EMG 24(8) #33634

Electromagnetics, 24:623–635, 2004Copyright © 2004 Taylor & Francis Inc.ISSN: 0272-6343 print/1532-527X onlineDOI: 10.1080/02726340490513356

Plane Wave Diffraction by Impedance PlanarSurfaces with a Step Discontinuity

BAHATTIN TÜRETKEN

The Scientific and Technical Research Council of Turkey (TÜBITAK)-UEKAEGebze, Kocaeli, Turkey

OSMAN YILDIRIM

Electronics Engineering DepartmentTurkish Air Force AcademyYesilyurt, Istanbul, Turkey

The diffraction of electromagnetic plane waves by a thick impedance half-plane lo-cated over a nonhomogeneous half-space backed by an impedance plane with a stepdiscontinuity is investigated rigorously by using the Fourier transform technique inconjunction with the mode matching method. The hybrid method of formulation givesrise to a scalar Wiener–Hopf equation of the second kind, the solution of whichcontains infinitely many constants satisfying an infinite system of linear algebraicequations. A numerical solution of this system is obtained for various values of theplate impedance, plate thickness, permittivity of the dielectric region, and the inci-dence angle wherefrom the effect of these parameters on the diffraction phenomenonis studied.

Keywords electromagnetic wave diffraction, planar surfaces, step discontinuity,Weiner–Hopf technique

Introduction

Recently, in the progress of the field analysis of electromagnetic wave scattering, atten-tion has been given to considering the scattering of plane waves by a series of planarsurfaces. Because of the geometrical and physical complexity of most planar surfaces, itis not always possible to obtain rigorous analytical solutions to diffraction problems. TheWiener–Hopf method is applicable to plane discontinuities like two- or three-part planeproblems, open structures like an infinite set of parallel half-planes, and closed structureslike parallel plate waveguide. The scattering by a parallel plate waveguide (Weinstein,1969; Mittra & Lee, 1971; Carlson & Heins, 1947) and by an infinite grating was re-solved in the early days. The diffraction of plane waves by three parallel infinitely thinsoft half-planes has been considered first by D. S. Jones, who formulated the problem as a

Received 2 December 2003; accepted 12 June 2004.The authors are indebted to the referees for their constructive criticisms, which led to the

improvement of this work.Address correspondence to Osman Yildirim, Electronics Engineering Department, Turkish Air

Force Academy—Hava Harp Okulu, 34800 Yesilyurt, Istanbul, Turkey. E-mail: o.yildirim@hho.edu.tr

623

624 B. Türetken and O. Yildirim

three-dimensional matrix Wiener–Hopf equation (Jones, 1986). These equations are con-verted into a pair involving a two-dimensional matrix and a scalar Wiener–Hopf equation.The three parallel half-planes problem was also considered by Abrahams (1987), who pre-sented a simpler approach to achieve the Wiener–Hopf factorization of the kernel matrix.

Solutions with the plates having finite thickness give much more realistic resultsabout the diffraction of electromagnetic waves by these kind of geometries. All thefindings presented above are based on basic assumptions of infinitely thin waveguide wallsand perfect conductivity. In practice, walls of waveguide are neither infinitely thin norperfectly conducting, therefore it is desirable to discuss the diffraction characteristics of aparallel plate waveguide with certain wall thicknesses and satisfying impedance boundaryconditions (Büyükaksoy & Polat, 1998). The aim of the present work is to produce ananalysis of the case where the semi-infinite plates are neither infinitely thin nor perfectlyconducting with dielectric loaded. To this end, we consider the diffraction of Ez-polarizedplane waves by a thick impedance half-plane located over a nonhomogeneous half-spacebacked by an impedance plane with a step discontinuity. The traditional formulation ofthis problem leads to the Wiener–Hopf equation, which cannot be solved by consideringthe known techniques. In this work, the scattered field in the waveguide region is expandedinto series of normal modes, and the Fourier integral representation is used elsewhere(Büyükaksoy & Polat, 1998). This yields a scalar modified Wiener–Hopf equation of thesecond kind, which can be solved by using the standard techniques. Numerical solutionof this system is obtained for various values of the plate impedance, plate thickness,permittivity of the dielectric region and the incidence angle, through which the effect ofthese parameters on the diffraction phenomenon is studied. A time factor e−iωt with ω

being the angular frequency is assumed and suppressed throughout the paper.

