Electromagnetic diffraction at skew incidence by a wedge with anisotropic impedance faces

Authors


Abstract

[1] The asymptotic expressions for the total field excited by a plane wave obliquely incident on a wedge with anisotropic impedance faces are represented in the framework of the uniform geometrical theory of diffraction. The problem of decoupling the functional difference equations is solved by applying the perturbation technique on the basis of two analytical solutions: normal incidence and grazing to the edge incidence. The numerical results agree well with those presented in published papers and provide a uniform and continuous behavior of the field dependent upon the skewness of the incident wave outside the wedge region.

1. Introduction

[2] Among many electromagnetic (EM) issues (radiation, scattering, and propagation), the wedge-shaped objects and their special case, the half plane, have been always acted as classic forms and structures of these types of issues, and they are most important objects in the study of EM problems. In recent years, with increasing interest in anisotropic composites, considerable attention has been drawn to the study of the electromagnetic scattering properties of surfaces coated with these materials. The anisotropic impedance boundary conditions [Senior and Volakis, 1991] are the simplest and most effective way to describe these problems. While in the case of skew incidence, so far, there do not exist good solutions to the difference equations generated by the boundary conditions in Maliuzhinets' approach [Maliuzhinets, 1958]. The main difficulty is the lack of techniques to solve the four coupled functional difference equations [Syed and Volakis, 1995]. To the authors' knowledge, those that can be solved analytically are limited to specific face impedances or geometrical configurations, [Vaccaro, 1980; Senior and Volakis, 1986; Rojas, 1988; Lyalinov, 1994; Lyalinov and Zhu, 1999, 2003; Manara et al., 1996; Manara and Nepa, 2000; Bernard, 1987, 1998]. Pelosi et al. [1998] has given an asymptotic solution using a perturbation technique based on the solution for normal incident. However, this asymptotic solution is valid only in the region deviating at most 30° from normal incidence. It is worth noting that in the case of very skew incidence, few researches have been carried out to provide a satisfactory solution.

[3] This paper analyzes the electromagnetic scattering of a plane wave obliquely incident on a wedge of an arbitrary open angle with anisotropic impedance faces, in which the impedance tensor has its principal anisotropy axes along the directions parallel and perpendicular to the edge. As indicated by Pelosi et al. [1998], first, the first-order anisotropic impedance boundary conditions are introduced to prescribe the wedge faces, and then followed the well-known Sommerfeld-Maliuzhinets method, the longitudinal components of the total fields are represented as Sommerfeld spectral integrals. Next, the boundary conditions are imposed to yield a group of functional difference equations. The approximate integral expressions of the spectral functions are then determined by applying the perturbation technique which is based on two analytical solutions (normal incidence [Maliuzhinets, 1958] and grazing to the edge incidence [Lyalinov and Zhu, 1999]) to decoupling the functional difference equations. Then by the residue theorem, the asymptotic evaluation of the spectral integrals is performed in the framework of the uniform geometrical theory of diffraction (UTD), with the high-frequency representation for the fields diffracted by anisotropic impedance wedge valid at close to normal incidence and almost grazing to the edge incidence. Finally, some examples of the numerical results are given and compared with those obtained also by the perturbation method [Pelosi et al., 1998] and the analytical solutions [Manara and Nepa, 2000] to demonstrate the uniform and continuous behavior of the fields in the absence of the wedge region.

[4] Though this work does not exactly solve the whole problem because of the limitation of the perturbation technique, the solution given here includes all the possible cases which can utilize the perturbation technique to derive the diffraction field for an anisotropic impedance wedge of an arbitrary open angle at skew incident, and should be at least as a practical way to obtain a comparatively good approximation to the field behavior at the general case of skew incidence, and comparison with other pure numerical or hybrid methods.

