A note on plane wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium

M. Ayub; M. Ramzan; A. B. Mann

doi:10.1364/OE.16.013203

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

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

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
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)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

More Like This

Diffraction of homogeneous and inhomogeneous plane waves by a planar junction between perfectly...
Yusuf Z. Umul
J. Opt. Soc. Am. A 24(6) 1786-1792 (2007)

Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting...
L. E. R. Petersson, et al.
J. Opt. Soc. Am. A 19(11) 2265-2280 (2002)

Material parameter retrieval procedure for general bi-isotropic metamaterials and its application...
Do-Hoon Kwon, et al.
Opt. Express 16(16) 11822-11829 (2008)

Related Topics
Table of Contents Category
- Diffraction and Gratings
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.

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

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.

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, -l≤x≤0. In the Drude-Born-Fedorov representation [5,34], the bi-isotropic medium is characterized by the following equations

D = ε E + ε α \nabla \times E

B = μ H + μ β \nabla \times 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 α≠β 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

\nabla \times Q_{1} = γ_{1} Q_{1},

\nabla \times Q_{2} = {- γ}_{2} Q_{2} .

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

γ_{1} = \frac{k}{(1 - k^{2} α β)} {\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} + \frac{k (α + β)}{2}},

and

γ_{2} = \frac{k}{(1 - k^{2} α β)} {\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} - \frac{k (α + β)}{2}},

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

Q_{1} = \frac{η_{1}}{η_{1} + η_{2}} (E + i η_{2} H),

and

Q_{2} = \frac{i}{η_{1} + η_{2}} (E - i η_{1} H),

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

η_{1} = \frac{η}{\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} + \frac{k (α - β)}{2}},

and

η_{2} = η {\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} - \frac{k (α - β)}{2}},

where $k = ω \sqrt{ε μ}$ and $η = \sqrt{\frac{μ}{ε}}$ .

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].

Q_{1} = Q_{1 t} + y Q_{1 y},

with

Q_{1 t} = Q_{1 x} i + Q_{1 z} k .

and

Q_{2} = Q_{2 t} + y Q_{2 y} .

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:

∣ \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ Q_{1 x} & Q_{1 y} & Q_{1 z} \end{matrix} ∣ = γ_{1} (Q_{1 x} i + Q_{1 y} j + Q_{1 z} k) .

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

Q_{1 x} = \frac{1}{k_{1 xz}^{2}} [{ik}_{y} \frac{\partial Q_{1 y}}{\partial x} - γ_{1} \frac{\partial Q_{1 y}}{\partial z}],

and

Q_{1 z} = \frac{1}{k_{1 xz}^{2}} [{ik}_{y} \frac{\partial Q_{1 y}}{\partial z} + γ_{1} \frac{\partial Q_{1 y}}{\partial x}],

where

k_{1 xz}^{2} = γ_{1}^{2} - k_{y}^{2} .

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

Q_{2 x} = \frac{1}{k_{2 xz}^{2}} [{ik}_{y} \frac{\partial Q_{2 y}}{\partial x} + γ_{2} \frac{\partial Q_{2 y}}{\partial z}],

Q_{2 z} = \frac{1}{k_{2 xz}^{2}} [{ik}_{y} \frac{\partial Q_{2 y}}{\partial z} - γ_{2} \frac{\partial Q_{2 y}}{\partial x}],

with

k_{2 xz}^{2} = γ_{2}^{2} - k_{y}^{2} .

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=iω B-K and ∇×H=-iω D+J may be written as:

\nabla \times Q_{1} - γ_{1} Q_{1} = S_{1},

\nabla \times Q_{2} - γ_{2} Q_{2} = S_{2},

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

S_{1} = \frac{η_{1}}{η_{1} + η_{2}} (\frac{i γ_{1}}{ω ε} J - (1 + α γ_{1}) K),

S_{2} = \frac{η_{1}}{η_{1} + η_{2}} (\frac{i γ_{2}}{ωμ} K - (1 + β γ_{2}) J) .

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

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} \frac{γ_{2}}{ω μ},

η_{2} + ω μ β = (1 - k^{2} α β) \frac{γ_{1}}{ω ε},

η_{1} - ω μ β = (1 - k^{2} α β) η_{1} (1 - β γ_{2}) .

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 E_x=E_y=0, for z=0, -l≤x≤0. Using this fact in Eqs. (7) and (8), the boundary conditions on the finite plane take the form

Q_{1 y} - i η_{2} Q_{2 y} = 0, z = 0, - l \leq x \leq 0,

and

Q_{1 x} - i η_{2} Q_{2 x} = 0, z = 0, - l \leq x \leq 0 .

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

\frac{1}{k_{1 xz}^{2}} [i k_{y} \frac{\partial Q_{1 y}}{\partial x} - γ_{1} \frac{\partial Q_{1 y}}{\partial z}] - i η_{2} \frac{1}{k_{2 xz}^{2}} [i k_{y} \frac{\partial Q_{2 y}}{\partial x} - γ_{2} \frac{\partial Q_{2 y}}{\partial z}] = 0, z = 0, - l \leq x \leq 0 .

