Elliptic Cylindrical Invisibility Cloak, a Semianalytical Approach Using Mathieu Functions

Elliptic Cylindrical Invisibility Cloak, a Semianalytical Approach Using Mathieu Functions E. Cojocaru Department of Theoretical Physics, IFIN HH, Bucharest-Magurele MG-6, Romania∗ (Dated: today) Abstract An elliptic cylindrical wave expansion method by using Mathieu functions is developed to obtain the scattering field for a two-dimensional elliptic cylindrical invisibility cloak. The cloak material parameters are obtained from the spatial transformation approach. A near-ideal model of the invisibility cloak is set up to solve the boundary problem at the inner boundary in the cloak shell. The proposed design provides a more practical cloak geometry when compared to previous designs of elliptic cylindrical cloaks. PACS numbers: 41.20.Jb, 42.25.Fx, 42.25.Gy 0 arXiv:0808.1498v1 [physics.comp-ph] 11 Aug 2008 Recently, achieving the invisibility of objects has received increased attention. Designs based on spatial coordinate transformations have been proposed [1, 2, 3, 4, 5, 6, 7, 8]. The inhomogeneous and anisotropic cloaks obtained from the spatial transformation bend the incoming waves around the cloaked region, so that the fields after emerging from the cloak are the same as if the incident waves had passed through the free space. The cloaking princi- ple was demonstrated experimentally in the microwave regime [4]. Cloaks designs have been reported and investigated mostly for spherical and circular cylindrical geometries primarily due to the simplicity of analysis for structures that possess radial and axial symmetries. Recently, elliptic cylindrical cloaks have been reported based on non-orthogonal coordinate systems, the cloak being described by permittivity and permeability tensors with non-zero off-diagonal terms and verification being done numerically by full-wave finite-element simu- lations [9, 10]. In this Letter, we will study the scattering for an elliptic cylindrical cloak by using a semianalytical approach that is based on Mathieu functions. By taking the advantage of the elliptic cylindrical geometry of the structure, we focus our analysis on the two-dimensional (2D) elliptic cylindrical cloak, because the wave equation can be simplified in comparison with the three-dimensional (3D) case. As in the circular cylindrical case [8], we introduce a small perturbation into the ideal cloak to avoid extreme values (zero or infinity) of the material parameters at the cloak’s inner surface. First, let us look at the wave equation inside the elliptic cylindrical cloak. In terms of the rectangular coordinates (x, y, z), the elliptic cylindrical coordinates (u, v, z) are defined by the following relations [11, 12] x = f cosh u cos v, y = f sinh u sin v, z = z, (1) with 0 ≤u < ∞, 0 ≤v ≤2π, and f the semifocal length of the ellipse. The contours of constant u are confocal ellipses, and of constant v are confocal hyperbolas. The z axis coincides with the cylinder axis. Similarly to the circular cylindrical cloak [1, 8], a simple transformation (u′, v′, z′) with u′ = b −a b u + a, v′ = v, z′ = z (2) can compress space from the elliptic cylinder region 0 < u < b into the elliptic shell a < u′ < b, u′ = a being the inner confocal ellipse. Note that u, u′, a, and b are dimensionless 1 quantities. Let the position vector in the original system be written as ⃗r = ˆxx + ˆyy + ˆzz and in the transformed system as ⃗r ′ = ˆxx′ + ˆyy′ + ˆzz′. Following the notations in [13], the scale factors Qj(j = 1, 2, 3) of the transformation [Eq. (2)] are found to be Q1 ≡| ∂⃗r/∂u′ | | ∂⃗r ′/∂u′ | = (cosh 2u −cos 2v′)1/2 T(cosh 2u′ −cos 2v′)1/2, Q2 ≡| ∂⃗r/∂v′ | | ∂⃗r ′/∂v′ | = (cosh 2u −cos 2v′)1/2 (cosh 2u′ −cos 2v′)1/2, (3) Q3 ≡| ∂⃗r/∂z′ | | ∂⃗r ′/∂z′ | = 1, where T = (b −a)/b. Finally, the permittivity and permeability tensor components for the cloak shell can be given as ϵu′ ϵ0 = µu′ µ0 = T, ϵv′ ϵ0 = µv′ µ0 = 1 T , ϵz′ ϵ0 = µz′ µ0 = cosh 2u −cos 2v′ T(cosh 2u′ −cos 2v′), (4) where air is assumed for the ambient environment and the interior regions. In the following we consider the transverse-electric (TE) polarized electromagnetic field (i.e., the electri- cal field only exists in the z′ direction). One obtains the following general wave equation governing Ez′ field in the cloak’s elliptic cylindrical coordinates 2 f 2 (cosh 2u′ −cos 2v′) ( ∂ ∂u′  1 µv′ ∂Ez′ ∂u′  + ∂ ∂v′  1 µu′ ∂Ez′ ∂v′  ) + ϵz′  1 ϵ0µ0 ∂2Ez′ ∂z′2 −∂2Ez′ ∂t2  = 0. (5) An exp(−iωt) time dependence is assumed, where ω is the circular frequency. If we substitute Eq. (4) for ϵz′, µu′, and µv′ we find T ∂2Ez′ ∂u′2 + 1 T ∂2Ez′ ∂v′2 + f 2kt 2 2T (cosh 2u −cos 2v′) = 0, (6) where kt 2 = k0 2 −kz 2, k0 2 = ω2ϵ0µ0, and kz is the z-component of the wave vector (for 2D case, kz = 0). Equation (6) can be solved by a separation of variables Ez′ = S(v′)R(u′) and the introduction of a separation constant c, n d2 dv′2 + (c −2q cos 2v′) o S(v′) = 0, (7) ( T 2 d2 du′2 −  c −2q cosh 2u′ −a T  ) R(u′) = 0, (8) where q = f 2k2 t /4. Outside the cloak Eq. (7) is the same, whereas Eq. (8)

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