2. Formulation of the Problem

[5] The problem to be analyzed is the EM scattering by a wedge with impedance faces as shown in Figure 1. The wedge has its edge along z axis of a cylindrical reference frame. nπ refers to the exterior wedge angle (in particular, n = 1 corresponding to a two-part plane with arbitrary impedance surfaces, n = 3/2 corresponding to an exterior right-angled wedge, and n = 2 corresponding to a half plane). An arbitrary polarization harmonic plane wave impinges on the edge from the direction determined by the angles β0 and ϕ0. The angle β0 is a measure of the incident skewness with respect to the edge of the wedge (expressly, β0 = π/2 corresponding to normal incidence and β0 = 0 corresponding to grazing to the edge incidence). The angle ϕ0 and ϕ are measured from the ϕ = 0 face. The ISB and RSB correspond to the incident shadow boundary and the reflecting shadow boundary, respectively. Suppressing the time dependence exp(jωt), the longitudinal components of the incident fields can be represented as

equation image
Figure 1.

Scattering of a plane wave by a wedge at skew incidence.

[6] In the cylindrical reference frame, we consider the impedance tensor of the wedge surface has its principal anisotropy axes along directions parallel and perpendicular to the edge, the surface impedance is in the form by the tensors as equation image, with equation image to meet the restriction of the passive anisotropic impedance. By the first-order Impedance Boundary Conditions (IBCs) [Senior and Volakis, 1991]: equation image × (equation image × equation image) = −equation images (equation image × equation image), where for ϕ = 0 face equation image = equation image, and for ϕ = nπ face equation image = −equation image, the anisotropic impedance boundary conditions for each wedge face can be expressed in the cylindrical coordinate as

equation image

[7] According to the Maxwell equations, all the field components transverse to the z axis can be represented in terms of their longitudinal ones as follows

equation image
equation image
equation image
equation image

[8] Rewriting equation (2) in terms of the longitudinal field components Ez and ηHz in (3), two groups of boundary conditions for corresponding faces of the third kind are obtained

equation image
equation image

[9] We seek solutions which satisfy the scalar Helmholtz equation (∇t2 + k2 sin2 β0) [Ez, η Hz]T = 0 in the region outside the wedge, the boundary condition equations (4), the Meixner conditions at the edge ρ = 0, and the radiation conditions at infinity. A proper representation of the solution is the superposition of plane waves in the form of Sommerfeld integrals as follows

equation image
equation image

where γ is the Sommerfeld double loop contour (Figure 2). The spectral functions Se (α) and Sh (α) are analytical inside the strip ∣Re[α]∣ ≤ nπ/2 in order to satisfy the radiation condition, except for one first-order pole at α = nπ/2 − ϕ0, whose residue just reproduces the incident field, and when ∣Im[α]∣ → ∞, Se (α) = O [exp (Im α)] and Sh (α) = O [exp (Im α)]. The problem then reduces to the key step of determining the spectral functions.

Figure 2.

Integration contour γ and the steepest descent paths SDP(±π).

3. Determination of the Spectral Functions

[10] As apparent, Ez and Hz are coupled in the boundary condition equations (4), each will depends on the other, we cannot derive the spectral functions Se (α) and Sh (α) directly, so we resort to the perturbation technique based on two special incidences with the analytical solutions: normal incidence and grazing to the edge incidence.

3.1. Almost Normal Incidence

[11] In this case we directly substitute the Sommerfeld integral representations of Ez and ηHz in (5) into the boundary condition (4), and integrate by part to obtain a set of coupled integral equations

equation image
equation image

where kt = ksin β0, sin θh0,n = sin β0 Zρρ0,n/η, sin θe0,n = sin β0 η/Zzz0,n.

[12] According to the Maliuzhinets' theorem, the necessary and sufficient condition to satisfy (6) is that the integrands must be even functions of α. A mathematic description is

equation image
equation image

It is clear that as β0 → π/2, the right side of (7) goes to zero, Ez and Hz would be decoupled. For this type of homogeneous equations, Maliuzhinets [1958] has given a detailed solution. Hence, to approximately solve the inhomogeneous equations (7), we can introduce the perturbation method, based on Maliuzhinets' solution with respect to β0 close to π/2, to spread the spectral functions Se,h (α) as the cos β0 series

equation image

where Ψe,h (α) = Ψe,h (α, θe,h0, θe,hn) are the special functions defined by Maliuzhinets [1958], which are analytical in the strip ∣Reα∣ ≤ nπ/2 and also the partial solutions of the corresponding homogeneous equations of (7).

equation image

is a meromorphic function with a single pole α = nπ/2 − ϕ0 in [0, nπ] and well known from Sommerfeld's solutions for a perfectly conducting wedge, whose residue just accounts for the incident field.