Analysis

We consider the diffraction of an Ez-polarized plane wave by two parallel thick and in-finitely thin semi-infinite impedance half-plates defined by S1 = {(x, y, z); x ∈ (−∞, 0],y ∈ [a, b], z ∈ (−∞,∞)}, S2 = {(x, y, z); x ∈ (−∞, 0], y ∈ [0, c], z ∈ (−∞,∞)},S3 = {(x, y, z); x ∈ [0,∞), y = 0, z ∈ (−∞,∞)}, as depicted in Figure 1. The regionbetween half-planes S1 and S2 consists of nonmagnetic and nonconducting simple mate-rial having permittivity ε1. The region outside of the waveguide formed by S1 and S2 isconsidered as a free space. The surface impedances of the horizontal walls y = c, y = a,y = b, and x < 0 and the surface impedance of the vertical walls x = y, y ∈ (0, c)are denoted by z = ηZ0. The surface impedance of the half-plane S3 is denoted byz1 = η1Z0, with Z0 being the characteristic impedance of the free space.

For analysis purposes, it is convenient to express the total field as follows:

UT (x, y) =

ui(x, y)+ ur(x, y)+ u1(x, y), y > b,

u2(x, y), c < y < a, x < 0,

u3(x, y), 0 < y < b, x > 0.

(1a)

Here, ui is the incident field given by

ui(x, y) = exp(−ik(x cosφ0 + y sin φ0)), (1b)

while ur denotes the field reflected from the plane y = b, namely,

ur(x, y) = η sin φ0 − 1

η sin φ0 + 1exp(−ik(x cosφ0 − (y − 2b) sin φ0)). (1c)

Wave Diffraction by Impendance Planar Surfaces 625

Figure 1. Geometry of the problem.

In (1b), (1c), k is the free space wave number, which is assumed to have a small positiveimaginary part. By considering k1 as the wave number of the dielectric region, uj ,j = 1, 2, 3, which satisfy the Helmholtz equation, are to be determined with the aid ofthe following boundary and continuity relations:(

1 + η

ik

∂

∂y

)u1(x, b) = 0, x < 0, (2a)

(1 − η

ik1

∂

∂y

)u2(x, a) = 0, x < 0, (2b)

(1 + η

ik1

∂

∂y

)u2(x, c) = 0, x < 0, (2c)

(1 + η1

ik

∂

∂y

)u3(x, 0) = 0, x > 0, (2d)

u1(x, b)− u3(x, b) = − 2η sin φ0

1 + η sin φ0e−ikb sin φ0e−ikx cosφ0 , x > 0, (2e)

∂u1

∂y(x, b)− ∂u3

∂y(x, b) = 2ik sin φ0

1 + η sin φ0e−ikb sin φ0e−ikx cosφ0 , x > 0, (2f)

u2(0, y) = u3(0, y), c < y < a, (2g)

∂u2

∂x(0, y) = ∂u3

∂x(0, y), c < y < a, (2h)

(1 + η

ik

∂

∂x

)u3(0, y) = 0, y ∈ {(0, c) ∪ (a, b)}, (2i)

626 B. Türetken and O. Yildirim

where

η = Z/Z0 and η1 = Z1/Z0. (2j)

Since u1(x, y) satisfies the Helmholtz equation in the range x ∈ (−∞,∞), its Fouriertransform with respect to x gives

[d2

dy2+ (k2 − α2)

]F(α, y) = 0, (3a)

with

F(α, y) = F+(α, y)+ F−(α, y), (3b)

where

F±(α, y) = ±∫ ±∞

0u1(x, y)e

iαxdx. (3c)

By taking into account the following asymptotic behaviors of u1 for x → ±∞:

u1(x, y) ={

0(e−iαx), x → −∞,

0(e−ikx cosφ0), x → +∞.(4)

One can show that F+(α, y) and F−(α, y) are regular functions of α in the half-planesIm{α} > Im{k cosφ0} and Im{α} < Im{k}, respectively. The general solution of (3a)satisfying the radiation condition for y → ∞ reads as follows:

F+(α, y)+ F−(α, y) = A(α)eiK(α)(y−b), (5a)

with

K(α) =√k2 − α2. (5b)

The square-root function is defined in the complex α-plane cut along α = k to α = k+i∞and α = −k to α = −k − i∞ such that K(0) = k.

In the Fourier transform domain (2a) takes the form

F−(α, b)+ η

ikF−(α, b) = 0, (6)

where the dot specifies the derivative with respect to y. By using the derivates of (5a)with respect to y and (6), we get

R+(α) = K(α)

kχ(α)A(α), (7a)

where

R+(α) = F+(α, b)+ η

ikF+(α, b) (7b)

Wave Diffraction by Impendance Planar Surfaces 627

and

χ(α) =[η + k

K(α)

]−1

. (7c)

In the region 0 < y < b and x > 0, u3(x, y) satisfies the Helmholtz equation

(∂2

∂x2+ ∂2

∂y2+ k2

)u3(x, y) = 0. (8)

The half-range Fourier transform of (8) yields

[d2

dy2+K2(α)

]G+(α, y) = [f (y)+ αg(y)], (9a)

with

f (y) = ∂

∂su3(0, y), (9b)

g(y) = −iu3(0, y); (9c)

G+(α, y), which is defined by

G+(α, y) =∫ ∞

0u3(x, y)e

iαxdx, (10)

is a function regular in the half-plane Im{α} > Im{−k}. The general solution of (9a)satisfying the impedance boundary condition at y = 0 reads

G+(α, y) = B(α)

[cosK(α)y − ik

η1

sinK(α)y

K(α)

]

+ 1

K(α)

∫ y

0[f (t)+ αg(t)] sin{K(α)(y − t)} dt.

(11)

Combining (2e) and (2f), we get

R+(α) = G+(α, b)+ η

ikG+(α, b), (12)

and B(α) can be solved uniquely to give

M(α)B(α) = R+(α)−∫ b

0[f (t)+ αg(t)]

×[

sin[K(α)(b − t)]K(α)

+ η

ikcos[K(α)(b − t)]

]dt

(13a)

with

M(α) = cos[K(α)b] − η

ikK(α) sin[K(α)b] − ik

η1

sin[K(α)b]K(α)

− η

η1cos[K(α)b]. (13b)

628 B. Türetken and O. Yildirim

Replacing (13a) into (11) we get

G+(α) = 1

M(α)

{cos[K(α)y] − ik

η1

sin[K(α)y]K(α)

}

×

R+(α)−

∫ b

0[f (t)+ αg(t)]

×[

sin[K(α)(b − t)]K(α)

+ η

ikcos[K(α)(b − t)]

]dt

+ 1

K(α)

∫ y

0[f (t)+ αg(t)] sin{K(α)(y − t)} dt.

(14)

Although the left-hand side of (14) is regular in the upper half-plane Im(α) > Im(−k),the regularity of the right-hand side is violated by the presence of simple poles occurringat zeros of M(α), namely at α = αm, satisfying

M(αm) = 0, Im(αm) > Im(k), m = 1, 2, . . . . (15)

These poles can be eliminated by imposing that their residues are zero. This gives

R+(αm) = sinKmb

2Km

[1 − η2

k2(Km)

2]vm[fm + αmgm], (16a)

where Km, vm, fm, and gm specify

Km = K(αm), (16b)

vm = b + η

iksin2[Kmb], (16c)

[fm

gm

]= 2

vm

∫ b

0

[f (t)

g(t)

]cos[Kmt] dt. (16d)

Since vm/2 is the square of the norm related to the eigenfunctions cos[Kmy], it shouldbe positive; for Im(k) → 0, this requirement is satisfied if

Im(η) > − kb

sin2[Kmb]and Re(η) = 0.

