Inhomogeneous electromagnetic plane wave diffraction by a perfectly electric conducting wedge at oblique incidence

Authors


Abstract

[1] The diffraction of an inhomogeneous electromagnetic plane wave obliquely incident on the edge of a perfectly conducting wedge is analyzed, to extend the results for the normal incidence case reported by Kouyoumjian et al. (1996). Uniform high-frequency expressions are obtained for the diffracted field in the format of the uniform geometrical theory of diffraction (UTD). The generalized dyadic diffraction coefficient in the standard UTD ray-fixed coordinate system is represented by a 2 × 2 full matrix with the extra-diagonal terms accounting for a coupling between the two components of the incident and diffracted electric field parallel and perpendicular to the edge-fixed incidence and diffraction plane, respectively. The latter characteristic of the diffraction matrix is due to both the vectorial properties of the evanescent incident electric field and the inclusion of higher-order terms in the asymptotic expression of the edge diffracted field. The introduction of higher-order terms in the asymptotic evaluation also guarantees an improved accuracy for smaller values of the large parameter, namely, for observation points closer to the diffracting edge, as compared with the ordinary UTD.

1. Introduction

[2] High-frequency methods are based on the definition of a suitable set of diffraction coefficients, which are determined by analytically solving specific canonical scattering problems. Each canonical problem is characterized by simple geometries (as for instance wedges, tips, corners), which however well approximate the exterior surface of the scattering body in the vicinity of the pertinent scattering center.

[3] In this context, the present paper deals with a high-frequency analysis of the diffraction of an inhomogeneous plane wave, impinging on the edge of a perfectly electric conducting (PEC) wedge at oblique incidence. The incident plane wave is arbitrarily polarized and exhibits an exponential variation of the field amplitude along an arbitrary direction in the plane perpendicular to the propagation direction. This situation may arise in practice when the incident field is a surface wave, a leaky wave, a lateral wave, or when propagation occurs in a lossy medium. For instance, a homogeneous plane wave impinging from air onto a planar interface between air and a lossy medium (ground) generates an inhomogeneous plane wave into the lossy medium. Consequently, a diffraction coefficient for inhomogeneous plane waves can be considered an important tool in the analysis of the interaction between electromagnetic waves and metallic objects buried underground [Bertoncini et al., 2001, 2004].

[4] A uniform geometrical theory of diffraction (UTD) solution [Kouyoumjian and Pathak, 1974] for inhomogeneous plane wave scattering by a PEC wedge has been presented by Kouyoumjian et al. [1996], when both the incidence and the exponential decay directions lie in the plane transverse to the edge. Here, the above analysis is extended to treat the more general case of skew incidence and completely arbitrary orientation of the exponential decay direction of the evanescent incident field. First, a representation of the incident field in terms of a plane wave with complex incidence angles is provided for a completely arbitrary incident inhomogeneous plane wave. Then, an exact integral representation for the total field is obtained by analytically continuing the Sommerfeld solution to complex incidence angles. The exact integral representation of the field is asymptotically evaluated by deforming the original Sommerfeld integration contour onto the two steepest descent paths (SDPs) through the saddle points at π and −π. The residue contributions of the poles that are captured in the contour deformation process are associated with the geometrical optics (GO) field contributions. The diffracted field contribution is obtained by asymptotically evaluating the integrals along the SDPs. To obtain the total field continuity at both the shadow boundaries (SBs) of the incident and reflected fields, all terms of order K−1/2 must be retained in the asymptotic approximation of the longitudinal field components, where K is the large parameter. Uniform asymptotic expressions are then derived for all the scattered field components, which are needed to construct a generalized dyadic diffraction coefficient. The asymptotic solution contains the conventional UTD transition function that is extended to complex arguments, as outlined by Kouyoumjian et al. [1996].

