# Scattering of Electromagnetic Plane Waves from a Coated PEMC Circular Cylinder Placed under PEC Wide Double Wedge

Authors: Muhammad Naveed; Shakeel Ahmed; Qaisar Abbas Naqvi
Journal: Mathematical Problems in Engineering
 (2010)

---

## Abstract

Scattering of electromagnetic plane waves from a coated perfect electromagnetic conductor (PEMC) circular
cylinder placed under perfect electric conducting (PEC) wide double wedge is presented. It is assumed that
the distance between the two wedges is large as compared to the wavelength. Therefore, the field at an
observation point can be considered to be composed of the incident field plus a response field from each of
the edges of double wedge and the cylinder. PEMC cylinder is taken to be infinite along its axis and has
been coated with a double positive (DPS) or double negative (DNG) material. The transmission coefficient
and diffraction pattern of PEC wide double wedge in the presence of both coated and uncoated PEMC
cylinder are studied. Results of special cases for PEMC cylinder, compared with the published work, are
found to be in fairly good agreement. The techniques of Clemmow, and Karp and Russek have been used to
investigate the transmission coefficient and diffraction pattern of the double wedge in the presence of both
coated and un-coated PEMC circular cylinder.


---

## Body

## 1. Introduction

Scattering from multiple objects has been investigated by many researchers [1–6]. A possible technique is to use fictitious line sources, located according to the geometry of each scatterers. This technique was used by Clemmow [7], Karp and Russek [8] for the diffraction of electromagnetic plane waves by a wide slit. Elsherbeni and Hamid [9, 10] further extended it for wide double wedge and perfect electric conductor (PEC) cylinder. In this paper the same technique has been applied to perfect electromagnetic conductor (PEMC) circular cylinder coated with homogeneous, isotropic, and linear material placed under PEC wide double wedge.The concept of PEMC as the generalization of (PEC) and a perfect magnetic conductor (PMC) has been studied quite recently by Lindell and Sihvola [11]. It has attracted the attention of many researchers [12–19]. The PEMC boundary conditions are of the general form (1.1)n⃗×(H⃗+ME⃗)=0n⃗⋅(D⃗-MB⃗)=0,
where M denotes the admittance of the PEMC boundary. Here, PMC corresponds to M=0, while PEC corresponds to M=±∞. In the recent years, there has been an increased interest in different classes of materials called meta materials like Double-Negative (DNG), Double Positive (DPS), Epsilon Negative (ENG) and Mu Negative (MNG). Veselago [20] characterized these mediums by negative real part of the permitivity as well as permeability. Scattering of electromagnetic plane waves by a conducting cylinder coated with meta material is investigated by Shen and Li  [21] and Ahmed and Naqvi [22–26].The electromagnetic scattering from an infinite PEMC cylinder coated with homogeneous, isotropic and linear DPS or DNG material placed under a PEC wide double wedge is studied. The known solutions for the scattered field by an isolated PEC wedge and an isolated coated PEMC cylinder are utilized. It is assumed that the field at any point is composed of the incident field and a response field from each of the double wedges and the cylinder. The response field consists of scattered field by the three scatterers due to the original plane wave plus an interaction field which will be represented by the three fictitious line sources located at the wedge edges and at the cylinder. The time dependence is assumed to beexp(jωt) and it is suppressed throughout the analysis.


## 2. Formulation of the Problem

For a single wedge, the two half planes of the wedge can be defined asϕ=0 and ϕ=2Φ. The geometry of the problem is shown in Figure 1(a) where two parallel wedges loaded with coated PEMC cylinder is shown. The radius of the inner cylinder is a and that of coated cylinder is b. The problem is two-dimensional since all the fields are uniform in the z-direction. By considering an E-polarized plane wave incident on an isolated PEC wedge and isolated PEMC cylinder, the known results of scattered field are presented in this section. The transmission coefficients for both coated and Uncoated cylinder are given. Figure 1(b) shows the geometry containing an Uncoated PEMC cylinder placed under PEC wide double wedge.(a) Geometry of the Problem-Coated PEMC cylinder placed under PEC wide double wedge. (b) Uncoated PEMC cylinder placed under PEC wide double wedge.
(a)(b)


