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

Beltrami flows were first introduced in the late 19th century [1]. There was no significant work on Beltrami flows for next 60 years. However, in 1950s and onwards it gained wide application in fluid mechanics and other related areas. Chandrasekhar [2], reintroduced Beltrami flows and worked on force free magnetic fields. Lakhtakia [3] compiled a catalogue on contemporary works.

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] was the pioneer and his work was enhanced by Lakhtakia [5]. Lakhtakia [6], and Lakhtakia and Weiglhofer [7] worked on the application of Beltrami field to time dependent electromagnetic field. On chiral wedges, Fisanov [8] and Przezdziecki [9] did exceptional job. Asghar and Lakhtakia [10] 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.

A Beltrami magnetostatic field exerts no Lorentz force on an electrically charged particle, and for this reason the concept has been extensively used in astrophysics as well as magneto-hydrodynamics [11,12]. Beltrami fields also occur as the circularly polarized plane waves in electromagnetic theory [13]. Although circularly polarized plane waves in free space and natural, optically active media [14,15] have been known since the time of Fresenel, their theoretical value is best expressed in biisotropic media [16–21]. In recent years, propagation of plane waves with negative phase velocity and its related applications in isotropic chiral materials can be found in [22–25].

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] can be deduced by taking an appropriate limit l →∞ whereas this is not possible in [31]. 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 [5,34], the bi-isotropic medium is characterized by the following equations

D=εE+εα×E
B=μH+μβ×H

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] 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]. 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,
×Q2=γ2Q2.

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

γ1=k(1k2αβ){1+k2(αβ)24+k(α+β)2},

and

γ2=k(1k2αβ){1+k2(αβ)24k(α+β)2},

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

Q1=η1η1+η2(E+iη2H),

and

Q2=iη1+η2(Eiη1H),

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,

and

η2=η{1+k2(αβ)24k(αβ)2},

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].

Q1=Q1t+yQ1y,

with

Q1t=Q1xi+Q1zk.

and

Q2=Q2t+yQ2y.

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).

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],

and

Q1z=1k1xz2[ikyQ1yz+γ1Q1yx],

where

k1xz2=γ12ky2.

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

Q2x=1k2xz2[ikyQ2yx+γ2Q2yz],
Q2z=1k2xz2[ikyQ2yzγ2Q2yx],

with

k2xz2=γ22ky2.

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,
×Q2γ2Q2=S2,

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

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

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

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

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,

and

Q1xiη2Q2x=0,z=0,lx0.

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

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

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,

where

δ=γ2k1xz2+γ1k2xz2iky(k2xz2k1xz2).

It is worthwhile to note that the boundary conditions (28) are of the same form as impedance boundary conditions [29]. 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,
Q1y(x,z+)z=Q1y(x,z)z;<x<l,x>0,z=0.

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+,
Q1y(x,0)=O(1)andQ1y(x,0)z=O(x+l)12asxl.

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),

with

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

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

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

where

k1xz2=k1x2+k1z2=γ12ky2.

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

(xδz)Q1yinc+(xδz)Q1ysca=0,z=0±,lx0,

and

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

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),

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.

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]],

and its derivative is defined as

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

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

(d2dz2+κ2)Ψ¯(υ,z)=0,

where

κ2=k1xz2υ2,
Ψ¯1(υ,0+)=iυδ[Ψ¯1(υ,0+)+Ψ¯0(υ,0)]Ψ¯0(υ,0),
Ψ¯1(υ,0)=iυδ[Ψ¯1(υ,0)+Ψ¯0(υ,0)]Ψ¯0(υ,0),

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).

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

Ψ¯(υ,z)={A(υ)eiκzifz>0,C(υ)eiκzifz<0.

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

Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0+)=A(υ),
Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0)=C(υ),
Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0+)=iκA(υ),
Ψ¯+(υ,0)+eiυlΨ¯(υ,0)+Ψ¯1(υ,0)=iκC(υ).

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,

and

C(υ)=J1(υ,0)+J1(υ,0)iκ,

where

J1(υ,0)=12[Ψ¯1(υ,0+)Ψ¯1(υ,0)],

and

J1(υ,0)=12[Ψ¯1(υ,0+)Ψ¯1(υ,0)].

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

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.

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,

where

L(υ)=(1+υδκ).