[5] The organization of the paper is described next. The canonical problem relevant to an inhomogeneous electromagnetic plane wave obliquely incident on a PEC wedge is formulated in section 2, where a detailed description of the electric and magnetic incident field is also given. An exact eigenfunction solution is provided in section 3. High-frequency uniform expressions for both the longitudinal and transverse components of the diffracted field are presented in section 4, and a generalized UTD dyadic diffraction coefficient is derived in section 5. Finally, samples of numerical results are shown in section 6, both to demonstrate the continuity of the total field across the GO SBs and to examine the accuracy of this high-frequency solution through comparisons with calculations based on the exact eigenfunction solution, as a function of the distance of the observation point from the diffracting edge.

2. Formulation of the Electromagnetic Problem

[6] The geometry for the scattering of an inhomogeneous plane wave by a PEC wedge is shown in Figure 1. In the figure, WA = (2 − n)π, with 1 ≤ n ≤ 2, denotes the interior wedge angle and P is the observation point which is located at (ρ, ϕ, z) in a cylindrical reference system with the z-axis overlapped to the edge of the wedge. An ejωt time dependence is assumed and suppressed in the following.

Figure 1.

Geometry for the scattering of an inhomogeneous plane wave impinging at oblique incidence on a PEC wedge.

[7] The wedge is illuminated by an inhomogeneous plane wave whose electric and magnetic fields can be cast in the following form:

equation image

where r = ρequation image + zequation image represents the observation point position vector and k is the complex wave vector relevant to the incident inhomogeneous plane wave.

[8] Following Clemmow [1996], the evanescence angle ψ is introduced (which is assumed to be non-negative), so that:

equation image

where equation image1 and equation image2 denote unit vectors pointing along the wave propagation direction and the direction of exponential field amplitude decay, respectively, and k denotes the wave number of the medium surrounding the wedge.

[9] The plane of incidence formed by the edge of the wedge and the real component of the wave vector of the incident plane wave is at equation image = equation image0. β0 is the angle between the wave propagation direction equation image1 and the edge of the wedge, measured on the above incidence plane. The direction of equation image2 in the plane perpendicular to the wave propagation direction is defined by the angle α0. In particular, when α0 = 0 it means that equation image2 lies in the incidence plane, and when α0 = π/2 it means that equation image2 is aligned with the unit vector equation image′ normal to the incidence plane.

[10] Introducing the third unit vector equation image3 = equation image1 × equation image2 and projecting the ray-fixed coordinate system unit vectors, equation image1, equation image2, equation image3, onto the cylindrical reference system, it follows:

equation image

where equation image and equation image are the unit vectors of the cylindrical reference system associated to the incidence plane.

[11] By using equation (3), the dot product k · r in equation (1) becomes

equation image

By introducing the two complex angles equation image and equation image so that:

equation image

the incident field can be expressed as

equation image

In particular, the incident field can be interpreted as a uniform plane wave impinging on the edge along a direction denoted by the two complex angles equation image and equation image0 + equation image. Moreover, Maxwell's equations require that

equation image

where Z represents the intrinsic impedance of the medium surrounding the wedge.

[12] By denoting with AE and AH the projections of E0 and H0 along the direction of equation image3, respectively, E0 and H0 can be expressed in terms of the above complex amplitudes [Clemmow, 1996] as follows:

equation image

[13] Equation (5) shows that an arbitrary inhomogeneus plane wave is completely determined once equation image and equation image are known, together with the projections of E0 and H0 in the direction perpendicular to the plane containing both equation image and equation image. It is also apparent that any inhomogeneous plane wave can be decomposed into a couple of contributions, which are characterized by (AE ≠ 0, AH = 0) and (AE = 0, AH ≠ 0), respectively. In the limit case of a homogeneous incident wave, i.e.ψ = 0, the above two contributions coincide with those relevant to the TE and TM polarization cases, respectively, when α0 = 0 (or vice versa when α0 = π/2). It is worth noting that, when ψ ≠ 0, the incident field also exhibits a non vanishing component along the propagation direction equation image. To summarize, the incident plane wave is completely identified once the angles α0, ϕ0, β0 and the evanescence angle ψ are known, together with the amplitude coefficients AE e AH.