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

\frac{\partial Q_{1 y}}{\partial x} \mp δ \frac{\partial Q_{1 y}}{\partial z} = 0, z = 0^{\pm}, - l \leq x \leq 0,

where

δ = \frac{γ_{2} k_{1 xz}^{2} + γ_{1} k_{2 xz}^{2}}{i k_{y} (k_{2 xz}^{2} - k_{1 xz}^{2})} .

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

Q_{1 y} (x, z^{+}) = Q_{1 y} (x, z^{-}); - \infty < x < - l, x > 0, z = 0,

\frac{\partial Q_{1 y} (x, z^{+})}{\partial z} = \frac{\partial Q_{1 y} (x, z^{-})}{\partial z}; - \infty < 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

Q_{1 y} (x, 0) = O (1) and \frac{\partial Q_{1 y} (x, 0)}{\partial z} = O (x^{- \frac{1}{2}}) as x \to 0^{+},

Q_{1 y} (x, 0) = O (1) and \frac{\partial Q_{1 y} (x, 0)}{\partial z} = O {(x + l)}^{- \frac{1}{2}} as x \to - l .

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

Q_{1 y} (x, z) = Q_{1 y}^{inc} (x, z) + Q_{1 y}^{sca} (x, z),

with

Q_{1 y}^{inc} (x, y, z) = \exp [i (k_{y} y + k_{1 x} x + k_{1 z} z)],

and scattered field Q^sca _1y satisfies the following homogeneous Helmholtz equation

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial z^{2}} + k_{1 xz}^{2}) Q_{1 y}^{sca} = 0 .

where

k_{1 xz}^{2} = k_{1 x}^{2} + k_{1 z}^{2} = γ_{1}^{2} - k_{y}^{2} .

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

(\frac{\partial}{\partial x} \mp δ \frac{\partial}{\partial z}) Q_{1 y}^{inc} + (\frac{\partial}{\partial x} \mp δ \frac{\partial}{\partial z}) Q_{1 y}^{sca} = 0, z = 0^{\pm}, - l \leq x \leq 0,

and

Q_{1 y}^{sca} (x, z^{+}) = Q_{1 y}^{sca} (x, z^{-}); - \infty < x < - l, x > 0, z = 0,

\frac{\partial}{\partial z} Q_{1 y}^{sca} (x, z^{+}) = \frac{\partial}{\partial z} Q_{1 y}^{sca} (x, z^{-}); - \infty < 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:

\bar{Ψ} (υ, z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} Q_{1 y}^{sca} (x, z) e^{i υ x} d x = {\bar{Ψ}}_{+} (υ, z) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, z) + {\bar{Ψ}}_{1} (υ, z),

where

{\bar{Ψ}}_{+} (υ, z) = \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} Q_{1 y}^{sac} (x, z) e^{i υ x} d x,

Note that $\bar{Ψ}$ -(υ,z) is regular for Imυ<Imk _1xz, and $\bar{Ψ}$ ₊(υ,z) is regular for Imυ>-Imk _1xz and $\bar{Ψ}$ ₁(υ,z) is analytic in the common region-Imk _1xz<Imυ<Imk _1xz. The Fourier transform of Eq. (32a) in the region -l≤x≤0,z=0 gives

{\bar{Ψ}}_{0} (υ, 0) = \frac{i}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]],

and its derivative is defined as

{\bar{Ψ}}_{0}^{/} (υ, 0) = \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]] .

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

(\frac{d^{2}}{d z^{2}} + κ^{2}) \bar{Ψ} (υ, z) = 0,

where

κ^{2} = k_{1 xz}^{2} - υ^{2},

{\bar{Ψ}}_{1}^{'} (υ, 0^{+}) = - \frac{i υ}{δ} [{\bar{Ψ}}_{1} (υ, 0^{+}) + {\bar{Ψ}}_{0} (υ, 0)] - {\bar{Ψ}}_{0}^{'} (υ, 0),

{\bar{Ψ}}_{1}^{'} (υ, 0^{-}) = \frac{i υ}{δ} [{\bar{Ψ}}_{1} (υ, 0^{-}) + {\bar{Ψ}}_{0} (υ, 0)] - {\bar{Ψ}}_{0}^{'} (υ, 0),

and

{\bar{Ψ}}_{-} (υ, 0^{+}) = {\bar{Ψ}}_{-} (υ, 0^{-}) = {\bar{Ψ}}_{-} (υ, 0),

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

\bar{Ψ} (υ, z) = {\begin{matrix} A (υ) e^{i κ z} & if z > 0, \\ C (υ) e^{- i κ z} & if z < 0 . \end{matrix}

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

{\bar{Ψ}}_{+} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{+}) = A (υ),

{\bar{Ψ}}_{+} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{-}) = C (υ),

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{+}) = i κ A (υ),

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{-}) = - i κ C (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) = J_{1} (υ, 0) + \frac{J_{1}^{'} (υ, 0)}{i κ},

and

C (υ) = {- J}_{1} (υ, 0) + \frac{J_{1}^{'} (υ, 0)}{i κ},

where

J_{1} (υ, 0) = \frac{1}{2} [{\bar{Ψ}}_{1} (υ, 0^{+}) - {\bar{Ψ}}_{1} (υ, 0^{-})],

and

J_{1}^{'} (υ, 0) = \frac{1}{2} [{\bar{Ψ}}_{1}^{'} (υ, 0^{+}) - {\bar{Ψ}}_{1}^{'} (υ, 0^{-})] .

