## Abstract

Transformation optics opens an exciting gateway to design electromagnetic “invisibility” cloaks with anisotropic and inhomogeneous medium. In this paper, we establish a generalized transformation procedure to highly improve the flexibilities for the design of two-dimensional (2D) cloaks. The general expressions for the complex medium parameters are developed, which can be readily applied to design 2D cloaks with arbitrary geometries. An example of 2D cloak with irregular cross section is designed and studied by full-wave simulations. The Huygens’ Principle is applied to quantitatively evaluate its unusual electromagnetic behaviors. All the theoretical and numerical results verify the effectiveness of the proposed approach. The generalization in this Paper makes a great step forward for the flexible design of electromagnetic cloaks with arbitrary shapes.

©2008 Optical Society of America

## 1. Introduction

The methods of reducing the electromagnetic (EM) scattering of objects have always been a subject of keen interest in microwave and optical society. Recently, an intriguing idea was proposed by Pendry *et. al.* for the design of invisible cloaking devices based on a forminvariant transformations of Maxwell’s equations [1–4]. The basic principle is to squeeze a volume in a virtual space into a shell with complex medium parameters in the physical space, which excluding EM waves in the concealment volume. This has been demonstrated by experiment in microwave regime using a composite metamaterial made of split ring resonators (SRRs) [2]. At optical wavelengths, a nonmagnetic cloak design has also been proposed that incorporates metallic nanowires [5]. Lately, extensive researches were performed about the theoretical and practical possibility of this ‘invisibility’ [6–9]. For example, the sensitivity of the field to the tiny perturbations of the cloak’s parameters was studied in Ref. 6 and 7, and the method to extending the bandwidth of the cloaks was studied in Ref. 8. The transformation approach has also been extended to other electromagnetic applications, such as the concentrators [10] and rotating fields [11]. Up to date, all of the previous cloak designs have symmetrical or regular shapes. The mostly often studied structures are the spherical and circular cylindrical cloaks [1–9] with rotational symmetry. The elliptic and eccentric elliptic cloaks [12,13] were latterly investigated to reduce the symmetry and increase the flexibility of the cloak shapes. The square cloak with flat surfaces and sharp corners was also developed to exemplify the strength of the transformation method [10].

Toward the practical and flexible realizations of EM cloaks, this Paper presents a general transformation method for designing cylindrical cloaks with arbitrarily regular or irregular 2D cross sections. The fundamental strategy is to introduce a new coordinate system conformal to the surface of the scatterer to be concealed. The general expressions for the complex medium parameters are deduced. A 2D cloak device with irregular geometry is designed. Full-wave simulation results are provided for verification. The cloaking performance is then quantitatively evaluated based on the Huygens’ Principle, in which the scatter width is calculated from the simulated near field. The generalization introduced here can be readily specialized to most of the previously designed cloaks, and represents an important step toward the realization of arbitrarily shaped cloaks.

## 2. Medium transformation for 2D electromagnetic cloaks with arbitrary geometries

In the original space, a point can be described by (x, y, z) in Cartesian coordinate or (r, *θ*, z) in Cylindrical coordinate with relationship,

For a cylinder with an arbitrary cross section enclosed by a contour *r*=R_{0}(*θ*), we first introduce a new coordinate variable defined as

to normalize the length of the position vector *r⃑* in terms of R_{0}(*θ*). Then we obtain

Lines of constant-*ρ* represent a family of contours with similar shapes as *r*=R_{0}(*θ*). Hence, (*ρ*, *θ*, *z*) forms a conformal coordinate system with the contour *r*=R_{0}(*θ*), as shown in Fig.1(a). To create a cloak, we define a spatial transformation that compresses the cylindrical volume with 0<*ρ*<1 in the (*ρ*, *θ*, *z*) system into an annular volume with *τ*<*ρ*′<1 (*τ*<1) in the (*ρ*′, *θ*′, *z*′) system via

Outside this domain the identity transformation is adopted. Finally,

is applied to complete the whole transformation. The transformed coordinate system is shown in Fig.1(b), where the cylindrical volume with *ρ*′<*τ* is completely excluded. Based on above transformation, we can derive the relationship between the transformed Cartesian coordinate (*x*′, *y*′, *z*′) and the original Cartesian coordinate (*x*, *y*, *z*) as

According to the procedure stated by Pendry *et al.*, the associated permittivity and permeability tensors of the transformation media become [3]

where ${\Lambda}_{i}^{{i}^{\prime}}=\frac{\partial {q}_{{i}^{\prime}}}{\partial {q}_{i}}$ is the Jacobian transformation matrix between the transformed and the original coordinates. After a nontrivial derivation, the tensor components of the transformation media can be written as

and *ε*′_{xz}=*ε*′_{yz}=*ε*′_{zx}=*ε*′_{zy}=0. The permeability tensor
$\stackrel{\overrightarrow{\to}}{\mu}\prime $
is equal to
$\stackrel{\overrightarrow{\to}}{\epsilon}\prime $
. Here
$\frac{d{R}_{0}\left(\theta \prime \right)}{d\theta \prime}$
represents the first order derivative of R_{0}(*θ*′) over *θ*′, and *τ* represents the linear compressing ratio. Eq.(8) gives the general expressions of the medium parameters for 2D cloaks with outer boundary defined by *r*′=R_{0}(*θ*′) and inner boundary defined by *r*′=*τ* R_{0}(*θ*′). For the special case R_{0}(*θ*′)=*b* and *τ*=*a*/*b*, the tensors in Eq.(8) are simplified to the medium parameters of the cylindrical-symmetry cloak [1–4]. For elliptical cylindrical cloaks with semi-major axis *b* and semi-minor axis *a*, we can insert the contour equation

and a desired compressing ratio *τ* into Eq.(8) to get the media parameters. In fact, R_{0}(*θ*′) can be chosen as arbitrary continuous functions with period 2π to represent closed contours with arbitrary shapes. It can be generally expressed by a Fourier series as

