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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13203–13217
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A note on plane wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium

M. Ayub, M. Ramzan, and A. B. Mann  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13203-13217 (2008)
http://dx.doi.org/10.1364/OE.16.013203


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Abstract

We studied the problem of diffraction of an electromagnetic plane wave by a perfectly conducting finite strip in a homogeneous bi-isotropic medium and obtained some improved results which were presented both mathematically and graphically. The problem was solved by using the Wiener-Hopf technique and Fourier transform. The scattered field in the far zone was determined by the method of steepest decent. The significance of present analysis was that it recovered the results when a strip was widened into a half plane.

© 2008 Optical Society of America

1. Introduction

A Beltrami field is proportional to its own curl everywhere in a source-free region and can be either left-handed or right-handed. For the analysis of time-harmonic electromagnetic fields in isotropic chiral and bi-isotropic media, Bohren [4

4. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458 (1974). [CrossRef]

] was the pioneer and his work was enhanced by Lakhtakia [5

5. A. Lakhtakia, Beltrami Fields in Chiral Media, (World Scientific, Singapore, 1994). [CrossRef]

]. Lakhtakia [6

6. A. Lakhtakia, “Time-dependent scalar Beltrami-Hertz potentials in free space,” Int. J. Infrared Millim. Waves 15, 369–394 (1994). [CrossRef]

], and Lakhtakia and Weiglhofer [7

7. A. Lakhtakia and W. S. Weiglhofer, “Covariances and invariances of the Beltrami-Maxwell postulates,” IEE Proc. Sci. Meas. Technol. 142, 262–26 (1995). [CrossRef]

] worked on the application of Beltrami field to time dependent electromagnetic field. On chiral wedges, Fisanov [8

8. V. V. Fisanov, “Distinctive features of edge fields in a chiral medium,” Sov. J. Commun. Technol. Electronics 37, (3), 93.(1992).

] and Przezdziecki [9

9. S. Przezdziecki, “Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium,” Acta Physica Polonica A, 83739 (1993).

] did exceptional job. Asghar and Lakhtakia [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

] showed that the concept of Beltrami fields can be exploited to calculate the diffraction of only one scalar field and the rest can be obtained thereof.

In this paper, the diffracted field due to a plane wave by a perfectly conducting finite strip in a homogeneous bi-isotropic medium is obtained in an improved form by solving two uncoupled Wiener-Hopf equations. The significance of the present analysis is that the results of half plane [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

] can be deduced by taking an appropriate limit l →∞ whereas this is not possible in [31

31. S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , 9, 39–51 (1998).

]. It is found that the two edges of the strip give rise to two diffracted fields (one from each edge) and an interaction field (double diffraction of two edges).

2. Formulation of the problem

Let us assume the scattering of a plane electromagnetic wave with the assumption that all space is occupied by a homogeneous bi-isotropic medium except for a perfectly conducting strip z=0, -lx≤0. In the Drude-Born-Fedorov representation [5

5. A. Lakhtakia, Beltrami Fields in Chiral Media, (World Scientific, Singapore, 1994). [CrossRef]

,34

34. F. I. Fedorov, Theory of Gyrotropy, (Minsk: Nauka i Tehnika), 1976).

], the bi-isotropic medium is characterized by the following equations

D=εE+εα×E
(1)
B=μH+μβ×H
(2)

where ε and µ are the permittivity and the permeability scalars, respectively, while α and β are the bi-isotropy scalars. D is the electric displacement, H is the magnetic field strength, B is the magnetic induction, and E is the electric field strength. The bi-isotropic medium with α=β is reciprocal and is then called a chiral medium. Recently, it has been proved [26

26. A. Lakhtakia and W. S. Weiglhofer, “Constraint on linear, homogeneous constitutive relations,” Phys. Rev. E , 50, 5017–5019 (1994). [CrossRef]

] that non-reciprocal bi-isotropic media are not permitted by the structure of modern electromagnetic theory. Certainly in the MHz-PHz regime, this statement has not been experimentally challenged yet, although in the ¡ 1 kHz regime there is some experimental evidence to the contrary which has not been independently confirmed [33

33. Mackay and Lakhtakia, “Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121–209 (2008). [CrossRef]

]. However, in the mathematical study the case αβ may also be considered for generality.

Let us assume the time dependence of Beltrami fields to be of the form exp(-iωt), where ω is the angular frequency. The source free Maxwell curl postulates in the bi-isotropic medium can be set up as

×Q1=γ1Q1,
(3)
×Q2=γ2Q2.
(4)

The two wave numbers γ 1and γ 2 are given by

γ1=k(1k2αβ){1+k2(αβ)24+k(α+β)2},
(5)

and

γ2=k(1k2αβ){1+k2(αβ)24k(α+β)2},
(6)

where Beltrami fields in terms of the electric field E and the magnetic field H, as given in [27

27. A. Lakhtakia and B. Shanker, “Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model,” Int. J. Appl. Electromagn. Mater. 4, 65–82 (1993).