Consider now the region c < y < a, x < 0, where the total field can be expressed interms of Fourier series as

u2(x, y) =∞∑n=1

Cn

[sin[ξn(y − c)] − η

ik1ξn cos[ξn(y − c)]

]e−iβnx (17a)

with

tan[ξn(c − a)] = 2ik1ηξn

k21 + η2(ξn)2

, n = 1, 2, . . . (17b)

Wave Diffraction by Impendance Planar Surfaces 629

and

βn =√k2

1 − (ξn)2. (17c)

From the boundary conditions (2g), (2h), (2i) and the relations (9b), (9c) we get

u2(0, y) = ig(y), (18a)

∂

∂xu2(0, y) = f (y), c < y < a. (18b)

Using (2g), (2h), and (2i) we may write

(1 + η

ik

∂

∂x

)u3(0, y) =

0, a < y < b,(1 + η

ik1

∂

∂x

)u2(0, y), c < y < b,

0, 0 < y < c,

(19)

η

ikf (y)+ ig(y) =

0, a < y < b,(1 + η

ik1

∂

∂x

)u2(0, y), c < y < b,

0, 0 < y < c.

(20)

Owing to (16d), f (y) and g(y) can be expanded into a Fourier cosine series as follows:

[f (y)

g(y)

]=

∞∑m=1

[fmgm

]cos[Kmt] dt. (21)

Substituting (17a) and (21) into (18b) and (20), we obtain

∞∑m=1

fm cos[Kmy] = −i∞∑n=1

Cnβn

[sin[ξn(y − c)] − η

ik1ξn cos[ξn(y − c)]

], c < y < a,

(22)

and

∞∑m=1

[gm − η

kfm

]cos[Kmy] =

0, a < y < b,

−i∞∑n=1

Cn

(1 − η

k1βn

)2n(y), c < y < a,

0, a < y < c,

(23a)

where

2n(y) =[

sin[ξn(y − c)] − η

ik1ξn cos[ξn(y − c)]

]. (23b)

630 B. Türetken and O. Yildirim

Let us multiply both sides of (22) by 2l(y) and integrate from y = c to y = a to get

cl = − k

ηβlµl

∞∑m=1

5ml

vmlfm, l = 1, 2, . . . (24a)

with µl , 5ml , and vml being defined by

µl =(

1 − η2ξ2l

k21

)(a − c)− 2η

ik, (24b)

5ml =(

1 − η2ξ2l

k21

)[cos(Kma)+ η

ikKm sin(Kma)

]sin[ξ1(a − c)]

− 2ηξlik1

[cos(Kmc)− η

ikKm sin(Kmc)

],

(24c)

vml = (Km)2 − (ξl)

2. (24d)

Similarly, the multiplication of both sides of (23a) by cos[Kly] and its integration fromy = 0 to y = b yields

gl − η

kfl = k

ηvl

∞∑m=1

5ln

vln

(1 − η

k1βn

)cn, l = 1, 2, . . . . (25)

By using the continuity relations and evaluating the resultant integral, one obtains the fol-lowing modified Wiener–Hopf equation of the second kind valid in the strip Im(k cosφ0) <

Im(α) < Im(k):

ik

η1χ1(α)

χ(α)

N(α)R+(α)− F−(α, b) = − 2k sin φ0

(1 + η sin φ0)

e−ikb sin φ0

(α − k cosφ0)+ 1

η1(η1 − η)

×∞∑m=1

η21K

2m − k2

Km

(fm + αgm)

(α + αm)(α − αm)sin(Kmb),

(26)

where

χ1(α) =[η1 + k

K(α)

]−1

(27a)

and

N(α) = eiK(α)bM(α). (27b)

Wave Diffraction by Impendance Planar Surfaces 631

The formal solution of (17a) can easily be obtained through the classical Wiener–Hopfprocedure. The result is

ik

η1

χ+(α)χ1+(α)N+(α)

R+(α)

= − 2k sin φ0

(1 + η sin φ0)

N−(k cosφ0)χ1−(k cosφ0)

χ−(k cosφ)

e−ikb sin φ0

(α − k cosφ)

− 1

η1(η1 − η)

∞∑m=1

{[η2

1K2m − k2]Km

sin(Kmb)

2αm

N+(αm)χ1+(αm)χ+(αm)

× (fm − αmgm)

(α + αm)

}.

