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ISRN Optics
Volume 2012 (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

Received 13 February 2012; Accepted 27 March 2012

Academic Editors: A. Danner, M. Midrio, and A. Tervonen

Copyright © 2012 Naoki Okada and James 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.

Abstract

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

An optical cloak enables objects to be concealed from electromagnetic detection. Pendry et al. developed a method to design cloaks via coordinate transformations [1]. The coordinate transformation is such that light is guided around the cloak region. Material parameters (permittivity and permeability) can be obtained in the transformed coordinate system and put into Maxwell’s equations. This approach enables one to design not only cloaks but also other metamaterials that can manipulate light flow. For example, concentrators [2], rotation coatings [3], polarization controllers [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 have been realized in the optical region [18–25], because material parameters given by coordinate transformations have complicated anisotropies.

Numerical simulations are useful to analyze complicated metamaterial structures. In this paper, we present a finite-difference time-domain (FDTD) analysis of a cloak. The FDTD method has gained popularity for several reasons: it is easy to implement, it works in the time domain, and its arbitrary shapes can be calculated [26–29]. FDTD modelings of cloaks with a diagonal (uniaxial) permittivity tensor have been demonstrated [30–38], but a cloak with a nondiagonal permittivity tensor has never been calculated by the FDTD method. The diagonal case can be stably calculated by mapping material parameters having values less than one to a dispersion model [31]. However, we found that mapping the nondiagonal elements to dispersion models causes the computation to diverge.

In this paper, we analyze the numerical stability for a cloak with a nondiagonal permittivity tensor and derive the FDTD formulation. We apply our method to simulate light propagation in the vicinity of an elliptic-cylindrical cloak. To the best of authors’ knowledge, this is the first time that a cloak with a nondiagonal anisotropy has been calculated using the FDTD method.

2. Numerical Stability for Nondiagonal Permittivity Tensor

In the stability analysis, we confirm that the FDTD method for a cloak with a diagonal permittivity tensor cannot directly be extended to the nondiagonal case. Under a coordinate transformation for a cloak [39], material parameters can be expressed as 𝜀 𝑖 𝑗 = 𝜇 𝑖 𝑗 √ = ± 𝑔 𝑔 𝑖 𝑗 , ( 1 ) where 𝜀 𝑖 𝑗 is the relative permittivity, 𝜇 𝑖 𝑗 is the relative permeability, 𝑔 𝑖 𝑗 is the metric tensor, and 𝑔 = d e t 𝑔 𝑖 𝑗 . Because 𝜀 𝑖 𝑗 , 𝜇 𝑖 𝑗 are constructed from the symmetric metric tensor 𝑔 𝑖 𝑗 , they are symmetric. Consequently, 𝜀 𝑖 𝑗 , 𝜇 𝑖 𝑗 have real eigenvalues with orthogonal eigenvectors and are thus diagonalizable. The eigenvalues, 𝜆 , of 𝜀 𝑖 𝑗 , 𝜇 𝑖 𝑗 for an eigenvector 𝐕 are defined by 𝜀 𝑖 𝑗 𝐕 = 𝜇 𝑖 𝑗 𝐕 = 𝜆 𝐕 . ( 2 ) The phase velocity of light in a material is given by 𝑐 = 𝑐 0 / 𝜆 ( 𝑐 0 = vacuum light speed), and the Courant-Friedrichs-Lewy (CFL) stability limit becomes Δ 𝑡 ≤ 𝜆 ℎ 𝑐 0 √ 𝑑 , ( 3 ) where Δ 𝑡 is the time step, ℎ is the grid spacing, and 𝑑 = 1, 2, and 3 dimensions. Since the FDTD stability depends on the eigenvalues of 𝜀 𝑖 𝑗 and 𝜇 𝑖 𝑗 , to analyze nondiagonal cases, we must first find the eigenvalues and diagonalize 𝜀 𝑖 𝑗 and 𝜇 𝑖 𝑗 . After the diagonalization, the FDTD method for diagonal cases [31–38] can be applied. For diagonal 𝜀 𝑖 𝑗 and 𝜇 𝑖 𝑗 , elements having values less than one are replaced by dispersive quantities to avoid violating the causality and numerical stability [40–44].

In summary, the FDTD modeling for nondiagonal 𝜀 𝑖 𝑗 and 𝜇 𝑖 𝑗 requires three steps:(1)find the eigenvalues and eigenvectors and diagonalize the material parameters,(2)map the eigenvalues having values less than one to a dispersion model,(3)solve Maxwell’s equations using the dispersive FDTD method.