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

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{i v l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{+}) = i κ [{\bar{Ψ}}_{+} (υ, 0) + e^{- i v l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{+})],

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i v l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{-}) = i κ [{\bar{Ψ}}_{+} (υ, 0) + e^{- i v l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{-})] .

By eliminating $\bar{Ψ}$ ′₁(υ,0⁺) from Eqs. (49a) and (45) and $\bar{Ψ}$ ′₁(υ,0^-) from Eqs. (49b) and (46) and then by adding the resultant equations, we get

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- ivl} {\bar{Ψ}}_{-}^{'} (υ, 0) - i κ L (υ) J_{1} (υ) + \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]] = 0 .

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

- i υ {\bar{Ψ}}_{+} (υ, 0) - i υ e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + δ L (υ) J_{1}^{'} (υ) + \frac{k_{1 x}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]] = 0,

where

L (υ) = (1 + \frac{υ}{δ κ}) .

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 + \frac{υ}{δ κ}) = 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 J′₁(υ,0) from Eqs. (50) and (51) into Eqs. (45) and (46), we get

A (υ) = \frac{1}{i κ L (υ)} {{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]]

C (υ) = \frac{1}{i κ L (υ)} {{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]]

where $δ_{1} = \frac{1}{δ}$ . 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

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} \bar{Ψ}_{-}^{'} (υ, 0) + {S (υ) J}_{1} (υ) = \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [1 - \exp [- i (k_{1 x} + υ) l]],

and

- i υ {\bar{Ψ}}_{+} (υ, 0) - i υ e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + δ L_{+} (υ) L_{-} (υ) J_{1}^{'} (υ) = \frac{k_{1 x}}{\sqrt{2 π} (k_{1 x} + υ)} [1 - \exp [- i (k_{1 x} + υ) l]],

where

S (υ) = - i κ (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 $k_{1 x z} r (r = \sqrt{x^{2} + z^{2}})$ . Thus, following the procedure given in [29] (Sec. 5.5, p. 196), we obtain

{\bar{Ψ}}_{+}^{'} (υ, 0) = \frac{k_{1 z} S_{+} (υ)}{\sqrt{2 π}} [G_{1} (υ) + T (υ) C_{1}],

{\bar{Ψ}}_{-}^{'} (υ, 0) = \frac{k_{1 z} S_{-} (υ)}{\sqrt{2 π}} [G_{2} (- υ) + T (- υ) C_{2}],

{\bar{Ψ}}_{+}^{'} (υ, 0) = \frac{{i L}_{+} (υ)}{\sqrt{2 π} υ} [G_{1}^{'} (υ) + T (υ) C_{1}^{'}],

and

\bar{Ψ}_(υ, 0) = \frac{- i L_(υ)}{\sqrt{2 π υ}} [G_{2}^{'} (- υ) - T (- υ) C_{2}^{'}],

where

S_{+} (υ) = {(k_{1 xz} + υ)}^{\frac{1}{2}} L_{+} (υ),

and

S_(υ) = e^{\frac{- i π}{2}} {(k_{1 xz} - υ)}^{\frac{1}{2}} L_(υ),

G_{1} (υ) = \frac{1}{(υ + k_{1 x})} [\frac{1}{S_{+} (υ)} - \frac{1}{S_{+} ({- k}_{1 x})}] - e^{- {ilk}_{1 x}} R_{1} (υ),

G_{2} (υ) = \frac{e^{- {ilk}_{1 x}}}{(υ - k_{1 x})} [\frac{1}{S_{+} (υ)} - \frac{1}{S_{+} (k_{1 x})}] - R_{2} (υ),

C_{1} = S_{+} (k_{1 xz}) [\frac{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}) [\frac{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) = \frac{υ}{(υ + k_{1 x})} [\frac{1}{L_{+} (υ)} - \frac{1}{L_{+} (- k_{1 x})}] - e^{- {ilk}_{1 x}} R_{1} (υ),

G_{2}^{'} (υ) = \frac{e^{- {ilk}_{1 x}}}{(υ - k_{1 x})} [\frac{υ}{L_{+} (υ)} + \frac{k_{1 x}}{L_{+} (k_{1 x})}] - R_{2} (υ),