[13] Substituting (8) into (7), and equating the terms of the same power of cos β0, we obtain the recurrent equations with constant coefficients

equation image
equation image

Letting ξe,h−1 (α) = 0, the leading term, which satisfies the homogeneous equations (10), is

equation image

Then substituting ξe,h0 (α) into the recurrent equations (10), the other order terms can be obtained by applying the modified Fourier transformation [Thuzhilin, 1973; Bernard, 1987] with integration along the imaginary axis.

[14] The first-order term takes the form

equation image
equation image

where σj (t, α) = equation image.

[15] It is worth observing that in (11), ξe0 (α) and ξh0 (α) are proportional to Uei and Uhi, respectively. However, in (12), ξe1 (α) and ξh1 (α) are proportional to Uhi andUei, respectively. This just implies that in the case of skew incidence, the longitudinal components of the total fields will consist of two parts, the copolar part (the contribution of the even-order terms) and the cross-polar part (the contribution of the odd-order terms).

[16] For the case of small deviation from normal incidence (∣cos β0∣ ≪ 1), the spectral function Se,h (α) can be approximated by the first two leading terms of the cos β0 series as

equation image

where Πe,h (α) = Ψe,h (α) [ξe,h0 (α) + ξe,h1 (α) cos β0 + O (cos2 β0)]

3.2. Almost Grazing to the Edge Incidence

[17] However, in this case we have to first transform the boundary condition (4) as follows to acquire the coupled Maliuzhinets' equations.

[18] By (4a)j × (4b), we obtain

equation image

and by (4b)j × (4a), we obtain

equation image

Introducing two new components P and T, with

equation image

the boundary condition on each wedge face can be represented by P and T as

equation image

[19] As P and T are the linear combination of Ez andηHz, they also satisfy all the constraints whichEz and ηHz do. So the solutions of P and T should also be in the form of the Sommerfeld integral as

equation image
equation image

[20] Following the same method used to derive equations (6) and (7) in section 3.1, we obtain the coupled Maliuzhinets' equations

equation image
equation image

where

equation image

[21] Dividing the equations (18) by sin β0 and introducing thereafter a new parameter Θ, such that sin Θ = jcos β0/sin β0, and cos Θ = 1/sin β0, where Θ∈ [0, j∞), β0 ∈ [0, π/2]. Equations (18) then reduce to the form

equation image
equation image

where sin σ0,n = A10,n.

[22] Lyalinov and Zhu [1999] had solved these equations by introducing a newly defined meromorphic function when the right side of equations (19) turns to zero. Followed his method, we introduce two new unknown spectral functions ζP (α) and ζT (α) with

equation image

where

equation image

[23] χ(z) is the solution of the functional equation χ(z + 2Φ) = cos (z/2) χ(z − 2Φ), which was first introduced by Bobrovnikov and Fisanov [1988] during their study of the wave diffraction and has been studied extensively [Bobrovnikov and Fisanov, 1988; Avdeev, 1994; Lyalinov and Zhu, 1999]. equation image (α) is the same as has been defined in (9).

[24] Substituting (20) into (19), we obtain

equation image
equation image

[25] In (22) we find that when β0 → 0, sin (α ± Θ) → ∞, the right side of (22) goes to zero. So we can also introduce the perturbation technique when the incident wave is almost grazing to the edge to spread the ζP,T (α) as

equation image

[26] Substituting (23) into (22), and equating the terms of the same power of sin β0, we obtain the recurrent equations with constant coefficients

equation image
equation image

[27] Letting ζP,T−1 (α) = 0, the leading term, which is also the solution of (22) with right side of it equal to zero, is

equation image
equation image

[28] Then substituting ζP,T0 (α) into the recurrent equations (24), the other-order terms can be obtained by applying the modified Fourier transformation with integration along the imaginary axis.

