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

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.

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 𝑛×𝐻+𝑀𝐸=0𝑛⋅𝐷−𝑀𝐵=0, (1.1) 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.

2. 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.

(a)

(b)

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.

2.1. PEC Wedge Excited by a Plane Wave

For the plane wave incident on the edge of the wedge at an angle 𝜙₀ with respect to the negative 𝑥-axis the incident field is given as 𝐸^𝑖_𝑧𝐻^𝑖_𝑧=𝐸₀𝐻₀exp𝑗𝑘𝑥cos𝜙₀+𝑦sin𝜙₀. (2.1) The uniform expression for the field diffracted from a PEC wedge has the form [6] 𝐸^𝑑_𝑧𝐻^𝑑_𝑧=exp−𝑗𝑘𝜌√𝜌𝐷^𝑠_ℎ𝜌,𝜙,𝜙₀𝐸;𝑛^𝑖𝑧𝐻^𝑖_𝑧. (2.2) The diffraction coefficient for the PEC wedge is 𝐷^𝑠_ℎ𝜌,𝜙,𝜙₀;𝑛=−√𝜌−sgnsin𝜋+𝜙−𝜙₀2𝑛cos𝜋+𝜙−𝜙₀2𝑛×𝐹2𝑘𝜌𝑛sin𝜋+𝜙−𝜙₀2𝑛−sgnsin𝜋−𝜙−𝜙₀2𝑛cos𝜋−𝜙−𝜙₀2𝑛×𝐹2𝑘𝜌𝑛sin𝜋−𝜙−𝜙₀2𝑛±sgnsin𝜋+𝜙+𝜙₀2𝑛cos𝜋+𝜙+𝜙₀2𝑛×𝐹2𝑘𝜌𝑛sin𝜋+𝜙+𝜙₀2𝑛±sgnsin𝜋−𝜙+𝜙₀2𝑛cos𝜋−𝜙+𝜙₀2𝑛×𝐹2𝑘𝜌𝑛sin𝜋−𝜙+𝜙₀,2𝑛 (2.3)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 𝐹(𝑥)=1𝜋exp𝑗𝑥²𝜋+𝑗4^∞_𝑥exp−𝑗𝜇²𝑑𝜇. (2.4)