C_{1}^{'} = L_{+} (k_{1 xz}) [\frac{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}) [\frac{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) = \frac{E_{- 1} [W_{- 1} {- i (k_{1 xz} \mp k_{1 x}) l} - W_{- 1} {- i (k_{1 xz} + υ) l}]}{2 π i (υ \pm k_{1 x})},

T (υ) = \frac{1}{2 π i} E_{- 1} W_{- 1} {- i (k_{1 xz} + υ) l},

E_{- 1} = 2 e^{i \frac{π}{4}} e^{{ik}_{1 xz} l} {(l)}^{\frac{1}{2}} {(i)}^{- 1} h_{- 1},

and

W_{n - \frac{1}{2}} (p) = \int_{0}^{\infty} \frac{u^{n} e^{- u}}{u + p} du = Γ (n + 1) e^{\frac{z}{2}} p^{\frac{1}{2} n - \frac{1}{2}} W_{- \frac{1}{2} (n + 1), \frac{1}{2} n} (p),

where p=-i(k _1xz+υ)l and $n = \frac{- 1}{2}$ .W_m,n is known as a Whittaker function. Now, making use of Eqs. (58–61) in Eqs. (53) and (54), we get

\begin{matrix} A (υ) \\ C (υ) \end{matrix}} = \frac{k_{1 z} sgn (z)}{\sqrt{2 π} i κ L (υ)} {\begin{matrix} S_{+} (υ) G_{1} (υ) + S_{+} (υ) T (υ) C_{1} + e^{- i υ l} S_{-} (υ) \\ \times [G_{2} (- υ) + T (- υ) C_{2}] - \frac{(1 - e^{- il (k_{1 x} + υ)})}{(k_{1 x} + υ)} \end{matrix}}

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

Q_{1 y}^{sca} (x, z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} {\begin{matrix} A (υ) \\ C (υ) \end{matrix}} \exp (i κ ∣ z ∣ - i υ x) d υ,

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

Q_{1 y}^{sca} (x, z) = Ψ^{sep} (x, z) + Ψ^{int} (x, z),

where

Ψ^{sep} (x, z) = \frac{k_{1 z} sgn (z)}{2 π} \int_{- \infty}^{\infty} \frac{S_{+} (υ) \exp (i κ ∣ z ∣ - i υ x)}{i κ L (υ) S_{+} (- k_{1 x}) (k_{1 x} + υ)} d υ

and

Ψ^{int} (x, z) = - \frac{k_{1 z} sgn (z)}{2 π} \int_{- \infty}^{\infty} \frac{1}{ikL (υ)} [S_{+} (υ) R_{1} (υ) e^{- {ilk}_{1 x}} - C_{1} S_{+} (υ) T (υ)

Here, Ψ^sep(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 Ψ^int (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+iξ), (0<ϑ<π, -∞<ξ<∞). Hence, for large k _1xz r, Eqs. (76), (78) and (79) become

Q_{1 y}^{sca} (x, y) = \frac{i k_{1 xz}}{\sqrt{2 π}} {(\frac{π}{2 k_{1 xz} r})}^{\frac{1}{2}} {\begin{matrix} A (- k_{1 xz} \cos ϑ) \\ C (- k_{1 xz} \cos ϑ) \end{matrix}} \sin (ϑ) \exp (i k_{1 xz} r + i \frac{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)]

and

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)]

where A(-k _1xz cosϑ) and C(-k _1xz cosϑ) can be found from Eq. (75), while

f_{1} (- k_{1 xz} \cos ϑ) = \frac{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) = \frac{1}{(k_{1 x} - k_{1 xz} \cos ϑ)} [\frac{ϑ_{1} e^{- il (k_{1 x} - k_{1 xz} \cos ϑ)}}{L (- k_{1 xz} \cos ϑ)} - \frac{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})}

f_{2} (- k_{1 x z} \cos ϑ) = \frac{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}}

and

g_{2} (- k_{1 x z} \cos ϑ) = \frac{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}} .

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 ${L t}_{l \to \infty} (\frac{e^{i k l}}{\sqrt{k l}})$ 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) = \frac{1}{\sqrt{2 π}} [\frac{{- k}_{1 z} κ_{+} (υ) L_{+} (υ)}{i κ (υ) L (υ) (k_{1 x} + υ) κ_{+} (- k_{1 x}) L_{+} (- k_{1 x})} + \frac{δ_{1} υ L_{+} (υ)}{κ (υ) L (υ) (k_{1 x} + υ) L_{+} (- k_{1 x})}] .

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:

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).
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 $θ_{0} = \frac{π}{4}, k_{1 xz} = 1$ , l=1.

Download Full Size | PDF

Fig. 2. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at $θ_{0} = \frac{π}{4}, k_{1 xz} = 1$ , l=50.

Download Full Size | PDF

Fig. 3. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at $θ_{0} = \frac{π}{4}, k_{1 xz} = 1$ , l=100.

