Radio Science

An incremental theory of double edge diffraction

Authors


  • 2 May 2005

Abstract

[1] A novel general procedure for defining incremental field contributions for double diffraction at a pair of perfectly conducting (PEC) wedges in an arbitrary configuration is presented. The new formulation provides an accurate first-order asymptotic description of the interaction between two edges, which is valid both for skewed separate wedges and for edges joined by a common PEC face. It also includes a double incremental slope diffraction augmentation, which provides the correct dominant high-frequency incremental contribution at grazing aspect of incidence and observation. This new formulation is obtained by applying to both edges, the wedge-shaped incremental dyadic diffraction coefficients for single edge diffraction. The total doubly diffracted field is obtained from a double spatial integration along each of the two edges on which consecutive diffractions occur. It is found that this distributed field representation precisely recovers the doubly diffracted field predicted by the uniform theory of diffraction (UTD) and that may be applied to complement ray field methods close to and at caustics. It can be applied as well in all those situations in which a stationary phase condition is not yet well established. Numerical examples are presented and compared with those calculated from both Method of Moment solution and second-order UTD ray techniques. Excellent agreement was found in all cases examined.

1. Introduction

[2] It has been widely recognized that high-frequency ray methods, such as the Geometrical Theory of Diffraction in its uniform version (UTD) [Kouyoumjian and Pathak, 1974], are useful in efficiently describing scattered or diffracted fields originating from complex objects. Complex structures may include two interacting edges in which case contributions involving field of rays doubly diffracted by the two interacting edges need to be introduced to improve the field estimate [Tiberio and Kouyoumjian, 1979; Tiberio et al., 1989; Ivrissimtzis and Marhefka, 1991, 1992; Capolino et al., 1997; Albani, 2005]. Furthermore, difficulties may occur in applying ray methods close to and at caustics, and one is required to introduce distributed incremental field representation to overcome these difficulties. One advantage of the incremental representation is that it may improve upon the field estimate whenever the stationary phase condition leading to the ray field regime, is not yet well established. In particular, it is desirable to use incremental descriptions of diffracted fields that naturally and smoothly blend into the ray field estimate. The Incremental Theory of Diffraction (ITD) [Tiberio and Maci, 1994; Tiberio et al., 2004] provides a self-consistent, high-frequency description of a wide class of scattering phenomena within a unified framework. The ITD may be classified as a field-based method, since the incremental field contributions are directly deduced from the solution of locally tangent canonical problems that, generally, have a uniform cylindrical configuration. An important feature of the ITD is that it smoothly recovers the ray-field description and explicitly satisfies reciprocity, a feature that is typical of field-based methods. Satisfaction of reciprocity is a very desirable property, and is embedded in Maxwell equations. Let us recall the ITD procedure for defining incremental fields. This procedure, which consists of two steps, is based on the relationship between the incremental field description of the scattered or diffracted field and the solution of the canonical problem. First, the spectral integral representation for the exact solution of the relevant local canonical problem is cast in a convenient form as an inverse Fourier transform (FT) of the product of two spectral functions. By means of Fourier analysis, it is found that the same spectral integral may be represented as a spatial integral convolution along the longitudinal coordinate of the local cylindrical configuration. Second, the incremental field contribution is then naturally localized at the tangential point on the actual object. This is referred to as the ITD FT-convolution process. Explicit dyadic expressions of the incremental diffraction coefficients for wedge-shaped configurations are presented by Tiberio et al. [2004]. The asymptotic analysis performed here yields high-frequency, closed form expressions which explicitly satisfy reciprocity, and are well behaved at any incident and observation aspects, including those close to and at the longitudinal coordinate axis of the cylindrical configuration. An analytical verification for the case of a disk is provided by Erricolo et al. [2007]. It is also found that, for the scalar case, the same result was obtained by Rubinowicz [1965], by means of an entirely different method.

