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

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.


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 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 be exp jωt and it is suppressed throughout the analysis.

Formulation of the Problem
For a single wedge, the two half planes of the wedge can be defined as φ 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.

PEC Wedge Excited by a Plane Wave
For the plane wave incident on the edge of the wedge at an angle φ 0 with respect to the negative x-axis the incident field is given as The uniform expression for the field diffracted from a PEC wedge has the form 6

2.2
The diffraction coefficient for the PEC wedge is D s and D h 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 It is assumed that point of observation is far from the edge of the wedge.For large argument approximation, Fresnel integral simplifies as Hence diffraction coefficient for E-polarized plane wave, incident on the edge of the wedge simplifies to

2.6
The angles between the incident and diffracted rays and normal to the screen are φ and φ 0 , respectively.

Circular Cylinder Excited by a Plane Wave
A circular cylinder is defined by the surface ρ a, while its axis coincides with the z-axis.
The radiated fields due to plane wave incident on a circular cylinder 27 are where where the Neumann number n 1 for n 0 and 2 for n > 0. In 2.8 , T n is the transmission coefficient.Its values for both co and cross-polarized components of Uncoated PEMC cylinder 18 is whereas the transmission coefficient for coated PEMC cylinder 22 is where

2.11
In above equations J n • is the Bessel function of order n and H n • is the Hankel function of second kind of order n.Primes indicate the derivative with respect to the whole argument.

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 .

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 ρ 0 , φ 0 parallel to the z-axis, its field in the absence of the wedge is given as where R is the distance between the line source and the field point, k is the wave number, and H 2 0 • 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 where

3.3
Here, again the angles between the incident and diffracted rays and normal to the screen are φ and φ 0 , respectively.

PEMC-Coated Circular Cylinder Excited by a Cylindrical Wave
The scattered field due to cylindrical wave incident on circular cylinder 27 is where where T n 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 .

Interaction Contributions of the Geometry
There are two conducting wedges separated by a distance 2s, 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 noninteraction 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 where n 1 2π − α /π and n 2 2π − β /π and c 1 , c 2 and c 3 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 θ 0 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, wellknown far field conditions are used in which φ 0 To determine c 1 , c 2 and c 3 , 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  The transmission coefficient T for plane wave incidence is calculated using the expression given by Karp and Russek 8 as where E is E θ, s, d, n 1 , n 2 , a in the limit as θ approaches θ 0 .the transmission coefficient for the slit when a uncoated PEMC cylinder is placed under the slit.Comparison of T c with both the copolarized T co and cross-polarized T cross components of uncoated PEMC cylinder is studied.In all the cases cylinder radius ka is taken as 0.5.show exactly the same behavior as that of T c when Mη 0 ∞.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 Mη 0 ∞.In Figure 3, a comparison of T cross 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 Mη 0 0. In Figures 4 and 5, a comparison of T co and T cross for Mη 0 0 and Mη 0 ±1 at kd 0 and kd 5 are presented, respectively.In Figure 4 a , it can be seen that when the cylinder, with Mη 0 0, is at kd 0, T co is less than that of T cross , but when it is shifted below the center of the aperture plane, say at kd 5, T co becomes larger than T cross 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, T cross remains larger at kd 5 when Mη 0 ±1, as shown in Figure 5 b , whereas at kd 0T cross is less than T co as shown in Figure 5 a .Moreover, it can be seen that when kd 0, T co shows almost similar behavior as that of T c but T cross is larger than T c .When the cylinder is shifted to kd 5, both T co and T cross are larger than T c .To further highlight the effect of Mη 0 on T cross , Figure 6 shows that T cross is maximum when Mη 0 ±1 and decreases for other values of Mη 0 .Similarly Figure 7 shows the effect of variation of ka on T cross at Mη 0 ±1.Obviously the value of T cross is larger for ka 0.5 and decreases for smaller values of ka.Both these figures are for kd 0. The behavior of T co and T cross for obliquely incident plane wave at θ 0 15 0 and θ 0 30 0 for ka 0.1, kd 0 and Mη 0 ±1 is shown in Figures 8 a and  8 b .At θ 0 15 0 , T cross is higher than unity in the lower range of ks ks ≤ 3 and is larger than T co .For the same cylinder parameters but with θ 0 30 0 , both T co and T cross becomes less than unity.However, T cross oscillates with greater amplitude as compared to T co .Hence incident angle effects the peak locations of T co and T cross .To see the effect of interior wedge angle on the transmission coefficients for Mη 0 ±1, ka 0.1, it is observed that as the wedge angle is increased, the amplitude of oscillation in both T co and T cross is increased, that is, the interior wedge angle effects the levels of maxima and minima of the oscillation in both the  In the third part of discussion, the transmission coefficient of coated PEMC cylinder is presented.Behavior of both copolarized T c co and cross-polarized T c cross 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 T c co and T c cross for Mη 1 ±1 at kd 0 and kd 5 with relative permitivity r −1.5 and relative permeability μ r −1, are shown in Figures 14 a and 14 b , respectively.It can be seen that in both the cases, T c co is larger than T c cross , which is contrary to Uncoated PEMC cylinder in which T co remains less than T cross for Mη 0 ±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 T c co and T c cross are greater than unity whereas T c , in general, remains less than unity.The variation in the radius of coated cylinder also effects the behavior of T c co and T c cross as shown in Figure 15. Figure 15 b shows that T c co oscillates with greater amplitude as the the value of b is increased.However, T c cross does not show considerable change in behavior with the increase in radius b as hi-lighted in Figure 15 a .The behavior of T c co and T c cross for oblique incidence is shown in Figure 16 for incident angles θ 0 20 0 and θ 0 30 0 .Figure 16 a shows that T c cross gets less than unity as the angle of incidence is increased from zero, whereas the amplitude of oscillation for T c co decreases with the increase of incidence angle θ 0 as shown in Figure 16 b .All the plots of Figures 15 and 16 are for for Mη 1 ±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 T c co and T c cross .In case of T c cross , as shown in Figure 17 a , the oscillations are always around unity and decreases with increasing ks whereas in case of T c co 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.In the last part of discussion, diffraction pattern of wide double wedge in the presence of coated PEMC cylinder is presented.In Figure 18, the effect of Mη

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 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 of T co and T cross of an Uncoated PEMC cylinder and T c co and T c cross 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.

Figure 1 :
Figure 1: a Geometry of the Problem-Coated PEMC cylinder placed under PEC wide double wedge.b Uncoated PEMC cylinder placed under PEC wide double wedge.

Figure 1 aMathematicalFigure 10 :
Figure 1 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 T c for PEC double wedge with zero wedge angle, loaded with PEC cylinder is made with