It is assumed that point of observation is far from the edge of the wedge. For large argument approximation, Fresnel integral simplifies as 𝐹(𝑥)≈12√𝜋𝜋𝑥exp−𝑗4. (2.5) Hence diffraction coefficient for 𝐸-polarized plane wave, incident on the edge of the wedge simplifies to 𝐷𝜙,𝜙₀≈(),𝑛sin𝜋/𝑛𝜋𝜋𝑛cos𝑛−cos𝜙−𝜙₀𝑛⁻¹−𝜋cos𝑛−cos𝜙+𝜙₀𝑛⁻¹. (2.6) 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 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 𝐸^𝐶_𝑝=𝜋𝐻2𝑗⁰⁽²⁾𝑘𝜌𝐺_𝑝𝜙,𝜙₀,𝑎, (2.7) where 𝐺_𝑝𝜙,𝜙₀,𝑎=−2𝑗𝜋^∞_𝑛=0𝜖_𝑛(−1)^𝑛𝑇_𝑛cos𝑛𝜙−𝜙₀, (2.8) where the Neumann number 𝜖_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] is𝑇_𝑛=𝐻_𝑛⁽²⁾(𝑘𝑎)𝐽^/_𝑛(𝑘𝑎)+𝑀²𝜂²₀𝐽_𝑛(𝑘𝑎)𝐻^(2)/_𝑛(𝑘𝑎)1+𝑀²𝜂²₀𝐻_𝑛⁽²⁾(𝑘𝑎)𝐻^(2)/_𝑛(𝑘𝑎),Copolarized2𝑀𝜂₀𝜋𝑘𝑎1+𝑀²𝜂²₀𝐻_𝑛⁽²⁾(𝑘𝑎)𝐻^(2)/_𝑛(𝑘𝑎),Crosspolarized (2.9) whereas the transmission coefficient for coated PEMC cylinder [22] is 𝑇_co=𝐽^′_𝑛𝑘₀𝑏/𝜂₀(𝐴)−𝐽_𝑛𝑘₀𝑏/𝜂₁(𝐵)𝐻_𝑛⁽²⁾𝑘₀𝑏/𝜂₁(𝐵)−𝐻_𝑛^(2)′𝑘₀𝑏/𝜂₀(𝐴),𝑇_cross=𝑗𝑀𝜂₁𝐻_𝑛⁽¹⁾𝑘₁𝑎−𝐻_𝑛⁽²⁾𝑘₁𝑎𝐻_𝑛^(1)′𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑎𝐽_𝑛𝑘₀𝑏𝐻_𝑛^(2)′𝑘₀𝑏−𝐽^′_𝑛𝑘₀𝑏𝐻_𝑛⁽²⁾𝑘₀𝑏𝜂₀𝐻_𝑛⁽²⁾𝑘₀𝑏(𝐵)−𝐻_𝑛^(2)′𝑘₀𝑏(𝐴), (2.10) where 𝐴=𝑎𝑐𝑏+𝑑−𝑗𝑀𝜂₁𝐻_𝑛⁽²⁾𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑎𝑒𝑎𝑏+𝑓𝐻_𝑛⁽¹⁾𝑘₁𝑏−𝐻_𝑛⁽²⁾𝑘₁𝑏𝐻_𝑛^(1)′𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑎+𝑗𝑀𝜂₁𝐻_𝑛⁽²⁾𝑘₁𝑏𝐻_𝑛^(2)′𝑘₁𝑎𝑒𝑎𝑏+𝑓𝐻_𝑛⁽¹⁾𝑘₁𝑎−𝐻_𝑛⁽²⁾𝑘₁𝑎𝐻_𝑛^(1)′𝑘₁𝑎𝐻^(2)′_𝑛𝑘₁𝑎,𝐵=𝐻_𝑛^(1)′𝑘₁𝑏−𝐻_𝑛^(2)′𝑘₁𝑏𝐻_𝑛^(1)′𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑎𝑎𝑐𝑏+𝑑−𝑗𝑀𝜂₁𝐻_𝑛⁽²⁾𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑎𝑒𝑎𝑏+𝑓+𝑗𝑀𝜂₁𝐻_𝑛^(2)′𝑘₁𝑏𝐻_𝑛^(2)′𝑘₁𝑎𝑒𝑎𝑏+𝑓𝐻_𝑛⁽¹⁾𝑘₁𝑎−𝐻_𝑛⁽²⁾𝑘₁𝑎𝐻_𝑛^(1)′𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑎,𝐻𝑎=^𝑛(2)𝑘₀𝑏𝜂₀−𝐻_𝑛⁽²⁾𝑘₁𝑏𝐻_𝑛^(2)′𝑘₀𝑏𝜂₁𝐻_𝑛^(2)′𝑘₁𝑏,1𝑏=𝜂₁𝐻_𝑛⁽¹⁾𝑘₁𝑏−𝐻_𝑛⁽²⁾𝑘₁𝑏𝐻_𝑛^(2)′𝑘₁𝑏𝐻_𝑛^(2)′𝑘₁𝑏,𝑐=𝐻_𝑛⁽¹⁾𝑘₁𝑎−𝐻_𝑛⁽²⁾𝑘₁𝑎𝐻_𝑛^(1)′𝑘₁𝑏𝐻_𝑛^(2)′𝑘₁𝑏,𝐻𝑑=^𝑛(2)𝑘₁𝑎𝐻_𝑛^(2)′𝑘₀𝑏𝐻_𝑛^(2)′𝑘₁𝑏,𝑒=𝑗𝑀𝜂₁𝐻_𝑛^(1)′𝑘₁𝑎−𝐻_𝑛^(1)′𝑘₁𝑏𝐻_𝑛^(2)′𝑘₁𝑎𝐻^(2)′_𝑛𝑘₁𝑏,𝑓=𝑗𝑀𝜂₁𝐻_𝑛^(2)′𝑘₀𝑏𝐻_𝑛^(2)′𝑘₁𝑎𝐻_𝑛^(2)′𝑘₁𝑏. (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.

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 𝐸^𝑖=𝜋𝐻2𝑗⁰⁽²⁾(𝑘𝑅), (3.1) 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 𝜋𝐸=𝐻2𝑗⁰⁽²⁾𝑘𝜌𝐹𝜙,𝜌₀,𝜙₀,𝑛, (3.2) where 𝐹𝜙,𝜌₀,𝜙₀≈𝐻,𝑛⁰⁽²⁾𝑘𝜌₀exp−𝑗(𝜋/2)sin(𝜋/𝑛)×𝜋𝜋𝑛cos𝑛−cos𝜙−𝜙₀𝑛⁻¹−𝜋cos𝑛−cos𝜙+𝜙₀𝑛⁻¹. (3.3) 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 𝐸^𝐶_𝑙=𝜋𝐻2𝑗⁰𝑘𝜌𝐺_𝑙𝜙,𝜌₀,𝜙₀,𝑎, (3.4) where 𝐺_𝑙𝜙,𝜌₀,𝜙₀,𝑎=−^∞_𝑛=0𝜖_𝑛𝑗^𝑛𝑇_𝑛𝐻_𝑛𝑘𝜌₀cos𝑛𝜙−𝜙₀, (3.5) 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).