], are :

Q1=η1η1+η2(E+iη2H),
(7)

and

Q2=iη1+η2(Eiη1H),
(8)

where Q 1 is the left-handed Beltrami field and Q 2 is the right-handed Beltrami field. In Eqs. (7) and (8), the two impedances η 1and η 2 are given by

η1=η1+k2(αβ)24+k(αβ)2,
(9)

and

η2=η{1+k2(αβ)24k(αβ)2},
(10)

where k=ωεμ and η=με .

Since we are interested in scattering of electromagnetic waves with a prescribed y-variation, therefore, it is appropriate to decompose the Beltrami fields as [28

28. W. S. Weiglhofer, “Isotropie chiral media and scalar Hertz potential,” J. Phys. A,21,2249 (1988).

].

Q1=Q1t+yQ1y,
(11)

with

Q1t=Q1xi+Q1zk.
(12)

and

Q2=Q2t+yQ2y.
(13)

where the fields Q 1t and Q 2t lie in the xz-plane and j is a unit vector along the y-axis such that j.Q 1t=0 and j.Q 2t=0. Now, the Eq. (3) can be written as:

ijkxyzQ1xQ1yQ1z=γ1(Q1xi+Q1yj+Q1zk).
(14)

Assuming all the field vectors having an explicit exp(ikyy) dependence on the variable y and comparing x and z components on both sides of the above equation, we obtain

Q1x=1k1xz2[ikyQ1yxγ1Q1yz],
(15)

and

Q1z=1k1xz2[ikyQ1yz+γ1Q1yx],
(16)

where

k1xz2=γ12ky2.
(17)

Similarly, from Eq. (4), with explicit exp(ikyy) dependence on the variable y, we may obtain

Q2x=1k2xz2[ikyQ2yx+γ2Q2yz],
(18)
Q2z=1k2xz2[ikyQ2yzγ2Q2yx],
(19)

with

k2xz2=γ22ky2.
(19a)

It is sufficient to explore the scattering of the scalar field Q 1y and Q 2y because the other components of Q 1 and Q 2 can then be completely determined by using Eqs. (15–19).

Now using the constitutive relations (1) and (2), the Maxwell curl postulates ∇×E= B-K and ∇×H=- D+J may be written as:

×Q1γ1Q1=S1,
(20a)
×Q2γ2Q2=S2,
(20b)

where S 1 and S 2 are the corresponding source densities and are given by

S1=η1η1+η2(iγ1ωεJ(1+αγ1)K),
(21a)
S2=η1η1+η2(iγ2ωμK(1+βγ2)J).
(21b)

In deriving Eqs. (21a) and (21b), we have used the following relations

1+ωεαη2=(1k2αβ)(1+αγ),
(22)
1ωεαη1=(1k2αβ)η1γ2ωμ,
(23)
η2+ωμβ=(1k2αβ)γ1ωε,
(24)
η1ωμβ=(1k2αβ)η1(1βγ2).
(25)

Furthermore, Q 1 is E like and Q 2 is H like. Similarly S 1 is K like and S 2 is J like where J and K are the electric and magnetic source current densities, respctively. The boundary condition which is necessary is that the tangential component of the electric field must vanish on perfectly conducting finite plane. This implies that Ex=Ey=0, for z=0, -lx≤0. Using this fact in Eqs. (7) and (8), the boundary conditions on the finite plane take the form

Q1yiη2Q2y=0,z=0,lx0,
(26a)

and

Q1xiη2Q2x=0,z=0,lx0.
(26b)

With the help of Eqs. (16) and (17), Eq. (26b) becomes

1k1xz2[ikyQ1yxγ1Q1yz]iη21k2xz2[ikyQ2yxγ2Q2yz]=0,z=0,lx0.
(27)

Thus the scalar fields Q 1y and Q 2y satisfy the boundary conditions (26a) and (27). Now, eliminating Q 2y from Eqs. (26a) and (27), we obtain

Q1yxδQ1yz=0,z=0±,lx0,
(28)

where

δ=γ2k1xz2+γ1k2xz2iky(k2xz2k1xz2).
(28a)

It is worthwhile to note that the boundary conditions (28) are of the same form as impedance boundary conditions [29

29. B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon, London, (1958).

]. We observe that there is no boundary for -∞<x<-l,x>0,z=0. Therefore the continuity conditions are given by

Q1y(x,z+)=Q1y(x,z);<x<l,x>0,z=0,
(29)
Q1y(x,z+)z=Q1y(x,z)z;<x<l,x>0,z=0.
(30)

The edge conditions (local properties) on the field that invoke the appropriate physical constraint of finite energy near the edges of the boundary discontinuities require that

Q1y(x,0)=O(1)andQ1y(x,0)z=O(x12)asx0+,
(31a)
Q1y(x,0)=O(1)andQ1y(x,0)z=O(x+l)12asxl.
(31b)