[3] This article was motivated by several actual scattering and diffraction problems in which the dominant field of a singly diffracted ray is shadowed by a second edge, or is near to the shadowing. The introduction of doubly diffracted rays becomes necessary in order to compensate for such a discontinuity. It is well known that the consecutive application of the ordinary UTD diffraction coefficients fails when the edge of the second wedge lies within the transition region of the field diffracted by the first wedge [Tiberio et al., 1989]. This is due to the rapid spatial variation and to the non-ray-optical behavior of the field diffracted by the first wedge, which impinges on and illuminates the second wedge. The same limitation is expected to affect also the ITD representation, thus preventing a simple subsequent application of the coefficients for single diffraction. Therefore, it is necessary to develop a double diffraction coefficient which uniformly accounts for the different transitions that may occur. In some UTD formulations available in the literature [Tiberio et al., 1989; Albani, 2005] these coefficients are derived by solving a proper canonical problem.

[4] In this paper, an ITD dyadic coefficient is presented for double diffraction between a pair of skewed separate wedges, as well as for double diffraction between the edges of joined wedges that have a common face. In both cases, the new formulation provides an accurate, first-order asymptotic description of the interaction between the edges by introducing an augmented incremental slope diffracted field. The ITD representation for the incremental doubly diffracted field (ITD-DD) requires a twofold numerical integration in the space domain, which arises from the ITD line integral representation for the diffracted field at each edge. It is also worth noting that the present ITD-DD description is valid in the general case where the edges are not necessarily coplanar.

[5] This paper is organized as follows. The ITD formulation for single edge diffraction is summarized in section 2. The extension to the case of double diffraction between a pair of skew or joined wedges is presented in section 3, where the question of how the even and odd components of the spectral Green's function contribute to the incremental field it is also discussed. There, it is shown that the double slope diffraction augmentation can be accounted for by the product of the odd parts of the ITD dyadic diffraction coefficients. Discussion about singularities and discontinuity compensation mechanisms of the proposed formulation is provided in section 4 for both skewed separate and joined wedges. In section 5, some numerical examples are presented and discussed.

2. Summary of ITD Formulation for Single Diffraction

[6] Let us consider an arbitrarily shaped edge discontinuity in a perfectly conducting (PEC) object, which is large in terms of wavelength (Figure 1). The shape of the edge is described by the curved line l. Also, let be the exterior wedge angle at each point along the edge l; in general, n may not be uniform along l. According to the ITD formulation [Tiberio and Maci, 1994; Tiberio et al., 2004], the electric field diffracted by the actual edge at an observation point P when illuminated by an electromagnetic point source at S is represented by the following spatial integral along l

equation image

where dEd(P, Ql) represents the vector electromagnetic incremental field contribution diffracted at P by each local point Ql along l. According to a well- established locality principle at high frequency, the incremental field contribution dEd(P, Ql) is deduced from the solution of a canonical wedge problem locally tangent at Ql, which has an infinite uniform cylindrical configuration, with exterior angle . To this end, a local rectangular coordinate system with its origin at Ql and z-axis tangent to the line l is defined, which has a local spherical (r, β, equation image) coordinate system associated to it (Figure 1). By using this reference system, the incremental diffracted field contribution at P ≡ (r, β, equation image) that is due to an electromagnetic point source located at S ≡ (r′, β′, ϕ′), may be conveniently expressed as,

equation image

where

equation image

is the incident electromagnetic field at Ql and

equation image

is the diffraction dyadic in which the elements are the diffraction coefficients for soft (s) and hard (h) boundary conditions (b.c.), respectively. In (3) and (4), (equation image, equation image) and (equation image′, equation image′) are recognized as the ray-fixed unit vectors associated with the diffraction and the incidence, respectively [Kouyoumjian and Pathak, 1974]. The coefficients in (4) are obtained after applying the ITD localization process to the spectral representation of the canonical Green's Function for the wedge problem [Tiberio et al., 2004]. This leads to the diffraction coefficients

equation image

in which the minus and plus signs apply to either soft or hard boundary conditions on the faces of the wedge, respectively, and Φ = (ϕ ∓ ϕ′). Furthermore, in (5)

equation image

where υ is the value of the spectral variable as defined by Tiberio et al. [2004], and obtained by