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),

and

κ(υ)=κ+(υ)κ(υ),

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]. 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]],
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]],

where δ1=1δ . In [31], 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] 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]],

and

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

where

S(υ)=iκ(υ)L(υ)=S+(υ)S(υ),

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

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

and

Ψ¯_(υ,0)=iL_(υ)2πυ[G2(υ)T(υ)C2],

where

S+(υ)=(k1xz+υ)12L+(υ),

and

S_(υ)=eiπ2(k1xzυ)12L_(υ),
G1(υ)=1(υ+k1x)[1S+(υ)1S+(k1x)]eilk1xR1(υ),
G2(υ)=eilk1x(υk1x)[1S+(υ)1S+(k1x)]R2(υ),
C1=S+(k1xz)[G2(k1xz)+S+(k1xz)G1(k1xz)T(k1xz)1S+2(k1xz)T2(k1xz)],
C2=S+(k1xz)[G1(k1xz)+S+(v)G2(k1xz)T(k1xz)1S+2(k1xz)T2(k1xz)],
G1(υ)=υ(υ+k1x)[1L+(υ)1L+(k1x)]eilk1xR1(υ),
G2(υ)=eilk1x(υ-k1x)[υL+(υ)+k1xL+(k1x)]R2(υ),
C1=L+(k1xz)[G2(k1xz)+L+(k1xz)G1(k1xz)T(k1xz)1L+2(k1xz)T2(k1xz)],
C2=L+(k1xz)[G1(k1xz)+L+(k1xz)G2(k1xz)T(k1xz)1L+2(k1xz)T2(k1xz)],
R1,2(υ)=E1[W1{i(k1xzk1x)l}W1{i(k1xz+υ)l}]2πi(υ±k1x),
T(υ)=12πiE1W1{i(k1xz+υ)l},
E1=2eiπ4eik1xzl(l)12(i)1h1,

and

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

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)},

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]. 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,

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),

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,

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.

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]. 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),
Q1ysca(sep)(x,y)=[ik1zsgn(z)f1(k1xzcosϑ)+g1(k1xzcosϑ)]
×14πk1xz(1k1xzr)12exp(ik1xzr+iπ4),

and

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

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

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].

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

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)].

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]. 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] 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.

Download Full Size | PPT Slide | PDF

 

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.

Download Full Size | PPT Slide | PDF

 

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.

Download Full Size | PPT Slide | PDF

 

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.

Download Full Size | PPT Slide | PDF

 

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.

Download Full Size | PPT Slide | PDF

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] by taking an appropriate limit. In [31], 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]. 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).

References

  • View by:
  • |
  • |
  • |

  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).

2008 (1)

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

2005 (1)

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

2004 (2)

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

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]

1998 (2)

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]

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).

1995 (1)

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

1994 (5)

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

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.

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

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).

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

1993 (2)

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).

S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).

1992 (2)

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

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

1991 (3)

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

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

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

1990 (3)

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

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

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

1988 (1)

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

1987 (1)

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]

1974 (1)

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

1958 (1)

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]

1956 (1)

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

Asghar, B.

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]

Asghar, S.

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).

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]

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).

Barajas, O.

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

Beltrami, E.

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.

Bohren, C. F.

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

Chambers, L. I. G.

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]

Chandrasekhar, S.

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

Diamond, J. R.

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

Fisanov, V. V.

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

Hayat, T.

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]

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).

Lakhtakia, A.

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

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]

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

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

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

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).

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

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).

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

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

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

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]

Lindell, I. V.

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

Mackay, T. G.

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

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

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]

McKelvey, J. P.

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

Monzon, J. C.

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

Przezdziecki, S.

S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).

Shanker, B.

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).

Sihvola, A. H.

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

Varadan, V. K.

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]

Varadan, V. V.

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]

Weiglhofer, W. S.

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

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

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

Zaghloul, H.

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

Acta Physica Polonica A (1)

S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).

Amer. J. Phys. (2)

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

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

Asrtophys. J (1)

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

Chem. Phys. Lett. (1)

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

Czech J. Phys. (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.

Czech. J. Phys. (1)

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

IEE Proc. Sci. Meas. Technol. (1)

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

IEEE Trans. Antennas Propagat (1)

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

Int. J. Appl. Electromagn. Mater. (2)

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).

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).

Int. J. Appl. Electromagn. Mechanics (1)

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).

Int. J. Infrared Millim. Waves (2)

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

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

J. Mod. Opt. (1)

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]

J. Phys. A (1)

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

J. Phys. A: Math. Gen. (1)

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]

Microwave Opt Technol. Lett. (1)

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]

Microwave Opt. Technol. Lett (1)

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

Microwave Opt. Technol. Lett. (2)

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

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

Phys. Rev. E (2)

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

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

Prog. Opt. (1)

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

Quart. J. Mech. Appl. Math (1)

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]

Sov. J. Commun. Technol. Electronics (1)

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

Speculat. Sci. Technol. (1)

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

Other (7)

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).