It is to be noted that the field Q 2y also satisfies Eqs. (28–30). Finally, the scattered field must satisfy the radiation conditions in the limit (x 2+z 2)1/2→∞. We must also observe at this juncture that, in effect, we need to consider the diffraction of only one scalar field, that is either Q 1y or Q 2y, at a time, but the presence of the other scalar field is reflected in the complicated nature of the boundary condition (28). If we set the incident field to be a plane wave, then

Q1y(x,z)=Q1yinc(x,z)+Q1ysca(x,z),
(32)

with

Q1yinc(x,y,z)=exp[i(kyy+k1xx+k1zz)],
(32a)

and scattered field Qsca 1y satisfies the following homogeneous Helmholtz equation

(2x2+2z2+k1xz2)Q1ysca=0.
(33)

where

k1xz2=k1x2+k1z2=γ12ky2.
(33a)

Also the boundary conditions (28) to (30) will take the following form

(xδz)Q1yinc+(xδz)Q1ysca=0,z=0±,lx0,
(34)

and

Q1ysca(x,z+)=Q1ysca(x,z);<x<l,x>0,z=0,
(35a)
zQ1ysca(x,z+)=zQ1ysca(x,z);<x<l,x>0,z=0.
(35b)

For the solution of Eq. (33) subject to the boundary conditions (34–35b), we introduce the Fourier transform w.r.t variable x as:

Ψ¯(υ,z)=12πQ1ysca(x,z)eiυxdx=Ψ¯+(υ,z)+eiυlΨ¯(υ,z)+Ψ¯1(υ,z),
(36)

where

Ψ¯+(υ,z)=12π0Q1ysac(x,z)eiυxdx,
Ψ¯(υ,z)=12πlQ1ysac(x,z)eiυ(x+l)dx,
Ψ¯1(υ,z)=12πl0Q1ysac(x,z)eiυxdx.
(37)

Note that Ψ¯ -(υ,z) is regular for Imυ<Imk 1xz, and Ψ¯ +(υ,z) is regular for Imυ>-Imk 1xz and Ψ¯ 1(υ,z) is analytic in the common region-Imk 1xz<Imυ<Imk 1xz. The Fourier transform of Eq. (32a) in the region -lx≤0,z=0 gives

Ψ¯0(υ,0)=i2π(k1x+υ)[1+exp[i(k1x+υ)l]],
(38)

and its derivative is defined as

Ψ¯0/(υ,0)=k1z2π(k1x+υ)[1+exp[i(k1x+υ)l]].
(38a)

The Fourier transform of Eqs. (34–35b), respectively, yields

(d2dz2+κ2)Ψ¯(υ,z)=0,
(39)

where

κ2=k1xz2υ2,
(39a)
Ψ¯1(υ,0+)=iυδ[Ψ¯1(υ,0+)+Ψ¯0(υ,0)]Ψ¯0(υ,0),
(40)
Ψ¯1(υ,0)=iυδ[Ψ¯1(υ,0)+Ψ¯0(υ,0)]Ψ¯0(υ,0),
(41)

and

Ψ¯(υ,0+)=Ψ¯(υ,0)=Ψ¯(υ,0),
Ψ¯+(υ,0+)=Ψ¯+(υ,0)=Ψ¯+(υ,0),
Ψ¯(υ,0+)=Ψ¯(υ,0)=Ψ¯(υ,0),
Ψ¯+(υ,0+)=Ψ¯+(υ,0)=Ψ¯+(υ,0).
(42)

The solution of Eq. (39) satisfying radiation condition is given by

Ψ¯(υ,z)={A(υ)eiκzifz>0,C(υ)eiκzifz<0.
(43)

By substituting Eqs. (36) and (42) to Eq. (43), we get

Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0+)=A(υ),
(44a)
Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0)=C(υ),
(44b)
Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0+)=iκA(υ),
(44c)
Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0)=iκC(υ).
(44d)

Subtracting Eq. (44b) from Eq. (44a) and Eq. (44d) from Eq. (44c) and then by adding and subtracting the resultant equations, we obtain

A(υ)=J1(υ,0)+J1(υ,0)iκ,
(45)

and

C(υ)=J1(υ,0)+J1(υ,0)iκ,
(46)

where

J1(υ,0)=12[Ψ¯1(υ,0+)Ψ¯1(υ,0)],
(47)

and

J1(υ,0)=12[Ψ¯1(υ,0+)Ψ¯1(υ,0)].
(48)

Making use of Eq. (44a) in Eq. (44c) and Eq. (44b) in Eq. (44d), we can write

Ψ¯+(υ,0)+eivlΨ¯(υ,0)+Ψ¯1(υ,0+)=iκ[Ψ¯+(υ,0)+eivlΨ¯(υ,0)+Ψ¯1(υ,0+)],
(49a)
Ψ¯+(υ,0)+eivlΨ¯(υ,0)+Ψ¯1(υ,0)=iκ[Ψ¯+(υ,0)+eivlΨ¯(υ,0)+Ψ¯1(υ,0)].
(49b)