equation image

The present ITD formulation provides a description of the incremental field, which is completely symmetrical with respect to ϕ and ϕ′, as well as with respect to β and β′, thus, explicitly satisfying reciprocity. Moreover, these expressions are well behaved at any incidence and observation aspects, including β = 0 and/or β′ = 0. In particular, a vanishing field contribution is predicted when the aspect of incidence or observation approaches the axis of the locally tangent configuration. The transitional behavior of the total diffracted field, which is expected to compensate for the discontinuities of the Geometrical Optics (GO) field, is reconstructed by a numerical integration of the incremental contributions along the actual edge. The discontinuities of the total diffracted field are caused by the singularities of the incremental field, that occur at the shadow boundaries of the GO field (β = β′; ϕ ∓ ϕ′ = π). To provide an estimate of the singly diffracted field by the actual wedge, the incremental field contribution defined in (2) is distributed and then spatially integrated along its edge. When the ray field regime is well established, this estimate precisely recovers the proper ray-optical UTD diffracted field. Moreover, when the stationary phase condition is not well established or when the observation point lies within the shadow boundary transition regions, the spatial integral representation also provides a correct estimate of the diffracted field even if it does not have a ray-optical nature. This allows us to extend the ITD description to double wedge configurations, using a proper subsequent application of the ITD description for the singly diffracted field.

Figure 1.

Geometry at a locally tangent wedge.

3. ITD Formulation for Double Diffraction

[7] Let us consider a pair of wedges with curved edges l1 and l2, arranged in an arbitrary configuration as depicted in Figure 2. In particular, they may or not be coplanar, and, in the former case, they may or may not share a common face. The ITD formulation is applied to describe the doubly diffracted field mechanisms that arise from the diffraction at l2 of the diffracted field from l1. For this purpose, the incremental field diffracted at each point of the edge l2 by the edge l1 is used as the incident field on l2. A pictorial representation of this incremental distributed approach as applied to the double diffraction mechanism is given in Figure 2. According to the ITD formulation described in section 1, each Q1 point along the l1 edge, which is illuminated by an incident field from a source point S, is the source of an incremental diffracted field that exists everywhere in the surrounding space. In particular, the incremental diffracted field from Q1 impinges on each Q2 point along the l2 edge. There, it undergoes a second diffraction, which gives rise to a further incremental diffracted field, propagating from Q2 to any arbitrary observation point P. Therefore, the double diffraction mechanism can be associated with the incremental raypath SQ1Q2P, where Q1 and Q2 are all the possible pairs of incremental diffraction points along l1 and l2, respectively. The total double diffracted field E12dd(P) at P is then reconstructed by superimposing in integral form all the incremental doubly diffracted field contributions dE12dd(P, Q1, Q2) at P, for any Q1 and Q2 along both l1 and l2, and namely,

equation image

In order to obtain an explicit dyadic expression for dE12dd, it is useful to define an edge-fixed local spherical coordinate system (ri, βi, ϕi) at each edge (i = 1, 2), with their origins at the two local Qi edge points. The relevant local geometry is depicted in Figure 3. According to the ITD formulation, the incident field E1d(Q2) on each local Q2 point of the second edge, which is singly diffracted by the first edge when illuminated by an incident field Ei(Q1) from the source point S, may be represented as spatial integral along the first edge l1, i.e.,

equation image

where

equation image

Since the two edges are generally not coplanar, a rotation angle γ12 is defined between the ray-fixed unit vectors (equation image1, equation image1) and (equation image2′, equation image2′) associated with the diffraction at the first edge and with the illumination at the second edge, respectively. Consequently, the incremental field contribution localized at Q2 and double diffracted at the observation point P by the second edge when illuminated by the incremental field dE1d(Q1, Q2) is obtained as

equation image

where

equation image

and equation image(γ12) is the transformation matrix between the two local spherical systems at Q1 and Q2. It is worth pointing out that the term equation image(γ12) · dE1d(Q1, Q2) is the incremental incident field from edge l1 on each local point Q2, which is expressed in the local reference system associated to l2. Finally, the total double diffracted field E12dd(P) in (8) is obtained by using there the incremental double diffraction contributions (11).

Figure 2.

Pictorial representation of the incremental double diffraction mechanism.

Figure 3.

Local geometry for a pair of skew separated wedges.