Download Full Size | PDF

Fig. 4. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at $θ_{0} = \frac{π}{4}, k_{1 xz} = 1$ , l10²².

Download Full Size | 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 $θ_{0} = \frac{π}{4}$ , k _1xz=1.

Download Full Size | PDF

For all the situations, θ ₀=45°, the graphs (1), (2), (3), (4) and (5) show that the field, in the region 0<θ≤π, 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:

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.
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).

Previous Article Next Article

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Fig. 1. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at

θ_{0} = \frac{π}{4}, k_{1 xz} = 1

, l=1.

View in Article | Download Full Size | PDF

Fig. 2. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at

θ_{0} = \frac{π}{4}, k_{1 xz} = 1

, l=50.

View in Article | Download Full Size | PDF

Fig. 3. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at

θ_{0} = \frac{π}{4}, k_{1 xz} = 1

, l=100.

View in Article | Download Full Size | PDF

Fig. 4. Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ ₁ at

θ_{0} = \frac{π}{4}, k_{1 xz} = 1

, l10²².

View in Article | Download Full Size | 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

θ_{0} = \frac{π}{4}

, k _1xz=1.

View in Article | Download Full Size | PDF

Equations (129)

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

D = ε E + ε α \nabla \times E

B = μ H + μ β \nabla \times H

\nabla \times Q_{1} = γ_{1} Q_{1},

\nabla \times Q_{2} = {- γ}_{2} Q_{2} .

γ_{1} = \frac{k}{(1 - k^{2} α β)} {\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} + \frac{k (α + β)}{2}},

γ_{2} = \frac{k}{(1 - k^{2} α β)} {\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} - \frac{k (α + β)}{2}},

Q_{1} = \frac{η_{1}}{η_{1} + η_{2}} (E + i η_{2} H),

Q_{2} = \frac{i}{η_{1} + η_{2}} (E - i η_{1} H),

η_{1} = \frac{η}{\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} + \frac{k (α - β)}{2}},

η_{2} = η {\sqrt{1 + \frac{k^{2} {(α - β)}^{2}}{4}} - \frac{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} .

∣ \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ Q_{1 x} & Q_{1 y} & Q_{1 z} \end{matrix} ∣ = γ_{1} (Q_{1 x} i + Q_{1 y} j + Q_{1 z} k) .

Q_{1 x} = \frac{1}{k_{1 xz}^{2}} [{ik}_{y} \frac{\partial Q_{1 y}}{\partial x} - γ_{1} \frac{\partial Q_{1 y}}{\partial z}],

Q_{1 z} = \frac{1}{k_{1 xz}^{2}} [{ik}_{y} \frac{\partial Q_{1 y}}{\partial z} + γ_{1} \frac{\partial Q_{1 y}}{\partial x}],

k_{1 xz}^{2} = γ_{1}^{2} - k_{y}^{2} .

Q_{2 x} = \frac{1}{k_{2 xz}^{2}} [{ik}_{y} \frac{\partial Q_{2 y}}{\partial x} + γ_{2} \frac{\partial Q_{2 y}}{\partial z}],

Q_{2 z} = \frac{1}{k_{2 xz}^{2}} [{ik}_{y} \frac{\partial Q_{2 y}}{\partial z} - γ_{2} \frac{\partial Q_{2 y}}{\partial x}],

k_{2 xz}^{2} = γ_{2}^{2} - k_{y}^{2} .

\nabla \times Q_{1} - γ_{1} Q_{1} = S_{1},

\nabla \times Q_{2} - γ_{2} Q_{2} = S_{2},

S_{1} = \frac{η_{1}}{η_{1} + η_{2}} (\frac{i γ_{1}}{ω ε} J - (1 + α γ_{1}) K),

S_{2} = \frac{η_{1}}{η_{1} + η_{2}} (\frac{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} \frac{γ_{2}}{ω μ},

η_{2} + ω μ β = (1 - k^{2} α β) \frac{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 \leq x \leq 0,

Q_{1 x} - i η_{2} Q_{2 x} = 0, z = 0, - l \leq x \leq 0 .

\frac{1}{k_{1 xz}^{2}} [i k_{y} \frac{\partial Q_{1 y}}{\partial x} - γ_{1} \frac{\partial Q_{1 y}}{\partial z}] - i η_{2} \frac{1}{k_{2 xz}^{2}} [i k_{y} \frac{\partial Q_{2 y}}{\partial x} - γ_{2} \frac{\partial Q_{2 y}}{\partial z}] = 0, z = 0, - l \leq x \leq 0 .

\frac{\partial Q_{1 y}}{\partial x} \mp δ \frac{\partial Q_{1 y}}{\partial z} = 0, z = 0^{\pm}, - l \leq x \leq 0,

δ = \frac{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^{-}); - \infty < x < - l, x > 0, z = 0,

\frac{\partial Q_{1 y} (x, z^{+})}{\partial z} = \frac{\partial Q_{1 y} (x, z^{-})}{\partial z}; - \infty < x < - l, x > 0, z = 0 .