(28)

Here N+(α), χ+(α), χ1+(α) and N−(α), χ−(α), χ1−(α) are the split functions, regularand free of zero in the half-planes Im(α) > Im(−k) and Im(α) < Im(k), respectively,resulting from the Wiener–Hopf factorization of the kernel function as

χ(α)

χ1(α)N(α)= χ+(α)

χ1+(α)N+(α)χ−(α)

χ1−(α)N−(α). (29)

The explicit expression of N±(α) can be obtained by following the procedure outlinedin Mittra and Lee (1971):

N+(α) =[cos(kb)− η

isin(kb)

]1/2exp

{bK(α)

πln

(α + iK(α)

k

)}

× exp

{iαb

π

[1 − C + ln

(2π

kb

)+ i

π

2

]}×

∞∏m=1

(1 + α

αm

)exp

(iαb

mπ

),

(30a)

N−(α) = N+(α). (30b)

In (30a) C is the Euler constant given by C = 0.57721. . . . As to the split functionsχ±(α) and χ1±(α), they can be expressed explicitly in terms of the Maluizhinets function(Senior, 1975) as follows:

χ+(k cosφ) = 23/2

√2

ηsin

φ

2

{Mπ(3π/2 − φ − θ)Mπ(π/2 − φ + θ)

Mπ(π/2)

}2

×{[

1 + √2 cos

(π/2 − φ + θ

2

)][1 + √

2 cos

(3π/2 − φ − θ

2

)]}−1

,

(31a)

χ−(k cosφ) = χ+(k cosφ), (31b)

with

sin θ = 1

η(31c)

632 B. Türetken and O. Yildirim

and

Mπ = exp

{1

8π

∫ z

0

π sin u− 2√

2π sin(u/2)+ 2u

cos udu

}. (31d)

By considering (28) for α = α1, α2, . . . with (16a), (24a), and (25), one gets infinitelymany equations in an infinite number of unknowns, which yield the constants fr .

Analysis of the Diffracted Field

The scattered field in the region y > b can be obtained by taking the inverse Fouriertransforms of F(α, y):

u1(x, y) = 1

2π

∫L

A(α)eiK(α)(y−b)e−iαx dα. (32)

Here L is a straight line parallel to the real α-axis lying in the strip Im(k cosφ0) <

Im(α) < Im(k). The asymptotic evaluation of the integrals in (32) through the saddlepoint technique enables us to write for the diffraction field.

u1d(ρ, φ) ≈

u0D(φ, φ0)+ eiπ/4

√2π

sin φ

(1 + η sin φ)

N−(k cosφ)χ1−(k cosφ0)

χ−(k cosφ)

×∞∑m=1

[η21K

2m − k2]Km

sin[Kmb]2αm

N+(αm)χ1+(αm)χ+(αm)

(fm − αmgm)

(αm − k cosφ)

× eikρ√kρ

, (33a)

with

u0 = e−ikb sin φ0 (33b)

and

D(φ, φ0) = e−i3π/4

√2

π· sin φ0

(1 + η sin φ0)

sin φ

(1 + η sin φ)

N−(k cosφ0)χ1−(k cosφ0)

χ−(k cosφ0)

× N−(k cosφ)χ1−(k cosφ)

χ−(k cosφ)

1

(cosφ + cosφ0),

(33c)

where (ρ, φ) are the cylindrical polar coordinates defined by x = ρ cosφ, y−b = ρ sin φand u0 is the expression of the incident field at y = b, x = 0.

It can be checked easily that the result obtained by setting η = η1 = 0 and ε1 = ε0in (33a) is identical to that previously given in Alkumru (1998, formula 42(b)). Anothereffective check on the analysis can be made by showing numerically that the continuityrelation in (22) is satisfied. The absolute error is less than 1% for truncation numberbigger than 14.

In order to see the accuracy and the effectiveness of different values of the parametersrelated to geometry on the diffraction phenomenon, some numerical results concerningthe variation of the diffracted field (20 log |u1d

√kρ| versus the observation angle φ are

presented. Figure 2 shows the variation of the scattering field versus the observation angle

Wave Diffraction by Impendance Planar Surfaces 633

Figure 2. 20 log |u1d√kρ| versus the observation angle φ, for different values of (a/b) (a = 2c =

0.25λ, φ0 = 45◦, ε1 = 2ε0 = 118π 10−9 F/m, η = 3i, η1 = 1.5i).

for the thickness of the plate (b–a). As expected, the diffracted field increases with theincreasing values of the thickness.