[8] In (12), equation image(υi, ϕi, ϕ′i) with i = 1, 2, are the two diffraction dyadics as in (4) associated with the incremental single diffraction at the first and second edges, respectively. Before proceeding further, we should point out that the UTD technique for double diffraction between two wedges in an arbitrary configuration [Albani, 2005] when applicable, has been demonstrated to be a very powerful tool. It has been found that, in order to obtain an UTD double diffraction description that provides the proper asymptotic order in all the different transitions that may occur, in the spectral formulation it is best to retain the product of both the even and the odd parts of each cotangent associated with the spectral Green's Function of a single wedge [Felsen and Marcuwitz, 1994] Indeed, the nonvanishing contributions to the solution for the canonical problem of a pair of wedges obtained via spectral synthesis, are provided by the even parts of the final spectra. These latter are obtained from the product of both the two even and odd parts of each cotangent. On the other hand, the mixed terms, which are the results of the product between an even and an odd part of the spectrum, provide a vanishing contribution when integrated into the symmetrical spectral domain. This provides us the guidelines for defining the correct ITD double diffraction coefficients.

[9] From the preceding discussion, it is quite clear that, in order to obtain an incremental double diffraction formulation which remains valid even in the overlapping transition regions, as a first step it is useful to consider both the even and the odd parts of the incremental diffraction contributions in the ITD description of the singly diffracted field equation image(υi, ϕi, ϕ′i). We thus obtain, in place of (5), the expression

equation image

where the minus and plus signs apply for soft and hard boundary conditions. In (13),

equation image

where De(υii±) is defined as in (6), and

equation image

in which Φi = (ϕi ∓ ϕ′i) with i = 1, 2. Next, it is useful to retain in (12) only the contributions that are the product of the two even and odd parts of the dyadic associated with the single diffraction by each wedge. In fact, when the double diffracted incremental field (11) is distributed and integrated along the infinite straight edges of two canonical wedges, due to the symmetry of the spatial integration contour only the even parts, with respect to υ1 and υ2, give nonvanishing contributions to the total double diffracted field. On the other hand, the mixed odd parts, which are the product of an even and an odd part of the two dyadics, give vanishing contributions when integrated along the same symmetrical integration domain.

[10] Thus, the final form of the dyadic double diffraction coefficient is

equation image

where

equation image

in which

equation image

and

equation image

In (19), the minus and plus signs apply for soft and hard boundary conditions respectively, and De,o(υii) are defined as in (6) and (15).

4. Singularities and Discontinuities

[11] As previously pointed out, the field doubly diffracted by a pair of wedges can be obtained by distributing and integrating the incremental DD contributions (11) along the two edges of the wedges. The obtained doubly diffracted field is well behaved close to and at caustics, but also provides a correct compensation for the various discontinuities which may occur in the singly diffracted field contributions, as will be shown in the next section. The transitional behavior and the appropriate discontinuities of the doubly diffracted field near and at the shadow boundaries of the singly diffracted field, are provided by a numerical integration of the singularities of the dyadic (16). In this section, we briefly analyze these singularities in order to clarify the behavior of the proposed formulation under various regimes.

4.1. Separated Wedges

[12] Let us consider two canonical separated wedges with noncoplanar edges, and then consider a field diffracted by edge l1 that is successively diffracted by edge l2 at an observation point P. The same considerations may be applied for the DD contribution from edge l2 to edge l1. When the observation point is far from the shadow boundaries of the singly diffracted field, each even part of equation image12e(·) in (16) provides a complete description of the dominant field diffracted by a single wedge (section 2). Therefore, the integration of the even part equation image12e(·) accounts for double diffraction as a cascade of single edge diffractions, that yields the dominant contribution, with asymptotic order k−1, to the doubly diffracted field outside the overlapping transition regions. On the other hand, when integrated along the edge of a single canonical wedge, each odd part of equation image12o(·) in (16) provides a vanishing contribution to the singly diffracted field. Thus, as can be proved by an accurate asymptotic analysis of the corresponding double spectral integral representation [Capolino et al., 1997], the integration of the odd part equation image12o(·) provides a higher-order contribution, with asymptotic order k−2, compared to the dominant one provided by the even part. This contribution is associated with the slope diffraction that is due to the rapid spatial variation of the singly diffracted field at the second edge.

[13] Next, let us consider the case in which the observation point approaches the incident or the reflected shadow boundary of the singly diffracted field by the first edge. Both the even and odd parts of (16) exhibit singularities at υ2 = 0; (equation image2equation image2′) = π associated with diffraction by the second wedge. In this case, the expected transitional behavior of the doubly diffracted field from the entire canonical configuration across these shadow boundaries can be reconstructed from an integration of this singular incremental field along the two canonical edges. In particular, it is found that the integration of equation image12e(·) in (16), which is the leading diffraction term, provides a field contribution with asymptotic order of k−1/2 across reflected and incidence shadow boundaries, where the disappearance of the diffracted-reflected and the diffracted-incident contributions, respectively, occur. Again, the integration of the higher order term equation image12o(·) in (16) provides a field contribution with asymptotic order of k−1 which compensates for the discontinuity in the derivative of the total field across the same shadow boundaries.

