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ISRN Optics
Volume 2012 (2012), Article ID 536209, 7 pages
http://dx.doi.org/10.5402/2012/536209
Research Article

FDTD Modeling of a Cloak with a Nondiagonal Permittivity Tensor

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 [46], waveguides [711], wave shape conversion [12], object illusions [1315], and optical black holes [16, 17] have been designed. However, not many metamaterials have been realized in the optical region [1825], 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 [2629]. FDTD modelings of cloaks with a diagonal (uniaxial) permittivity tensor have been demonstrated [3038], 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 𝜀𝑖𝑗=𝜇𝑖𝑗=±𝑔𝑔𝑖𝑗,(1) where 𝜀𝑖𝑗 is the relative permittivity, 𝜇𝑖𝑗 is the relative permeability, 𝑔𝑖𝑗 is the metric tensor, and 𝑔=det𝑔𝑖𝑗. Because 𝜀𝑖𝑗,𝜇𝑖𝑗 are constructed from the symmetric metric tensor 𝑔𝑖𝑗, they are symmetric. Consequently, 𝜀𝑖𝑗,𝜇𝑖𝑗 have real eigenvalues with orthogonal eigenvectors and are thus diagonalizable. The eigenvalues, 𝜆, of 𝜀𝑖𝑗,𝜇𝑖𝑗 for an eigenvector 𝐕 are defined by𝜀𝑖𝑗𝐕=𝜇𝑖𝑗𝐕=𝜆𝐕.(2) The phase velocity of light in a material is given by 𝑐=𝑐0/𝜆 (𝑐0= vacuum light speed), and the Courant-Friedrichs-Lewy (CFL) stability limit becomesΔ𝑡𝜆𝑐0𝑑,(3) where Δ𝑡 is the time step, is the grid spacing, and 𝑑= 1, 2, and 3 dimensions. Since the FDTD stability depends on the eigenvalues of 𝜀𝑖𝑗 and 𝜇𝑖𝑗, to analyze nondiagonal cases, we must first find the eigenvalues and diagonalize 𝜀𝑖𝑗 and 𝜇𝑖𝑗. After the diagonalization, the FDTD method for diagonal cases [3138] can be applied. For diagonal 𝜀𝑖𝑗 and 𝜇𝑖𝑗, elements having values less than one are replaced by dispersive quantities to avoid violating the causality and numerical stability [4044].

In summary, the FDTD modeling for nondiagonal 𝜀𝑖𝑗 and 𝜇𝑖𝑗 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 𝜀𝑖𝑗 and 𝜇𝑖𝑗 in orthonormal elliptic-cylindrical coordinates [45, 46], and in the other 𝜀𝑖𝑗 and 𝜇𝑖𝑗 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 𝑘𝑎 and 𝑘𝑏 are depicted. The elliptic-cylindrical cloak is horizontal when 𝑘>1, and vertical when 𝑘<1. In the cloak region, 𝑘𝑎𝑥2+𝑘2𝑦2𝑘𝑏, the material parameters are expressed by𝜀𝑖𝑗=𝜇𝑖𝑗=𝜀𝑥𝑥𝜀𝑥𝑦0𝜀𝑥𝑦𝜀𝑦𝑦000𝜀𝑧𝑧,(4) where 𝜀𝑥𝑥=𝑟+𝑘𝑟𝑘𝑎2𝑎2𝑅22𝑘𝑎𝑟3(𝑟𝑘𝑎)𝑟5𝑥2,𝜀𝑥𝑦=𝑘2𝑎2𝑅2𝑘𝑎1+𝑘2𝑟3(𝑟𝑘𝑎)𝑟5𝜀𝑥𝑦,𝑦𝑦=𝑟+𝑘𝑟𝑘𝑎2𝑎2𝑅22𝑘3𝑎𝑟3(𝑟𝑘𝑎)𝑟5𝑦2,𝜀(5)𝑧𝑧=𝑏𝑏𝑎2𝑟𝑘𝑎𝑟,(6) where 𝑟=𝑥2+𝑘2𝑦2 and 𝑅=𝑥2+𝑘4𝑦2. From (4) to (6) we can obtain three eigenvalues 𝜆1=𝛼1𝛼+1,𝜆2=1𝜆1,𝜆3=𝜀𝑧𝑧,(7) where 𝛼=1+4𝑟5(𝑟𝑘𝑎)𝑘2𝑎2𝑅2𝑥2+𝑦2.(8) Since 𝜀𝑖𝑗 is symmetric, it is diagonalized by the eigenvalue matrix Λ and its orthogonal matrix 𝑃 as follows: 𝜀𝑖𝑗=𝑃Λ𝑃𝑇,(9) where 𝜆Λ=1000𝜆2000𝜆3𝜀,(10)𝑃=𝑥𝑦𝛽𝜆2𝜀𝑦𝑦𝛽0𝜀𝑦𝑦𝜆2𝛽𝜀𝑥𝑦𝛽0001,(11) where 𝛽=𝜀2𝑥𝑦+(𝜆2𝜀𝑦𝑦)2.