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

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

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

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

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

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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l=1.

Fig. 2.
Fig. 2.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l=50.

Fig. 3.
Fig. 3.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l=100.

Fig. 4.
Fig. 4.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l1022.

Fig. 5.
Fig. 5.

Variation of the amplitude of diffracted field in the half plane versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1xz =1.

Equations (129)

Equations on this page are rendered with MathJax. Learn more.

D = ε E + ε α × E
B = μ H + μ β × H
× Q 1 = γ 1 Q 1 ,
× Q 2 = γ 2 Q 2 .
γ 1 = k ( 1 k 2 α β ) { 1 + k 2 ( α β ) 2 4 + k ( α + β ) 2 } ,
γ 2 = k ( 1 k 2 α β ) { 1 + k 2 ( α β ) 2 4 k ( α + β ) 2 } ,
Q 1 = η 1 η 1 + η 2 ( E + i η 2 H ) ,
Q 2 = i η 1 + η 2 ( E i η 1 H ) ,
η 1 = η 1 + k 2 ( α β ) 2 4 + k ( α β ) 2 ,
η 2 = η { 1 + k 2 ( α β ) 2 4 k ( α β ) 2 } ,
Q 1 = Q 1 t + y Q 1 y ,
Q 1 t = Q 1 x i + Q 1 z k .
Q 2 = Q 2 t + y Q 2 y .
i j k x y z Q 1 x Q 1 y Q 1 z = γ 1 ( Q 1 x i + Q 1 y j + Q 1 z k ) .
Q 1 x = 1 k 1 xz 2 [ ik y Q 1 y x γ 1 Q 1 y z ] ,
Q 1 z = 1 k 1 xz 2 [ ik y Q 1 y z + γ 1 Q 1 y x ] ,
k 1 xz 2 = γ 1 2 k y 2 .
Q 2 x = 1 k 2 xz 2 [ ik y Q 2 y x + γ 2 Q 2 y z ] ,
Q 2 z = 1 k 2 xz 2 [ ik y Q 2 y z γ 2 Q 2 y x ] ,
k 2 xz 2 = γ 2 2 k y 2 .
× Q 1 γ 1 Q 1 = S 1 ,
× Q 2 γ 2 Q 2 = S 2 ,
S 1 = η 1 η 1 + η 2 ( i γ 1 ω ε J ( 1 + α γ 1 ) K ) ,
S 2 = η 1 η 1 + η 2 ( i γ 2 ωμ K ( 1 + β γ 2 ) J ) .
1 + ω ε α η 2 = ( 1 k 2 α β ) ( 1 + α γ ) ,
1 ω ε α η 1 = ( 1 k 2 α β ) η 1 γ 2 ω μ ,
η 2 + ω μ β = ( 1 k 2 α β ) γ 1 ω ε ,
η 1 ω μ β = ( 1 k 2 α β ) η 1 ( 1 β γ 2 ) .
Q 1 y i η 2 Q 2 y = 0 , z = 0 , l x 0 ,
Q 1 x i η 2 Q 2 x = 0 , z = 0 , l x 0 .
1 k 1 xz 2 [ i k y Q 1 y x γ 1 Q 1 y z ] i η 2 1 k 2 xz 2 [ i k y Q 2 y x γ 2 Q 2 y z ] = 0 , z = 0 , l x 0 .
Q 1 y x δ Q 1 y z = 0 , z = 0 ± , l x 0 ,
δ = γ 2 k 1 xz 2 + γ 1 k 2 xz 2 i k y ( k 2 xz 2 k 1 xz 2 ) .
Q 1 y ( x , z + ) = Q 1 y ( x , z ) ; < x < l , x > 0 , z = 0 ,
Q 1 y ( x , z + ) z = Q 1 y ( x , z ) z ; < x < l , x > 0 , z = 0 .
Q 1 y ( x , 0 ) = O ( 1 ) and Q 1 y ( x , 0 ) z = O ( x 1 2 ) as x 0 + ,
Q 1 y ( x , 0 ) = O ( 1 ) and Q 1 y ( x , 0 ) z = O ( x + l ) 1 2 as x l .