By eliminating Ψ¯1(υ,0+) from Eqs. (49a) and (45) and Ψ¯1(υ,0-) from Eqs. (49b) and (46) and then by adding the resultant equations, we get

Ψ¯+(υ,0)+eivlΨ¯(υ,0)iκL(υ)J1(υ)+k1z2π(k1x+υ)[1+exp[i(k1x+υ)l]]=0.
(50)

In a similar way, by eliminating Ψ¯ 1(υ,0+) from Eqs. (49a) and (40), Ψ¯ 1(υ,0-) from Eqs. (49b) and (41), and then subtracting the resulting equations, we get

iυΨ¯+(υ,0)iυeiυlΨ¯(υ,0)+δL(υ)J1(υ)+k1x2π(k1x+υ)[1+exp[i(k1x+υ)l]]=0,
(51)

where

L(υ)=(1+υδκ).
(51a)

Eqs. (50) and (51) are the standardWiener-Hopf equations. Let us proceed to find the solution for these equations.

3. Solution of theWiener-Hopf equations

For the solution of theWiener-Hopf equations, one can make use of the following factorization

L(υ)=(1+υδκ)=L+(υ)L(υ),
(52a)

and

κ(υ)=κ+(υ)κ(υ),
(52b)

where L +(υ) and κ+(υ) are regular for Imυ>-Im k 1xz, i.e., for upper half plane and L_(υ) and κ-(υ) are regular for Im υ<Imk 1xz, i.e., lower half plane. The factorization expression (52a) has been accomplished by Asghar et al [30

30. S. Asghar, T. Hayat, and B. Asghar, “Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium,” J. Mod. Opt. , 3, 515–528 (1998). [CrossRef]

]. By putting the values of J 1 (υ,0) and J1(υ,0) from Eqs. (50) and (51) into Eqs. (45) and (46), we get

A(υ)=1iκL(υ){Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+k1z2π(k1x+υ)[1+exp[i(k1x+υ)l]]
+υδ1κL(υ){Ψ¯+(υ,0)+eiυlΨ¯(υ,0)-k1z2π(k1x+υ)[1+exp[i(k1x+υ)l]],
(53)
C(υ)=1iκL(υ){Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+k1z2π(k1x+υ)[1+exp[i(k1x+υ)l]]
+υδ1κL(υ){Ψ¯+(υ,0)+eiυlΨ¯(υ,0)k1x2π(k1x+υ)[1+exp[i(k1x+υ)l]],
(54)

where δ1=1δ . In [31

31. S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , 9, 39–51 (1998).

], the terms of O(δ 1) are neglected while in the present analysis the δ 1 parameter is taken up to order one so that the results due to semi infinite barrier [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

] can be recovered by taking an appropriate limit. To accomplish this, we have to solve both theWiener- Hopf equations to find the values of unknown functions A(υ) and C(υ). For this we use Eqs. (52a) and (52b) in Eqs.(50) and (51), which gives

Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+S(υ)J1(υ)=k1z2π(k1x+υ)[1exp[i(k1x+υ)l]],
(55)

and

iυΨ¯+(υ,0)iυeiυlΨ¯(υ,0)+δL+(υ)L(υ)J1(υ)=k1x2π(k1x+υ)[1exp[i(k1x+υ)l]],
(56)

where

S(υ)=iκ(υ)L(υ)=S+(υ)S(υ),
(57)

and S +(υ) and S -(υ) are regular in upper and lower half plane, respectively. Equations of types (55) and (56) have been considered by Noble [29

29. B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon, London, (1958).

] and a similar analysis may be employed to obtain an approximate solution for large k1xzr(r=x2+z2) . Thus, following the procedure given in [29

29. B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon, London, (1958).

] (Sec. 5.5, p. 196), we obtain

Ψ¯+(υ,0)=k1zS+(υ)2π[G1(υ)+T(υ)C1],
(58)
Ψ¯(υ,0)=k1zS(υ)2π[G2(υ)+T(υ)C2],
(59)
Ψ¯+(υ,0)=iL+(υ)2πυ[G1(υ)+T(υ)C1],
(60)

and

Ψ¯_(υ,0)=iL_(υ)2πυ[G2(υ)T(υ)C2],
(61)

where

S+(υ)=(k1xz+υ)12L+(υ),
(62a)