[14] Because of the uniformity of the boundary conditions along the edge direction, all the fields components of the total field exhibit the same dependence on z as the incident field in equation (1). In particular, the ρ and ϕ components of the total electric and magnetic fields can be expressed as [Senior and Volakis, 1995]:

equation image

[15] Equation (6) shows that the transverse components of the total electric and magnetic fields can be expressed in terms of the electric and magnetic field components along the edge direction (longitudinal field components). Consequently, once an asymptotic approximation for the z-component of the electric and magnetic field has been determined, equation (6) can be used to derive approximate asymptotic expressions for the transverse components Eρ, Eϕ, Hρ and Hϕ.

3. Exact Eigenfunction Solution

[16] By analytically continuing the exact eigenfunction solution for the homogeneous plane wave scattering by a PEC wedge [Balanis, 1989], the z-components of the total fields E and H can be evaluated in terms of a series expansion containing the Bessel functions of the first type and fractional order, Jm/n, with a complex argument:

equation image

where

equation image

with equation image = k sin equation image, β± = equation image ± (ϕ0 + equation image) and

equation image

[17] Moreover, the longitudinal components (E0)z = E0 · equation image and (H0)z = H0 · equation image can be expressed in terms of AE and AH through equations (3) and (5) as

equation image

where the elements mij of the 2 × 2 matrix M are given by

equation image

[18] By using equations (7) and (8), the partial derivatives of I(ρ, β) with respect to ρ and ϕ can be expressed as

equation image

where the following property of the Bessel functions has been used:

equation image

[19] From equations (6), (7), and (10) it is now possible to calculate an exact eigenfunction expression for all the transverse components of the electric and magnetic fields. For instance, for Eρ and Eϕ the above mentioned procedure gives

equation image

Similar exact eigenfunction expressions can be easily derived for Hρ and Hϕ.

4. High-Frequency Asymptotic Analysis

[20] The series expansion in equation (8) converges rapidly for small values of ∣equation imageρ∣. On the contrary, for large values of ∣equation imageρ∣, a high-frequency asymptotic expansion for I(ρ, β) in inverse powers of ∣equation imageρ∣ is very useful, because of the slow convergence of (8).

[21] In order to derive an asymptotic expansion for I(ρ, β), the summation in equation (8) is first transformed into an integral along the Sommerfeld contour C′ − C [Balanis, 1989] and then evaluated asymptotically by the method of the steepest descent path. Each I(ρ, β) term is evaluated as a sum of two integrals along the steepest descent paths plus the residue contributions relevant to the poles that are captured in the contour deformation process:

equation image

where

equation image
equation image

The pole singularities of the function G(ξ) are located at

equation image

In equation (11), the residue summation and the SDP integrals account for the GO and the diffracted field contributions, respectively.

[22] Equations (11)–(13) coincide with those in Kouyoumjian et al. [1996] (also extended to the case of an impedance wedge in a lossy medium by Manara et al. [1998]), when the following substitutions are made

equation image

[23] On the basis of the above substitutions, the original vectorial problem reduces to the asymptotic evaluation of SDP integrals as those arising in Kouyoumjian et al. [1996], when the wedge is surrounded by a fictjous lossy medium exhibiting a complex wave number equation image. Although the saddle points of the SDPs remain at ξ = ±π, the presence of an immaginary part in equation image modifies the shape of the SDP integration paths in the complex plane ξ = x + jy, which are now obtained according to the equation

equation image

The angle denoting the SDP slope at the saddle point becomes equation imageπ + equation image and equation imageπ + equation image for ξ = +π and ξ = −π, respectively.