4. 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 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] 𝐸^𝑡=𝐸^𝑖+𝐸^𝑠, (4.1) where 𝐸^𝑠=𝐸^𝑠1+𝐸^𝑠2+𝐸^𝑠3,𝐸^𝑠1=𝜋𝐻2𝑗⁰𝑘𝜌₁exp−𝑗𝑘𝑠sin𝜃₀𝐷𝜙₁,𝜙₀₁,𝑛₁+𝑐₃𝐹𝜙₁,𝑠₁,𝜙₃₁,𝑛₁+𝑐₂𝐹𝜙₁,2𝑠,𝜙₂₁,𝑛₁,𝐸^𝑠2=𝜋𝐻2𝑗⁰𝑘𝜌₂exp+𝑗𝑘𝑠sin𝜃₀𝐷𝜙₂,𝜙₀₂,𝑛₁+𝑐₃𝐹𝜙₁,𝑠₁,𝜙₃₂,𝑛₁+𝑐₂𝐹𝜙₁,2𝑠,𝜙₁₂,𝑛₂,𝐸^𝑠3=𝜋𝐻2𝑗⁰𝑘𝜌₃exp−𝑗𝑘𝑑cos𝜃₀𝐷𝜙₃,𝜙₀₃,𝑎+𝑐₁𝐺𝜙₃,𝑠₁,𝜙₁₃,𝑎+𝑐₂𝐺𝜙₃,𝑠₂,𝜙₂₃,𝑎, (4.2) 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 determine 𝑐₁,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 another2𝑐₁−𝑐₂𝐹𝜙₃₁,2𝑠,𝜙₂₁,𝑛₁+𝐹𝜙₂₁,2𝑠,𝜙₂₁,𝑛₁−𝑐₃𝐹𝜙₃₁,𝑠₁,𝜙₃₁,𝑛₁+𝐹𝜙₂₁,𝑠₁,𝜙₃₁,𝑛₁=exp−𝑗𝑘𝑠sin𝜃₀𝐷𝜙₃₁,𝜙₀₁,𝑛₁+𝐷𝜙₂₁,𝜙₀₁,𝑛₁,2𝑐₂−𝑐₁𝐹𝜙₃₂,2𝑠,𝜙₁₂,𝑛₂+𝐹𝜙₁₂,2𝑠,𝜙₁₂,𝑛₂−𝑐₃𝐹𝜙₃₂,𝑠₂,𝜙₃₂,𝑛₂+𝐹𝜙₁₂,𝑠₂,𝜙₃₂,𝑛₂=exp𝑗𝑘𝑠sin𝜃₀𝐷𝜙₃₂,𝜙₀₂,𝑛₂+𝐷𝜙₁₂,𝜙₀₂,𝑛₂,2𝑐₃−𝑐₁𝐺𝜙₁₃,𝑠₁,𝜙₁₃,𝑎+𝐺𝜙₂₃,𝑠₁,𝜙₁₃,𝑎−𝑐₂𝐺𝜙₁₃,𝑠₂,𝜙₂₃,𝑎+𝐺𝜙₂₃,𝑠₂,𝜙₂₃,𝑎=exp−𝑗𝑘𝑑cos𝜃₀𝐷𝜙₁₃,𝜙₀₃,𝑎+𝐷𝜙₂₃,𝜙₀₃,𝑎. (4.3) By solving (4.3) for 𝑐₁, 𝑐₂, and 𝑐₃, the scattered field is found and is rewritten in the form 𝐸^𝑠=exp−𝑗𝑘𝜌𝐸𝜋𝑘𝜌𝜃,𝑠,𝑑,𝑛₁,𝑛₂,𝑎, (4.4) where the scattered field pattern 𝐸(𝜃,𝑠,𝑑,𝑛₁,𝑛₂,𝑎) is obtained from (4.4).

The transmission coefficient 𝑇 for plane wave incidence is calculated using the expression given by Karp and Russek [8] as 𝐸𝑇=Re1−𝑗2𝑘𝑠, (4.5) where 𝐸 is 𝐸(𝜃,𝑠,𝑑,𝑛₁,𝑛₂,𝑎) in the limit as 𝜃 approaches 𝜃₀.

