RCS of Chiral Elliptic Cylinder Embedded in Infinite Chiral Medium

Received Jan 29, 2017 Revised Jun 10, 2017 Accepted Aug 11, 2017 This paper presents an analytic solution to the scattering properties of chiral elliptic cylinder embedded in infinite chiral medium due to incident plane wave. The external electromagnetic fields as well as the internal electromagnetic fields are written in terms Mathieu functions and expansion coefficients. In order to obtain both the internal and external unknown field expansion coefficients, the boundary conditions are applied rigorously at the surface of different chiral/chiral material. Results are plotted graphically for the normalized scattering widths for elliptic cylinders of different sizes and chiral materials to show the effects of these parameters on scattering cross widths. It is shown numerically by adding the external chiral material to elliptic cylinder provides more parameters to control the RCS.

The elliptic cylinder is a geometry that has been extensively analyzed in the literature due to its ability to create cylindrical cross sections of different shapes by changing the axial ratio of the ellipse. Furthermore, since the elliptic cylindrical coordinate system is one of the coordinate systems in which the wave equation is separable, solutions to problems involving elliptic cylinders can be obtained in exact form.
In this paper, we present a solution to the problem of scattering of a right-circularly polarized (RCP) plane wave from a chiral elliptic cylinder of arbitrary axial ratio placed in a distinct infinite chiral medium, while in [21] the chiral cylinder was embedded in free space. Both chiral media in this problem are isotropic. The present solution will require expanding the fields in two different chiral regions, and the solution obtained will therefore supply more parameters to control the normalized bistatic scattering width when compared to that in [21].

THEORY
Consider a RCP plane wave that is propagating in an infinite isotropic chiral medium, being incident on an infinitely long elliptic cylinder at an angle i  with respect to the minus x-axis of a Cartesian coordinate system located at the center of a cross section of the cylinder with its z-axis along the axis of the where  is a unit vector in the direction of increasing ,  1 R k is the wavenumber of the incident plane wave, and ) , (   are the polar coordinates of an observation point with respect to the above mentioned Cartesian coordinate system. In accordance with the Drude-Born-Federov (DBF) constitutive relations that have been used in this paper [22], the wavenumber of the right circularly polarized wave is given by where 0 k is the wavenumber in free space, 1  is the chirality parameter of the external chiral medium, and 1 r  and 1 r  are the relative permeability and relative permittivity of the external chiral medium.
The incident electric field in (1) can now be expanded in terms of elliptical vector wave functions as functions, and is the same for the other field expansions given below too. Maxwell's equations can next be used to expand the incident magnetic field in terms of elliptical vector wave functions as Z denoting the free space wave impedance. As the elliptic cylinder is made up of an isotropic chiral material and is surrounded by another isotropic chiral medium, the scattered and transmitted electro-magnetic fields will have both co-polar and cross-polar components.
The fields transmitted inside the chiral elliptic cylinder can also be expressed as (2) and (9), respectively, by changing the subscript 1 in these Equations, to 2. The unknown expansion coefficients can be obtained by imposing the boundary conditions corresponding to the continuity of the tangential field components at the surface s    of the chiral elliptic cylinder [25], which may be expressed mathematically as (13) where ξ is the outward unit normal to the surface of the elliptic cylinder. Substituting the above developed expressions into the fields of (12) and (13), and applying the orthogonal property of the angular Mathieu functions, yield

H H H
) , The summations over m in (14) (14)- (17) can be expressed in matrix form and solved, to obtain the unknown field expansion coefficients. Using asymptotic expressions of the radial Mathieu functions of the fourth kind and their first derivatives, we can write the co-and cross polar scattered electric fields in the far zone as is obtained from (21), by changing B to C and R1 to L1. Explicit expressions for the normalized bistatic echo width of the right-and left-polarized waves can then be written as

RESULTS AND ANALYSIS
Numerical results are presented as normalized echo pattern widths for isotropic chiral elliptic cylinders of different axial ratios, embedded in another isotropic chiral medium of different relative permittivities and chirality parameters. First, we chose the parameters , 0 . . To validate the analysis and the software used for calculating the results, we computed the normalized echo pattern widths for the above chiral cylinder when it is excited by a plane wave that is transverse magnetically (TM) polarized in the axial z-direction, that is obtained by summing right-and leftcircularly polarized incident waves of equal field amplitudes. The results are in good agreement with those in [15] for an analogous chiral circular cylinder in free space, verifying the accuracy of the analysis and that of the software used for the calculations.  Figure 3 shows the right-and left-polarized echo-width patterns for the chiral elliptic cylinder in Figure 2, when it is placed in a chiral medium which is similar to that in Figure 2, but with 5 .
. When compared with the plots in Figure 2, we see that as the scattering angle increases from 0 o to 180 o , the reduction of the right-polarized echo-width magnitude is much higher. Also the left-polarized echo-width magnitude is much lower for all scattering angles, though its behavior is somewhat similar to that of the same in Figure 2.

CONCLUSION
An exact solution to the problem of scattering of a RCP plane wave by a chiral elliptic cylinder placed in another chiral medium, presented using the method of separation of variables. Results have been presented as normalized bistatic right-and left-polarized echo-width patterns for chiral elliptic cylinders of different axial ratios and chiral materials, to show the effects of these on scattering. It is seen that the presence of two different chiral materials could significantly influence the left-and right-polarized pattern widths.