### 2.1. PEC Wedge Excited by a Plane Wave

For the plane wave incident on the edge of the wedge at an angleϕ0 with respect to the negative x-axis the incident field is given as (2.1)(EziHzi)=(E0H0)exp⁡[jk(xcos⁡ϕ0+ysinϕ0)].
The uniform expression for the field diffracted from a PEC wedge has the form [6] (2.2)(EzdHzd)=exp⁡[-j(kρ)]ρDhs(ρ,ϕ,ϕ0;n)(EziHzi).
The diffraction coefficient for the PEC wedge is (2.3)Dhs(ρ,ϕ,ϕ0;n)=-ρ{-sgn⁡(sin(π+ϕ-ϕ02n))cos⁡(π+ϕ-ϕ02n)×F[2kρn|sin(π+ϕ-ϕ02n)|]-sgn⁡(sin(π-(ϕ-ϕ0)2n))cos⁡(π-(ϕ-ϕ0)2n)×F[2kρn|sin(π-(ϕ-ϕ0)2n)|]±sgn⁡(sin(π+ϕ+ϕ02n))cos⁡(π+ϕ+ϕ02n)×F[2kρn|sin(π+ϕ+ϕ02n)|]±sgn⁡(sin(π-(ϕ+ϕ0)2n))cos⁡(π-(ϕ+ϕ0)2n)×F[2kρn|sin(π-(ϕ+ϕ0)2n)|]},Ds and Dh are the diffraction coefficients of E- and H-polarization, respectively, and n=2Φ/π. Function sgn(·) is the signum function whereas F(·) is the Fresnel integral defined as (2.4)F(x)=1πexp⁡(jx2+jπ4)∫x∞exp⁡(-jμ2)dμ.It is assumed that point of observation is far from the edge of the wedge. For large argument approximation, Fresnel integral simplifies as(2.5)F(x)≈12πxexp⁡[-jπ4].
Hence diffraction coefficient for E-polarized plane wave, incident on the edge of the wedge simplifies to (2.6)D(ϕ,ϕ0,n)≈sin(π/n)πn{[cos⁡(πn)-cos⁡(ϕ-ϕ0n)]-1-[cos⁡(πn)-cos⁡(ϕ+ϕ0n)]-1}.
The angles between the incident and diffracted rays and normal to the screen are ϕ and ϕ0, respectively.


### 2.2. Circular Cylinder Excited by a Plane Wave

A circular cylinder is defined by the surfaceρ=a, while its axis coincides with the z-axis. The radiated fields due to plane wave incident on a circular cylinder [27] are (2.7)EpC=π2jH0(2)(kρ)Gp(ϕ,ϕ0,a),
where (2.8)Gp(ϕ,ϕ0,a)=-2jπ∑n=0∞ϵn(-1)nTncos⁡[n(ϕ-ϕ0)],
where the Neumann number ϵn=1 for n=0 and 2 for n>0. In (2.8), Tn is the transmission coefficient. Its values for both co and cross-polarized components of Uncoated PEMC cylinder [18] is(2.9)Tn={Hn(2)(ka)Jn/(ka)+M2η02Jn(ka)Hn(2)/(ka)(1+M2η02)Hn(2)(ka)Hn(2)/(ka),Copolarized2Mη0πka(1+M2η02)Hn(2)(ka)Hn(2)/(ka),Crosspolarized
whereas the transmission coefficient for coated PEMC cylinder [22] is (2.10)Tco⁡=(Jn′(k0b)/η0)(A)-(Jn(k0b)/η1)(B)(Hn(2)(k0b)/η1)(B)-(Hn(2)′(k0b)/η0)(A),Tcross=jMη1[Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1a)Hn(2)′(k1a)][Jn(k0b)Hn(2)′(k0b)-Jn′(k0b)Hn(2)(k0b)η0Hn(2)(k0b)(B)-Hn(2)′(k0b)(A)],
where
(2.11)A=[(acb+d)-jMη1Hn(2)(k1a)Hn(2)′(k1a)(eab+f)][Hn(1)(k1b)-Hn(2)(k1b)Hn(1)′(k1a)Hn(2)′(k1a)]+jMη1Hn(2)(k1b)Hn(2)′(k1a)(eab+f)[Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1a)Hn(2)′(k1a)],B=[Hn(1)′(k1b)-Hn(2)′(k1b)Hn(1)′(k1a)Hn(2)′(k1a)][(acb+d)-jMη1Hn(2)(k1a)Hn(2)′(k1a)(eab+f)]+jMη1Hn(2)′(k1b)Hn(2)′(k1a)(eab+f)[Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1a)Hn(2)′(k1a)],a=Hn(2)(k0b)η0-Hn(2)(k1b)Hn(2)′(k0b)η1Hn(2)′(k1b),b=1η1[Hn(1)(k1b)-Hn(2)(k1b)Hn(2)′(k1b)Hn(2)′(k1b)],c=Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1b)Hn(2)′(k1b),d=Hn(2)(k1a)Hn(2)′(k0b)Hn(2)′(k1b),e=jMη1[Hn(1)′(k1a)-Hn(1)′(k1b)Hn(2)′(k1a)Hn(2)′(k1b)],f=jMη1Hn(2)′(k0b)Hn(2)′(k1a)Hn(2)′(k1b).In above equationsJn(·) is the Bessel function of order n and Hn(·) is the Hankel function of second kind of order n. Primes indicate the derivative with respect to the whole argument.


