residues alni and blni, and excluding these poles, it vanishes as ν → ∞.

[26] When two different media are present, the function Fi(ν) can be replaced by the unknown function:

equation image

This function has its poles (with residues a and b) 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 Fs(ν) the scattered field function:

equation image

Excluding the poles ±n and ±νn (n = 0, 1, 2.), both functions Fs(ν) and g(ν) · Fs(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 Fs(ν) (see equations (38)).

[27] Since g(ν) is an entire function, the poles of g(ν) · Fs(ν) are the poles ν = ±νn (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) · Fs(ν) yields:

equation image

[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 Fs(ν):

equation image

Considering the expressions of g(n), alni and blni, the following series are involved in the solution (36):

equation image

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 Fis(ν), (i = 1,.4) of Fs(ν):

equation image

where:

equation image

It is immediate to verify that Fs(ν) vanishes when the two media are the same (Z2 = Z1), yielding the expected result:

equation image

Also, it is easily verified that Fs(ν) and g(ν) Fs(ν) vanish as v → ∞.

[30] In order to better understand the physical meaning of Fs(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 = ±νn of the characteristic equation:

equation image

The first set defines the poles of the primary field Fi(ν), while the second set defines the poles of the total field F(ν). Since F(ν) = Fi(ν) + Fs(ν) does not have the poles ν = ±n of the primary field, the residues of Fs(ν) must exactly compensate those of Fi(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 Fs(ν) about the points ν = ±n, (n = 1, 2,…).

[31] In the PEC case, we observe that when Z2 = 0 equations (38) reduce to:

equation image

By indicating with F3i(ν) and F4i(ν) the components of Fi(ν) relevant to the interior of the wedge, trigonometric manipulations yield:

equation image
equation image

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

[32] By indicating with F1(ν), F1i(ν) and F2(ν), F2i(ν) the first two components of F(ν) and Fi(ν), trigonometric manipulations yield:

equation image
equation image

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 F1(ν) and F2(ν) are only those of the PEC wedge:

equation image

The evaluation of the residues at these poles yields:

equation image
equation image

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

equation image

[33] In the general case, for n ≠ 0 the coefficients in the series (30) are expressed by the residues of F(ν) = Fs(ν) + Fi(ν) at the poles νn. Since F1i(ν) does not present these poles, we evaluate the residues of F1s(ν):

equation image

where:

equation image

Since F1(ν) and F3(ν) are regular at ν = 0, we obtain the expected result:

equation image

For F2(ν) and F4(ν), the pole located at the origin yields:

equation image

thus providing the value of Ez at the edge:

equation image

which vanishes for a PEC wedge (Z2 = 0) and yields Ez (0) = Eo when Z2 = Z1.

5. Conclusion

[34] An exact eigenfunction series solution is obtained for the boundary-value problem of electromagnetic scattering by an isorefractive wedge of arbitrary aperture angle. This new canonical solution yields the well-known MacDonald series solution for the particular case of a perfectly conducting wedge.

Appendix A:: Function F(ν) and the Sommerfeld Functions in the Angular Regions

[35] The exact MacDonald-type solution obtained in this paper is well suited for the calculation of near fields, but is not convenient for far-field evaluation. Far-field results are readily obtained if one knows the Sommerfeld functions involved in the two angular regions. In this Appendix A, we relate the function F(ν) obtained in section 4 with the rotating waves v1,2l (w) [Daniele, 2003] present in the two angular regions l = 1 and l = 2. The Sommerfeld functions sl (w) in these angular regions are expressed by:

equation image

The Fourier transform of F(ν) is defined by:

equation image

From (33), the residue theorem yields:

equation image

Alternative expressions of the four fi (w), (i = 1, 2.,4) components of f(w) can be obtained by considering the two rotating waves v1,2l (w) present in the two angular regions l = 1 and l = 2. Application of Laplace transforms to expression (3) provides the representations:

equation image
equation image

Letting equation imagel = 0 and comparing with equation A3 yields:

equation image
equation image

From these expressions, the relation between the Sommerfeld functions and the components Fi(ν) of the function F(ν) follows:

equation image

For rational values of the angle α, the evaluation of the integrals (A8) can be obtained in closed form by the residue theorem [Daniele, 2003]. To be concise for rational values of α, the substitution u = ejπνyields integrals having the form ∫0equation imagef(u)du where the function f(u) is rational in u and the parameter b is (linearly) related to equation imageo and assures the existence of the integral. Furthermore since we work with Fourier transforms defined in a distribution space, we observe that, in the presence of poles located on real half-axis 0 < Re[u] < ∞, this integral must be considered a Cauchy principal value integral. The evaluation of integrals having the form ∫0uequation imagef(u)du constitutes a classical exercise faced by the residue theorem, and we get:

equation image

where uk are the poles of f(u) located outside the real half-axis 0 < Re[u] < ∞ and xh the poles on the real half-axis. R(uk) and R(xk) are the residue of the integrand function equation imagef(u) in the poles uk and xk. The equation (A9) provides exact closed form expressions of the Sommerfeld functions equation image (w) for rational values of α. Irrational values of α can be studied by a perturbative approach such that used to obtain the asymptotic behaviors of the eigenvalues νn(α) (16).

[36] Given the Malyuzhinets functions, the structure of the far field can be obtained with well known standard techniques [Norris and Osipov, 1999; Senior and Volakis, 1995]. For instance the diffracted field is provided by the saddle point method. It yields:

equation image

Acknowledgments

[37] The authors are grateful to Daniele Monopoli for assistance in the preparation of the manuscript. This work was supported by NATO in the framework of the Science for Peace Program under grant CBP.MD.SFPP 982376.

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