Scattering by a Penetrable Wedge Structure

V. Daniele1 G. Lombardi 2

1 Dipartimento di Elettronica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail: vito.daniele@polito.it, tel.: +39 011. 564068 2 Dipartimento di Elettronica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail: guido.lombardi@polito.it, tel.: +39 011 564012.

Abstract This paper presents our recent results on the study of the scattering and diffraction of an incident plane wave by an isotropic penetrable wedge.

We have formulated the problem in terms of generalized Wiener-Hopf equations and the solution is obtained using analytical and numerical-analytical approaches that reduce the Wiener-Hopf factorization to Fredholm integral equations of second kind.

Several numerical tests at normal incidence are presented to validate the new technique.

1 INTRODUCTION

Since the mid-nineties the diffraction of a penetrable wedge has been a canonical diffraction problem of great interest for engineering, mathematical and physical communities, see references in [1].

Tentative approaches of solutions were based on different formulations with analytical and numerical schemes, some of them are reported in the following.

Radlow [2] proposed in 1964 an attempt of solution for the diffraction by the right-angled dielectric wedge solving a multidimensional Wiener-Hopf equation, but Kraut et al. ascertained that this solution was wrong in 1969 [3].

Integral equation formulation based on a standard perturbation technique was used in 1977 by Rawlins [4] for the dielectric wedge problem using a general integral equation formulation.

In 1991 Kim et al. extended the work of Rawlins by proposing an approximate solution of an arbitrary-angled dielectric wedge using physical optics [5-6] and they gave the results in terms of diffraction coefficients and far-field patterns.

The Sommerfeld-Malyuzhinets (SM) method was extended by Budaev in 1995 [7] to study the dielectric wedge problems.

In 1999 Croisille and Lebeau have presented a formulation of the problem in terms of singular integral equations in the Fourier domain, solved using the Galerkin collocation method [8].

The Kontorovich-Lebedev (KL) transform has been effectively used by Rawlins and Osipov et al. in [9-11] in a one-dimensional spectral.

Fig. 1: the dielectric wedge

In this paper we briefly present our recent results

on the study of the scattering and diffraction of an incident plane wave by an isotropic penetrable wedge [1].

We have formulated the problem in terms of generalized Wiener-Hopf equations and the solution is obtained using a numerical-analytical approach that reduces the Wiener-Hopf factorization to Fredholm integral equations of second kind.

The method constitutes an extension of the Wiener-Hopf formulation for impenetrable wedge problems that we have effectively used [12].

As in [12] we propose an approximate method to factorize the kernel of the generalized Wiener-Hopf equations that is based on the use of the Fredholm method introduced in [13].

In the framework of the penetrable wedge, applicative and theoretical aspects are detailed reported in [1, 14, 15].

2 MATHEMATICAL FORMULATION

For the sake of simplicity, let us consider an incident E-polarized plane wave impinging a dielectric wedge with relative permittivity r , see Fig. 1.

( )ojk Cosiz oE E e (1)

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where k is the propagation constant and o is the azimuthal angle of incidence. The Wiener-Hopf technique [16] for angular problems is based on the introduction of the following Laplace transforms:

0

0

( , ) ( , )

( , ) ( , )

jz z

j

V E e d

I H e d (2)

where the subscript + indicates plus functions, i.e. functions having regular half-planes of convergence that are upper half-planes in the -plane. From [1, 14-16], we obtain two uncoupled system of GWHE functional equations of the following kind (i=1,3):

11 ( 1) 1

1

2( 1) 2 ( 1) 2

2

( ) ( ) ( )

( ) ( ) ( )

i i i

i i i

Y X m X mn

Y X m X mn

(3)

where the unknowns are related to the physical quantities (2). Notice that the unknown are defined in three different complex planes: 1 2, ,m m . From (3), using the mapping proposed in [16], we obtain a new system of equations (4) that hold in a new unique complex plane i .

11 1 1( 1)

1

22 2 2( 1) ( 1)

2

( ) ( ) ( )

( ) ( ) ( )

i i i

i i i

Y X Xn

Y X Xn

(4)

Although the new system of equations is Classical Wiener-Hopf equations of dimension 2, the unknowns are defined in two different complex planes i . The solution of equations (4) can be obtained using the general procedure described in [1, 13-14], the Fredholm technique, that reduce the factorization of

CWHE to the solution of Fredholm integral equations of second kind, see (5)-(6). Moreover, in this case, we need to relate the unknowns defined in different complex planes i . This can be accomplished using the Cauchy formula [1,14], for instance:

12 1

1 21

( )1( )2

ii

X mX m dm

j m m (7)

The use of the angular plane w1 and w2 and of special warping improve the convergence of the numerical discretization of the equations (5)-(7). Further details on the procedure to get the solution and numerical results in terms of diffraction coefficients of a dielectric wedge will be available in [1] and discussed at the conference.

2 NUMERICAL RESULTS

The efficiency, the convergence and the validation of the proposed approximate solutions is illustrated through several test problems in [1]. Here we propose some numerical results in detail related to a particular test case. As in [1] we estimate the solutions in terms of total far-field, GTD component, UTD component, GO component. With reference to Fig. 1. the physical parameters of the problem are: =7 /8, r=3, o=13 /32, and Eo=1 V/m. The E-polarized incident plane wave impinges on the dielectric wedge and generates one face a ( ) reflected wave and one face a transmitted wave. The transmitted wave is totally reflected inside the wedge for two times and generates two evanescent transmitted waves through the two interfaces. This configuration allows defining three geometrical optics shadow boundaries (omitting the ones for the evanescent waves): incident shadow boundary, face areflected shadow boundary, face a transmitted and double totally reflected shadow boundary.