## 2.1. PEC Wedge Excited by a Plane Wave

For the plane wave incident on the edge of the wedge at an angleϕ0 with respect to the negative x-axis the incident field is given as (2.1)(EziHzi)=(E0H0)exp⁡[jk(xcos⁡ϕ0+ysinϕ0)].
The uniform expression for the field diffracted from a PEC wedge has the form [6] (2.2)(EzdHzd)=exp⁡[-j(kρ)]ρDhs(ρ,ϕ,ϕ0;n)(EziHzi).
The diffraction coefficient for the PEC wedge is (2.3)Dhs(ρ,ϕ,ϕ0;n)=-ρ{-sgn⁡(sin(π+ϕ-ϕ02n))cos⁡(π+ϕ-ϕ02n)×F[2kρn|sin(π+ϕ-ϕ02n)|]-sgn⁡(sin(π-(ϕ-ϕ0)2n))cos⁡(π-(ϕ-ϕ0)2n)×F[2kρn|sin(π-(ϕ-ϕ0)2n)|]±sgn⁡(sin(π+ϕ+ϕ02n))cos⁡(π+ϕ+ϕ02n)×F[2kρn|sin(π+ϕ+ϕ02n)|]±sgn⁡(sin(π-(ϕ+ϕ0)2n))cos⁡(π-(ϕ+ϕ0)2n)×F[2kρn|sin(π-(ϕ+ϕ0)2n)|]},Ds and Dh are the diffraction coefficients of E- and H-polarization, respectively, and n=2Φ/π. Function sgn(·) is the signum function whereas F(·) is the Fresnel integral defined as (2.4)F(x)=1πexp⁡(jx2+jπ4)∫x∞exp⁡(-jμ2)dμ.It is assumed that point of observation is far from the edge of the wedge. For large argument approximation, Fresnel integral simplifies as(2.5)F(x)≈12πxexp⁡[-jπ4].
Hence diffraction coefficient for E-polarized plane wave, incident on the edge of the wedge simplifies to (2.6)D(ϕ,ϕ0,n)≈sin(π/n)πn{[cos⁡(πn)-cos⁡(ϕ-ϕ0n)]-1-[cos⁡(πn)-cos⁡(ϕ+ϕ0n)]-1}.
The angles between the incident and diffracted rays and normal to the screen are ϕ and ϕ0, respectively.