3. FDTD Formulation of the Elliptic-Cylindrical Cloak

Two designs of elliptic-cylindrical cloaks have been proposed. One has diagonal 𝜀 𝑖 𝑗 and 𝜇 𝑖 𝑗 in orthonormal elliptic-cylindrical coordinates [45, 46], and in the other 𝜀 𝑖 𝑗 and 𝜇 𝑖 𝑗 are nondiagonal in Cartesian coordinates [47, 48]. We derive a FDTD formulation for the latter in the transverse magnetic (TM) polarization.

3.1. Diagonalization

Figure 1 shows an elliptic-cylindrical cloak in Figure 1(a) Cartesian coordinates and Figure 1(b) transformed coordinates. The inner axis 𝑎 , the outer axis 𝑏 , and the perpendicular axes 𝑘 𝑎 and 𝑘 𝑏 are depicted. The elliptic-cylindrical cloak is horizontal when 𝑘 > 1 , and vertical when 𝑘 < 1 . In the cloak region, √ 𝑘 𝑎 ≤ 𝑥 2 + 𝑘 2 𝑦 2 ≤ 𝑘 𝑏 , the material parameters are expressed by 𝜀 𝑖 𝑗 = 𝜇 𝑖 𝑗 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜀 𝑥 𝑥 𝜀 𝑥 𝑦 0 𝜀 𝑥 𝑦 𝜀 𝑦 𝑦 0 0 0 𝜀 𝑧 𝑧 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , ( 4 ) where 𝜀 𝑥 𝑥 = 𝑟 + 𝑘 𝑟 − 𝑘 𝑎 2 𝑎 2 𝑅 2 − 2 𝑘 𝑎 𝑟 3 ( 𝑟 − 𝑘 𝑎 ) 𝑟 5 𝑥 2 , 𝜀 𝑥 𝑦 = 𝑘 2 𝑎 2 𝑅 2 − 𝑘 𝑎 1 + 𝑘 2 𝑟 3 ( 𝑟 − 𝑘 𝑎 ) 𝑟 5 𝜀 𝑥 𝑦 , 𝑦 𝑦 = 𝑟 + 𝑘 𝑟 − 𝑘 𝑎 2 𝑎 2 𝑅 2 − 2 𝑘 3 𝑎 𝑟 3 ( 𝑟 − 𝑘 𝑎 ) 𝑟 5 𝑦 2 , 𝜀 ( 5 ) 𝑧 𝑧 = 𝑏 𝑏 − 𝑎 2 𝑟 − 𝑘 𝑎 𝑟 , ( 6 ) where √ 𝑟 = 𝑥 2 + 𝑘 2 𝑦 2 and √ 𝑅 = 𝑥 2 + 𝑘 4 𝑦 2 . From (4) to (6) we can obtain three eigenvalues 𝜆 1 = 𝛼 − 1 𝛼 + 1 , 𝜆 2 = 1 𝜆 1 , 𝜆 3 = 𝜀 𝑧 𝑧 , ( 7 ) where 𝛼 = 1 + 4 𝑟 5 ( 𝑟 − 𝑘 𝑎 ) 𝑘 2 𝑎 2 𝑅 2 𝑥 2 + 𝑦 2 . ( 8 ) Since 𝜀 𝑖 𝑗 is symmetric, it is diagonalized by the eigenvalue matrix Λ and its orthogonal matrix 𝑃 as follows: 𝜀 𝑖 𝑗 = 𝑃 Λ 𝑃 𝑇 , ( 9 ) where ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜆 Λ = 1 0 0 0 𝜆 2 0 0 0 𝜆 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜀 , ( 1 0 ) 𝑃 = 𝑥 𝑦 𝛽 𝜆 2 − 𝜀 𝑦 𝑦 𝛽 0 𝜀 𝑦 𝑦 − 𝜆 2 𝛽 𝜀 𝑥 𝑦 𝛽 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0 0 1 , ( 1 1 ) where 𝛽 = 𝜀 2 𝑥 𝑦 + ( 𝜆 2 − 𝜀 𝑦 𝑦 ) 2 .

fig1
Figure 1: Elliptic-cylindrical cloak. (a) Cartesian coordinates: 𝑎 , 𝑏 are inner and outer axes; 𝑘 𝑎 , 𝑘 𝑏 are the perpendicular axes. (b) Transformed coordinates.
3.2. Mapping Eigenvalues to a Dispersion Model