Q_{1 y} (x, 0) = O (1) and \frac{\partial Q_{1 y} (x, 0)}{\partial z} = O (x^{- \frac{1}{2}}) as x \to 0^{+},

Q_{1 y} (x, 0) = O (1) and \frac{\partial Q_{1 y} (x, 0)}{\partial z} = O {(x + l)}^{- \frac{1}{2}} as x \to - 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)],

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial 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} .

(\frac{\partial}{\partial x} \mp δ \frac{\partial}{\partial z}) Q_{1 y}^{inc} + (\frac{\partial}{\partial x} \mp δ \frac{\partial}{\partial z}) Q_{1 y}^{sca} = 0, z = 0^{\pm}, - l \leq x \leq 0,

Q_{1 y}^{sca} (x, z^{+}) = Q_{1 y}^{sca} (x, z^{-}); - \infty < x < - l, x > 0, z = 0,

\frac{\partial}{\partial z} Q_{1 y}^{sca} (x, z^{+}) = \frac{\partial}{\partial z} Q_{1 y}^{sca} (x, z^{-}); - \infty < x < - l, x > 0, z = 0 .

\bar{Ψ} (υ, z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} Q_{1 y}^{sca} (x, z) e^{i υ x} d x = {\bar{Ψ}}_{+} (υ, z) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, z) + {\bar{Ψ}}_{1} (υ, z),

{\bar{Ψ}}_{+} (υ, z) = \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} Q_{1 y}^{sac} (x, z) e^{i υ x} d x,

{\bar{Ψ}}_{-} (υ, z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{- l} Q_{1 y}^{sac} (x, z) e^{i υ (x + l)} d x,

{\bar{Ψ}}_{1} (υ, z) = \frac{1}{\sqrt{2 π}} \int_{- l}^{0} Q_{1 y}^{sac} (x, z) e^{i υ x} d x .

{\bar{Ψ}}_{0} (υ, 0) = \frac{i}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]],

{\bar{Ψ}}_{0}^{/} (υ, 0) = \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]] .

(\frac{d^{2}}{d z^{2}} + κ^{2}) \bar{Ψ} (υ, z) = 0,

κ^{2} = k_{1 xz}^{2} - υ^{2},

{\bar{Ψ}}_{1}^{'} (υ, 0^{+}) = - \frac{i υ}{δ} [{\bar{Ψ}}_{1} (υ, 0^{+}) + {\bar{Ψ}}_{0} (υ, 0)] - {\bar{Ψ}}_{0}^{'} (υ, 0),

{\bar{Ψ}}_{1}^{'} (υ, 0^{-}) = \frac{i υ}{δ} [{\bar{Ψ}}_{1} (υ, 0^{-}) + {\bar{Ψ}}_{0} (υ, 0)] - {\bar{Ψ}}_{0}^{'} (υ, 0),

{\bar{Ψ}}_{-} (υ, 0^{+}) = {\bar{Ψ}}_{-} (υ, 0^{-}) = {\bar{Ψ}}_{-} (υ, 0),

{\bar{Ψ}}_{+} (υ, 0^{+}) = {\bar{Ψ}}_{+} (υ, 0^{-}) = {\bar{Ψ}}_{+} (υ, 0),

{\bar{Ψ}}_{-}^{'} (υ, 0^{+}) = {\bar{Ψ}}_{-}^{'} (υ, 0^{-}) = {\bar{Ψ}}_{-}^{'} (υ, 0),

{\bar{Ψ}}_{+}^{'} (υ, 0^{+}) = {\bar{Ψ}}_{+}^{'} (υ, 0^{-}) = {\bar{Ψ}}_{+}^{'} (υ, 0) .

\bar{Ψ} (υ, z) = {\begin{matrix} A (υ) e^{i κ z} & if z > 0, \\ C (υ) e^{- i κ z} & if z < 0 . \end{matrix}

{\bar{Ψ}}_{+} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{+}) = A (υ),

{\bar{Ψ}}_{+} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{-}) = C (υ),

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{+}) = i κ A (υ),

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{-}) = - i κ C (υ) .

A (υ) = J_{1} (υ, 0) + \frac{J_{1}^{'} (υ, 0)}{i κ},

C (υ) = {- J}_{1} (υ, 0) + \frac{J_{1}^{'} (υ, 0)}{i κ},

J_{1} (υ, 0) = \frac{1}{2} [{\bar{Ψ}}_{1} (υ, 0^{+}) - {\bar{Ψ}}_{1} (υ, 0^{-})],

J_{1}^{'} (υ, 0) = \frac{1}{2} [{\bar{Ψ}}_{1}^{'} (υ, 0^{+}) - {\bar{Ψ}}_{1}^{'} (υ, 0^{-})] .

