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[1] A metamaterial lens consisting of a paraboloid of revolution that separates two portions of space having real and opposite refractive indexes but the same intrinsic impedance is considered, as well as a lens consisting of a paraboloidal double-negative radome. For both lenses, geometrical optics yields the exact electromagnetic field for an axially incident plane wave.

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[2]Pendry [2000] has shown that a planar slab of double-negative (DNG) metamaterial that has the opposite (real) refractive index and the same intrinsic impedance of the surrounding medium constitutes a perfect lens, in that it images a point onto another point without aberrations. In this work, we expand on Pendry's concept by considering other geometries that constitute lenses. One of the two media we consider has permittivity ε and permeability μ, both real and positive. It is in contact with a DNG material that has permittivity −ε and permeability −μ and therefore, as proven by Ziolkowski and Heyman [2001], has a refractive index that is negative and the opposite of the refractive index of the first medium; the two media have the same positive intrinsic impedance Z = Y^{−1}.

[3] In the first geometry under consideration, the surface that separates the two media is a paraboloid of revolution with the DNG material on its concave side and a plane electromagnetic wave axially incident from either side of the surface. In the second geometry, a paraboloidal radome made of DNG material and separated from the surrounding medium by two coaxial and confocal paraboloids of revolution is considered, again with an axially incident plane wave. In either case the geometrical optics solution satisfies Maxwell's equations and the boundary conditions exactly, and therefore represents the exact solution to the boundary-value problem.

[4] The geometries considered herein are not as general as Pendry's structure, because they are limited to specific locations of source and observation points. The crucial fact that geometrical optics is also the exact electromagnetic solution is a peculiar property of the paraboloid of revolution [Lee, 1975]. This property has been utilized in the past to obtain exact geometrical optics solutions for the convex metallic paraboloid by Schensted [1955], for the convex isorefractive paraboloid by Roy and Uslenghi [1997], and for the convex multilayer isorefractive paraboloidal radome by Liang and Uslenghi [2007a]. Preliminary results on DNG paraboloidal geometries were presented at a symposium [Liang and Uslenghi, 2007b].

2. Single Paraboloid

[5] The paraboloidal coordinates ξ, η, ϕ are related to the rectangular coordinates x, y, z by the relations

where 0 ≤ ξ < ∞, 0 ≤ η < ∞, and 0 ≤ ϕ < 2π. The separation surface η = η_{1} is a paraboloid with z as symmetry axis and focus at F(x = y = z = 0). The DNG medium fills the space on the concave side of the surface. The incident plane wave on the concave side carries energy in the negative z direction while its phase increases in the positive z direction. Its electric and magnetic fields, E^{i} and H^{i}, may be written as

where −k is the propagation constant. Upon incidence on the separation surface, the wave is transmitted without reflection and emerges on the convex side as a diverging beam whose electric and magnetic fields, E^{t} and H^{t}, are given by

where

[6] Note that

where = (x + y + z) / is the radial unit vector in spherical coordinates from the origin F.

[7] The diverging beam appears to emanate from the focus F. Thus the structure maps the point at z = +∞ onto the (virtual) point F. It can be verified that the geometrical optics fields (2) and (3) satisfy Maxwell's equations and the boundary conditions exactly, and therefore constitute the exact solution to the boundary-value problem. The geometry for the single DNG paraboloid of revolution is shown in Figure 1.

[8] With reference to Figure 1, the incident Poynting vector on the concave side of the paraboloid is

whereas the transmitted Poynting vector on the convex side is

[9] The case of a plane wave incident on the convex side of the separation surface may be treated similarly, resulting in a transmitted beam that converges onto the focus F, thus mapping the point at z = −∞ onto F. However, the structure of the fields in the vicinity of the focus would require a detailed investigation akin to that employed to describe a gaussian beam and would not result in an exact solution coincident with geometrical optics. Obviously, similar considerations would apply if the DNG medium were on the convex side of the separation surface.

3. Paraboloidal Radome

[10] The separation surfaces η = η_{1} and η = η_{2} < η_{1} are paraboloids of revolution with focus F and symmetry axis z. The space between the two surfaces is filled with DNG material. A plane wave carrying energy in the negative z direction and incident on the concave side of the structure is transmitted without reflection across η = η_{2} and appears in the DNG medium as a beam diverging from the focus F, that upon incidence on η = η_{1} is transmitted without reflection and appears as a plane wave on the convex side of the radome lens. The geometry of the structure is shown in Figure 2. The incident field is given by equation (2) with a change in the sign of the exponent (the phase now increases in the negative z direction). The field in the DNG medium is

[11] The field on the convex side of the structure is

[12] The geometrical optics fields (2), (11) and (12) satisfy both Maxwell's equations and the boundary conditions exactly, hence constitute the exact solution to the boundary value problem. The radome lens maps the point at z = +∞ onto the point z = −∞.

[13] The Poynting vector S^{i} on the concave side of the radome, S^{DNG} inside the radome layer, and S^{t} on the convex side of the radome are given by (9) and

[14] Analogous considerations apply to the case in which the energy propagation path is reversed; that is, the primary plane wave is incident on the convex side of the lens. For this latter case, the transient electric field magnitude is shown in Figure 3 for η_{1} = 3λ/2 and η_{2} = λ/2, λ being the wavelength; the ratio of the plane-wave transmitted to plane-wave incident Poynting vectors is (η_{1}/η_{2})^{2} = 9.

4. Conclusion

[15] A new canonical solution has been found for axial transmission of a plane electromagnetic wave across confocal and coaxial paraboloids of revolution that separate double-positive (DPS) regions of space from double-negative (DNG) metamaterial regions. All regions have the same intrinsic impedance.

[16] This boundary-value problem is remarkable because the exact solution is also the geometrical optics solution. The solution has been provided explicitly for a single paraboloidal surface and for a single-layer paraboloidal radome. However, it can be easily extended to multi-layer radomes.