## 2.2. Circular Cylinder Excited by a Plane Wave

A circular cylinder is defined by the surfaceρ=a, while its axis coincides with the z-axis. The radiated fields due to plane wave incident on a circular cylinder [27] are (2.7)EpC=π2jH0(2)(kρ)Gp(ϕ,ϕ0,a),
where (2.8)Gp(ϕ,ϕ0,a)=-2jπ∑n=0∞ϵn(-1)nTncos⁡[n(ϕ-ϕ0)],
where the Neumann number ϵn=1 for n=0 and 2 for n>0. In (2.8), Tn is the transmission coefficient. Its values for both co and cross-polarized components of Uncoated PEMC cylinder [18] is(2.9)Tn={Hn(2)(ka)Jn/(ka)+M2η02Jn(ka)Hn(2)/(ka)(1+M2η02)Hn(2)(ka)Hn(2)/(ka),Copolarized2Mη0πka(1+M2η02)Hn(2)(ka)Hn(2)/(ka),Crosspolarized
whereas the transmission coefficient for coated PEMC cylinder [22] is (2.10)Tco⁡=(Jn′(k0b)/η0)(A)-(Jn(k0b)/η1)(B)(Hn(2)(k0b)/η1)(B)-(Hn(2)′(k0b)/η0)(A),Tcross=jMη1[Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1a)Hn(2)′(k1a)][Jn(k0b)Hn(2)′(k0b)-Jn′(k0b)Hn(2)(k0b)η0Hn(2)(k0b)(B)-Hn(2)′(k0b)(A)],
where
(2.11)A=[(acb+d)-jMη1Hn(2)(k1a)Hn(2)′(k1a)(eab+f)][Hn(1)(k1b)-Hn(2)(k1b)Hn(1)′(k1a)Hn(2)′(k1a)]+jMη1Hn(2)(k1b)Hn(2)′(k1a)(eab+f)[Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1a)Hn(2)′(k1a)],B=[Hn(1)′(k1b)-Hn(2)′(k1b)Hn(1)′(k1a)Hn(2)′(k1a)][(acb+d)-jMη1Hn(2)(k1a)Hn(2)′(k1a)(eab+f)]+jMη1Hn(2)′(k1b)Hn(2)′(k1a)(eab+f)[Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1a)Hn(2)′(k1a)],a=Hn(2)(k0b)η0-Hn(2)(k1b)Hn(2)′(k0b)η1Hn(2)′(k1b),b=1η1[Hn(1)(k1b)-Hn(2)(k1b)Hn(2)′(k1b)Hn(2)′(k1b)],c=Hn(1)(k1a)-Hn(2)(k1a)Hn(1)′(k1b)Hn(2)′(k1b),d=Hn(2)(k1a)Hn(2)′(k0b)Hn(2)′(k1b),e=jMη1[Hn(1)′(k1a)-Hn(1)′(k1b)Hn(2)′(k1a)Hn(2)′(k1b)],f=jMη1Hn(2)′(k0b)Hn(2)′(k1a)Hn(2)′(k1b).In above equationsJn(·) is the Bessel function of order n and Hn(·) is the Hankel function of second kind of order n. Primes indicate the derivative with respect to the whole argument.


## 3. Cylindrical Wave Incident

In this section, the known solutions due to a line source excitation for scattered fields from isolated PEC wedge and from coated PEMC cylinder are presented. The purpose is to get the interaction contribution from each of the two wedges and a coated PEMC cylinder using the known solutions and by incorporating the techniques used by Clemmow [7], and Karp and Russek [8].


### 3.1. PEC Wedge Excited by a Cylindrical Wave

The field of a line source in the presence of a conducting wedge whose edge is parallel to the source is well known. If the source is of unit amplitude and is located at(ρ0,ϕ0) parallel to the z-axis, its field in the absence of the wedge is given as (3.1)Ei=π2jH0(2)(kR),
where R is the distance between the line source and the field point, k is the wave number, and H0(2)(·) is the Hankel function of the second kind of order zero. The diffracted field in the presence of the wedge is given asymptotically in [28]. Here the asymptotic expression of the Hankel function is replaced by the Hankel function itself, therefore, the diffracted field has the appearance of cylindrical wave emanating from a line source located at the edge of the wedge expressed in the form (3.2)E=π2jH0(2)(kρ)F(ϕ,ρ0,ϕ0,n),
where (3.3)F(ϕ,ρ0,ϕ0,n)≈H0(2)(kρ0)exp⁡[-j(π/2)]sin(π/n)πn×{[cos⁡(πn)-cos⁡(ϕ-ϕ0n)]-1-[cos⁡(πn)-cos⁡(ϕ+ϕ0n)]-1}.
Here, again the angles between the incident and diffracted rays and normal to the screen are ϕ and ϕ0, respectively.


### 3.2. PEMC-Coated Circular Cylinder Excited by a Cylindrical Wave

The scattered field due to cylindrical wave incident on circular cylinder [27] is (3.4)ElC=π2jH0(kρ)Gl(ϕ,ρ0,ϕ0,a),
where (3.5)Gl(ϕ,ρ0,ϕ0,a)=-∑n=0∞ϵnjnTnHn(kρ0)cos⁡[n(ϕ-ϕ0)],
where Tn is the transmission coefficient. Its values for co and cross-polarized components of both Uncoated and coated PEMC cylinders are given by (2.9) and (2.10).


## 3.1. PEC Wedge Excited by a Cylindrical Wave

The field of a line source in the presence of a conducting wedge whose edge is parallel to the source is well known. If the source is of unit amplitude and is located at(ρ0,ϕ0) parallel to the z-axis, its field in the absence of the wedge is given as (3.1)Ei=π2jH0(2)(kR),
where R is the distance between the line source and the field point, k is the wave number, and H0(2)(·) is the Hankel function of the second kind of order zero. The diffracted field in the presence of the wedge is given asymptotically in [28]. Here the asymptotic expression of the Hankel function is replaced by the Hankel function itself, therefore, the diffracted field has the appearance of cylindrical wave emanating from a line source located at the edge of the wedge expressed in the form (3.2)E=π2jH0(2)(kρ)F(ϕ,ρ0,ϕ0,n),
where (3.3)F(ϕ,ρ0,ϕ0,n)≈H0(2)(kρ0)exp⁡[-j(π/2)]sin(π/n)πn×{[cos⁡(πn)-cos⁡(ϕ-ϕ0n)]-1-[cos⁡(πn)-cos⁡(ϕ+ϕ0n)]-1}.
Here, again the angles between the incident and diffracted rays and normal to the screen are ϕ and ϕ0, respectively.