If
$\frac{d{R}_{0}\left(\theta \prime \right)}{d\theta \prime}$
is continuous, such as the cylindrical-symmetry cloak and the elliptical cylindrical cloak, the medium parameters will be continuously varying in the cloak region. If
$\frac{d{R}_{0}\left(\theta \prime \right)}{d\theta \prime}$
is discontinuous in certain *θ _{d}*, which means the cloaks have sharp corners, the medium parameters will also be discontinuous at the corresponding positions. Fortunately, it has been verified by the square cloak in Ref.10 that, such discontinuity does not break any fundamental cloaking properties. For a square cloak with a side length 2

*s*

_{1}for the inner square and a side length 2

*s*

_{2}for the outer square, we can substitute the contour equation

and *τ*=*s*
_{1}/*s*
_{2} into Eqs.(8) to obtain the medium parameters. The above discussion means the generalization in this Paper can be specialized to all of the formerly designed cloaks with conformal inner and outer boundaries.

## 3. Electromagntic properties of a 2D cloak with irregular geometry

To show the flexibility of the approach to design 2D irregular cloaks without any symmetry,

is chosen as an example. To verify the properties of the cloak, full wave simulations based on finite-element method (FEM) are performed. The exciting TM plane wave has an electric filed polarized in z direction with unit amplitude, and incident upon the cloak along the +x direction. The frequency of the harmonic wave is set to be 1 GHz. In simulation, we fill the inner region of the cloak with perfect electric conductor (PEC) and see whether it can be “seen” from outside. Perfect matched layers (PML) are applied to terminate the computational domain in ±x and ±y directions. Fig.2 shows the numerical results for the electric field distribution near the cloaked structure, which is computed with 98456 elements and 394480 unknowns. As can be seen, the wave is smoothly bent around the cloaked area and the phase fronts are perfectly restored when the wave exits the cloak.

Since the proposed cloak has no symmetry in any directions, it’s necessary to study its interaction with EM waves from different orientations. An effective way is to investigate its property under the illumination of a cylindrical wave, which can be decomposed to different planar wave components. In the simulation, a line source is set at the position *x*=-1m, *y*=-1m to generate the cylindrical wave. The results are given in Fig.3. It’s seen that the cylindrical wave is perfectly guided around the cloaked object without any obvious scattering.

To quantitatively evaluate the cloaking performance, the scatter width *σ*(the 2D equivalent of a radar cross section) is calculated based on the Huygens’ Principle. To determine *σ*, the scattered electric field in far field region is calculated by the integration of the simulated near field along the outer boundary of the scattering object or any other contours which enclosing the scattering object. The integration expression for *σ* in terms of the near field is

where *E⃑ _{c}* and

*H→*is the EM fields on the integration contour

_{s}*C*,

*r̂*

_{0}is the unit vector of the scattering direction,

*r→*′ is the position vector on the contour

*C*, and

*η*

_{0}is the free space wave impedance. Considering the non-symmetry of the cloak structure introduced above, the scatter widths for four different incident directions are calculated. The cases with and without cloak are both investigated. The scattering patterns are plotted in Fig.4. Table1 lists some parameters to describe and compare the scattering properties, including the averaged and the maximum scatter widths, and the ratios between the cases with and without cloaks. It’s seen that the cloak greatly reduces the scatter width in different scattering angles. The total scatter power (equivalent to the averaged scatter width) of the irregular PEC cylinder is reduced more than 20 times and the maximum scatter width is reduced more than 90 times. No doubt the ratios could be pushed even larger with finer meshes in simulation. A more interesting phenomenon is that the scattering power of the cloaked structure is almost isotropy over all the angles, which is very different from conventional scattering from objects with irregular shapes.

In this section, the invisibility of the cloaks to TM waves are numerically verified. Since the relative permeability tensor $\stackrel{\overrightarrow{\to}}{\mu}\prime $ is identical to the relative permittivity tensor $\stackrel{\overrightarrow{\to}}{\epsilon}\prime $ , the responses of the cloaks to TE waves are the dualtiy of their responses to TM waves. Hence the proposed cloak is also invisible to TE-porlarized waves. Here, the numerical results for TE cases are not included for brevity.

## 4. Conclusion

A general transformation procedure for designing 2D cloaks with arbitrarily geometries is demonstrated. The general expressions for the complex medium parameters are deduced, which can be readily specialized to most of the previously designed cloaks. A peculiar cloak device with irregular shape is designed as an example and studied by FEM simulations. The Huygens’ Principle is applied to quantitatively evaluate its cloaking properties. All the results verify the flexibility and effectiveness of the proposed method. Although we limit ourselves to 2D cases, the method can be readily extended to construct three dimensional (3D) cloaks with arbitrary shapes. The generalization proposed in this Paper represents an important step and provides a powerful tool toward the flexible design of EM cloaks with arbitrary shapes.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (60501018), the National Basic Research Program of China under Grant (2004CB719800), and the Knowledge Innovation Program of Chinese Academy of Sciences. The authors also acknowledge Master student Kan Yao for his contribution in the numerical simulations.

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