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

From (7) and (8), 𝜆1 and 𝜆3 have values less than one in the cloak region (𝑘𝑎𝑟𝑘𝑏). Thus, 𝜆1,𝜆3 must be replaced by dispersive quantities by using (for example) the Drude model 𝜆𝑖=𝜀𝑖𝜔2𝑝𝑖𝜔2𝑗𝜔𝛾𝑖,(𝑖=1,3),(12) where 𝜔 is the angular frequency, 𝜀𝑖 is the infinite-frequency permittivity, 𝜔𝑝𝑖 is the plasma frequency, and 𝛾𝑖 is the collision frequency. For simplicity, we consider the lossless case, 𝛾𝑖=0. Then the plasma frequencies are given by 𝜔𝑝𝑖=𝜔𝜀𝑖𝜆𝑖, where 𝜀𝑖=max(1,𝜆𝑖).

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,(13) 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 𝐸𝑥, 𝐸𝑦, and 𝐻𝑧 (𝐷𝑥, 𝐷𝑦, and 𝐵𝑧). The 𝐃- and 𝐁-update equations are obtained using Yee algorithm [2629] as follows: 𝐷𝑥𝑛+1=𝐷𝑛𝑥+Δ𝑡𝑑𝑦𝐻𝑧𝑛+1/2,𝐷𝑦𝑛+1=𝐷𝑛𝑦Δ𝑡𝑑𝑥𝐻𝑧𝑛+1/2,𝐵(14)𝑧𝑛+3/2=𝐵𝑧𝑛+1/2Δ𝑡𝑑𝑥𝐸𝑦𝑛+1𝑑𝑦𝐸𝑥𝑛+1,(15) where we simply write 𝐷𝑥(𝑡=𝑛Δ𝑡)𝐷𝑛𝑥 (𝑛= integer) and 𝑑𝑥, 𝑑𝑦 are the spatial difference operators defined by 𝑑𝑥𝑓(𝑥,𝑦)=𝑓𝑥+2,𝑦𝑓𝑥2,𝑑,𝑦𝑦𝑓(𝑥,𝑦)=𝑓𝑥,𝑦+2𝑓𝑥,𝑦2.(16)