## 3.2. PEMC-Coated Circular Cylinder Excited by a Cylindrical Wave

The scattered field due to cylindrical wave incident on circular cylinder [27] is (3.4)ElC=π2jH0(kρ)Gl(ϕ,ρ0,ϕ0,a),
where (3.5)Gl(ϕ,ρ0,ϕ0,a)=-∑n=0∞ϵnjnTnHn(kρ0)cos⁡[n(ϕ-ϕ0)],
where Tn is the transmission coefficient. Its values for co and cross-polarized components of both Uncoated and coated PEMC cylinders are given by (2.9) and (2.10).


## 4. Interaction Contributions of the Geometry

There are two conducting wedges separated by a distance2s, where 2ks≫1 and a coated PEMC circular cylinder of radius b whose axis is parallel to the edges of two parallel wedges as shown in Figure 1(a). All the three bodies are considered to be illuminated by a plane wave of unit amplitude. The field at any point is considered to be composed of the incident field plus a response field from each of the two wedges and the cylinder. The response field consists of scattered field by the three scatterers due to the original plane wave (the non-interaction field) plus an interaction field which will be represented by three fictitious line sources located at the wedge edges and at the cylinder in order to take into account multiple interaction between three objects. Consider, for example, edge of wedge A which is excited by direct plane wave plus line source fields of edge B (the second wedge)and edge C (the cylinder axis). The interaction can be conveniently expressed in terms of the response of edge A to the line source at the opposite edge and at the cylinder axis.If the plane wave is restricted such that the incident field does not illuminate the lower faces of the half planes, the total field in the forward direction is given by [10] (4.1)Et=Ei+Es,
where (4.2)Es=Es1+Es2+Es3,Es1=π2jH0(kρ1)[exp⁡(-jkssinθ0)]D(ϕ1,ϕ01,n1)+c3F(ϕ1,s1,ϕ31,n1)+c2F(ϕ1,2s,ϕ21,n1),Es2=π2jH0(kρ2)[exp⁡(+jkssinθ0)]D(ϕ2,ϕ02,n1)+c3F(ϕ1,s1,ϕ32,n1)+c2F(ϕ1,2s,ϕ12,n2),Es3=π2jH0(kρ3)[exp⁡(-jkdcos⁡θ0)]D(ϕ3,ϕ03,a)+c1G(ϕ3,s1,ϕ13,a)+c2G(ϕ3,s2,ϕ23,a),
where n1=(2π-α)/π and n2=(2π-β)/π and c1, c2 and c3 are the unknown strengths of the line sources at wedge edges and along the cylinder axis, respectively. Let the incoming plane wave be incident from above the slit and the observation point be below. Further, let θ0 between the incoming plane wave and the normal to the plane of the screen (measured from the positive y-axis), and let θ represent the angle between the observation point and normal to the screen (measured from the negative y-axis). All angles are considered positive if measured counterclockwise with respect to the normal and negative if clockwise. When the observation point is far from the edges as compared to the width of double wedge kρ/2s≫1, approximate relations between them can be simply stated. Therefore, well-known far field conditions are used in which ϕ0=ϕ01=ϕ03=π/2+θ0, ϕ02=π/2-θ0, ϕ1=ϕ3≃3π/2+θ, ϕ2≃3π/2-θ, ϕ12=ϕ21≃π, ϕ13=ψ≃tan-1(d/s), ϕ31=ϕ32≃π+ψ, ϕ23≃π-ψ, ρ1≃ρ-ssinθ, ρ2≃ρ+ssinθ, ρ3≃ρ-dcosθ, and s1 and s2 are the distances between the edges of the two wedges and the cylinder axis, respectively.To determinec1,c2 and c3, the analysis of Karp and Russek [8] has been followed by imposing the requirement that the fields scattered by the two wedges and the cylinder be consistent with one another(4.3)2c1-c2[F(ϕ31,2s,ϕ21,n1)+F(ϕ21,2s,ϕ21,n1)]-c3[F(ϕ31,s1,ϕ31,n1)+F(ϕ21,s1,ϕ31,n1)]=exp⁡(-jkssinθ0)[D(ϕ31,ϕ01,n1)+D(ϕ21,ϕ01,n1)],2c2-c1[F(ϕ32,2s,ϕ12,n2)+F(ϕ12,2s,ϕ12,n2)]-c3[F(ϕ32,s2,ϕ32,n2)+F(ϕ12,s2,ϕ32,n2)]=exp⁡(jkssinθ0)[D(ϕ32,ϕ02,n2)+D(ϕ12,ϕ02,n2)],2c3-c1[G(ϕ13,s1,ϕ13,a)+G(ϕ23,s1,ϕ13,a)]-c2[G(ϕ13,s2,ϕ23,a)+G(ϕ23,s2,ϕ23,a)]=exp⁡(-jkdcos⁡θ0)[D(ϕ13,ϕ03,a)+D(ϕ23,ϕ03,a)].
By solving (4.3) for c1, c2, and c3, the scattered field is found and is rewritten in the form (4.4)Es=exp⁡(-jkρ)πkρE(θ,s,d,n1,n2,a),
where the scattered field pattern E(θ,s,d,n1,n2,a) is obtained from (4.4).The transmission coefficientT for plane wave incidence is calculated using the expression given by Karp and Russek [8] as (4.5)T=Re[(1-j)Ẽ]2ks,
where Ẽ is E(θ,s,d,n1,n2,a) in the limit as θ approaches θ0.


