Effect of Dissipative and Dispersive DNG Material Coating on the Scattering Behavior of Parallel Nihility Circular Cylinders

Electromagnetic scattering from coated nihility circular cylinders, illuminated by E-polarized plane wave, is investigated using an iterative procedure. Cylinders are infinite in length. The boundary conditions are applied on the surface of each cylinder in an iterative procedure in order to solve for the field expansion coefficients. The effect of different types of the coating layers including double positive DPS and double negative DNG on the alteration of the forward and backward scattering has been observed. Specially, the effect of dispersive and dissipative DNG coating layer has been focused. Numerical verifications are presented to prove the validity of this formulation by comparison with the published literature.


Introduction
Scattering from multiple cylinders in free space has been studied by many researchers 1-14 .For the analysis of multiple cylinders, different techniques had been used to analyze the scattering between nearby conducting circular cylinders 1 .Twersky 2 extended the multiple scattering problem to an N number of cylinders for the first time.He expressed the total radiation field as an incident field plus a scattered field of various orders.Burke et al. 3 derived the multiple scattering solution for N cylinders as a series of Hankel functions.Ragheb and Hamid 4 studied the scattering of plane waves by N circular cylinders.They used conducting circular cylinders to simulate a cylindrical reflector 5 .cylinder defined by its radius, material type core, and coating layer and its center coordinate with respect to the global cylindrical coordinate system ρ, φ .In this analysis, e jωt , time dependence, has been used and suppressed throughout.Consider an E-polarized incident plane wave to be the incident field on "ith", cylinder is expressed in its local cylindrical coordinates ρ i , φ i , as where ρ i and φ i represent the location of "ith" cylinder with respect to origin, k 0 is the free space wave number, J n • is the Bessel function of first kind, and φ 0 is the angle of incidence of the plane wave with respect to the x-axis.The corresponding φ component of the magnetic field is given by Prime represents derivative with respect to the argument, and η 0 μ 0 / 0 is the impedanceof free space.The scattered fields by the "ith" cylinder in Region 0, that is, free space, may be expressed as k 0 ρ i e jn φ i −φ 0 .

2.3
While the transmitted fields in region 1, that is, inside the coating layer of the "ith" cylinder, may be written as and the fields inside the core cylinder η 2 μ 2 / 2 , k 2 ω √ μ 2 2 are given as where a in , b in , c in , and d in are the unknown scattering coefficients of the fields in different regions.

Solution of the Unknown Scattering Coefficients
The unknown scattering coefficients may be calculated by applying the boundary conditions at the surface of core and coating of the ith cylinder.The continuity of the tangential components of electric and magnetic fields at the surface of the core, that is, ρ a i of the ith cylinder, is given as

3.1
Using boundary conditions 3.1 and substituting the values of the field components, the unknowns may be expressed after some mathematical manipulation as  Now, the boundary conditions at the surface of coating layer of the ith cylinder, that is, at ρ b i , are given as

3.3
Since the fields given in equations 2.1 -2.5 all are expressed in the local coordinates of the ith cylinder, so addition theorem of Hankel functions is used to transfer these fields in terms of global coordinates.Generally, the transformation from qth coordinates to the pth coordinates is given for the ith as 8 ρ q sin φ q ≥ ρ p sin φ p , −cos −1 ρ q cos φ q − ρ p cos φ p d pq , ρ q sin φ q < ρ p sin φ p ,  where ρ q , φ q and ρ p , φ p are the origins of qth and pth coordinate systems, respectively.Solving the boundary conditions 3.3 for the unknowns a il while applying the addition theorem of Hankel functions, we get where 3.6 In the above equations, the integers l, n 0, ±1, ±2, . . ., N i and i, g 0, 1, 2, . . ., M. All the summations range from minus infinity to plus infinity theoretically, but the upper limits N i are truncated to a numerical value which is related to the size and nature of the cylinders and also to the distance between the centers of the cylinders.
Equations 3.5 give the unknown scattering coefficients for the ith cylinder.Following the same procedure on the rest of the cylinders, we obtain a matrix form solution as Solving 3.7 , we get the unknown scattering coefficients a for all the cylinders.After truncating the infinite series solutions for the field expressions, solution of 3.7 gives the unknown scattering coefficients a in .Applying the limiting procedure proposed by Lakhtakia 16,17 , the scattered field from parallel nihility cylinders is obtained.The H-polarized incidence case can be deduced from E-polarized case using the duality theorem.The expressions of H z and E φ fields in H-polarized case correspond to the E z and H φ field expressions in the E-polarized incidence case.