From (7) and (8), 𝜆 1 and 𝜆 3 have values less than one in the cloak region ( 𝑘 𝑎 ≤ 𝑟 ≤ 𝑘 𝑏 ). Thus, 𝜆 1 , 𝜆 3 must be replaced by dispersive quantities by using (for example) the Drude model 𝜆 𝑖 = 𝜀 ∞ 𝑖 − 𝜔 2 𝑝 𝑖 𝜔 2 − 𝑗 𝜔 𝛾 𝑖 , ( 𝑖 = 1 , 3 ) , ( 1 2 ) where 𝜔 is the angular frequency, 𝜀 ∞ 𝑖 is the infinite-frequency permittivity, 𝜔 𝑝 𝑖 is the plasma frequency, and 𝛾 𝑖 is the collision frequency. For simplicity, we consider the lossless case, 𝛾 𝑖 = 0 . Then the plasma frequencies are given by 𝜔 𝑝 𝑖 √ = 𝜔 𝜀 ∞ 𝑖 − 𝜆 𝑖 , where 𝜀 ∞ 𝑖 = m a x ( 1 , 𝜆 𝑖 ) .

3.3. FDTD Discretization

Using the diagonalized material parameters and eigenvalues mapped to the Drude model, we derive an FDTD formulation to solve Maxwell’s equations, 𝜕 𝐃 − 𝜕 𝑡 = ∇ × 𝐇 , 𝜕 𝐁 𝜕 𝑡 = ∇ × 𝐄 , ( 1 3 ) where 𝐃 is the electric flux density, 𝐇 is the magnetic field, 𝐁 is the magnetic flux density, and 𝐄 is the electric field. In the TM polarization, electromagnetic fields reduce to three nonzero components 𝐸 𝑥 , 𝐸 𝑦 , and 𝐻 𝑧 ( 𝐷 𝑥 , 𝐷 𝑦 , and 𝐵 𝑧 ). The 𝐃 - and 𝐁 -update equations are obtained using Yee algorithm [26–29] as follows: 𝐷 𝑥 𝑛 + 1 = 𝐷 𝑛 𝑥 + Δ 𝑡 ℎ 𝑑 𝑦 𝐻 𝑧 𝑛 + 1 / 2 , 𝐷 𝑦 𝑛 + 1 = 𝐷 𝑛 𝑦 − Δ 𝑡 ℎ 𝑑 𝑥 𝐻 𝑧 𝑛 + 1 / 2 , 𝐵 ( 1 4 ) 𝑧 𝑛 + 3 / 2 = 𝐵 𝑧 𝑛 + 1 / 2 − Δ 𝑡 ℎ 𝑑 𝑥 𝐸 𝑦 𝑛 + 1 − 𝑑 𝑦 𝐸 𝑥 𝑛 + 1 , ( 1 5 ) where we simply write 𝐷 𝑥 ( 𝑡 = 𝑛 Δ 𝑡 ) → 𝐷 𝑛 𝑥 ( 𝑛 = integer) and 𝑑 𝑥 , 𝑑 𝑦 are the spatial difference operators defined by 𝑑 𝑥 ℎ 𝑓 ( 𝑥 , 𝑦 ) = 𝑓 𝑥 + 2 ℎ , 𝑦 − 𝑓 𝑥 − 2 , 𝑑 , 𝑦 𝑦 ℎ 𝑓 ( 𝑥 , 𝑦 ) = 𝑓 𝑥 , 𝑦 + 2 ℎ − 𝑓 𝑥 , 𝑦 − 2 . ( 1 6 )