## 5. Results and Discussion

Figure1(a) shows the geometry which consists of PEC double wedge and a coated PEMC circular cylinder whereas Figure 1(b) contains an Uncoated PEMC cylinder placed under PEC wide double. In these figures, d represents the distance of the PEMC cylinder from the edge of PEC wedge. In the first part of discussion, a comparison of transmission coefficients (Tc) for PEC double wedge with zero wedge angle, loaded with PEC cylinder is made with the transmission coefficient for the slit when a uncoated PEMC cylinder is placed under the slit. Comparison of Tc with both the copolarized (Tco) and cross-polarized (Tcross) components of uncoated PEMC cylinder is studied. In all the cases cylinder radius (ka) is taken as 0.5. Figures 2(a) and 2(b) show the comparison of Tc with Tco when the cylinder is located at kd=0 and kd=5 from the slit, respectively. It can be observed that Tco, in both the cases, show exactly the same behavior as that of Tc when Mη0=∞. These results are in fairly good agreement with the Elsherbeni's results [10]. Therefore, it is quite obvious that the uncoated PEMC cylinder behaves like PEC cylinder at Mη0=∞. In Figure 3, a comparison of Tcross has been made with the transmission coefficient (T) for the slit when the cylinder radius ka=0, that is, an unloaded slit. It can be observed that the two coefficients have the same behavior. It is because the cross-polarized component of PEMC cylinder is zero at Mη0=0. In Figures 4 and 5, a comparison of Tco and Tcross for Mη0=0 and Mη0=±1 at kd=0 and kd=5 are presented, respectively. In Figure 4(a), it can be seen that when the cylinder, with Mη0=0, is at kd=0, Tco is less than that of Tcross, but when it is shifted below the center of the aperture plane, say at kd=5, Tco becomes larger than Tcross which is obvious from Figure 4(b). Moreover, the transmission coefficients oscillate with decreasing amplitude as expected and tend to unity as the slit width ks tends to infinity. But, contrary to this effect, Tcross remains larger at kd=5 when Mη0=±1, as shown in Figure 5(b), whereas at kd=0Tcross is less than Tco as shown in Figure 5(a). Moreover, it can be seen that when kd=0, Tco shows almost similar behavior as that of Tc but Tcross is larger than Tc. When the cylinder is shifted to kd=5, both T(co) and T(cross) are larger than Tc. To further highlight the effect of Mη0 on Tcross, Figure 6 shows that Tcross is maximum when Mη0=±1 and decreases for other values of Mη0. Similarly Figure 7 shows the effect of variation of ka on Tcross at Mη0=±1. Obviously the value of Tcross is larger for ka=0.5 and decreases for smaller values of ka. Both these figures are for kd=0. The behavior of Tco and Tcross for obliquely incident plane wave at θ0=150 and θ0=300 for ka=0.1, kd=0 and Mη0=±1 is shown in Figures 8(a) and 8(b). At θ0=150, Tcross is higher than unity in the lower range of ks(ks≤3) and is larger than Tco. For the same cylinder parameters but with θ0=300, both Tco and Tcross becomes less than unity. However, Tcross oscillates with greater amplitude as compared to Tco. Hence incident angle effects the peak locations of Tco and Tcross. To see the effect of interior wedge angle on the transmission coefficients for Mη0=±1, ka=0.1, it is observed that as the wedge angle is increased, the amplitude of oscillation in both Tco and Tcross is increased, that is, the interior wedge angle effects the levels of maxima and minima of the oscillation in both the cases, however this effect is more dominant in Tcross as compared to Tco as shown in Figures 9(a) and 9(b).(a) Slit transmission coefficient forθ0=0, kd=0, ka=0.5. (b) Slit transmission coefficient for θ0=0, kd=5, ka=0.5.