and

S_(υ)=eiπ2(k1xzυ)12L_(υ),
(62b)
G1(υ)=1(υ+k1x)[1S+(υ)1S+(k1x)]eilk1xR1(υ),
(63)
G2(υ)=eilk1x(υk1x)[1S+(υ)1S+(k1x)]R2(υ),
(64)
C1=S+(k1xz)[G2(k1xz)+S+(k1xz)G1(k1xz)T(k1xz)1S+2(k1xz)T2(k1xz)],
(65)
C2=S+(k1xz)[G1(k1xz)+S+(v)G2(k1xz)T(k1xz)1S+2(k1xz)T2(k1xz)],
(66)
G1(υ)=υ(υ+k1x)[1L+(υ)1L+(k1x)]eilk1xR1(υ),
(67)
G2(υ)=eilk1x(υ-k1x)[υL+(υ)+k1xL+(k1x)]R2(υ),
(68)
C1=L+(k1xz)[G2(k1xz)+L+(k1xz)G1(k1xz)T(k1xz)1L+2(k1xz)T2(k1xz)],
(69)
C2=L+(k1xz)[G1(k1xz)+L+(k1xz)G2(k1xz)T(k1xz)1L+2(k1xz)T2(k1xz)],
(70)
R1,2(υ)=E1[W1{i(k1xzk1x)l}W1{i(k1xz+υ)l}]2πi(υ±k1x),
(71)
T(υ)=12πiE1W1{i(k1xz+υ)l},
(72)
E1=2eiπ4eik1xzl(l)12(i)1h1,
(73)

and

Wn12(p)=0uneuu+pdu=Γ(n+1)ez2p12n12W12(n+1),12n(p),
(74)

where p=-i(k 1xz+υ)l and n=12 .Wm,n is known as a Whittaker function. Now, making use of Eqs. (58–61) in Eqs. (53) and (54), we get

A(υ)C(υ)}=k1zsgn(z)2πiκL(υ){S+(υ)G1(υ)+S+(υ)T(υ)C1+eiυlS(υ)×[G2(υ)+T(υ)C2](1eil(k1x+υ))(k1x+υ)}
+υδ12πκL(υ){L+(υ)G1(υ)+T(υ)L+(v)C1+eiυl×[(L(υ)G2(υ)+T(υ)L+(v)C2)](1eil(k1x+υ))(k1x+υ)},
(75)

where A(ν) corresponds to z>0 and C(ν) corresponds to z<0. We can see that the second term in the above equation was altogether missing in Eq. (70) of [31

31. S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , 9, 39–51 (1998).

]. This term includes the effect of δ 1 parameter in it which can be seen from the solution also. Now, Q sca 1y (x, z) can be obtained by taking the inverse Fourier transform of Eq. (43). Thus

Q1ysca(x,z)=12π{A(υ)C(υ)}exp(iκziυx)dυ,
(76)

where A(υ) and C(υ) are given by Eq. (75). Substituting the value of A(υ) and C(υ) from Eq. (75) into Eq. (76) and using the approximations (63–70), one can break up the field Ψ(x, z) into two parts

Q1ysca(x,z)=Ψsep(x,z)+Ψint(x,z),
(77)

where

Ψsep(x,z)=k1zsgn(z)2πS+(υ)exp(iκziυx)iκL(υ)S+(k1x)(k1x+υ)dυ
+k1zsgn(z)2πeil(k1x+υ)S(υ)exp(iκziυx)iκL(υ)S+(k1x)(k1x+υ)dυ
12πδ1eil(k1x+υ)exp(iκziυx)κL(υ)(k1x+υ)dυ+12πL(υ)eil(k1x+υ)exp(iκziυx)κL(υ)(k1x+υ)L+(k1x)dυ
+12πδ1exp(iκziυx)κL(υ)(k1x+υ)dυ,
(78)

and

Ψint(x,z)=k1zsgn(z)2π1ikL(υ)[S+(υ)R1(υ)eilk1xC1S+(υ)T(υ)
+S+(υ)eilυR2(υ)C2T(υ)S+(υ)eilυ]exp(iκziυx)dυ
12πδ1κL(υ)[T(υ)L+(υ)C1+T(υ)L_(υ)C2L+(υ)R1(υ)eilk1x
L_(υ)R2(υ)eilυ]exp(iκziυx)dυ.
(79)

Here, Ψsep(x,z) consists of two parts each representing the diffracted field produced by the edges at x=0 and x=-l, respectively, although the other edge were absent while Ψint (x,z) gives the interaction of one edge upon the other.

4. Far field solution

The far field may now be calculated by evaluating the integrals appearing in Eqs. (76), (78) and (79), asymptotically [32

32. E. T. Copson, Asymptotic Expansions, (Cambridge University Press, 1967).

]. For that we put x=rcosϑ, |z|=r sinϑ and deform the contour by the transformation υ=-k 1xzcos(ϑ+), (0<ϑ<π, -∞<ξ<∞). Hence, for large k 1xz r, Eqs. (76), (78) and (79) become