To find the 𝐄 -update equations, we consider the relation 𝐃 = 𝜀 0 𝜀 𝑖 𝑗 𝐄 , ( 1 7 ) where 𝜀 0 is the vacuum permittivity. From (9), we obtain 𝜀 0 𝜀 𝐄 = 𝑖 𝑗 − 1 𝐃 = 𝑃 Λ − 1 𝑃 𝑇 𝐃 . ( 1 8 ) Substituting (10) in (18) and multiplying 𝜆 1 𝜆 2 by both sides, we obtain 𝜀 0 𝜆 1 𝜆 2 𝐸 𝑥 = 𝜆 1 𝑡 2 2 + 𝜆 2 𝑡 2 1 𝐷 𝑥 + 𝑡 1 𝑡 2 𝜆 1 − 𝜆 2 𝐷 𝑦 , ( 1 9 ) 𝜀 0 𝜆 1 𝜆 2 𝐸 𝑦 = 𝜆 1 𝑡 2 1 + 𝜆 2 𝑡 2 2 𝐷 𝑦 + 𝑡 1 𝑡 2 𝜆 1 − 𝜆 2 𝐷 𝑥 , ( 2 0 ) where 𝑡 1 = 𝜀 𝑥 𝑦 / 𝛽 and 𝑡 2 = ( 𝜆 2 − 𝜀 𝑦 𝑦 ) / 𝛽 . Substituting the Drude model for 𝜆 1 as shown in (12) and using the inverse Fourier transformation rule, − 𝜔 2 → 𝜕 2 / 𝜕 𝑡 2 , (19) becomes 𝜀 0 𝜆 2 𝜀 ∞ 1 𝜕 2 𝜕 𝑡 2 + 𝜔 2 𝑝 1 𝐸 𝑥 = 𝜀 ∞ 1 𝑡 2 2 + 𝜆 2 𝑡 2 1 𝜕 2 𝜕 𝑡 2 + 𝜔 2 𝑝 1 𝑡 2 2 𝐷 𝑥 + 𝑡 1 𝑡 2 𝜀 ∞ 1 − 𝜆 2 𝜕 2 𝜕 𝑡 2 + 𝜔 2 𝑝 1 𝐷 𝑦 . ( 2 1 ) For the discretization, we use the central difference approximation and the central average operator, 𝜕 2 𝜕 𝑡 2 𝐸 𝑛 𝑥 = 𝐸 𝑥 𝑛 + 1 − 2 𝐸 𝑛 𝑥 + 𝐸 𝑥 𝑛 − 1 Δ 𝑡 2 , 𝐸 𝑛 𝑥 = 𝐸 𝑥 𝑛 + 1 + 2 𝐸 𝑛 𝑥 + 𝐸 𝑥 𝑛 − 1 4 . ( 2 2 ) The central average operator improves the stability and accuracy [40, 49, 50]. Similarly, 𝐷 𝑥 and 𝐷 𝑦 are discretized, and we obtain the 𝐸 𝑥 -update equation 𝐸 𝑥 𝑛 + 1 = − 𝐸 𝑥 𝑛 − 1 𝑎 + 2 − 1 𝑎 + 1 𝐸 𝑛 𝑥 + 1 𝜀 0 𝜆 2 𝑎 + 1 × 𝑏 + 1 𝐷 𝑥 𝑛 + 1 + 𝐷 𝑥 𝑛 − 1 − 2 𝑏 − 1 𝐷 𝑛 𝑥 + 𝑐 + 1 𝐷 𝑦 𝑛 + 1 + 𝐷 𝑦 𝑛 − 1 − 2 𝑐 − 1 𝐷 𝑛 𝑦 , ( 2 3 ) where 𝐷 𝑦 𝑛 + 1 , 𝐷 𝑛 𝑦 , 𝐷 𝑦 𝑛 − 1 must be spatially interpolated due to the staggered Yee cell [31], and 𝑎 ± 𝑖 = 𝜀 ∞ 𝑖 Δ 𝑡 2 ± 𝜔 2 𝑝 𝑖 4 , 𝑏 ± 𝑖 = 𝜀 ∞ 𝑖 𝑡 2 2 + 𝜆 2 𝑡 2 1 Δ 𝑡 2 ± 𝜔 2 𝑝 𝑖 𝑡 2 2 4 , 𝑐 ± 𝑖 = 𝑡 1 𝑡 2 𝜀 ∞ 𝑖 − 𝜆 2 Δ 𝑡 2 ± 𝜔 2 𝑝 𝑖 𝑡 2 2 4 . ( 2 4 ) Similarly, the 𝐸 𝑦 -update equation is obtained by exchanging 𝑡 1 ↔ 𝑡 2 , 𝐸 𝑥 ↔ 𝐸 𝑦 , and 𝐷 𝑥 ↔ 𝐷 𝑦 .

To find the 𝐻 𝑧 -update equation, we consider the relation 𝐵 𝑧 = 𝜇 0 𝜀 𝑧 𝑧 𝐻 𝑧 = 𝜇 0 𝜀 ∞ 3 − 𝜔 2 𝑝 3 𝜔 2 𝐻 𝑧 . ( 2 5 ) Analogously to the 𝐸 𝑥 field, the 𝐻 𝑧 -update equation can be obtained in the form 𝐻 𝑧 𝑛 + 3 / 2 = − 𝐻 𝑧 𝑛 − 1 / 2 𝑎 + 2 − 3 𝑎 + 3 𝐻 𝑧 𝑛 + 1 / 2 + 𝐵 𝑧 𝑛 + 3 / 2 − 2 𝐵 𝑧 𝑛 + 1 / 2 + 𝐵 𝑧 𝑛 − 1 / 2 𝜇 0 Δ 𝑡 2 𝑎 + 3 , ( 2 6 ) where 𝜇 0 is the vacuum permeability.