Simulations
Far-field patterns from an array of five identical nihility cylinders of circular cross section, coated by different coating layers, are shown in this section.The radii of all the cylinders are taken the same as a i 0.1λ 0 , while distance between their centers is taken to be d.The proposed geometries made by coated nihility cylinders are excited by plane wave propagating along positive x-axis.To check the validity of the formulation and numerical codes, scattered field patterns of the coated and uncoated parallel nihility cylinders have been compared with those obtained for the uncoated parallel perfect electric conducting PEC cylinders 8 .The code generated for the plots of the paper has been tested by comparison to plots for uncoated PEC cylinders 8 .
Figures 2-5 present the far-field patterns due to different configurations of coated and uncoated nihility circular cylinders.Figure 2 shows the far-zone scattered field patterns of the five equally spaced coated nihility cylinders arranged along x-axis.The center-to-center distance between the cylinders in this arrangement is taken as d 0.75λ 0 .From this figure, it is seen that there is a big difference between the scattered field patterns of the uncoated nihility and PEC cylinders.And the back-scattered field due to nihility cylinders is smaller than that for the PEC cylinders.It is also observed from this figure that back-scattered field pattern due to double positive DPS 1 5, μ 1 1 coated nihility cylinders is comparable to that of uncoated PEC cylinders except that in this case forward and backward fields are almost equal.Figure 3    case of DPS coated nihility cylinders, back-scattered field can be controlled when forwardscattered field increases.
Figure 4 contains the far-field patterns of the five nihility cylinders placed in corner reflector configuration, with d 0.75λ 0 as the spacing between their centers.It is observed from the figure that in the case of DPS coated nihility cylinders, back-scattered field is greater than forward-scattered field in contrast to uncoated cases of nihility and PEC corner reflector configurations.Figure 5 presents the far-field patterns of the five nihility cylinders placed in a star-like configuration at the origin, with d 0.5λ 0 as the spacing between their centers.In this case also, it is observed that in the case of DPS coated nihility cylinders, back-scattered field is greater than forward-scattered field in contrast to uncoated cases of nihility and PEC corner reflector configurations.−0.399 0.263i, μ 1 −1.397 0.623i materials.Figure 6 shows the reduction in forward and backward fields when coated by dissipative and dispersive DNG material.Figure 7 shows a very small variation due to lossless and dissipative and dispersive DNG materials.Figure 8 shows that in case of corner reflector formed by DNG coated nihility cylinders, forward-scattered field is reduced, while this backward-scattered field is increased for dissipative and dispersive DNG material.Figure 9 shows that in case of star-like configuration formed by DNG coated nihility cylinders, forward-scattered field is reduced, while the backward-scattered field is increased in case of dissipative and dispersive DNG material.

Conclusions
From the numerical results, it is observed that the forward-and backward-scattered fields can be altered or changed using different types of coating layers, on the nihility circular cylinders.Specially using DNG coating layers, a large contribution is obtained in the forward-scattered field as compared to the backward-scattered field.

Figures 6 - 1 − 5 , μ 1 − 1
9 present the far-zone scattered field patterns of different arrangements of nihility circular cylinders coated by lossless double-negative DNG and dissipative and dispersive DNG 1