Q 1 y ( x , z ) = Q 1 y inc ( x , z ) + Q 1 y sca ( x , z ) ,
Q 1 y inc ( x , y , z ) = exp [ i ( k y y + k 1 x x + k 1 z z ) ] ,
( 2 x 2 + 2 z 2 + k 1 xz 2 ) Q 1 y sca = 0 .
k 1 xz 2 = k 1 x 2 + k 1 z 2 = γ 1 2 k y 2 .
( x δ z ) Q 1 y inc + ( x δ z ) Q 1 y sca = 0 , z = 0 ± , l x 0 ,
Q 1 y sca ( x , z + ) = Q 1 y sca ( x , z ) ; < x < l , x > 0 , z = 0 ,
z Q 1 y sca ( x , z + ) = z Q 1 y sca ( x , z ) ; < x < l , x > 0 , z = 0 .
Ψ ¯ ( υ , z ) = 1 2 π Q 1 y sca ( x , z ) e i υ x d x = Ψ ¯ + ( υ , z ) + e i υ l Ψ ¯ ( υ , z ) + Ψ ¯ 1 ( υ , z ) ,
Ψ ¯ + ( υ , z ) = 1 2 π 0 Q 1 y sac ( x , z ) e i υ x d x ,
Ψ ¯ ( υ , z ) = 1 2 π l Q 1 y sac ( x , z ) e i υ ( x + l ) d x ,
Ψ ¯ 1 ( υ , z ) = 1 2 π l 0 Q 1 y sac ( x , z ) e i υ x d x .
Ψ ¯ 0 ( υ , 0 ) = i 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] ,
Ψ ¯ 0 / ( υ , 0 ) = k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] .
( d 2 d z 2 + κ 2 ) Ψ ¯ ( υ , z ) = 0 ,
κ 2 = k 1 xz 2 υ 2 ,
Ψ ¯ 1 ( υ , 0 + ) = i υ δ [ Ψ ¯ 1 ( υ , 0 + ) + Ψ ¯ 0 ( υ , 0 ) ] Ψ ¯ 0 ( υ , 0 ) ,
Ψ ¯ 1 ( υ , 0 ) = i υ δ [ Ψ ¯ 1 ( υ , 0 ) + Ψ ¯ 0 ( υ , 0 ) ] Ψ ¯ 0 ( 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 ) = Ψ ¯ + ( υ , 0 ) .
Ψ ¯ ( υ , z ) = { A ( υ ) e i κ z if z > 0 , C ( υ ) e i κ z if z < 0 .
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) = A ( υ ) ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) = C ( υ ) ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) = i κ A ( υ ) ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) = i κ C ( υ ) .
A ( υ ) = J 1 ( υ , 0 ) + J 1 ( υ , 0 ) i κ ,
C ( υ ) = J 1 ( υ , 0 ) + J 1 ( υ , 0 ) i κ ,
J 1 ( υ , 0 ) = 1 2 [ Ψ ¯ 1 ( υ , 0 + ) Ψ ¯ 1 ( υ , 0 ) ] ,
J 1 ( υ , 0 ) = 1 2 [ Ψ ¯ 1 ( υ , 0 + ) Ψ ¯ 1 ( υ , 0 ) ] .
Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) = i κ [ Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) ] ,
Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) = i κ [ Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) ] .
Ψ ¯ + ( υ , 0 ) + e ivl Ψ ¯ ( υ , 0 ) i κ L ( υ ) J 1 ( υ ) + k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] = 0 .
i υ Ψ ¯ + ( υ , 0 ) i υ e i υ l Ψ ¯ ( υ , 0 ) + δ L ( υ ) J 1 ( υ ) + k 1 x 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] = 0 ,
L ( υ ) = ( 1 + υ δ κ ) .
L ( υ ) = ( 1 + υ δ κ ) = L + ( υ ) L ( υ ) ,
κ ( υ ) = κ + ( υ ) κ ( υ ) ,
A ( υ ) = 1 i κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ]
+ υ δ 1 κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) - k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] ,