[24] A uniform asymptotic evaluation of the integrals along the SDP paths through the saddle points at ±π can be performed by resorting to Van der Waerden method [Van der Waerden, 1956]. In particular, by retaining all the terms of order (∣equation imageρ)−1/2, the longitudinal components of the diffracted field can be expressed as:

equation image

where:

equation image

Suitable expressions for the terms appearing in equation (18) can be directly derived from equations (40) and (41) given by Kouyoumjian et al. [1996], once the substitutions defined in (15) are made. It results that

equation image

where F[equation imageρa(β)] denotes the Kouyoumjian and Pathak transition function [Kouyoumjian and Pathak, 1974] extended to complex arguments and a(β) is a generalized distance parameter [Kouyoumjian et al., 1996]. The value of ξp in D+ and D− must be chosen on the basis of equation (14), by considering the closest pole to the SDP paths crossing the integration contour in the vicinity of the saddle points at +π and −π, respectively.

[25] It has been shown [Senior and Volakis, 1995] that in the limit case of a homogeneous incident plane wave (ψ = 0)

equation image

It follows that, close to the SBs, equation (17) reduces to the standard UTD dyadic diffraction coefficient for soft and hard polarization.

[26] Finally it is worth noting that the matrix in equation (17) is diagonal, as expected since the boundary conditions for the longitudinal field components are decoupled on the faces of a PEC wedge:

equation image

[27] In order to provide suitable asymptotic expressions also for the ρ and ϕ components of the diffracted field, equation (17) can be substituted into equation (6), as suggested by Büyükdura [1996]. It follows that:

equation image

where the elements tij of the 3 × 2 matrix T are given by

equation image
equation image
equation image
equation image
equation image
equation image

with

equation image
equation image

[28] The ρ and ϕ components of the diffracted field in equation (19) include a term of order k−1/2 plus other terms of higher-order (k−3/2) as already discussed by Büyükdura [1996] and by Kouyoumjian et al. [1983], where the homogeneous plane wave incidence case is analyzed. In the above cited papers, it is shown that the introduction of higher-order terms in the asymptotic analysis gives a solution for the diffracted field that not only compensates the jump discontinuities in the GO field but also the discontinuities in the GO field derivatives. Moreover, the solution with higher-order terms was found to be accurate for reduced values of the large parameter (normalized distance of the observation point from the diffracting edge) as compared to the ordinary UTD solution.

[29] For the sake of compactness, explicit expressions for the GO field are not given in this paper as they can be easily derived from those given by Kouyoumjian et al. [1996] for the 2D scattering problem. In particular, in section 4 of Kouyoumjian et al. [1996] it has been shown that the SBs are rotated with respect to the conventional optical boundaries. For the problem under analysis, the above displacement can be evaluated as

equation image

where y = −Im(−β±) and α has been defined in equation (13).

5. A Generalized UTD Dyadic Diffraction Coefficient

[30] In order to derive a generalized dydadic diffraction coefficient, equation (19) can be expressed in terms of the incident field components referred to the standard UTD ray-fixed coordinate system equation image1, equation image = equation image1 × equation image, equation image shown in Figure 2.

Figure 2.

Description of the locally inhomogeneous plane wave associated to the diffracted field.

[31] By considering that

equation image

the incident field complex amplitudes AE and AH can be expressed in terms of the UTD components, Eβ and Eϕ′, as follows

equation image

[32] Furthermore, by using equation (9), the longitudinal components of the electric and magnetic field can be cast in the form:

equation image

where P = MN.

[33] Also, as apparent from the exponential terms in equation (19), when ∣equation imageρequation image 1 the diffracted field can be interpreted as a locally inhomogeneous plane wave characterized by the complex wave vector

equation image

where

equation image

In the above relationship, equation image = cos βRequation image + sin βRequation image, with βR and βI representing the real and imaginary part of the skewness incidence angle equation image, respectively.

[34] Therefore, as in equation (2), kd can be expressed as:

equation image

where βI is the evanescence angle associated with the diffracted locally inhomogemeous plane wave and equation image = equation image × equation image. It is apparent that the two real vectors k and k are perpendicular to each other and ∣k2 − ∣k2 = k2, according to inhomogenous plane wave theory.