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{i v l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{+}) = i κ [{\bar{Ψ}}_{+} (υ, 0) + e^{- i v l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{+})],

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i v l} {\bar{Ψ}}_{-}^{'} (υ, 0) + {\bar{Ψ}}_{1}^{'} (υ, 0^{-}) = i κ [{\bar{Ψ}}_{+} (υ, 0) + e^{- i v l} {\bar{Ψ}}_{-} (υ, 0) + {\bar{Ψ}}_{1} (υ, 0^{-})] .

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- ivl} {\bar{Ψ}}_{-}^{'} (υ, 0) - i κ L (υ) J_{1} (υ) + \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]] = 0 .

- i υ {\bar{Ψ}}_{+} (υ, 0) - i υ e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + δ L (υ) J_{1}^{'} (υ) + \frac{k_{1 x}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]] = 0,

L (υ) = (1 + \frac{υ}{δ κ}) .

L (υ) = (1 + \frac{υ}{δ κ}) = L_{+} (υ) L_{-} (υ),

κ (υ) = κ_{+} (υ) κ_{-} (υ),

A (υ) = \frac{1}{i κ L (υ)} {{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]]

+ \frac{υ δ_{1}}{κ L (υ)} {{\bar{Ψ}}_{+} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) - \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]],

C (υ) = \frac{1}{i κ L (υ)} {{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-}^{'} (υ, 0) + \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]]

+ \frac{υ δ_{1}}{κ L (υ)} {{\bar{Ψ}}_{+} (υ, 0) + e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) - \frac{k_{1 x}}{\sqrt{2 π} (k_{1 x} + υ)} [- 1 + \exp [- i (k_{1 x} + υ) l]],

{\bar{Ψ}}_{+}^{'} (υ, 0) + e^{- i υ l} \bar{Ψ}_{-}^{'} (υ, 0) + {S (υ) J}_{1} (υ) = \frac{k_{1 z}}{\sqrt{2 π} (k_{1 x} + υ)} [1 - \exp [- i (k_{1 x} + υ) l]],

- i υ {\bar{Ψ}}_{+} (υ, 0) - i υ e^{- i υ l} {\bar{Ψ}}_{-} (υ, 0) + δ L_{+} (υ) L_{-} (υ) J_{1}^{'} (υ) = \frac{k_{1 x}}{\sqrt{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),

{\bar{Ψ}}_{+}^{'} (υ, 0) = \frac{k_{1 z} S_{+} (υ)}{\sqrt{2 π}} [G_{1} (υ) + T (υ) C_{1}],

{\bar{Ψ}}_{-}^{'} (υ, 0) = \frac{k_{1 z} S_{-} (υ)}{\sqrt{2 π}} [G_{2} (- υ) + T (- υ) C_{2}],

{\bar{Ψ}}_{+}^{'} (υ, 0) = \frac{{i L}_{+} (υ)}{\sqrt{2 π} υ} [G_{1}^{'} (υ) + T (υ) C_{1}^{'}],

\bar{Ψ}_(υ, 0) = \frac{- i L_(υ)}{\sqrt{2 π υ}} [G_{2}^{'} (- υ) - T (- υ) C_{2}^{'}],

S_{+} (υ) = {(k_{1 xz} + υ)}^{\frac{1}{2}} L_{+} (υ),

S_(υ) = e^{\frac{- i π}{2}} {(k_{1 xz} - υ)}^{\frac{1}{2}} L_(υ),

G_{1} (υ) = \frac{1}{(υ + k_{1 x})} [\frac{1}{S_{+} (υ)} - \frac{1}{S_{+} ({- k}_{1 x})}] - e^{- {ilk}_{1 x}} R_{1} (υ),

G_{2} (υ) = \frac{e^{- {ilk}_{1 x}}}{(υ - k_{1 x})} [\frac{1}{S_{+} (υ)} - \frac{1}{S_{+} (k_{1 x})}] - R_{2} (υ),

C_{1} = S_{+} (k_{1 xz}) [\frac{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}) [\frac{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) = \frac{υ}{(υ + k_{1 x})} [\frac{1}{L_{+} (υ)} - \frac{1}{L_{+} (- k_{1 x})}] - e^{- {ilk}_{1 x}} R_{1} (υ),

G_{2}^{'} (υ) = \frac{e^{- {ilk}_{1 x}}}{(υ - k_{1 x})} [\frac{υ}{L_{+} (υ)} + \frac{k_{1 x}}{L_{+} (k_{1 x})}] - R_{2} (υ),

C_{1}^{'} = L_{+} (k_{1 xz}) [\frac{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}) [\frac{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) = \frac{E_{- 1} [W_{- 1} {- i (k_{1 xz} \mp k_{1 x}) l} - W_{- 1} {- i (k_{1 xz} + υ) l}]}{2 π i (υ \pm k_{1 x})},

T (υ) = \frac{1}{2 π i} E_{- 1} W_{- 1} {- i (k_{1 xz} + υ) l},

E_{- 1} = 2 e^{i \frac{π}{4}} e^{{ik}_{1 xz} l} {(l)}^{\frac{1}{2}} {(i)}^{- 1} h_{- 1},