C ( υ ) = 1 i κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ]
+ υ δ 1 κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) k 1 x 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + S ( υ ) J 1 ( υ ) = k 1 z 2 π ( k 1 x + υ ) [ 1 exp [ i ( k 1 x + υ ) l ] ] ,
i υ Ψ ¯ + ( υ , 0 ) i υ e i υ l Ψ ¯ ( υ , 0 ) + δ L + ( υ ) L ( υ ) J 1 ( υ ) = k 1 x 2 π ( k 1 x + υ ) [ 1 exp [ i ( k 1 x + υ ) l ] ] ,
S ( υ ) = i κ ( υ ) L ( υ ) = S + ( υ ) S ( υ ) ,
Ψ ¯ + ( υ , 0 ) = k 1 z S + ( υ ) 2 π [ G 1 ( υ ) + T ( υ ) C 1 ] ,
Ψ ¯ ( υ , 0 ) = k 1 z S ( υ ) 2 π [ G 2 ( υ ) + T ( υ ) C 2 ] ,
Ψ ¯ + ( υ , 0 ) = i L + ( υ ) 2 π υ [ G 1 ( υ ) + T ( υ ) C 1 ] ,
Ψ ¯ _ ( υ , 0 ) = i L _ ( υ ) 2 π υ [ G 2 ( υ ) T ( υ ) C 2 ] ,
S + ( υ ) = ( k 1 xz + υ ) 1 2 L + ( υ ) ,
S _ ( υ ) = e i π 2 ( k 1 xz υ ) 1 2 L _ ( υ ) ,
G 1 ( υ ) = 1 ( υ + k 1 x ) [ 1 S + ( υ ) 1 S + ( k 1 x ) ] e ilk 1 x R 1 ( υ ) ,
G 2 ( υ ) = e ilk 1 x ( υ k 1 x ) [ 1 S + ( υ ) 1 S + ( k 1 x ) ] R 2 ( υ ) ,
C 1 = S + ( k 1 xz ) [ G 2 ( k 1 xz ) + S + ( k 1 xz ) G 1 ( k 1 xz ) T ( k 1 xz ) 1 S + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
C 2 = S + ( k 1 xz ) [ G 1 ( k 1 xz ) + S + ( v ) G 2 ( k 1 xz ) T ( k 1 xz ) 1 S + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
G 1 ( υ ) = υ ( υ + k 1 x ) [ 1 L + ( υ ) 1 L + ( k 1 x ) ] e ilk 1 x R 1 ( υ ) ,
G 2 ( υ ) = e ilk 1 x ( υ - k 1 x ) [ υ L + ( υ ) + k 1 x L + ( k 1 x ) ] R 2 ( υ ) ,
C 1 = L + ( k 1 xz ) [ G 2 ( k 1 xz ) + L + ( k 1 xz ) G 1 ( k 1 xz ) T ( k 1 xz ) 1 L + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
C 2 = L + ( k 1 xz ) [ G 1 ( k 1 xz ) + L + ( k 1 xz ) G 2 ( k 1 xz ) T ( k 1 xz ) 1 L + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
R 1 , 2 ( υ ) = E 1 [ W 1 { i ( k 1 xz k 1 x ) l } W 1 { i ( k 1 xz + υ ) l } ] 2 π i ( υ ± k 1 x ) ,
T ( υ ) = 1 2 π i E 1 W 1 { i ( k 1 xz + υ ) l } ,
E 1 = 2 e i π 4 e ik 1 xz l ( l ) 1 2 ( i ) 1 h 1 ,
W n 1 2 ( p ) = 0 u n e u u + p du = Γ ( n + 1 ) e z 2 p 1 2 n 1 2 W 1 2 ( n + 1 ) , 1 2 n ( p ) ,
A ( υ ) C ( υ ) } = k 1 z sgn ( z ) 2 π i κ L ( υ ) { S + ( υ ) G 1 ( υ ) + S + ( υ ) T ( υ ) C 1 + e i υ l S ( υ ) × [ G 2 ( υ ) + T ( υ ) C 2 ] ( 1 e il ( k 1 x + υ ) ) ( k 1 x + υ ) }
+ υ δ 1 2 π κ L ( υ ) { L + ( υ ) G 1 ( υ ) + T ( υ ) L + ( v ) C 1 + e i υ l × [ ( L ( υ ) G 2 ( υ ) + T ( υ ) L + ( v ) C 2 ) ] ( 1 e il ( k 1 x + υ ) ) ( k 1 x + υ ) } ,
Q 1 y sca ( x , z ) = 1 2 π { A ( υ ) C ( υ ) } exp ( i κ z i υ x ) d υ ,
Q 1 y sca ( x , z ) = Ψ sep ( x , z ) + Ψ int ( x , z ) ,