[35] Consequently, the components of the diffracted electric field can be expressed as

equation image

where

equation image

[36] Finally, by combining equations (19), (22), and (25), it follows:

equation image

where D = QTP. By defining Eβi(QD) = equation image · Ei and Eequation imagei(QD) = equation image · Ei as the components of the incident electric field parallel and perpendicular to the plane of incidence at the point of diffraction QD, respectively [Balanis, 1989], equation (26) can be rewritten as

equation image

where s = ρ/sinβR is the distance between the observation point and the diffraction point QD along the propagation direction of the diffracted field equation image. The latter lies on the surface of the generalized Keller diffraction cone, whose semi-angle of aperture is given by the real part of equation image, which reduces to β0 for the homogeneous plane wave incidence case, as expected.

[37] It is worth underlying that the asymptotic solution here derived is an accurate approximation of a rigorous integral solution of a wedge scattering problem where the incident field amplitude exhibits an exponential variation close to the diffracting edge. Therefore, the asymptotic solution intrinsically contains the effect on the scattered field of the incident field amplitude variation, which is usually accounted for by a slope diffraction term added to the ordinary UTD diffraction coefficient. It follows that the present solution can be used to account for complex incident fields once a systematic procedure is outlined to approximate the above incident field with one or more inhomogeneous plane waves. Alternatively, the high-frequency solution here derived can be used to extract a generalized slope diffraction term valid for any orientation of the amplitude decay direction with respect to the incidence direction. Work is in progress on both the above non trivial tasks and the results will be the subject of future papers. For example, since the direction of maximum amplitude decrease of the incident field can be arbitrarily oriented in the plane transverse to the direction of propagation, the present solution is being applied to isolate the slope diffraction terms related to the incident field variation along the directions perpendicular and parallel to the plane of incidence, and compare them with the results obtained by Lumholt [1999] and Breinbjerg et al. [1998]. Lumholt [1999] analyzed the specific case of a PEC half-plane illuminated at oblique incidence by an inhomogeneous plane wave exhibiting an exponential variation of the incident field amplitude in the direction parallel to the plane of incidence (α0 = 0). On the contrary, Breinbjerg et al. [1998] considered the same 2D scattering problem as in Kouyoumjian et al. [1996] but limited to the PEC half-plane geometrical configuration. In both Lumholt [1999] and Breinbjerg et al. [1998] an asymptotic solution of the half-plane canonical problem was derived by an analytic continuation of the Sommerfeld solution for the case of a complex skew incidence angle (corresponding to the angle equation image of the present paper) and a complex incidence angle corresponding to the angle equation image0 + equation image of this paper, respectively.

6. Numerical Results

[38] Samples of numerical results are shown in this section both to demonstrate the continuity of the total field across the SBs and the accuracy of the uniform high-frequency solution derived. In particular, the amplitude of all the total electric field components is plotted in Figures 3 and 4 as a function of the observation angle ϕ, at a given normalized distance from the edge ρ/λ. The total field is evaluated on the plane z = 0, namely the field is normalized with respect to the exponential term ejkzcos equation image. The amplitude of all the three total field components has been evaluated through the proposed asymptotic solution (continuos line) and compared against the eigenfunction solution (dots) as shown in Figure 3, when ρ = λ (Figure 3a) and ρ = λ/2 (Figure 3b) corresponding to ∣equation imageρ∣ = 4.6 and ∣equation imageρ∣ = 2.3, respectively. The evanescence angle is ψ = 10°, and α0 = 0, β0 = 45°, ϕ0 = 60°, WA = 30°. Moreover, the amplitude of the incident plane wave is set to (AE = 1, AH = 0) in Figure 3. Similar results are shown in Figure 4, where the incident plane wave is characterized by (AE = 0, AH = 1). All the figures show an excellent agreement between the asymptotic solution and the exact series expansion, even at the lowest distance from the edge. Also, the asymptotic solution is apparently continuous at the incidence and reflection SBs (ISB and RSB). It is worth noting that for the specific geometrical and electrical parameters considered in Figure 3 the direction of maximum decrease of the incident field amplitude lies in the incidence plane, while the incident electric field is completely perpendicular to the same plane. It follows that the longitudinal component of the total electric field exactly vanishes. Also, since in this particular case the imaginary component of the incidence angle, equation image, is equal to zero, the shadow and reflection boundaries coincide with the conventional optical boundaries at π + ϕ0 and π − ϕ0, respectively.