[29] The first-order term takes the form as

equation image
equation image

[30] In the same way, the spectral functions ζP,T (α) can be approximated by their first two leading terms of sin β0 series as

equation image

[31] Till now we have obtained the approximate expressions of the new spectral functions ζP (α) and ζT (α), when sin β0 ≪ 1, and by the relation ηHz = (P + jT)/2, Ez = (jP + T)/2, we can finally acquire the spectral function Se,h (α) = Πe,h (α) equation image (α) as in (13), but with

equation image

[32] From the expressions of Πe,h (α), ζP,T0 (α) and ζP,T1 (α), we also found that the spectral functions Se,h (α) are proportional to Uei and Uhi, which again shows that at skew incidence, the cross-polar components of the total fields will be produced by the impedance wedge.

4. UTD Solution

[33] Once obtaining the spectral functions, then by the residue theorem, the Sommerfeld integration can be deformed into the collective contribution of (1) the residues of the poles of Se,h (α), which account for the geometrical optic (GO) field (real poles' contribution due to equation image (α)) and the surface wave field (complex poles' contribution due to Ψe,h (α) or ΦP,T (α)), the latter will not be discussed here because of its minor contribution when talking about the fields in the far zone, and (2) two steepest descent path (SDP) integrals through the saddle points (±π) (Figure 2), which account for the diffraction field by the edge.

[34] So the solution can be represented as

equation image

[35] In this paper, we talk about the case that only the ϕ = 0 face is illuminated, viz. 0 ≤ ϕ0nπ − π. By the trigonometric identical transformation, the meromorphic function equation image (α) can be changed into the other form as

equation image

[36] Evidently, the functions equation image (α + nπ/2 − ϕ) have two real poles at α1 = ϕ − ϕ0 and α2 = ϕ + ϕ0, whose residues just reproduce the incident field and the reflected field by ϕ = 0 face, respectively, (A detailed description of real poles for both wedge faces are illuminated and only the ϕ = nπ face is illuminated has been given by Yuan and Zhu [2004]). The geometrical optic fields are given by the results of residues as

equation image

where Ree, Reh, Rhe, Rhh are the tensor reflected coefficients derived from the residues with

equation image
equation image
equation image
equation image
equation image

The shadow boundaries appear at ϕ = π ± ϕ0. Applying the modified Pauli-Clemmow's SDP method, the asymptotic solution of the diffraction fields can be finally obtained. An asymptotic UTD solution is in the form as

equation image

where F(x) = j2 equation image exp (jx) equation imageequation imagedt is the UTD transition function [Kouyoumjian and Pathak, 1974], which can eliminate the discontinuity of geometrical optical fields at the shadow boundaries.

[37] Compared with the expressions of diffraction fields derived by Pelosi et al. [1998] also with perturbation technique, (32) is not that complicated and has clearly physical meanings, in that the transition function appears with the factors π − (ϕ − ϕ0) and π − (ϕ + ϕ0), which, when the factors tend to zero, just are the incident shadow boundary and reflective shadow boundary respectively. In addition, this approach can evaluate the fields at very skew incidence.

[38] It is worth noting that when it comes to the normal incident case, (32) will be similar to the UTD formulation given by Tiberio et al. [1985], furthermore, for the case of perfectly electric conducting (PEC) wedges, (32) can retrogress to a form similar to the well known formulation given by Kouyoumjian and Pathak [1974]. Compared with the two expressions given by the two former, (32) is apparently more concise, because, here, we do not need to talk about the N±, but directly give the final results by determining the shadow boundaries from the meromorphic function equation image (α).

5. Numerical Examples

[39] To validate the method, we choose a right-angled wedge (n = 3/2) because of the fact that in this configuration, the Maliuzhinets function is known in a simple trigonometric functional form. Meanwhile, in all the numerical examples below, the independent factor exp (−jkz cos β0) is suppressed, and all the fields are evaluated at a normalized distance ktρ = 10 from the edge.

[40] Figure 3 shows the amplitude variation of the total longitudinal field ηHz with respect to the observation points. The incident plane wave is transverse electric (TE) polarized (equation image = 0, equation image = 1) and impinges on the edge from ϕ0 = 45° and β0 a parameter as 90°, 70°, 50° respectively. The impedance tensor is Zzz0/η = Zρρ0/η = j/2 (isotropic impedance), Zzzn = Zρρn = 0 (PEC). The numerical results are perfectly accordant with Pelosi et al. [1998, Figure 3a] (dots).

Figure 3.