Ψ sep ( x , z ) = k 1 z sgn ( z ) 2 π S + ( υ ) exp ( i κ z i υ x ) i κ L ( υ ) S + ( k 1 x ) ( k 1 x + υ ) d υ
+ k 1 z sgn ( z ) 2 π e il ( k 1 x + υ ) S ( υ ) exp ( i κ z i υ x ) i κ L ( υ ) S + ( k 1 x ) ( k 1 x + υ ) d υ
1 2 π δ 1 e il ( k 1 x + υ ) exp ( i κ z i υ x ) κ L ( υ ) ( k 1 x + υ ) d υ + 1 2 π L ( υ ) e il ( k 1 x + υ ) exp ( i κ z i υ x ) κ L ( υ ) ( k 1 x + υ ) L + ( k 1 x ) d υ
+ 1 2 π δ 1 exp ( i κ z i υ x ) κ L ( υ ) ( k 1 x + υ ) d υ ,
Ψ int ( x , z ) = k 1 z sgn ( z ) 2 π 1 ikL ( υ ) [ S + ( υ ) R 1 ( υ ) e ilk 1 x C 1 S + ( υ ) T ( υ )
+ S + ( υ ) e il υ R 2 ( υ ) C 2 T ( υ ) S + ( υ ) e il υ ] exp ( i κ z i υ x ) d υ
1 2 π δ 1 κ L ( υ ) [ T ( υ ) L + ( υ ) C 1 + T ( υ ) L _ ( υ ) C 2 L + ( υ ) R 1 ( υ ) e ilk 1 x
L _ ( υ ) R 2 ( υ ) e il υ ] exp ( i κ z i υ x ) d υ .
Q 1 y sca ( x , y ) = i k 1 xz 2 π ( π 2 k 1 xz r ) 1 2 { A ( k 1 xz cos ϑ ) C ( k 1 xz cos ϑ ) } sin ( ϑ ) exp ( i k 1 xz r + i π 4 ) ,
Q 1 y sca ( sep ) ( x , y ) = [ i k 1 z sgn ( z ) f 1 ( k 1 xz cos ϑ ) + g 1 ( k 1 xz cos ϑ ) ]
× 1 4 π k 1 xz ( 1 k 1 xz r ) 1 2 exp ( ik 1 xz r + i π 4 ) ,
Q 1 y sca ( int ) ( x , y ) = [ i k 1 z sgn ( z ) f 2 ( k 1 xz cos ϑ ) + g 2 ( k 1 xz cos ϑ ) ]
× 1 4 π k 1 xz ( 1 k 1 xz r ) 1 2 exp ( ik 1 xz r + i π 4 ) ,
f 1 ( k 1 xz cos ϑ ) = S + ( k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) S + ( k 1 x ) ( k 1 x k 1 xz cos ϑ )
e il ( k 1 x k 1 xz cos ϑ ) S + ( k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) S + ( k 1 x ) ( k 1 x k 1 xz cos ϑ ) ,
g 1 ( k 1 xz cos ϑ ) = 1 ( k 1 x k 1 xz cos ϑ ) [ ϑ 1 e il ( k 1 x k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) L + ( k 1 xz cos ϑ ) e il ( k 1 xz k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) L + ( k 1 x )
δ 1 L ( k 1 xz cos ϑ ) ] ,
f 2 ( k 1 x z cos ϑ ) = 1 L ( k 1 x z cos ϑ ) [ S + ( k 1 x z cos ϑ ) R 1 ( k 1 x z cos ϑ ) e i l k 1 x
+ S + ( k 1 x z cos ϑ ) e i l k 1 x z cos ϑ R 2 ( k 1 x z cos ϑ )
C 1 S + ( k 1 x z cos ϑ ) T ( k 1 x z cos ϑ )
C 2 T ( k 1 x z cos ϑ ) S + ( k 1 x z cos ϑ ) e i l k 1 x z cos ϑ ] ,
g 2 ( k 1 x z cos ϑ ) = 1 L ( k 1 x z cos ϑ ) [ L + ( k 1 x z cos ϑ ) R 1 ( k 1 x z cos ϑ ) e i l k 1 x .
+ L + ( k 1 x z cos ϑ ) R 2 ( k 1 x z cos ϑ ) e i l k 1 x z cos ϑ
T ( k 1 x z cos ϑ ) L + ( k 1 x z cos ϑ ) C 1
T ( k 1 x z cos ϑ ) L + ( k 1 x z cos ϑ ) C 2 ] .
A ( υ ) = 1 2 π [ k 1 z κ + ( υ ) L + ( υ ) i κ ( υ ) L ( υ ) ( k 1 x + υ ) κ + ( k 1 x ) L + ( k 1 x ) + δ 1 υ L + ( υ ) κ ( υ ) L ( υ ) ( k 1 x + υ ) L + ( k 1 x ) ] .

Metrics