To find the 𝐄-update equations, we consider the relation 𝐃=𝜀0𝜀𝑖𝑗𝐄,(17) where 𝜀0 is the vacuum permittivity. From (9), we obtain 𝜀0𝜀𝐄=𝑖𝑗1𝐃=𝑃Λ1𝑃𝑇𝐃.(18) Substituting (10) in (18) and multiplying 𝜆1𝜆2 by both sides, we obtain 𝜀0𝜆1𝜆2𝐸𝑥=𝜆1𝑡22+𝜆2𝑡21𝐷𝑥+𝑡1𝑡2𝜆1𝜆2𝐷𝑦,(19)𝜀0𝜆1𝜆2𝐸𝑦=𝜆1𝑡21+𝜆2𝑡22𝐷𝑦+𝑡1𝑡2𝜆1𝜆2𝐷𝑥,(20) where 𝑡1=𝜀𝑥𝑦/𝛽 and 𝑡2=(𝜆2𝜀𝑦𝑦)/𝛽. Substituting the Drude model for 𝜆1 as shown in (12) and using the inverse Fourier transformation rule, 𝜔2𝜕2/𝜕𝑡2, (19) becomes 𝜀0𝜆2𝜀1𝜕2𝜕𝑡2+𝜔2𝑝1𝐸𝑥=𝜀1𝑡22+𝜆2𝑡21𝜕2𝜕𝑡2+𝜔2𝑝1𝑡22𝐷𝑥+𝑡1𝑡2𝜀1𝜆2𝜕2𝜕𝑡2+𝜔2𝑝1𝐷𝑦.(21) For the discretization, we use the central difference approximation and the central average operator, 𝜕2𝜕𝑡2𝐸𝑛𝑥=𝐸𝑥𝑛+12𝐸𝑛𝑥+𝐸𝑥𝑛1Δ𝑡2,𝐸𝑛𝑥=𝐸𝑥𝑛+1+2𝐸𝑛𝑥+𝐸𝑥𝑛14.(22) The central average operator improves the stability and accuracy [40, 49, 50]. Similarly, 𝐷𝑥 and 𝐷𝑦 are discretized, and we obtain the 𝐸𝑥-update equation 𝐸𝑥𝑛+1=𝐸𝑥𝑛1𝑎+21𝑎+1𝐸𝑛𝑥+1𝜀0𝜆2𝑎+1×𝑏+1𝐷𝑥𝑛+1+𝐷𝑥𝑛12𝑏1𝐷𝑛𝑥+𝑐+1𝐷𝑦𝑛+1+𝐷𝑦𝑛12𝑐1𝐷𝑛𝑦,(23) where 𝐷𝑦𝑛+1,𝐷𝑛𝑦,𝐷𝑦𝑛1 must be spatially interpolated due to the staggered Yee cell [31], and 𝑎±𝑖=𝜀𝑖Δ𝑡2±𝜔2𝑝𝑖4,𝑏±𝑖=𝜀𝑖𝑡22+𝜆2𝑡21Δ𝑡2±𝜔2𝑝𝑖𝑡224,𝑐±𝑖=𝑡1𝑡2𝜀𝑖𝜆2Δ𝑡2±𝜔2𝑝𝑖𝑡224.(24) Similarly, the 𝐸𝑦-update equation is obtained by exchanging 𝑡1𝑡2, 𝐸𝑥𝐸𝑦, and 𝐷𝑥𝐷𝑦.

To find the 𝐻𝑧-update equation, we consider the relation 𝐵𝑧=𝜇0𝜀𝑧𝑧𝐻𝑧=𝜇0𝜀3𝜔2𝑝3𝜔2𝐻𝑧.(25) Analogously to the 𝐸𝑥 field, the 𝐻𝑧-update equation can be obtained in the form 𝐻𝑧𝑛+3/2=𝐻𝑧𝑛1/2𝑎+23𝑎+3𝐻𝑧𝑛+1/2+𝐵𝑧𝑛+3/22𝐵𝑧𝑛+1/2+𝐵𝑧𝑛1/2𝜇0Δ𝑡2𝑎+3,(26) where 𝜇0 is the vacuum permeability.

In summary, the electromagnetic fields are iteratively updated in the following sequence:(1)update the components of 𝐃𝑛+1 according to (14),(2)update the components of 𝐄𝑛+1 according to the sample given in (23),(3)update the components of 𝐁𝑛+3/2 according to (15),(4)update the components of 𝐇𝑛+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 𝜆0= 750 nm (400 THz) is in the TM polarization; the grid spacing is = 10 nm (𝜆0/= 75); and the time step is given by the CFL limit, Δ𝑡=/(𝑐02). 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 (: 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 𝐻𝑧 field distributions using = 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)=lim𝑟||𝐸2𝜋𝑟𝑠||(𝜙)2||𝐸0||2,(27) where 𝜙 is the scattering angle, |𝐸𝑠(𝜙)|2 is the scattered power in far field, and |𝐸0|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/𝜆0, scattered by a PEC (without cloak) and cloak using different grid spacings, = 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) 𝐻𝑧 field distributions using = 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, = 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 by𝜎𝑡=𝜎(𝜙)𝑑𝜙.(28) 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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