[14] A particular case occurs whenever the second edge is located in the transition region of the field singly diffracted from the first edge and the double diffracted field is observed in the transition regions associated with the incidence or reflected shadow boundaries of the second edge. In this case, both terms in (16) may exhibit singularities when both β1′ = β1(υ1 = 0); (ϕ1 ∓ ϕ1′) = π and β2′ = β2(υ2 = 0); (ϕ2 ∓ ϕ2′) = π. The nonray optical behavior of the field incident on the second edge after being diffracted from the first is correctly reconstructed by an integration of the singular ITD-DD incremental field along the first edge. At the same time, when observed close to the shadow boundaries, the transitional behavior of the field diffracted from the second edge is correctly reconstructed by an integration of the same singular ITD-DD incremental field along the second edge. Since also the GO contribution is shadowed in this situation, both the higher and the dominant doubly diffracted contributions become of the same order as the GO field, and jointly compensate for the above double discontinuity. When the stationary phase point are focalized on the two edges, this phenomenology is the same as the one obtained from an suitable asymptotic analysis of the double spectral integral representation of the doubly diffracted field. The resulting UTD formulation for doubly diffracted rays, which is clearly discussed by Capolino et al. [1997] and Albani [2005], involves the use of appropriate transition functions that resort to generalized Fresnel integrals.

[15] Finally, as already noted in section 2, it can be observed that the ITD-DD contribution tends to vanish when the aspect of incidence or observation approaches the axis of a locally tangent configuration associated with the single incremental edge diffraction. This property implies that the ITD formulation for double edge diffraction predicts a zero interaction between two points along a straight edge, as well as a vanishing interaction when two points of diffraction along the same curved edge tend to collapse. Indeed in both cases a twofold zero field contribution occurs in (16).

4.2. Joined Wedges

[16] Let us consider the particular case of two coplanar wedges joined by a common PEC face, as in a polygonal plate (Figure 4). Since the two edges are coplanar, the rotation angle γ12 between the ray-fixed unit vectors (equation image1, equation image1) and (equation image2′, equation image2′) assumes the values γ12 = 0, π. Consequently, equation image12e,o(P, Q2, Q1, S) in (17) becomes a diagonal matrix in which the nonvanishing terms are

equation image

Moreover, for this specific geometry, the incremental field which is diffracted at the point Q1 and impinges on each point Q2, propagates at grazing along the common face of the double wedge. As a consequence, ϕ1 = (0, ) and ϕ2′ = (0, ). Now, it is well-known that at grazing incidence, the field incident on the second edge (which is diffracted by the first edge) is only one half of the total field that propagates along the common face of the wedges [Tiberio and Kouyoumjian, 1979; Ivrissimtzis and Marhefka, 1991]. This occurs because the incident and reflected fields tend to merge into a single field structure (the total field), as the direction of incidence approaches grazing. Therefore, when applying the ITD-DD formulation to a pair of wedges with a common PEC face, at grazing incidence only one half of the incremental field diffracted at Q1 constitutes the incident field on each Q2 point of the second edge.

Figure 4.

Local geometry for a pair of joined wedges with a common PEC face.

[17] Next, it should be noted that, since a common PEC face has been assumed, the leading term Dse(Q1, Q2, P)Dse(S, Q1, Q2) of the equation image1equation image2 component vanishes. The tangential component of the total electric field tends to vanish, but exhibits a rapid spatial variation. Thus, a strong dominant slope diffraction effect is expected. This effect is properly accounted for by the higher order term Dso(Q1, Q2, P)Dso(S, Q1, Q2) retained in the formulation, which exhibits singularities when the observation point crosses the shadow boundary associated with the second edge. The spatial integration of this incremental contribution provides a smoother transition from the lit onto the shadow region of the singly diffracted field from the first edge. Analogously, it is worth noting that the higher order term Dho(Q1, Q2, P)Dho(S, Q1, Q2) of the equation image1equation image2 component vanishes. However, the leading term Dhe(Q1, Q2, P)Dhe(S, Q1, Q2) exhibits proper singularities when the observation point crosses the shadow boundary associated with the second edge. Again, the spatial integration of this incremental contribution provides a smoother transition from the lit into the shadow region of the singly diffracted from the first edge. On the other hand, the even terms provide a nonzero leading contribution to the incremental doubly diffracted field component that is perpendicular to the common face of the double joined wedges. It can also be observed that the higher-order slope contributions in (20) vanish due to hard polarization.

