Radiation From a Line Source Shielded by a Confocal Elliptic Layer of DNG Metamaterial

The two-dimensional problem of electromagnetic radiation from a line source located inside an elliptic-cylinder confocal sheath of double-negative metamaterial is solved exactly by separation of variables. Numerical results are presented and discussed for the radiation pattern and the field inside the sheath, as functions of frequency, location of the line source, and sheath thickness and eccentricity.


I. INTRODUCTION
A Confocal elliptic cylindrical layer made of doublenegative (DNG) metamaterial is considered. The space inside and outside the layer is filled with a lossless homogeneous double-positive (DPS) medium characterized by a real positive electric permittivity ϵ 1 and a real positive magnetic permeability µ 1 , whereas the DNG sheath is characterized by a real negative permittivity −ϵ 2 and a real negative permeability −µ 2 . The isorefractive condition ϵ 1 µ 1 = ϵ 2 µ 2 is assumed. Consequently, the wavenumber is where ω is the angular frequency, and ℓ = 1 in the DPS regions of space and ℓ = 2 inside the DNG sheath, meaning that the real positive wavenumber in the DPS regions has the same magnitude but opposite sign to the real negative wavenumber in the DNG layer. The real positive intrinsic impedances are The primary field is an electric or magnetic line source parallel to the generators of the cylindrical sheath and located anywhere inside the DPS medium encased by the sheath. The analysis is conducted in phasor domain with the timedependence factor exp(+jωt) omitted throughout. In all regions, the field components are expressed as infinite series of Mathieu functions whose modal expansion coefficients are found analytically by imposing the boundary conditions. This novel analytical solution is evaluated numerically for different values of the parameters involved: dimensions and thickness of the sheath in terms of wavelength, eccentricity of the elliptical cross-section of the sheath, ratio of the intrinsic impedances of the DPS and DNG media, location of the line source. Particular attention is devoted to the radiated far field and to the field inside the DNG layer.
O. Akgol, D. Erricolo  A simplified solution is obtained when the intrinsic impedances of the DPS and DNG media are equal and the source is located in the plane containing the focal lines of the elliptical sheath. In particular, a comparison with a geometrical-optics solution is performed when the source coincides with a focal line; this simplified solution was presented at a conference [1]. Previous works have examined the case of an elliptic cylinder completely filled with DNG metamaterial and excited by a line source located at a focal line, both from an optical [2] and an electromagnetic [3] viewpoint.

II. GEOMETRY OF THE PROBLEM
A cross-section of the structure is shown in Fig. 1. The generators of the elliptic-cylindrical sheath are parallel to the z-axis, and the elliptic cylinder coordinates (u, v, z) are related to the rectangular coordinates (x, y, z) by and d is the interfocal distance between the focal lines F 1 and F 2 . The dimensionless parameters where λ is the wavelength, are introduced. The inner and outer surfaces of the DNG sheath correspond to u = u 1 and u = u 2 , respectively. The position of the line source is indicated by S in Fig. 1. Because of symmetries, we may restrict its location to the first quadrant, corresponding to coordinates u 0 and v 0 where H (2) 0 is the Hankel function of the second kind and R is the distance of the observation point from the line source. This field may be expanded in an infinite series of products of radial and angular Mathieu functions [4]: where ui < (u > ) is the smaller (larger) between u and u 0 . The total electric field in the DPS region enclosed by the DNG sheath is where The electric field radiated into the infinite DPS space surrounding the structure is The electric field inside the DNG layer is where the fact that the angular functions Se m and So m are even functions of the parameter c has been taken into account. The modal expansion coefficients a where l = 1(2) for DPS (DNG) media, and here and in the following the prime means derivative with respect to u. It is found that where The coefficients a as should be expected. With reference to circular cylindrical coordinates (ρ, ϕ, z) (see Fig.1), the radiated far field is where the far-field coefficient is In the absence of the DNG sheath, the far-field coefficient is that of the line source (6), and is equal to one. Hence, the farfield power pattern normalized to the power pattern without the DNG sheath is Simplifications in the formulas of this section occur when the line source is located either in the x = 0 plane or in the y = 0 plane of Fig.1. In particular, if the source coincides with the focal line F 1 of Fig.1, so that u 0 = v 0 = 0, the far-field coefficient (18) simpifies to: The solution for an isotropic magnetic line source is obtained by duality from the solution for an electric line source.
IV. A GEOMETRICAL OPTICS SOLUTION Consider the particular case of an isotropic line source located at the focal line F 1 . With reference to the cross-section of Fig. 2, the optical ray F 1 A is totally transmitted across the interface u = u 1 in the direction AB, as though it were produced by an anisotropic virtual line source located at the other focal line F 2 . Along AB, the normalized power emitted per unit angle is [2] where β is the angle that the direction F 2 AB of propagation forms with the positive x-axis. The eccentricity of the crosssectional ellipse u = u ℓ is where a ℓ is the semi-major axis of the ellipse. The optical ray AB is totaly transmitted across the interface u = u 2 in the direction F 1 BC. If the virtual line source at F 2 were isotropic, the normalized power emitted per unit angle in the direction BC would be where α is the angle that the direction F 1 BC forms with the positive x-axis. Therefore, the optical power emitted per unit angle along BC, normalized to the power emitted isotropically per unit angle by the line source at F 1 , is To complete the optical solution, it remains to express β as a function of α. After some geometrical manipulations, it is found that cos β = (1 + χ 2 2 ) cos α + 2χ 2 1 + χ 2 2 + 2χ 2 cos α) . (25)

V. NUMERICAL RESULTS
We first examine comparisons between the radiated power pattern given by eq. (19) and the geometrical optical solution of eq. (24). The results are parameterized according to the ratio r = 2a 1 /λ. A smaller value of the ratio correspond to an electrically small cross-section, while a large value of the ratio is associated with an electrically large cross-section, up to infinity that corresponds to the geometrical optics case. Referring to Fig. 3, when r = 0.05 the radiated power pattern is a slightly flattened circle; when r = 0.1, the flattening becomes more prominent; and, when r = 4.3 the power radiated pattern approaches the geometrical optics solution. Next, we examine the effect of changing ζ, while keeping constant all other parameters. Fig. 4 indicates quite significant changes of the normalized radiated power pattern of (19) when ζ = 0.1, 1, 10.  The last numerical example is the behavior of the magnitude of the the electric field within the DNG, which is represented in Fig. 5 for various values of the ratio r. For smaller values of r there are fewer oscillations within the DNG layer, while the behavior becomes more complex for larger values of r.
The computations of the Mathieu functions were carried out with an extension of the software described in [5] and the acceleration technique described in [6].

VI. CONCLUSION
The novel two-dimensional boundary value problem of radiation from a line source shielded by a confocal elliptic layer of DNG metamaterial has been solved exactly. Numerical data for the radiated far-field and the field inside the sheath have been exhibited.