Figure 3.

Amplitude of all the components of the total electric field, normalized to the amplitude of the incident electric field (∣E0∣), for two different values of the distance from the edge: (a) ρ = λ and (b) ρ = λ/2. The other geometrical and electrical parameters are: WA = 30°, α0 = 0°, β0 = 45°, ϕ0 = 60°, ψ = 10°, AE = 1 V/m and ZAH = 0.

Figure 4.

Amplitude of all the components of the total electric field, normalized to the amplitude of the incident electric field (∣E0∣), for two different values of the distance from the edge: (a) ρ = λ and (b) ρ = λ/2. The other geometrical and electrical parameters are: WA = 30°, α0 = 0°, β0 = 45°, ϕ0 = 60°, ψ = 10°, AE = 0 V/m and ZAH = 1 V/m.

[39] Further numerical results are shown in Figure 5, where the amplitude of the four entries of the generalized UTD dyadic diffraction coefficient in equation (26) are plotted as a function of the observation angle ϕ, at ρ = 2λ. The inhomogeneous incident plane wave impinges on the wedge from the direction β0 = 60°, ϕ0 = 30°, α0 = 45°. The interior wedge angle is WA = 90°. A set of different values is considered for the evanescence angle: ψ = 0° (continuous line), ψ = −5° (dashed line) and ψ = −10° (dashed-dotted line). The amplitude of the extra diagonal terms is apparently smaller than that of the diagonal terms, which are related to a co-polar coupling effect. Also, when the evanescence angle does not vanish, they exhibit a jump discontinuity at the SBs, which must compensate the corresponding discontinuity in the evanescent GO field. When ψ = 0 (homogeneous incident plane wave), the extra diagonal terms are continouous at the SBs, as expected, but they are not smooth since they have to compensate a discontinuity in the derivative of the GO field [Büyükdura, 1996].

Figure 5.

Amplitude of the entries of the generalized UTD dyadic diffraction coefficient: (a) Dββ; (b) Dβϕ; (c) Dϕβ; (d) Dϕϕ. The geometrical parameters are: WA = 90°, α0 = 45°, β0 = 60°, ϕ0 = 30°, ρ = 2λ.

7. Conclusions

[40] A uniform high-frequency solution for the diffraction of an inhomogeneous electromagnetic plane wave by a PEC wedge at oblique incidence has been presented in the framework of UTD. It extends the previous solution given by Kouyoumjian et al. [1996] which is valid at normal incidence and when both the real and imaginary components of the incident wave vector lie in the plane transverse to the edge (two-dimensional problem). Numerical comparisons with the exact eigenfunction solution have shown that the asymptotic solution of the diffraction canonical problem remains accurate even at distances from the edge less than a wavelength. The exponential variation of the incident field close to the edge determines a rotation of the shadow boundaries with respect to the conventional GO shadow boundaries, as already discussed by Kouyoumjian et al. [1996], as well as a variation of the semi-angle aperture of the generalized Keller diffraction cone. The diffracted field amplitude can be derived by resorting to a generalized dyadic diffraction coefficient which is not diagonal. The analytical solution of the canonical diffraction problem has been carried out through the introduction of complex angles, so requiring an accurate asymptotic evaluation of the diffraction integrals since the GO pole singularities result to be complex and cross the steepest descent paths away from the corresponding saddle points. The high-frequency solution contains the standard UTD transition function extended to complex arguments.

[41] Finally, it has been shown that the uniform asymptotic approximation accurately describes the effects due to the rapid spatial variation of the incident field close to the diffracting edge. Work is in progress to extract from it a generalized slope diffraction coefficient, as well as to derive a systematic procedure to approximate any incident field with one or more inhomogeneous plane waves, so that the generalized diffraction coefficient derived here can be used to extend the applicability of ray techniques to more complex incident fields.

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