5. Numerical Results

[18] In this section, several numerical results are presented that apply the ITD formulation for double edge diffraction to scattering problems in which some typical PEC objects are illuminated by an electric dipole. In order to check the effectiveness of the ITD-DD formulation, comparisons with both Method of Moment (MoM) calculations and UTD ray-field descriptions, when applicable, are provided.

[19] As a first example, we considered the field diffracted by a perfectly conducting strip of width w = 0.5 λ when illuminated by an electric dipole placed at finite distance. The geometry of the problem is shown in the insets of Figures 5 and 6. The dipole was placed at α = 5o from the plane containing the strip, and at a distance r′ = 1.75 λ from the strip axis so that the second edge l2 was illuminated by the transition region field diffracted by the first edge. Observations were made at distance r = 1λ. The curves of the copolar components of the total field in which the dipole was either orthogonal (hard case) or parallel (soft case) to the edges, are shown in Figures 5 and 6, respectively. The results obtained by ITD-DD (continuous line) are compared with those obtained by first-order (singly diffracted) ITD solution (dashed line), and with the MoM calculations (dot-dashed line) [Capolino et al., 1997]. Very good accuracy between MoM and ITD-DD calculations is achieved for all observation aspects, including those in which the transition regions of the two edge overlap, due to the near grazing incidence. There, the ITD double diffraction mechanism provides the proper correcting field contributions. Results relevant to the same configurations are also shown in Figures 7 and 8, where calculations for ITD and UTD techniques are compared, for hard and soft illumination, respectively. Excellent agreement is observed for both first and second order estimates, which was one of the aims of the proposed incremental representation. It is quite clear that in the hard case, after diffracting from the first edge, the field propagating along the surface experiences a discontinuity at grazing aspects of observation, which is expected to be compensated for by the doubly diffracted contribution. Indeed, the first-order ITD and UTD fields predictions exhibits a noticeable discontinuity at grazing aspect, and this is precisely compensated for by the second-order diffraction mechanism. The final outcome is in very good agreement with the MoM solution. For the soft case illumination, the field propagating along the surface vanishes but exhibits a rapid spatial variation. This results in a strong dominant slope diffraction effect. At grazing observation aspects, the singly diffracted field illuminating the second edge vanishes, and first-order ITD and UTD calculations predicts a null at grazing. The continuous slope contribution predicted by MoM calculation is correctly estimated by introducing both ITD and UTD doubly diffracted contributions.

Figure 5.

Normalized amplitude of the copolar component of the total field for a PEC strip illuminated by an electric dipole: hard case.

Figure 6.

Normalized amplitude of the copolar component of the total field for a PEC strip illuminated by an electric dipole: soft case.

Figure 7.

Comparison between first- and second-order ITD and UTD estimates of copolar total field: hard case.

Figure 8.

Comparison between first- and second-order ITD and UTD estimates of copolar total field: soft case.

[20] Let us now consider a perfectly conducting circular cylinder with radius a = 2.8 λ and height h = 8 λ which was illuminated by a dipole placed on the axis at a distance h = 2λ. The geometry of the problem is shown in the insets of Figures 9 and 10. In Figure 9, the magnitude of the copolar θ-component of the scattered electric far-field is plotted for a z-directed electric dipole. The results obtained from the present ITD-DD formulation (solid line) is compared with those from a first order ITD solution (dashed line) and from MoM calculations (dotted line). It is observed that the first-order ITD solution exhibits a discontinuity at observation aspects near grazing on the top face. This result is expected since in this case, a singly diffracted field is shadowed by the structure. Here, a significant improvement is obtained by introducing the doubly diffracted contributions from the top of the cylinder, which causes the scattered field to compare very well with the MoM calculation. For observation aspects in the region θ = 90°–180°, noticeable differences between first-order ITD and MoM calculations appear. When both the doubly diffracted contributions from the top face and between the top and the bottom edges of the of the cylinder faces are included, the agreement between MoM and ITD-DD formulations become very good, and confirms the importance of the doubly diffracted field contributions, particularly in the forward scattering directions. A residual discrepancy with the MoM prediction is also observed near the axis at the forward scattering aspects. Indeed, it is well known that, at grazing aspects, diffracted fields even beyond the second order may play a significant role.