Amplitude variation of the total longitudinal field ηHz versus the observation points in the absence of a right-angled isotropic impedance wedge with Zzz0/η = Zρρ0/η = j/2, Zzzn = Zρρn = 0. The incident plane wave is TE polarized (equation image = 0, equation image = 1) and impinges on the edge from ϕ0 = 45° and β0 = 90° (solid line), β0 = 70° (dotted line), and β0 = 50° (dash-dotted line). The dots are results given by the method of Pelosi et al. [1998]. The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[41] Figure 4 shows the amplitude variation of the total longitudinal field Ez with respect to the observation points, when a transverse magnetic (TM) polarized plane wave (equation image = 1, equation image = 0) impinges on the edge of wedge with both faces coated by anisotropic impedance. In this example, likewise, ϕ0 = 45° and β0 = 90°, 70°, 50° respectively. The impedance tensor is Zzz0,n/η = 1, Zρρ0,n/η = 2. The numerical results agree very well with Pelosi et al. [1998, Figure 4a] (dots).

Figure 4.

Amplitude variation of the total longitudinal field Ez versus the observation points in the absence of a right-angled anisotropic impedance wedge with Zzz0,n/η = 1, Zρρ0,n/η = 2. The incident plane wave is TM polarized (equation image = 1, equation image = 0) and impinges on the edge from ϕ0 = 45° and β0 = 90° (solid line), β0 = 70° (dotted line), and β0 = 50° (dash-dotted line). The dots are results given by the method of Pelosi et al. [1998]. The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[42] In Figure 5, the impedance is also anisotropic, but at the direction parallel the edge, the impedance is zero, viz. Zzz0,n = 0 and Zρρ0,n/η = (1 ± j)/2. A TE polarized plane wave (equation image = 0, equation image = 1) incident from ϕ0 = 30° and β0 = 90°, 70°, 50°, respectively. The amplitude of the total field's variation is identical with Pelosi et al. [1998, Figure 6] (dots).

Figure 5.

Amplitude variation of the total longitudinal field ηHz versus the observation points in the absence of a right-angled anisotropic impedance wedge with Zzz0,n = 0, Zρρ0,n/η = (1 ± j)/2. The incident plane wave is TE polarized (equation image = 0, equation image = 1) and impinges on the edge from ϕ0 = 30° and β0 = 90° (solid line), β0 = 70° (dotted line), and β0 = 50° (dash-dotted line). The dots are results given by the method of Pelosi et al. [1998]. The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[43] The field behavior dependent upon the parameters of the impedance is demonstrated in Figure 6 by a TM polarized plane wave (Ezi = 1, ηHzi = 0) incident from β0 = 45°, ϕ0 = 30°. The face illuminated is characterized by an anisotropic impedance Zzz0/η = − j(0.6 − δ), Zρρ0/η = − j(0.6 + δ), with δ as a parameter: 0.4,0.2, 0.02, respectively, while the shaded face (ϕ = nπ) is PEC. The numerical results are consistent with the exact solutions (dots) [Manara and Nepa, 2000].

Figure 6.

Amplitude variation of the total longitudinal field Ez versus the observation points in the absence of a right-angled anisotropic impedance wedge with the face ϕ = nπ perfectly conducting. The incident plane wave is TM polarized (Ezi = 1, ηHzi = 0) and impinges on the edge from ϕ0 = 30° and β0 = 45°. The surface impedances are characterized by Zzz0/η = −j(0.6 − δ), Zρρ0/η = −j(0.6 + δ). For the dotted line, δ = 0.4; for the dash-dotted line, δ = 0.2; and for the solid line, δ = 0.02. The dots are results given by the method of Manara and Nepa [2000]. The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[44] A further example is reported in Figure 7 to demonstrate the field behavior dependent upon the parameters of the impedance. The incident wave is TE polarized (Ezi = 0, ηHzi = 1) and incident from β0 = 45°, ϕ0 = 60°. The face illuminated is anisotropic Zzz0/η = j, Zρρ0/η = jδ and the other is PEC, with δ = 0.5, 0.1, 0. The numerical results are accordant with the exact solutions (dots) [Manara and Nepa, 2000].

Figure 7.