Figure 3 shows the variation of the diffracted field with the observation angle φ, fordifferent values of the impedance half-plane Z1 = η1Z0. From Figure 3 one concludes thatthe diffracted field decreases with the decreasing values of |η1|. This variation continuesuntil the observation angle equals to 120◦. Figure 4 gives the variation of the diffractedfield versus the observation angle φ for different values of the incidence angle φ0 in-crease. And finally, Figure 5 shows the variation of the diffracted field versus the obser-vation angle φ for different values of the permittivity ε1. It is observed that the diffractedfield is not greatly affected by the permittivity for observation angle between (105◦–180◦).

Concluding Remarks

The diffraction of electromagnetic plane waves by a thick impedance half-plane locatedover a nonhomogeneous half-space backed by an impedance plane with a step disconti-nuity is investigated rigorously by using the Fourier transform technique in conjunctionwith the mode matching method. This solution contains a set of infinitely many constantssatisfying an infinite system of linear algebraic equations. The most important contribu-tion of the present work is obtaining the accurate formal solutions of the problem usingthe Wiener–Hopf technique in a transform domain. This solution is the exact solutionunder some restrictions. The first restriction is due to the truncation of the series withinfinite number of terms at a specific limit. And the second restriction stems from thecalculation of the diffracted field by using saddle point technique based on asymptoticapproximation (see (32)–(33)).

634 B. Türetken and O. Yildirim

Figure 3. 20 log |u1d√kρ| versus the observation angle φ, for different values of Z1 = η1Z0

(b = 2a = 4c = 0.5λ, φ0 = 45◦, ε1 = ε0 = 136π 10−9 F/m, η = −2i).

Figure 4. 20 log |u1d√kρ| versus the observation angle φ, for different values of φ0 (b = 2a =

4c = 0.5λ, φ0 = 45◦, ε1 = ε0 = 136π 10−9 F/m, η = −1.8i, η1 = −1.2i).

Wave Diffraction by Impendance Planar Surfaces 635

Figure 5. 20 log |u1d√kρ| versus the observation angle φ, for different values of ε1 (b = 2a =

4c = 0.5λ, φ0 = 45◦, ε0 = 136π 10−9 F/m, η = 3i, η1 = 1.5i).

The results obtained in this paper, in the case where η = η1 = 0 and ε1 = ε0,have been compared with the asymptotic solution related to Alkumru (1998) and theagreement was found to be very satisfactory. Furthermore, it has been shown numericallythat the error in the fulfillment of the continuity relation (22) is satisfactory. This can beconsidered as a good check for the reliability of the analysis made in this paper.

A numerical solution of this system is obtained for various values of the plateimpedance, plate thickness, permittivity of the dielectric region and the incidence anglewherefrom the effect of these parameters on the diffraction phenomenon is studied.

References

Abrahams, I. D. 1987. Scattering of sound by three semi-infinite planes. J. Sound Vibration 112(2):396–398.

Alkumru, A. 1998. Plane wave diffraction by three parallel thick impedance half-planes. Journalof Electromagnetic Waves and Applications 12:801–819.

Büyükaksoy, A., and B. Polat. 1998. Plane wave diffraction by a thick walled parallel plateimpedance waveguide. IEEE Antennas and Propag. 46 (Special Issue).

Carlson, J. F., and A. E. Heins. 1947. The reflection of an electromagnetic plane wave by an infiniteset of plates I. Quart. Appl. Math. 5:82–88.

Heins, A. E., and J. F. Carlson. 1947. The reflection of an electromagnetic plane wave by an infiniteset of plates II. Quart. Appl. Math. 5:82–88.

Jones, D. S. 1986. Diffraction by three semi-infinite planes. J. Sound. A404:299–321.Mittra, R., and S. W. Lee. 1971. Analytical techniques in the theory of guided waves. New York:

The MacMillan Company.Senior, T. B. A. 1975. Half-plane edge diffraction. Radio Sci. 10:645–650.Weinstein, L. A. 1969. The theory of diffraction and factorisation method. Boulder, CO: Golem Press.