1. Introduction

Controlling electromagnetic fields using material properties given by the coordinate transformation approach [1, 2, 3, 4, 5, 6, 7] has recently drawn extensive attention in the research community. The coordinate transformation approach has been discussed in a general coordinate system [1, 2, 3] and in Cartesian coordinates [4]. A coordinate transformation method in the form of optical conformal mapping [5, 6] in arbitrary coordinates has also been reported. It has been demonstrated, for example, in Ref. [3], that the transformation properties of Maxwell’s equations under certain coordinate transformations can yield material properties that have interesting functionalities, such as invisibility cloak [2, 3, 4], perfect lens [3], and magnification [3, 7], to name a few.

Due to its impressive applications, the invisibility cloak has received a more intensive study. The permittivity and permeability tensors have been given for anisotropic and inhomogeneous spherical [2, 4] and infinite cylindrical [3, 8, 9] shells that give absolute zero scattering cross section subject to external incident radiation when an arbitrary object is embedded inside. Both analytical and numerical calculations have been performed to verify the cloaking effects. Ray-tracing simulations [2, 4] in the geometric optics limit have been reported for a cloaked sphere with consistent results. Rigorous solutions to Maxwell’s equations in the spherical case [10] and in the 2-D cylindrical case [11] have been reported, confirming the cloaking effects on objects with these geometries. Numerical simulations using the full-wave finite-element method were performed to study the effects of cloaking material perturbations on the propagation of the incident waves associated with 2-D invisibility cloaks of various shapes, such as cylinders [8], squares [12], elliptical cylinders [13], and eccentric elliptical cylinders [14]. Numerical simulations using the Discrete Dipole Approximation method [15] were also reported in a 3-D spherical case. In all of these numerical studies, simulated local field distribution in the vicinity of the cloaking object has been found to be the same as that of the incident radiation. With the help of metamaterial technology, the required material specification can be practically implemented. For example, 2-D cylindrical cloaking has been realized using artificially structured metamaterials at microwave frequencies [9] and at optical frequencies [16].

The interaction between a spherical invisibility cloak and incident radiation from internal sources has also been investigated [17]. The electromagnetic field solution for a spherical cloak with an active device inside shows that extraordinary surface voltages induced at the inner boundary of the cloak prevent electromagnetic waves from going out.

The Discrete Dipole Approximation (DDA) [18, 19, 20] method is a robust numerical technique for the computation of elastic scattering and absorption by dielectric particles with arbitrary shapes. The conventional DDA formalism has been generalized to computations pertaining to particles with permeabilities μ≠1 [15] by representing the particle by a cubic array of both electric and magnetic oscillating dipoles. This formalism has been proven to be quite useful for simulating the field distributions and scattering cross sections of cloaked spheres.

To date, studies on invisibility cloaks have been limited to spheres and several 2-D irregular particles. To the authors’ best knowledge, the application of invisibility cloak to 3-D particles with irregular shapes has not been reported in the literature.

In this study, we first apply the coordinate transformation approach to ellipsoids, 3-D rounded cuboids, and 3-D rounded cylinders, and give the material properties required for the cloaks with these geometries. Furthermore, we simulate the electric field distributions and the scattering cross sections associated with cloaked objects using the DDA method.

2. Transformation equations

In Cartesian coordinates, a point in space is described by its Cartesian components xⁱ, with i=1,2,3. Under a coordinate transformation that maps point xⁱ to point $x^{i\prime }=x^{i\prime }\left(x^i\right)$ , a rank 2 contravariant tensor T^ij is transformed as follows:

where

is the Jacobian transformation matrix. As rank 2 contravariant tensors that describe material properties, the permittivity tensor ε^ij and the permeability tensor μ^ij are transformed as

Pendry et al. [2] and Schurig et al. [4] suggested that the left hand sides of Eq.(3a) and Eq.(3b) be interpreted as either the properties of the same material in the transformed coordinate system (the topological interpretation), or the permittivity and permeability of an actual material or metamaterial in a flat Cartesian coordinate system (the material interpretation).

As has been discussed in Ref. [4], the material properties of an invisibility cloak can be determined as illustrated in Fig. 1. Consider a closed domain that is to be transformed (region (I) in Fig. 1(a)). The internal and external regions of this domain are assumed to be vacuum. Next, consider a coordinate transformation that maps this region into a region (region (I′) in Fig. 1(b)) that has the same outer boundary, but contains a hole (region (III)) bounded by an inner boundary. Then, determine the material properties ${\epsilon }^{i\prime j\prime }$ and ${\mu }^{i\prime j\prime }$ in region (I′), which no longer represent a vacuum, on the basis of Eqs.(3a) and (3b). Meanwhile, the external region (region (II)) remains undistorted, and the material properties ${\epsilon }^{i\prime j\prime }$ and ${\mu }^{i\prime j\prime }$ in this region still represent a vacuum. In this manner, we have defined a cloaking region. Subject to radiation from outside sources, this object compresses the radiation field in region (I) into region (I′), the cloaking region, and leaves region (III), the cloaked region, radiation-free. This implies that there is no interaction between anything in the cloaked region and sources in the outside domain. Outside of the cloaking region, the radiation fields remain unchanged, as if neither the cloaking material nor any cloaked object exists. In principle, the regions (I), (I′), and (III) can be of arbitrary shapes, although the transformation matrix Eq.(2) associated with irregular shapes may be quite complicated.

