International Scholarly Research Network ISRN Optics Volume 2012, Article ID 536209, 7 pages doi:10.5402/2012/536209 Research Article FDTD Modeling of a Cloak with a Nondiagonal Permittivity Tensor Naoki Okada and James B. Cole Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan Correspondence should be addressed to Naoki Okada, [email protected] Received 13 February 2012; Accepted 27 March 2012 Academic Editors: A. Danner, M. Midrio, and A. Tervonen Copyright © 2012 N. Okada and J. B. Cole. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We demonstrate a finite-difference time-domain (FDTD) modeling of a cloak with a nondiagonal permittivity tensor. Numerical instability due to material anisotropies is avoided by mapping the eigenvalues of the material parameters to a dispersion model. Our approach is implemented for an elliptic-cylindrical cloak in two dimensions. Numerical simulations demonstrated the stable calculation and cloaking performance of the elliptic-cylindrical cloak. 1. Introduction method. The diagonal case can be stably calculated by mapping material parameters having values less than one to An optical cloak enables objects to be concealed from a dispersion model [31]. However, we found that mapping electromagnetic detection. Pendry et al. developed a method the nondiagonal elements to dispersion models causes the to design cloaks via coordinate transformations [1]. The computation to diverge. coordinate transformation is such that light is guided around In this paper, we analyze the numerical stability for a the cloak region. Material parameters (permittivity and cloak with a nondiagonal permittivity tensor and derive the permeability) can be obtained in the transformed coordinate FDTD formulation. We apply our method to simulate light system and put into Maxwell’s equations. This approach propagation in the vicinity of an elliptic-cylindrical cloak. enables one to design not only cloaks but also other To the best of authors’ knowledge, this is the first time that metamaterials that can manipulate light flow. For example, a cloak with a nondiagonal anisotropy has been calculated concentrators [2], rotation coatings [3], polarization con- using the FDTD method. trollers [4–6], waveguides [7–11], wave shape conversion [12], object illusions [13–15], and optical black holes [16, 17] have been designed. However, not many metamaterials 2. Numerical Stability for Nondiagonal have been realized in the optical region [18–25], because Permittivity Tensor material parameters given by coordinate transformations have complicated anisotropies. In the stability analysis, we confirm that the FDTD method Numerical simulations are useful to analyze complicated for a cloak with a diagonal permittivity tensor cannot directly metamaterial structures. In this paper, we present a finite- be extended to the nondiagonal case. Under a coordinate difference time-domain (FDTD) analysis of a cloak. The transformation for a cloak [39], material parameters can be FDTD method has gained popularity for several reasons: it expressed as is easy to implement, it works in the time domain, and its εij = μij =± ggij,(1) arbitrary shapes can be calculated [26–29]. FDTD modelings of cloaks with a diagonal (uniaxial) permittivity tensor have where εij is the relative permittivity, μij is the relative permea- been demonstrated [30–38], but a cloak with a nondiagonal bility, gij is the metric tensor, and g = det gij.Becauseεij, μij permittivity tensor has never been calculated by the FDTD are constructed from the symmetric metric tensor gij, they 2 ISRN Optics y b a x O ka kb (a) (b) Figure 1: Elliptic-cylindrical cloak. (a) Cartesian coordinates: a, b are inner and outer axes; ka, kb are the perpendicular axes. (b) Transformed coordinates. are symmetric. Consequently, εij, μij have real eigenvalues 3. FDTD Formulation of with orthogonal eigenvectors and are thus diagonalizable. the Elliptic-Cylindrical Cloak The eigenvalues, λ,ofεij, μij for an eigenvector V are defined by Two designs of elliptic-cylindrical cloaks have been pro- posed. One has diagonal εij and μij in orthonormal elliptic- ij ij εijV = μijV = λV. (2) cylindrical coordinates [45, 46], and in the other ε and μ are nondiagonal in Cartesian coordinates [47, 48]. We derive a FDTD formulation for the latter in the transverse magnetic = The phase velocity of light in a material is given by c c0/λ (TM) polarization. (c0 = vacuum light speed), and the Courant-Friedrichs-Lewy (CFL) stability limit becomes 3.1. Diagonalization. Figure 1 shows an elliptic-cylindrical cloak in Figure 1(a) Cartesian coordinates and Figure 1(b) λh Δt ≤ √ ,(3)transformed coordinates. The inner axis a, the outer axis c0 d b, and the perpendicular axes ka and kb are depicted. The elliptic-cylindrical cloak is horizontal when k>1, and Δ = where t is the time step, h is the grid spacing, and d vertical when k<1. In the cloak region, ka ≤ x2 + k2 y2 ≤ 1, 2, and 3 dimensions. Since the FDTD stability depends ij ij kb, the material parameters are expressed by on the eigenvalues of ε and μ , to analyze nondiagonal ⎡ ⎤ cases, we must first find the eigenvalues and diagonalize ⎢εxx εxy 0 ⎥ εij and μij. After the diagonalization, the FDTD method ⎢ ⎥ ij = ij = ⎢ε ε 0 ⎥ for diagonal cases [31–38] can be applied. For diagonal εij ε μ ⎣ xy yy ⎦,(4) ij and μ , elements having values less than one are replaced 00εzz by dispersive quantities to avoid violating the causality and numerical stability [40–44]. where In summary, the FDTD modeling for nondiagonal εij and r k2a2R2 − 2kar3 ij ε = + x2, μ requires three steps: xx r − ka (r − ka)r5 (1) find the eigenvalues and eigenvectors and diagonalize k2a2R2 − ka 1+k2 r3 εxy = xy, (5) the material parameters, (r − ka)r5 2 2 2 − 3 3 (2) map the eigenvalues having values less than one to a r k a R 2k ar 2 εyy = + y , dispersion model, r − ka (r − ka)r5 b 2 r − ka (3) solve Maxwell’s equations using the dispersive FDTD ε = , (6) method. zz b − a r ISRN Optics 3 where r = x2 + k2 y2 and R = x2 + k4 y2.From(4)to(6) D-andB-update equations are obtained using Yee algorithm we can obtain three eigenvalues [26–29] as follows: Δ n+1 = n t n+1/2 α − 1 1 Dx Dx + dyHz , λ1 = , λ2 = , λ3 = εzz,(7) h α +1 λ1 (14) Δt Dn+1 = Dn − d Hn+1/2, where y y h x z Δ 4r5(r − ka) n+3/2 = n+1/2 − t n+1 − n+1 α = 1+ . (8) Bz Bz dxEy dyEx , (15) k2a2R2 x2 + y2 h = Δ → n = where we simply write Dx(t n t) Dx (n integer) and ij Since ε is symmetric, it is diagonalized by the eigenvalue dx, dy are the spatial difference operators defined by matrix Λ and its orthogonal matrix as follows: P h h dx f x, y = f x + , y − f x − , y , ij = Λ T 2 2 ε P P ,(9) (16) h h d f x, y = f x, y + − f x, y − . where y 2 2 ⎡ ⎤ To find the E-update equations, we consider the relation ⎢λ1 00⎥ ⎢ ⎥ Λ = ⎢ ⎥ D = ε εijE, (17) ⎣ 0 λ2 0 ⎦, (10) 0 00λ3 where ε0 is the vacuum permittivity. From (9), we obtain ⎡ ⎤ −1 ij −1 T − ε0E = ε D = PΛ P D. (18) ⎢ εxy λ2 εyy ⎥ ⎢ 0⎥ ⎢ ⎥ ⎢ β β ⎥ Substituting (10)in(18) and multiplying λ1λ2 by both sides, = ⎢ − ⎥ P ⎢ εyy λ2 ε ⎥, (11) we obtain ⎢ xy 0⎥ ⎣ ⎦ = 2 2 − β β ε0λ1λ2Ex λ1t2 + λ2t1 Dx + t1t2(λ1 λ2)Dy, (19) 001 = 2 2 − ε0λ1λ2Ey λ1t1 + λ2t2 Dy + t1t2(λ1 λ2)Dx, (20) = 2 − 2 where β εxy +(λ2 εyy) . where t1 = εxy/β and t2 = (λ2 − εyy)/β. Substituting the Drude model for λ1 as shown in (12) and using the inverse 3.2. Mapping Eigenvalues to a Dispersion Model. From (7) Fourier transformation rule, −ω2 → ∂2/∂t2,(19)becomes and (8), λ1 and λ3 have values less than one in the cloak 2 region (ka ≤ r ≤ kb). Thus, λ1, λ3 must be replaced by ∂ 2 ε0λ2 ε∞1 + ωp1 Ex dispersive quantities by using (for example) the Drude model ∂t2 2 ω2 2 2 ∂ 2 2 pi = ε∞1t + λ2t + ω t Dx (21) λ = ε∞ − , (i = 1, 3), (12) 2 1 2 p1 2 i i ω2 − jωγ ∂t i 2 ∂ 2 where ω is the angular frequency, ε∞i is the infinite-frequency + t t (ε∞ − λ ) + ω D . 1 2 1 2 ∂t2 p1 y permittivity, ωpi is the plasma frequency, and γi is the collision frequency. For simplicity, we consider the lossless For the discretization, we use the central difference approxi- = = case, γi 0. Then the plasma frequencies are given by ωpi mation and the central average operator, ω ε∞ − λ ,whereε∞ = max(1, λ ). i i i i ∂2 En+1 − 2En + En−1 En = x x x , ∂t2 x Δt2 3.3. FDTD Discretization. Using the diagonalized material (22) − parameters and eigenvalues mapped to the Drude model, we En+1 +2En + En 1 En = x x x . derive an FDTD formulation to solve Maxwell’s equations, x 4 The central average operator improves the stability and ∂D =∇×H, accuracy [40, 49, 50]. Similarly, Dx and Dy are discretized, ∂t (13) and we obtain the Ex-update equation ∂B − − =∇×E, n+1 =− n−1 a1 n 1 ∂t Ex Ex +2 + Ex + + a1 ε0λ2a1 where D is the electric flux density, H is the magnetic field, − − × b+ Dn+1 + Dn 1 − 2b Dn (23) B is the magnetic flux density, and E is the electric field.