Figure 9.

Amplitude of the equation image-component of the scattered electric field for a PEC cylinder illuminated by a vertical electric dipole (a = 2.8 λ, h = 8 λ, d = 2 λ).

Figure 10.

Amplitude of the equation image-component of the scattered electric field for a PEC cylinder illuminated by an horizontal y-directed electric dipole (a = 2.8 λ, h = 8 λ, d = 2 λ).

[21] It is important to point out that, in this example, the use of a ray method, such as UTD, is computationally more efficient but fails near and at backscatter (θ ≃ 0°) and forward scatter aspects (θ ≃ 180°), where caustics of singly and doubly diffracted rays are present. On the other hand, the use of the incremental solution presented here provides a proper estimate of the scattered field especially at these observation aspects. Moreover, a ray tracing technique to identify the diffraction points, when they exists, is not required. However, in order to reduce the computational effort, the application of a hybrid UTD-ITD technique may be conceived, since the proposed solution tends to recover the singly and doubly diffracted ray predictions when a stationary phase condition is well established.

[22] The final example refers to the case in which the dipole was parallel to the top face of the cylinder (y-directed), as shown in the inset. Results for the ϕ-component (Figure 10) of the scattered electric far field obtained from the present ITD-DD formulation (solid line) are compared with those obtained from a first order ITD solution (dot-dashed line) and from MoM calculations (dashed line). The curves in Figure 10 show very good agreement between first-order ITD and MoM calculations, except near the z-axis at the forward scattering aspects, where the second-order augmentation is needed in order to obtain a good estimate of the scattered field. Again, it is worth noting that this estimate is obtained at a caustic direction of both singly and doubly diffracted rays, where a conventional ray method fails. These results confirm the importance of the incremental calculations of these contributions, close to and at caustics directions.

6. Concluding Remarks

[23] An incremental theory of double diffraction has been presented, that provides an accurate first-order asymptotic description of the interaction between two edges, for both skewed separated and joined wedges sharing a common PEC face. The formulation has been obtained by applying the wedge-shaped ITD dyadic diffraction coefficients for single edge diffraction to both edges. The use of both the even and odd parts of the spectra in the ITD coefficients, made it possible to obtain incremental double diffraction coefficients which includes slope diffraction augmentation, thus providing the correct dominant high-frequency incremental contributions at grazing aspects of incidence and observation. The final form of double diffracted incremental field explicitly satisfies reciprocity, and leads to a well-behaved description of the field at any incident and observation aspects, including those close to and at the longitudinal coordinate axes of the local canonical configurations.

[24] To obtain the total double diffracted field from an actual structure, the corresponding incremental coefficients have to be distributed and integrated in the space domain along each one of the interacting edges. As an example of the application of the proposed solution, calculations have been presented for examples of a strip and a circular cylinder illuminated by an electric dipole. The results, which has been obtained by applying the proposed technique, are found to be in very good agreement with those obtained from both MoM solution and second-order UTD ray techniques. Although these latter are found to be computationally very efficient in estimating the scattered or diffracted field, the numerical results have shown that the application of incremental techniques becomes fundamental in order to obtain a correct estimate of the field at all those aspects in which caustics of diffracted or doubly diffracted rays are present. Moreover, the application of ray tracing techniques is not required in order to identify the diffraction points, when these exists. However, as mentioned above, significant improvements in the accuracy of the estimate, by maintaining a good computational efficiency, may be obtained with the application of a hybrid UTD-ITD tecniques. This is because the proposed solution tends to recover the singly and doubly diffracted ray predictions when a stationary phase condition is well established.

Acknowledgments

[25] Roberto Tiberio was the main source of inspiration for this work, to which he dedicated himself until passing away prematurely in 2005. All the colleagues of the Electromagnetics group at the University of Siena, Italy, will always be grateful to Roberto Tiberio for having shared his extraordinary scientific skills and his generous friendship with them.

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