 

Fig. 1. Illustration of the coordinate transformation used to determine the material properties of a cloak.

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3. Coordinate transformations for ellipsoidal cloaks

The coordinate transformation used in the studies of spherical cloaks [2, 4] can be readily generalized to ellipsoids. For an ellipsoid with its semi-axes lying along x-, y-, and z-directions with lengths α ₁ b,α ₂ b, and α ₃ b, respectively, the outer boundary is described by

where α ₁, α ₂, and α ₃ are positive numbers describing the aspect ratio of the ellipsoid, and do not transform as a vector. Inside this boundary, we start with the coordinate transformation determined by Eq.(14) in Ref. [4],

where ${\delta }_i^{i\prime }$ is the Kronecker tensor, $r=\sqrt{{\left(x^1\right)}^2+{\left(x^2\right)}^2+{\left(x^3\right)}^2}$ is the distance between the point and the origin. Furthermore, we re-define r as a scaled distance

This definition of r guarantees that r=b for any point xⁱ on the outer boundary. Therefore, $x^{i\prime }=x^i$ . This means that the outer boundary is not changed by the transformation.

By writing the coordinate transformation in the fashion of Eq.(5), we have a singularity at the origin. Therefore, to investigate how the origin transforms, we need to avoid the singularity and study an infinitesimal ellipsoid bounded by the surface described by

where ε is a small positive number. In the limit when ε approaches 0, this surface becomes the origin. A little algebra will show that the transformed surface becomes

which is the boundary of a smaller concentric ellipsoid. Therefore, the coordinate transformation given by Eq.(5) and Eq.(6) transforms a closed ellipsoid with lengths of the outer semi-axes α ₁ b,α ₂ b, and α ₃ b into an ellipsoidal shell with the same lengths of the outer semi-axes and lengths of the inner semi-axes α ₁ a,α ₂ a, and α ₃ a. Outside this region, we assume the identity transformation. In the following discussions, all equations apply only to the internal region.

In Fig. 2 we show the application of the transformation Eq.(5) to a closed ellipsoidal region. Here we characterize the size of an irregular region in terms of the size parameter of an equal-volume sphere x=2πa _eff/λ where a _eff=(3V/4π)^1/3 with V the volume of the irregular region, and λ is the wavelength of some incident radiation. The ellipsoid shown in Fig. 2 has a size parameter of x=8, with α ₁=α ₂=1, and α ₃=2, or b=2^1/38λ/(2π). This ellipsoid is practically a spheroid with its axis of symmetry lying along the z direction.

 

Fig. 2. Application of the coordinate transformation to a spheroidal region subject to a plane-wave radiation propagating in the z-direction. Shown in the figures is the situation in the x-z plane, with the x- and z-axes as depicted in Fig. (a). The scales are in unit of λ. Figure (a) shows the closed region (green region) in Cartesian coordinates, with vacuum both inside and outside. The light rays (dark red lines) and wave fronts (dark blue lines) of the incident radiation are also shown. Figure (b) shows the transformed region according to the transformation Eq.(5) with a=0.5b, with the material properties in the cloaking region determined by Eq.(3a) and Eq.(3b). The light rays and wave fronts transform accordingly.

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Figure 2(a) shows the ellipsoidal region, along with the Cartesian coordinate grid, before the coordinate transformation. As we assume a vacuum everywhere, the coordinate grid lines can also be interpreted as light rays and wave fronts (loci of points having the same phase) if the incident radiation propagates in the z-direction. A coordinate transformation Eq.(5) with a=0.5b transforms this closed region into the cloaking region shown in Fig. 2(b). The grid lines in the region transform accordingly, and the transformed grid lines, which are geodesics in the transformed coordinates, can be interpreted as light rays and wave fronts in the presence of the cloak. As expected, the radiation field never penetrates into the cloaked region. Evidently, the light rays and wave fronts beyond the outer boundary remain the same, as an identity transformation is applied in this region.

Note that the coordinate transformation does not depend on the orientation of the incident radiation beam with respect to the cloak. Therefore, it also works for an incident radiance making an arbitrary angle θ to the z-direction. We just need to do the coordinate transformation in the same Cartesian coordinates spanned by the three semi-axes of the ellipsoid as can be seen in Fig. 3, where we show the transformation applied to the same ellipsoid, but subject to an incident radiance in the x-z plane making an angle of θ=30° to the z-axis.