(a)(b)Figure 3
Slit transmission coefficient forθ0=0, kd=0, ka=0.(a) Slit transmission coefficient forθ0=0, kd=0, ka=0.5. (b) Slit transmission coefficient for θ0=0, kd=5, ka=0.5.
(a)(b)(a) Slit transmission coefficient forθ0=0, kd=0, ka=0.5. (b) Slit transmission coefficient for θ0=0, kd=5, ka=0.5.
(a)(b)Figure 6
Slit transmission coefficient forθ0=0, kd=0, ka=0.5.Figure 7
Slit transmission coefficient forθ0=0, kd=0, ka=0.5.(a) Slit transmission coefficient forkd=0, ka=0.1. (b) Slit transmission coefficient for kd=0, ka=0.1.
(a)(b)(a) Slit transmission coefficient forkd=0, ka=0.1. (b) Slit transmission coefficient for kd=0, ka=0.1.
(a)(b)In the second part of discussion, normalized diffraction pattern of the slit loaded with PEC cylinder(Dc) compared with the corresponding normalized diffraction patterns in the presence of uncoated PEMC cylinder is presented. Comparison between copolarized (D(co)) and cross-polarized (D(cross)) components of PEMC cylinder for different values of Mη0 is made. In all the cases cylinder radius ka is taken as 0.5. Figure 10(a) presents Dc compared with D(co) for kd=0 and ks=8. The solid curve in the figure represents Dc. It is observed that D(co) shows similar behavior as that of Dc for Mη0=∞. Moreover, in Figure 10(b) it can be observed that D(cross) for both Mη0=0 and Mη0=∞ gives the same diffraction patterns as that of an unloaded slit (D) which is in good agreement with the published work [6]. This shows that cross-polarized component exists only for Mη0=±1 and becomes zero for other values of Mη0. To further investigate the effect of Mη0 on D(co) and D(cross), Figure 11 shows the comparison of both these diffraction patterns for Mη0=±1. It can be seen that the beam width for cross-polarized component is less than the beam width of copolarized component. To see the effect of slit width on D(co) and D(cross), the plots for different values of ks with kd=0 and Mη0=±1 are shown in Figures 12(a) and 12(b). It is observed that the number of side lobes increases with the increase in slit width for both D(co) and D(cross). When the cylinder is shifted to kd=1.5 for ks=8 and Mη0=±1, comparison of D(co) and D(cross) shown in Figure 13, reflects almost the similar behavior.(a) Slit diffraction pattern forθ0=00, kd=0, ka=0.5, ks=8. (b) Slit diffraction pattern for θ0=00, kd=0, ka=0.5, ks=8.
(a)(b)Figure 11
Slit diffraction pattern forθ0=00, kd=0, ka=0.5, ks=8.(a) Slit diffraction pattern forθ0=00, kd=0, ka=0.5. (b) Slit diffraction pattern for θ0=00, kd=0, ka=0.5.
(a)(b)Figure 13
Slit diffraction pattern forθ0=00, kd=1.5, ka=0.5, ks=8.In the third part of discussion, the transmission coefficient of coated PEMC cylinder is presented. Behavior of both copolarized(T(co)c) and cross-polarized (T(cross)c) components of coated PEMC cylinder is discussed. In all the plots radius of Uncoated cylinder is taken as a=0.2 cm and that of coated cylinder as b=0.3 cm. The validity of the code has been checked by making the coating equal to zero. Results are found to be in agreement with Uncoated PEMC cylinder. Comparison of Tcoc and Tcrossc for Mη1=±1 at kd=0 and kd=5 with relative permitivity ϵr=-1.5 and relative permeability μr=-1, are shown in Figures 14(a) and 14(b), respectively. It can be seen that in both the cases, Tcoc is larger than Tcrossc, which is contrary to Uncoated PEMC cylinder in which Tco remains less than Tcross for Mη0=±1. Furthermore, it is observed that the transmission coefficient is large in the presence of coated PEMC cylinder as compared to PC cylinder. In both the cases Tcoc and Tcrossc are greater than unity whereas Tc, in general, remains less than unity. The variation in the radius of coated cylinder also