W_{n - \frac{1}{2}} (p) = \int_{0}^{\infty} \frac{u^{n} e^{- u}}{u + p} du = Γ (n + 1) e^{\frac{z}{2}} p^{\frac{1}{2} n - \frac{1}{2}} W_{- \frac{1}{2} (n + 1), \frac{1}{2} n} (p),

\begin{matrix} A (υ) \\ C (υ) \end{matrix}} = \frac{k_{1 z} sgn (z)}{\sqrt{2 π} i κ L (υ)} {\begin{matrix} S_{+} (υ) G_{1} (υ) + S_{+} (υ) T (υ) C_{1} + e^{- i υ l} S_{-} (υ) \\ \times [G_{2} (- υ) + T (- υ) C_{2}] - \frac{(1 - e^{- il (k_{1 x} + υ)})}{(k_{1 x} + υ)} \end{matrix}}

+ \frac{υ δ_{1}}{\sqrt{2 π} κ L (υ)} {\begin{matrix} L_{+} (υ) {G'}_{1} (υ) + T (υ) {L_{+} (v) C'}_{1} + e^{- i υ l} \\ \times [({L_{-} (υ) G'}_{2} (- υ) + T (- υ) L_{+} {(v) C'}_{2})] - \frac{(1 - e^{- il (k_{1 x} + υ)})}{(k_{1 x} + υ)} \end{matrix}},

Q_{1 y}^{sca} (x, z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} {\begin{matrix} A (υ) \\ C (υ) \end{matrix}} \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) = \frac{k_{1 z} sgn (z)}{2 π} \int_{- \infty}^{\infty} \frac{S_{+} (υ) \exp (i κ ∣ z ∣ - i υ x)}{i κ L (υ) S_{+} (- k_{1 x}) (k_{1 x} + υ)} d υ

+ \frac{k_{1 z} sgn (z)}{2 π} \int_{- \infty}^{\infty} \frac{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

- \frac{1}{2 π} \int_{- \infty}^{\infty} \frac{δ_{1} e^{- il (k_{1 x} + υ)} \exp (i κ ∣ z ∣ - i υ x)}{κ L (υ) (k_{1 x} + υ)} d υ + \frac{1}{2 π} \int_{- \infty}^{\infty} \frac{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

+ \frac{1}{2 π} \int_{- \infty}^{\infty} \frac{δ_{1} \exp (i κ ∣ z ∣ - i υ x)}{κ L (υ) (k_{1 x} + υ)} d υ,

Ψ^{int} (x, z) = - \frac{k_{1 z} sgn (z)}{2 π} \int_{- \infty}^{\infty} \frac{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

- \frac{1}{2 π} \int_{- \infty}^{\infty} \frac{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) = \frac{i k_{1 xz}}{\sqrt{2 π}} {(\frac{π}{2 k_{1 xz} r})}^{\frac{1}{2}} {\begin{matrix} A (- k_{1 xz} \cos ϑ) \\ C (- k_{1 xz} \cos ϑ) \end{matrix}} \sin (ϑ) \exp (i k_{1 xz} r + i \frac{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)]

\times \frac{1}{4 π k_{1 xz}} {(\frac{1}{k_{1 xz} r})}^{\frac{1}{2}} \exp ({ik}_{1 xz} r + i \frac{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)]

\times \frac{1}{4 π k_{1 xz}} {(\frac{1}{k_{1 xz} r})}^{\frac{1}{2}} \exp ({ik}_{1 xz} r + i \frac{π}{4}),

f_{1} (- k_{1 xz} \cos ϑ) = \frac{S_{+} (- k_{1 xz} \cos ϑ)}{L (- k_{1 xz} \cos ϑ) S_{+} (- k_{1 x}) (k_{1 x} - k_{1 xz} \cos ϑ)}

- \frac{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) = \frac{1}{(k_{1 x} - k_{1 xz} \cos ϑ)} [\frac{ϑ_{1} e^{- il (k_{1 x} - k_{1 xz} \cos ϑ)}}{L (- k_{1 xz} \cos ϑ)} - \frac{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})}

- \frac{δ_{1}}{L (- k_{1 xz} \cos ϑ)}],

f_{2} (- k_{1 x z} \cos ϑ) = \frac{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) = \frac{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) = \frac{1}{\sqrt{2 π}} [\frac{{- k}_{1 z} κ_{+} (υ) L_{+} (υ)}{i κ (υ) L (υ) (k_{1 x} + υ) κ_{+} (- k_{1 x}) L_{+} (- k_{1 x})} + \frac{δ_{1} υ L_{+} (υ)}{κ (υ) L (υ) (k_{1 x} + υ) L_{+} (- k_{1 x})}] .

Abstract

1. Introduction

2. Formulation of the problem

3. Solution of theWiener-Hopf equations

4. Far field solution

5. Remarks

6. Graphical results

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (129)

Optics Express