Amplitude variation of the total longitudinal field ηHz versus the observation points in the absence of a right-angled anisotropic impedance wedge with the face ϕ = nπ perfectly conducting. The incident plane wave is TE polarized (Ezi = 0, ηHzi = 1) and impinges on the edge from ϕ0 = 60° and β0 = 45°. The surface impedances are characterized by Zzz0/η = j, Zρρ0/η = jδ. For the dash-dotted line, δ = 0.5; for the dotted line, δ = 0.1; and for the solid line, δ = 0.0. The dots are results given by the method of Manara and Nepa [2000]. The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[45] Figure 8 gives an example of the total longitudinal field Ez scattered by a very obliquely incident plane wave. The incident wave is TM polarized (equation image = 1, equation image = 0) and impinges on the edge from ϕ0 = 45° and β0 = 5°, 10°, 15°, respectively. The impedance tensor is Zzz0/η = j/4, Zρρ0/η = −j/4, Zzzn/η = (1 − j)/4, Zρρn/η = (1 + j)/4, respectively.

Figure 8.

Amplitude variation of the total longitudinal field Ez versus the observation points in the absence of a right-angled anisotropic impedance wedge with Zzz0/η = j/4, Zρρ0/η = −j/4, Zzzn/η = (1 − j)/4, Zρρn/η = (1 + j)/4. The incident plane wave is TM polarized (equation image = 1, equation image = 0) and impinges on the edge from ϕ0 = 45° and β0 = 15° (solid line), β0 = 10° (dotted line), and β0 = 5° (dash-dotted line). The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[46] Plots for the amplitude of the total longitudinal field ηHz of TE polarized (equation image = 0, equation image = 1) are shown in Figure 9. This time the incident plane wave also impinges on the edge from ϕ0 = 45° and β0 a parameter as 5°, 10°, 15°. The impedance tensor is Zzz0/η = 1/4, Zρρ0/η = 1 − j, Zzzn/η = 2j, Zρρn/η = j/2, respectively.

Figure 9.

Amplitude variation of the total longitudinal field ηHz versus the observation points in the absence of a right-angled anisotropic impedance wedge with Zzz0/η = 1/4, Zρρ0/η = 1 − j, Zzzn/η = 2j, Zρρn/η = j/2. The incident plane wave is TE polarized (equation image = 0, equation image = 1) and impinges on the edge from ϕ0 = 60° and β0 = 15° (solid line), β0 = 10° (dotted line), and β0 = 5° (dash-dotted line). The fields are evaluated at a normalized distance ktρ = 10.

[47] Finally, Figure 10 demonstrates the amplitude variation of the total longitudinal field Ez by a TM polarized (equation image = 1, equation image = 0) plane wave incident on a PEC wedge from direction ϕ0 = 30° and β0 a parameter as 60°, 45°, 20° respectively.

Figure 10.

Amplitude variation of the total longitudinal field Ez versus the observation points in the absence of a right-angled perfectly conducting wedge. The incident plane wave is TM polarized (equation image = 1, equation image = 0) and impinges on the edge from ϕ0 = 30° and β0 = 60° (solid line), β0 = 45° (dotted line), and β0 = 20° (dash-dotted line). The fields are evaluated at a normalized distance ktρ = 10 from the edge.

[48] We note that, as apparent in all numerical examples presented in this section, the discontinuity of the diffraction field at the incident shadow boundary and the reflecting shadow boundaries just compensates the discontinuity of GO fields so as to make the total field smooth and continuous everywhere in the absence of the wedge region.

6. Conclusions

[49] An asymptotic formulation (in the UTD framework) for total fields diffracted by an anisotropic impedance wedge has been presented, when the wedge is illuminated by an arbitrarily polarized plane wave impinging at skew incidence on its edge. The surface impedance tensor principal anisotropy axes are along directions parallel and perpendicular to the edge. The solution is derived by resorting to the perturbation technique based on two solved special cases: normal incidence and grazing to the edge incidence. The numerical examples give a good agreement with those in published papers, and provide a uniform and continuous behavior of the field with respect to the skewness of the incident wave not only in close to normal incidence, but also in almost grazing to the edge incidence. Further, the method can be applied to the isotropic case with the arbitrary impedance and the PEC case as well.

Acknowledgments

[50] The research in this paper is supported by the second Academy of China–Aerospace Science and Industry Corporation, China.

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