Here, a rotated coordinate grid represents the light rays and the wave fronts. Again, we notice from Fig. 3(b) that the light rays and wave fronts inside the boundary are transformed such that the cloaked region is never reached.

 

Fig. 3. Same as Fig. 2, but for an incident radiation in the x-z plane making an angle of θ=30° to the z-axis.

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The material properties ε^ij and μ^ij in the cloaking region can be determined by substituting Eqs.(5) and (6) into Eq.(2), leading to a transformation matrix as follows:

in Cartesian coordinates, where the Einstein summation convention has been assumed for indices i and k. For the vacuum, ε^ij=μ^ij=δ^ij. Therefore, a combination of Eq.(1) and Eq.(9) gives the material properties in the cloaking region as follows:

As we take the material interpretation, we have dropped all primes and relabeled the variables as follows: xⁱ→x̃ⁱ, r→r̃, $x^{i\prime }\to x^i$ , and r′→r, with r given by Eq.(6), r̃ and x̃ⁱ given by

Therefore, we have obtained the permittivity and permeability tensors of an ellipsoidal cloak. In the special case of spheres, α_i=1, Eq.(10) reduces to the simple results described by the Eq.(20) in Ref. [4].

4. Coordinate transformations for rounded-cuboids

Rahm et al. [12] have studied the application of the coordinate transformation approach to 2-D squares. The transformation they used can be easily generalized to 3-D cases. This transformation, however, leads to discontinuities in the variance of material properties of the cloak. We will present material properties of cloaks for rounded-cuboids with arbitrary aspect ratio, determined by an alternative coordinate transformation. In our approach, discontinuities are not involved.

We seek the possibility of representing a cuboid approximately by a superellipsoid [22, 23] bounded by the surface

which is well known in computer graphics and used to model a wide range of shapes, including rounded-cubes and rounded-cylinders [22]. Wriedt [23] introduced this shape to the light scattering community in his study of the T-Matrix method.

To approximate a cuboid, we let e=m=2/n. Therefore, Eq.(12) becomes

Eq.(13) is a direct generalization of Eq.(4) by replacing the exponent 2 with an integer n. We call this shape an order-n-cuboid for convenience. As can be seen in Fig. 4(a), an order-n-cuboid approaches a cube as the exponent n increases. In Fig. 4(b) we show the scattering patterns associated with an order-10-cuboidal scattering particle with α ₁=α ₂=α ₃ and with a cubical scattering particle. As can be seen in this figure, an order-n-cuboid with n=10 is already a good approximation of a cuboid as far as light scattering is concerned.

 

Fig. 4. (a) Approximations of a cube by order-n-cuboids with various n values, and α ₁=α ₁=α ₃. (b) Comparison of phase functions pertaining to an order-10-cuboidal scattering particle with α ₁=α ₂=1, α ₃=2 and pertaining to a cubical scattering particle with the same aspect ratio. Both particles are homogeneous and have the same size parameter x=4 and the same permittivity ε=1.44.

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To determine the material property tensors of an order-n-cuboidal cloak, we can still use the coordinate transformation Eq.(5), and redefine the scaled magnitude r again as

It can be easily verified that the boundary transforms to itself and the origin transforms to an inner boundary with b replaced by a.

Substituting Eq.(5) and Eq.(14) into Eq.(2), we obtain a transformation matrix as follows:

and a combination of Eq.(1) and Eq.(15) gives the permittivity and permeability tensors for an order-n-cuboidal cloak

with r given by Eq.(14), r̃ and x̃ⁱ given by Eq.(11).

In Fig. 5, we show a rounded cuboidal cloak, approximated by an order-10-cuboid with α ₁=α ₂=1, α ₃=2, and size parameter x=8. The light rays and wave fronts in the vicinity of the cloak as predicted by the coordinate transformation are shown. Two particle orientations were considered. Again, the light rays and wave fronts deviate from the cloaked region. Therefore, any object in the cloaked region is hidden from outside observers.

 

Fig. 5. Same as Fig. 2(b), but for a rounded cuboidal cloak approximated by an order- 10-cuboid with x=8, a=0.5b, α ₁=α ₂=1, and α ₃=2, subject to incident radiation propagating in various directions. (a) The incident radiation is along the z-axis; (b) The incident radiation is in the x-z plane, making an angle of θ=30° to the z-axis.

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5. Coordinate transformations for rounded cylinders

To approximate an elliptic cylinder of finite height, we let e=1 and m=2/n. Therefore, Eq.(12) becomes

which defines a shape we will call an order-n-cylinder. As Fig. 6(a) shows, an order-n-cylinder approaches a cylinder as the exponent n increases. Figure 6(b) implies that an order-n-cylinder with n=10 gives a good approximation to a cylinder as far as light scattering is concerned.