effects the behavior of Tcoc and Tcrossc as shown in Figure 15. Figure 15(b) shows that Tcoc oscillates with greater amplitude as the the value of b is increased. However, T(cross)c does not show considerable change in behavior with the increase in radius b as hi-lighted in Figure 15(a). The behavior of Tcoc and Tcrossc for oblique incidence is shown in Figure 16 for incident angles θ0=200 and θ0=300. Figure 16(a) shows that Tcrossc gets less than unity as the angle of incidence is increased from zero, whereas the amplitude of oscillation for Tcoc decreases with the increase of incidence angle θ0 as shown in Figure 16(b). All the plots of Figures 15 and 16 are for for Mη1=±1. Further, it can be seen in Figure 17 that interior wedge angle effects the peak-to-peak values of the oscillations both in the case of Tcoc and Tcrossc. In case of Tcrossc, as shown in Figure 17(a), the oscillations are always around unity and decreases with increasing ks whereas in case of Tcoc as shown in Figure 17(b), the oscillations are larger and are greater than unity. The plots for DPS-coated cylinder show almost similar behavior as that of DNG coated cylinder.(a) Slit diffraction pattern forθ0=00, kd=0, a=0.2 cm, b=0.3 cm. (b) Slit diffraction pattern for θ0=00, kd=5, a=0.2 cm, b=0.3 cm.
(a)(b)(a) Slit transmission coefficient forθ0=0, kd=0. (b) Slit transmission coefficient for θ0=0, kd=0.
(a)(b)(a) Slit transmission coefficient forθ0=0, kd=0, a=0.2 cm, b=0.3 cm. (b) Slit transmission coefficient for θ0=0, kd=0, a=0.2 cm, b=0.3 cm.
(a)(b)(a) Slit transmission coefficient forθ0=0, kd=0, a=0.2 cm, b=0.3 cm. (b) Slit transmission coefficient for θ0=0, kd=0, a=0.2 cm, b=0.3 cm.
(a)(b)In the last part of discussion, diffraction pattern of wide double wedge in the presence of coated PEMC cylinder is presented. In Figure18, the effect of Mη1 on the diffraction pattern is shown. Behavior of both copolarised (D(co)c) and cross-polarized (D(cross)c) components of coated PEMC cylinder with ϵr=1.5 and μr=1 for kd=0 and ks=8 is studied. In both the cases, it can be seen that D(co)c and D(cross)c show slight different behavior for Mη1=1 as compared to other values of Mη1. Figure 19 shows the variation in D(co)c and D(cross)c for ϵr=1.5, μr=1 and kd=0 with respect to the slit width. It is observed that both D(co)c and D(cross)c show different behavior for different values of slit widths.(a) Slit diffraction pattern forθ0=00, ks=8, kd=0, a=0.2, b=0.3. (b) Slit diffraction pattern for θ0=00, ks=8, kd=0, a=0.2, b=0.3.
(a)(b)(a) Slit diffraction pattern forθ0=00, kd=0, a=0.2, b=0.3. (b) Slit diffraction pattern for θ0=00, kd=0, a=0.2, b=0.3.
(a)(b)


## 6. Conclusion

The transmission coefficient and the diffraction pattern of three scatterers, that is, two PEC parallel wedges in the presence of a coated PEMC cylinder are presented. The results show that the transmission coefficient has a high value in the presence of a PEMC cylinder as compared to PEC cylinder. Furthermore, it is observed that the transmission coefficient varies under particular conditions such as by either shifting the cylinder below the center of the aperture plane of double wedge or by coating the PEMC cylinder with DPS or DNG materials. Variations in the transmission coefficient with respect to different values of admittance parameter for Uncoated and coated PEMC cylinder is also studied. It is found that the behavior ofT(co) and T(cross) of an Uncoated PEMC cylinder and T(co)c and T(cross)c of coated PEMC cylinder not only varies with the incident angles of the original plane wave but also show cosiderable change in the behavior by changing the interior wedge angles.



---

*Source: 254025-2010-12-28.xml*