Q1ysca(x,y)=ik1xz2π(π2k1xzr)12{A(k1xzcosϑ)C(k1xzcosϑ)}sin(ϑ)exp(ik1xzr+iπ4),
(80)
Q1ysca(sep)(x,y)=[ik1zsgn(z)f1(k1xzcosϑ)+g1(k1xzcosϑ)]
×14πk1xz(1k1xzr)12exp(ik1xzr+iπ4),
(81)

and

Q1ysca(int)(x,y)=[ik1zsgn(z)f2(k1xzcosϑ)+g2(k1xzcosϑ)]
×14πk1xz(1k1xzr)12exp(ik1xzr+iπ4),
(82)

where A(-k 1xz cosϑ) and C(-k 1xz cosϑ) can be found from Eq. (75), while

f1(k1xzcosϑ)=S+(k1xzcosϑ)L(k1xzcosϑ)S+(k1x)(k1xk1xzcosϑ)
eil(k1xk1xzcosϑ)S+(k1xzcosϑ)L(k1xzcosϑ)S+(k1x)(k1xk1xzcosϑ),
(83)
g1(k1xzcosϑ)=1(k1xk1xzcosϑ)[ϑ1eil(k1xk1xzcosϑ)L(k1xzcosϑ)L+(k1xzcosϑ)eil(k1xzk1xzcosϑ)L(k1xzcosϑ)L+(k1x)
δ1L(k1xzcosϑ)],
(84)
f2(k1xzcosϑ)=1L(k1xzcosϑ)[S+(k1xzcosϑ)R1(k1xzcosϑ)eilk1x
+S+(k1xzcosϑ)eilk1xzcosϑR2(k1xzcosϑ)
C1S+(k1xzcosϑ)T(k1xzcosϑ)
C2T(k1xzcosϑ)S+(k1xzcosϑ)eilk1xzcosϑ],
(85)

and

g2(k1xzcosϑ)=1L(k1xzcosϑ)[L+(k1xzcosϑ)R1(k1xzcosϑ)eilk1x.
+L+(k1xzcosϑ)R2(k1xzcosϑ)eilk1xzcosϑ
T(k1xzcosϑ)L+(k1xzcosϑ)C1
T(k1xzcosϑ)L+(k1xzcosϑ)C2].
(86)

The expressions (84) and (86) are additional terms including the effect of δ 1 parameter, which were altogether missing in the analysis of [31

31. S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , 9, 39–51 (1998).

].

5. Remarks

Mathematically we can derive the results of the half plane problem in the following manner: For the analysis purpose, in Eq. (75), it is assumed that the wave number k 1xz has positive imaginary part and using the L Hospital rule successively, the value of E -1, reduces to Ltl(eiklkl) which becomes zero and in turn result the quantities T (υ), R 1,2 (υ), G′ 2 (υ), C′ 1 and G 2 (υ) in zero. The third term in Eqs. (63), (64) and (67) also becomes zero as l→∞. The Eq. (75), after these eliminations reduces to

A(υ)=12π[k1zκ+(υ)L+(υ)iκ(υ)L(υ)(k1x+υ)κ+(k1x)L+(k1x)+δ1υL+(υ)κ(υ)L(υ)(k1x+υ)L+(k1x)].
(87)

Using the factorization

L(υ)=L +(υ)L -(υ),

and

κ(υ)=κ+(υ)κ-(υ).

and substituting the pole contribution υ=-k 1x, the above result reduces to Eq. (26a) of the Half Plane [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

]. Subsequently, Eq. (82), i.e., the interacted field vanishes by adopting the same procedure as in case of Eq. (75), while the separated field results into the diffracted field [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

] as the strip is widened to half plane.

6. Graphical results

A computer program MATHEMATICA has been used for graphical plotting of the separated field given by the expression (81). The values of parameter δ 1 are taken from 0.2 to 0.4. The following situations are considered:

  1. When the source is fixed in one position (for all values of δ 1) relative to the finite barrier, (θ 0=45°, l and θ are allowed to vary).
  2. When the source is fixed in one position, relative to the infinite barrier (θ 0=45°, l and θ are allowed to vary).
Fig. 1. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ0=π4,k1xz=1 , l=1.
Fig. 2. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ0=π4,k1xz=1 , l=50.
Fig. 3. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ0=π4,k1xz=1 , l=100.
Fig. 4. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ0=π4,k1xz=1 , l1022.
Fig. 5. Variation of the amplitude of diffracted field in the half plane versus observation angle ϑ, for different values of δ 1 at θ0=π4 , k 1xz=1.

For all the situations, θ 0=45°, the graphs (1), (2), (3), (4) and (5) show that the field, in the region 0<θπ, is most affected by the changes in δ 1, l and k 1xz. The main features of the graphical results, some of which can be seen in graphs (1), (2), (3), (4) and (5) are as follows:

  1. In graphs (1), (2) and (3) by increasing the value of strip length l and δ 1, the number of oscillations increases and the amplitude of the separated field decreases, respectively.
  2. The graphs of the diffracted field corresponding to the half plane is given in fig. (5). It is observed that the figs. (1)–(4) are in comparison with fig. (5) for various values of the different parameters.