In summary, the electromagnetic fields are iteratively updated in the following sequence:(1)update the components of 𝐃 𝑛 + 1 according to (14),(2)update the components of 𝐄 𝑛 + 1 according to the sample given in (23),(3)update the components of 𝐁 𝑛 + 3 / 2 according to (15),(4)update the components of 𝐇 𝑛 + 3 / 2 according to (26).

4. Simulation of the Elliptic-Cylindrical Cloak

We calculate electromagnetic propagation for the elliptic-cylindrical cloak using the FDTD formulation shown in Section 3. Figure 2 shows the simulation setup: the computational domain is terminated with a perfectly matched layer in the 𝑥 -direction, and a periodic boundary condition in the 𝑦 -direction [29]; the inside of the cloak is covered with a perfect electric conductor (PEC); a plane wave source of wavelength 𝜆 0 = 750 nm (400 THz) is in the TM polarization; the grid spacing is ℎ = 10 nm ( 𝜆 0 / ℎ = 75); and the time step is given by the CFL limit, Δ 𝑡 = ℎ / ( 𝑐 0 √ 2 ) . Simulation parameters are listed in Table 1.

tab1
Table 1: Simulation parameters for the elliptic-cylindrical cloak.
536209.fig.002
Figure 2: Simulation setup of the elliptic-cylindrical cloak for incident TM polarization. Inside of the cloak is covered with a perfect electric conductor (PEC). The gray elliptic region represents the cloak ( ℎ : grid spacing).

Figure 3 shows the FDTD results for the elliptic-cylindrical cloak at the steady state (50 wave periods). Figure 3(a) shows calculated 𝐻 𝑧 field distributions using ℎ = 10 nm. The wave propagates without significant disturbance around the cloak, and the calculation is stable. The small ripples on phase planes are purely numerical errors and can be made to vanish by reducing the grid spacing. This can be confirmed by calculating the radar cross section (RCS) [29, 51]. In two dimensions, the RCS is defined by 𝜎 ( 𝜙 ) = l i m 𝑟 → ∞ | | 𝐸 2 𝜋 𝑟 𝑠 | | ( 𝜙 ) 2 | | 𝐸 0 | | 2 , ( 2 7 ) where 𝜙 is the scattering angle, | 𝐸 𝑠 ( 𝜙 ) | 2 is the scattered power in far field, and | 𝐸 0 | 2 is the incident power. If there is no significant disturbance by the object, 𝜎 approaches zero. Figure 3(b) shows normalized RCSs on dB scale, 𝜎 / 𝜆 0 , scattered by a PEC (without cloak) and cloak using different grid spacings, ℎ = 20, 10, and 5 nm. The PEC or cloak using a coarse grid spacing scatters strongly, but the RCS of the cloak rapidly decreases as the grid spacing is reduced.

fig3
Figure 3: FDTD results for the elliptic-cylindrical cloak. (a) 𝐻 𝑧 field distributions using ℎ = 10 nm (black lines are inside and outside shells of the cloak). (b) Radar cross section (RCS) scattered by a PEC (without cloak) and cloak using different grid spacings, ℎ = 20, 10, and 5 nm (radial coordinate is dB).

Finally, we examine the cloaking performance of the elliptic-cylindrical cloak using the Drude model. Simulation parameters are the same as Table 1 and the cloak is optimized to a wavelength of 750 nm. In the wavelength band, 600 nm–900 nm, we calculate the total cross section (TCS) defined by 𝜎 𝑡 = 𝜎 ( 𝜙 ) 𝑑 𝜙 . ( 2 8 ) Figure 4(a) shows the calculated TCS spectrum. The TCS rapidly increases with wavelength shifts off the optimal. Figure 4(b) shows the RCS for several wavelengths, A: 730 nm, B: 750 nm, and C: 830 nm (normalized to the RCS for 750 nm). For wavelengths A and C, the scattering is much stronger than the optimal wavelength B.

fig4
Figure 4: Cloaking performance of the elliptic-cylindrical cloak using the Drude model. (a) Total cross section (TCS) spectrum. (b) Sampled RCSs of incident wavelengths, A: 730 nm, B: 750 nm, and C: 830 nm.

5. Conclusion

We describe a stable FDTD modeling procedure for a cloak with a nondiagonal permittivity tensor. When the eigenvalues of the material parameters are less than one, they must be mapped to a dispersion model in order to maintain numerical stability. We implement our method for an elliptic-cylindrical cloak in the TM mode. Numerical calculations demonstrated stable results and the cloaking performance.

Acknowledgment

The authors deeply appreciate the financial support of Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows.

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