We can then use the same coordinate transformation Eq.(5) to determine material property tensors of an order-n-cylindrical cloak by redefining the scaled magnitude r as

Again, the boundary transforms to itself and the origin transforms to an inner boundary with

 

Fig. 6. (a) Approximations of a cylinder by order-n-cylinders with various n values, and α ₁=α ₁=α ₃. (b) Same as Fig. 4(b), but for an order-10-cylindrical scattering particle of the same size parameter and aspect ratio, and a cylindrical scattering particle.

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b replaced by a. The corresponding transformation matrix will then be

and the permittivity/permeability tensor for an order-n-cylindrical cloak is a hybrid of the corresponding tensors for an ellipsoidal cloak and for an order-n-cuboidal cloak, given by

with r given by Eq.(18), r̃ and x̃ⁱ given by Eq.(11).

Note that when z=0, Eq.(17) and Eq.(4) are identical, and Eq.(18) and Eq.(6) are identical; When x=0 or y=0, Eq.(17) and Eq.(13) are identical, and Eq.(18) and Eq.(14) are identical. Namely, for an order-n-cylindrical cloak, the wave fronts and light rays behave the same as those associated with an ellipsoidal cloak in the x-y plane (similar to the situation shown in Fig. 2(b) and Fig. 3(b)), and the same as those associated with an order-n-cuboidal cloak in the x-z plane and the y-z plane (same as the situation shown in Fig. 5).

To conclude this section, in Fig. 7 we present 3-D views of light rays and wave fronts associated with the three irregular invisibility cloaks we discussed.

 

Fig. 7. 3-D views of light rays and wave fronts associated with the three irregular invisibility cloaks: (a) ellipsoid, (b) rounded-cuboid, and (c) rounded cylinder.

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6. DDA simulations

To check if these coordinate transformations really work, we simulated the light scattering pertaining to spheroidal cloaks, rounded cuboidal cloaks, and rounded cylindrical cloaks with the permittivity and permeability tensors given by Eq.(10), Eq.(16), and Eq.(20), respectively, using the DDA formalism discussed in Ref. [15]. The situations shown in Fig. 2(b), Fig. 3(b), and Fig. 5 are considered. Figure 8 shows the simulated electric-field distributions in the vicinity of an ellipsoidal cloak. DDA simulations for electric field distributions in the vicinity of a rounded cuboidal cloak are presented in Fig. 9. Both figures show the situation in the x-z plane. The DDA simulations in the x-z plane for a rounded cylindrical cloak look similar to Fig. 9. It can be noticed that the plane-wave feature outside of the cloak is perfectly kept. The field in the cloaking region is compressed with patterns that are consistent with the predictions of the coordinate transformation approach as can be seen in Fig. 2(b), Fig. 3(b), and Fig. 5. The field in the cloaked region is close to 0 with a leakage of about 10% of the radiating field, or about 1% of the radiative energy, into the cloaked region due to the discretization of material properties in the DDA calculations.

As a numerical method, the DDA is not expected to give exact zero scattering. In our calculations for the three cloaks with size parameter x=8, cloak parameter a=0.5b, and two particle orientations, all simulated scattering efficiencies Q _sca=C _sca/(πa ² _eff) are on the order of 10^-3, which is 3 orders lower than that for a regular dielectric particle. The simulated scattering efficiencies are on the order of 10^-5 for smaller cloaks (x=5) applied to smaller cloaked regions (a=0.3b).

 

Fig. 8. The DDA simulations of the electric-field distribution in the vicinity of an ellipsoidal cloak in the x-z plane.

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Fig. 9. The DDA simulations of the electric-field distribution in the vicinity of an order-10- cuboidal cloak in the x-z plane.

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7. Conclusions

Invisibility cloaks for particles with three irregular geometries have been studied on the basis of the coordinate transformation approach. The permittivity and permeability tensors have been given for ellipsoidal cloaks, 3-D rounded-cuboidal cloaks approximated by order-n-cuboidal cloaks, and 3-D rounded-cylindrical cloaks approximated by order-n-cylindrical cloaks. Numerical calculations using a generalized Discrete Dipole Approximation method were carried out to simulate the light scattering of plane-wave incident radiation associated with such cloaking objects. The simulated electric-field distributions in the vicinity of the cloak and scattering efficiencies suggest that these cloaks do not change the electric field outside at all. Therefore, outside observers cannot detect any object embedded within the cloaked region. Moreover, simulated scattering efficiencies of as low as 10^-5 have been observed.

Unlike spherical cloaks, for which the permittivity and permeability tensors are rotationally uniaxial, cloaks for irregular particles have more complicated material properties. This may cause extra difficulties in the fabrication of these cloaks using metamaterial or other materials.