7. Conclusion

The diffracted field due to a plane wave by a perfectly conducting finite strip in a homogeneous bi-isotropic medium is obtained in an improved form. It is found that the two edges of the strip give rise to two diffracted fields (one from each edge) and an interaction field (double diffraction of two edges). This seems to be the first attempt in this direction as we can deduce the results of half plane [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

] by taking an appropriate limit. In [31

31. S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , 9, 39–51 (1998).

], the δ parameter was not taken into account which ends up in an equation from which one cannot deduce the results for semi infinite barrier [10

10. S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

]. This has been proved mathematically as well as numerically which can be considered as check of the validity of the analysis in this paper. Thus, the new solution can be regarded as a correct solution for a perfectly conducting barrier.

Acknowledgments

The authors are grateful to the referee for his valuable suggestions. These suggestions were found useful in enhancing the quality of the paper. One of the authors, M. Ramzan, gratefully acknowledges the financial support provided by the Higher Education Commission (HEC) of Pakistan.

References and links

1.

E. Beltrami, “Considerazioni idrodinamiche,” Rend. Inst. Lombardo Acad. Sci. Lett.22,122–131(1889). An English translation is available: Beltrami, E. “Considerations on hydrodynamics,” Int. J. Fusion Energy 3(3),53–57(1985). V. Trkal, “Paznámka hydrodynamice vazkych tekutin,” C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48,302–311 1919. An English translation is available: V. Trkal, “A note on the hydrodynamics of viscous fluids,” Czech J. Phys. 44,97–106 1994.

2.

S. Chandrasekhar, “Axisymmetric Magnetic Fields and Fluid Motions,” Asrtophys. J ,. 124, 232 (1956). [CrossRef]

3.

A. Lakhtakia, “Viktor Trkal, Beltrami fields, and Trkalian flows,” Czech. J. Phys. 44, 89 (1994). [CrossRef]

4.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458 (1974). [CrossRef]

5.

A. Lakhtakia, Beltrami Fields in Chiral Media, (World Scientific, Singapore, 1994). [CrossRef]

6.

A. Lakhtakia, “Time-dependent scalar Beltrami-Hertz potentials in free space,” Int. J. Infrared Millim. Waves 15, 369–394 (1994). [CrossRef]

7.

A. Lakhtakia and W. S. Weiglhofer, “Covariances and invariances of the Beltrami-Maxwell postulates,” IEE Proc. Sci. Meas. Technol. 142, 262–26 (1995). [CrossRef]

8.

V. V. Fisanov, “Distinctive features of edge fields in a chiral medium,” Sov. J. Commun. Technol. Electronics 37, (3), 93.(1992).

9.

S. Przezdziecki, “Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium,” Acta Physica Polonica A, 83739 (1993).

10.

S. Asghar and A. Lakhtakia, “Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium,” Int. J. Appl. Electromagn. Mater. 5, 181–188, (1994).

11.

J. P. McKelvey, “The case of the curious curl,” Amer. J. Phys ,. 58, 306 (1990). [CrossRef]

12.

H. Zaghloul and O. Barajas, “Force- free magnetic fields,” Amer. J. Phys ,. 58, 783 (1990). [CrossRef]

13.

V. K. Varadan, A. Lakhtakia, and V. V. Varadan, “A comment on the solutions of the equation ∇×a=ka,” J. Phys. A: Math. Gen ,. 20, 2649 (1987). [CrossRef]

14.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, (Springer, Heidelberg, 1989).

15.

A. Lakhtakia, “Recent contributions to classical electromagnetic theory of chiral media,” Speculat. Sci. Technol. .14, 2–17 (1991).

16.

B. D. H. Tellegen, Phillips Res. Rep.3, 81 (1948).; errata: M. E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Stroudsberg, PA: Dowden, Hutchinson and Ross, 1974).

17.

L. I. G. Chambers, “Propagation in a gyrational medium,” Quart. J. Mech. Appl. Math,. 9, 360 (1956).; addendum: Quart. J. Mech. Appl. Math, 11, 253–255, (1958). [CrossRef]

18.

J. C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propagat ,. 38, 227.(1990), [CrossRef]

19.

A. H. Sihvola and I. V. Lindell, “Theory of nonreciprocal and nonsymmetric uniform transmission lines,” Microwave Opt. Technol. Lett. 4, 292 (1991).

20.

A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves , 12, 1167–1174 (1991). [CrossRef]

21.

A. Lakhtakia, “Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium,” Microwave Opt. Technol. Lett , 5, 163 (1992). [CrossRef]

22.

A. Lakhtakia and T. G. Mackay, “Infinite phase velocity as the boundary between positive and negative phase velocities,” Microwave Opt Technol Lett , 20, 165–166 (2004). [CrossRef]

23.

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, Negative phase velocity mediums, W. S. Weiglhofer and A. Lakhtakia (Eds.), Introduction to complex mediums for electromagnetics and optics, SPIE Press., W. A Bellingham, (2003).

24.

T. G. Mackay, “Plane waves with negative phase velocity in isotropic chiral mediums,” Microwave Opt. Technol. Lett. , 45, 120–121 (2005). [CrossRef]

25.

T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E , 69, 026602 (2004). [CrossRef]

26.