1 12 12 2 2 2

1 111 1 1 1 2 1 1 12 2

1 11

' ( ')'1( ) ( ) ( ) ' ( )

2 '

o o

o oo

o

k kX

k kkY X X d n

jk (5)

1 2 1 22 22 2 2 2

1 2 1 21 22 2 1 2 2 2 22 2

2 21 2

' ( ')'1( ) ( ) ( ) ' 0

2 '

k kX

k kkY X X d

jk (6)

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As a consequence, there are five GO regions: 1) incident wave, 2) incident wave, face a reflected wave, evanescent wave through face a, 3) face a transmitted wave, face b ( ) reflected wave from face a transmitted wave, double reflected wave from face a transmitted wave, 4) face a transmitted wave, face b reflected wave from face a transmitted wave, 5) evanescent wave through face b. The GO field, the UTD component and, the total far-field pattern are reported in Fig. 2 at the distance k =10 from the edge of the wedge.

Fig. 2: the GO field, the UTD component and, the

total far-field pattern at k =10 with the 5 GO regions labeled.

The solution of the problem is obtained applying the discretized method reported in [1] where the Fredholm factorization method is applied to the GWHE with discretization parameters A=10, h=0.05. Fig. 3 reports the absolute value of the total GTD diffraction coefficient (in dB) for each observation angle . The peaks of the GTD diffraction coefficients occur for the GO angles. Comparison with the PEC wedge is also shown.

Acknowledgments

This work was partially supported by the Italian Ministry of Education, University and Research (MIUR) under the PRIN grant 20097JM7YR.

Fig. 3: absolute value of the total GTD diffraction

coefficient (dB)

References

[1] V. Daniele, and G. Lombardi, “The Wiener-Hopf solution of the isotropic penetrable wedge problem: diffraction and total field,” IEEE Trans. Antennas Propagat., in press, Oct. 2011.

[2] J. Radlow, “Diffraction by a right-angled dielectric wedge,”' Intern. J. Enging Sci., vol. 2, n.3, pp. 275-290, Jun. 1964.

[3] E.A. Kraut, and G.W. Lehaman, “Diffraction of electromagnetic waves by a right-angled dielectric wedge,” J. Math. Phys., vol. 10, pp. 1340-1348, 1969.

[4] A.D. Rawlins, “Diffraction by a dielectric wedge,” J. Inst. Math. Appl., vol. 19, pp. 231-279, 1977.

[5] S.Y. Kim, J.W. Ra, and S.Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: Part I - Physical optics,” IEEE Trans. Antennas Propagat., vol. 39, n. 9, pp. 1272-1281, Sept. 1991.

[6] S.Y. Kim, J.W. Ra, and S.Y. Shin, “Diffraction by an arbitrary-angled dielectric wedge: Part II - Correction to Physical Optics Solution,” IEEE Trans. Antennas Propagat.,} vol. 39, n. 9, pp. 1282-1292, Sept. 1991.

[7] B. Budaev, Diffraction by wedges, London, UK: Longman Scient., 1995.

[8] J.P. Croisille, and G. Lebeau, Diffraction by an immersed elastic wedge, Lecture notes in math. 1723, Berlin, Germany: Springer-Verlag, 1999.

[9] A.D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proceedings Royal Society. Mathematical, Physical and Engineering Sciences., vol. 455, pp. 2655- 2686, 1999.

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[10] A.V. Osipov, “On the method of Kontorovich-Lebedev integrals in problems of wave diffraction in sectorial media,” in Problems of diffraction and propagation of waves, Vol. 25, pp. 173–-219, St Petersburg University Publications, 1993.

[11] M.A. Salem, A. Kamel, and A.V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proceedings of Royal Society: Mathematical, Physical and Engineering Sciences, vol. 462, pp. 2503-2522, 2006.

[12] V. Daniele, and G. Lombardi, “Wiener-Hopf Solution for Impenetrable Wedges at Skew Incidence,” IEEE Trans. Antennas Propagat., vol. 54, n. 9, pp. 2472-2485, Sept. 2006, doi: 10.1109/TAP.2006.880723

[13] V.G. Daniele, and G. Lombardi, “Fredholm Factorization of Wiener-Hopf scalar and matrix kernels,” Radio Science, vol. 42: RS6S01, 2007, doi:10.1029/2007RS003673

[14] V.G. Daniele, “The Wiener-Hopf Formulation of the Penetrable Wedge Problem: Part I,” Electromagnetics, Vol.30, n.8, pp.625–643, 2010.

[15] V.G. Daniele, “The Wiener-Hopf Formulation of the Penetrable Wedge Problem: Part II,” Electromagnetics, Vol.31, n.1, pp.1–17, 2011.

[16] V.G. Daniele, “The Wiener-Hopf technique for impenetrable wedges having arbitrary aperture angle,” SIAM Journal of Applied Mathematics, vol. 63, n. 4, pp. 1442-1460, 2003

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