5. Results and Discussion

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 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. Figures 2(a) and 2(b) show the comparison of 𝑇_𝑐 with 𝑇_co when the cylinder is located at 𝑘𝑑=0 and 𝑘𝑑=5 from the slit, respectively. It can be observed that 𝑇_co, in both the cases, show exactly the same behavior as that of 𝑇_𝑐 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 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 𝑀𝜂₀=0. In Figures 4 and 5, a comparison of 𝑇_co and 𝑇_cross 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, 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 𝑀𝜂₀=±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 𝑀𝜂₀ on 𝑇_cross, Figure 6 shows that 𝑇_cross is maximum when 𝑀𝜂₀=±1 and decreases for other values of 𝑀𝜂₀. Similarly Figure 7 shows the effect of variation of 𝑘𝑎 on 𝑇_cross at 𝑀𝜂₀=±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 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, 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 𝜃₀=30⁰, 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 𝑀𝜂₀=±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 cases, however this effect is more dominant in 𝑇_cross as compared to 𝑇_co as shown in Figures 9(a) and 9(b).

(a)

(b)

Figure 2 

(a) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0, 𝑘𝑎=0.5. (b) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=5, 𝑘𝑎=0.5.

Figure 3 

Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0, 𝑘𝑎=0.

(a)

(b)

Figure 4 

(a) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0, 𝑘𝑎=0.5. (b) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=5, 𝑘𝑎=0.5.

(a)

(b)

Figure 5 

(a) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0, 𝑘𝑎=0.5. (b) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=5, 𝑘𝑎=0.5.

Figure 6 

Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0, 𝑘𝑎=0.5.

Figure 7 

Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0, 𝑘𝑎=0.5.

(a)

(b)

Figure 8 

(a) Slit transmission coefficient for 𝑘𝑑=0, 𝑘𝑎=0.1. (b) Slit transmission coefficient for 𝑘𝑑=0, 𝑘𝑎=0.1.

(a)

(b)

Figure 9 

(a) Slit transmission coefficient for 𝑘𝑑=0, 𝑘𝑎=0.1. (b) Slit transmission coefficient for 𝑘𝑑=0, 𝑘𝑎=0.1.

In the second part of discussion, normalized diffraction pattern of the slit loaded with PEC cylinder (𝐷_𝑐) 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 𝐷_𝑐 compared with 𝐷_(co) for 𝑘𝑑=0 and 𝑘𝑠=8. The solid curve in the figure represents 𝐷_𝑐. It is observed that 𝐷_(co) shows similar behavior as that of 𝐷_𝑐 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)

(b)

Figure 10 

(a) Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑘𝑎=0.5, 𝑘𝑠=8. (b) Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑘𝑎=0.5, 𝑘𝑠=8.

Figure 11 

Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑘𝑎=0.5, 𝑘𝑠=8.

(a)

(b)

Figure 12 

(a) Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑘𝑎=0.5. (b) Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑘𝑎=0.5.

Figure 13 

Slit diffraction pattern for 𝜃0=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^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 𝑀𝜂₁=±1 at 𝑘𝑑=0 and 𝑘𝑑=5 with relative permitivity 𝜖_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 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 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 𝜃₀=20⁰ and 𝜃₀=30⁰. 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 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 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.

(a)

(b)

Figure 14 

(a) Slit diffraction pattern for 𝜃0=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)

Figure 15 

(a) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0. (b) Slit transmission coefficient for 𝜃0=0, 𝑘𝑑=0.

(a)

(b)

Figure 16 

(a) Slit transmission coefficient for 𝜃0=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)

Figure 17 

(a) Slit transmission coefficient for 𝜃0=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.

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 𝑀𝜂₁ on the diffraction pattern is shown. Behavior of both copolarised (𝐷^𝑐_(co)) and cross-polarized (𝐷^𝑐_(cross)) components of coated PEMC cylinder with 𝜖_𝑟=1.5 and 𝜇_𝑟=1 for 𝑘𝑑=0 and 𝑘𝑠=8 is studied. In both the cases, it can be seen that 𝐷^𝑐_(co) and 𝐷^𝑐_(cross) show slight different behavior for 𝑀𝜂₁=1 as compared to other values of 𝑀𝜂₁. Figure 19 shows the variation in 𝐷^𝑐_(co) and 𝐷^𝑐_(cross) for 𝜖_𝑟=1.5, 𝜇_𝑟=1 and 𝑘𝑑=0 with respect to the slit width. It is observed that both 𝐷^𝑐_(co) and 𝐷^𝑐_(cross) show different behavior for different values of slit widths.

(a)

(b)

Figure 18 

(a) Slit diffraction pattern for 𝜃0=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)

Figure 19 

(a) Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑎=0.2, 𝑏=0.3. (b) Slit diffraction pattern for 𝜃0=00, 𝑘𝑑=0, 𝑎=0.2, 𝑏=0.3.

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 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.