A. Lakhtakia and W. S. Weiglhofer, “Constraint on linear, homogeneous constitutive relations,” Phys. Rev. E , 50, 5017–5019 (1994). [CrossRef]

27.

A. Lakhtakia and B. Shanker, “Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model,” Int. J. Appl. Electromagn. Mater. 4, 65–82 (1993).

28.

W. S. Weiglhofer, “Isotropie chiral media and scalar Hertz potential,” J. Phys. A,21,2249 (1988).

29.

B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon, London, (1958).

30.

S. Asghar, T. Hayat, and B. Asghar, “Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium,” J. Mod. Opt. , 3, 515–528 (1998). [CrossRef]

31.

S. Asghar and T. Hayat, “Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium,” Int. J. Appl. Electromagn. Mechanics , 9, 39–51 (1998).

32.

E. T. Copson, Asymptotic Expansions, (Cambridge University Press, 1967).

33.

Mackay and Lakhtakia, “Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121–209 (2008). [CrossRef]

34.

F. I. Fedorov, Theory of Gyrotropy, (Minsk: Nauka i Tehnika), 1976).

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(290.5838) Scattering : Scattering, in-field

ToC Category:
Diffraction and Gratings

History
Original Manuscript: June 10, 2008
Revised Manuscript: July 4, 2008
Manuscript Accepted: July 18, 2008
Published: August 13, 2008

Citation
M. Ayub, M. Ramzan, and A. B. Mann, "A note on plane wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," Opt. Express 16, 13203-13217 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13203


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References

  1. E. Beltrami, "Considerazioni idrodinamiche," Rend. Inst. Lombardo Acad. Sci. Lett.22,122 -131(1889). An English translation is available: Beltrami, E. "Considerations on hydrodynamics," Int. J. Fusion Energy 3,53- 57(1985). V. Trkal, "Paznámka hydrodynamice vazkych tekutin," C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48, 302 -311 1919. An English translation is available: V. Trkal, "A note on the hydrodynamics of viscous fluids," Czech J. Phys. 44, 97-1061994.
  2. S. Chandrasekhar, "Axisymmetric Magnetic Fields and Fluid Motions," Asrtophys. J,  124, 232 (1956). [CrossRef]
  3. A. Lakhtakia, "Viktor Trkal, Beltrami fields, and Trkalian flows," Czech. J. Phys. 44, 89 (1994). [CrossRef]
  4. C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458 (1974). [CrossRef]
  5. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994). [CrossRef]
  6. A. Lakhtakia, "Time-dependent scalar Beltrami-Hertz potentials in free space," Int. J. Infrared Millim. Waves 15, 369-394 (1994). [CrossRef]
  7. A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995). [CrossRef]
  8. V. V. Fisanov, "Distinctive features of edge fields in a chiral medium," Sov. J. Commun. Technol. Electronics 37, 93 (1992).
  9. S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).
  10. S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).
  11. J. P. McKelvey, "The case of the curious curl," Amer. J. Phys. 58, 306 (1990). [CrossRef]
  12. H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990). [CrossRef]
  13. V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987). [CrossRef]
  14. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, Heidelberg, 1989).
  15. A. Lakhtakia, "Recent contributions to classical electromagnetic theory of chiral media," Speculat. Sci. Technol. 14, 2-17 (1991).
  16. B. D. H. Tellegen, Phillips Res. Rep. 3, 81 (1948).; errata: M. E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Stroudsberg, PA: Dowden, Hutchinson and Ross, 1974).
  17. L. I. G. Chambers, "Propagation in a gyrational medium," Quart. J. Mech. Appl. Math. 9, 360 (1956), addendum: Quart. J. Mech. Appl. Math 11, 253-255 (1958). [CrossRef]
  18. J. C. Monzon, "Radiation and scattering in homogeneous general biisotropic regions," IEEE Trans. Antennas Propagat,  38, 227 (1990). [CrossRef]
  19. A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).
  20. A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991). [CrossRef]
  21. A. Lakhtakia, "Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium," Microwave Opt. Technol. Lett,  5, 163 (1992). [CrossRef]
  22. A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004). [CrossRef]
  23. A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, "Negative phase velocity mediums," W. S. Weiglhofer and A. Lakhtakia (Eds.), Introduction to complex mediums for electromagnetics and optics, (SPIE Press. Bellingham, W. A, 2003).
  24. T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005). [CrossRef]
  25. T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004). [CrossRef]
  26. A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994). [CrossRef]
  27. A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).
  28. W. S. Weiglhofer, "Isotropie chiral media and scalar Hertz potential," J. Phys. A 21, 2249 (1988).
  29. B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon, London, 1958).
  30. S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998). [CrossRef]
  31. S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).
  32. E. T. Copson, Asymptotic Expansions (Cambridge University Press, 1967).
  33. Mackay and Lakhtakia, "Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121-209 (2008). [CrossRef]
  34. F. I. Fedorov, Theory of Gyrotropy (Minsk: Nauka i Tehnika), 1976).

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