The material properties of toroidal invisibility cloaks are derived based on the coordinate transformation method. The permittivity and permeability tensors for toroidal cloaks are substantially different from those for spherical cloaks, but quite similar to those for 2D cylindrical cloaks because a singularity is involved at the inner boundary in both the cases. The cloaking effect is confirmed by the electric field distribution in the vicinity of toroidal cloaks simulated from the generalized discrete-dipole approximation (DDA) method. This study extends the concept of electromagnetic cloaking of arbitrarily-shaped objects to a complex geometry.
©2009 Optical Society of America
Transformation optics [1, 2, 3, 4, 5, 6, 7] has recently become a powerful tool to control the propagation of electromagnetic waves by altering material properties as determined by coordinate transformations in a general coordinate system [1, 2, 3] or in Cartesian coordinates , or by optical conformal mapping [5, 6] in an arbitrary coordinate system. The new tool makes it possible to design optical devices with a variety of interesting functionalities. The invisibility cloak is one of the optical device designs using the coordinate transformation approach and has drawn a great deal of attention due to its potential applications. An invisibility cloak reroutes the incident electromagnetic radiation in the cloaking layer and inhibits it from penetrating into the cloaked region, while the electromagnetic fields outside of the cloaking region remain unchanged. Therefore, the cloaking layer and any object inside it are invisible. The material properties, i.e., the permittivity and permeability tensors, of an invisibility cloak derived from the coordinate transformation approach are anisotropic and inhomogeneous. These material properties were first given for spherical [2, 4] and 2D infinite cylindrical [3, 8] cloaks. Rigorous solutions to Maxwell’s equations have been reported in the spherical and cylindrical cases [9, 10] to confirm the cloaking effects for these geometries. Numerical simulations have been carried out to study the cloaking effects of invisibility cloaks of various shapes, including 2D cylinders [8, 11, 12, 13] and squares , and 3D ellipsoids, cylinders and cuboids . Recently, designs of invisibility cloaks with arbitrary shapes have been studied in both 2D [16, 17, 18, 19] and 3D [20, 21, 22] cases. With recent advances in the metamaterial technology, simplified specifications of the required material properties have been practically implemented in the case of 2D cylinders. The cloaking effect has been observed at both microwave  and optical frequencies .
It has been discussed in Ref.  that the coordinate transformations utilized to design invisibility cloaks fall into two categories: point transformations, which transform a point into a cloaked region; and, line transformations, which transform a line into a cloaked region. Note that at the inner boundary of a line-transformed cloak, one tangential component of the material property tensors has a singularity. According to this classification scheme, all arbitrarily-shaped 3D cloaks reported in the literature are point-transformed cloaks, and all 2D cloaks are line-transformed cloaks.
These point-transformed “arbitrarily-shaped” 3D cloaks, however, do not represent all possible geometries in 3D space. From a topological perspective, they have a genus-zero surface, meaning that the surface possesses no “hole”. Surfaces exist with a genus greater than zero. The simplest genus one surface is a torus, which has one hole in the surface. The volume bounded by a torus, a toroid, is the simplest genus one object. To make a toroidal object invisible, we can simply put it into a genus-zero cloak, for example, a spherical cloak, but the volume of the genus-zero cloak may be unnecessarily large if the toroid has a long thin tube. In such situations, a toroidal cloak is preferred as it uses less cloaking material. Obviously, toroidal cloaks are line-transformed cloaks, since the toroidal cloaked region can only be transformed from a line. A toroidal cloak can be made by cutting out a segment from a line transformed 2D cylindrical cloak and gluing the two ends together. To date, invisibility cloaks for toroids have not been reported in the literature.
In this study, first we apply the coordinate transformation method to the toroidal geometry and derive the material properties required for toroidal cloaks. Second, we compare the material properties for toriodal cloaks with those for spherical cloaks and 2D cylindrical cloaks. Third, we show simulated electric field distributions in the vicinity of toroidal cloaks to confirm the cloaking effect.
2. Coordinate transformations for toroidal cloaks
Figure 1(a) shows a torus generated by revolving a circle about an axis in the same plane as itself in 3D space, as shown in Fig. 1(b), where it is assumed that the axis about which the circle revolves is the z-axis. Figure 1(b) also shows that Cartesian coordinates (x, y, z) of any point in a toroid can be expressed in terms of three parameters r, u and v in the form
where R is the distance from the center of the circle to the coplanar axis, r ∈ [0,b] is the distance from the point to the center of the circle with b the radius of the circle, and angles u, v ∈ [0,2π]. Unit vectors er, eu and ev span the toroidal coordinates. Angle u is the angle between er and the z-direction, and angle v is the angle between the er × eu plane and the x-direction. In Figs. 1(a) and 1(b), it is assumed that b = R/2.
To create a toroidal cloaked region, we consider the following coordinate transformation
where b and a are the radii of the circles that form the outer and inner boundaries of the cloaking region, respectively. A toroidal cloak with b = R/2 and a = b/2 is illustrated in Fig. 1(c). Half of the outer boundary of the cloaking region (the dark green surface) is removed to expose the cloaked region. The transformation equation, Eq. (2), is identical to the coordinate transformations used for spherical cloaks [2, 4] and for 2D cylindrical cloaks [3, 8] in the sense that it involves a linear compression in the “radial” direction with the space in the angular directions remaining intact. However, the radius r is no longer the distance from a point in the space to the origin or to the z-axis, but the distance from the point to the center of the revolving circle, which itself forms a circle in the x-y plane when revolving about the z-axis.
Figure 2 illustrates a toroidal cloak subject to an incident plane wave. The incident light rays (blue lines) and wave fronts (red lines) in the cloaking region are determined by the coordinate transformation Eq. (2). They are compressed into both inner and outer sides of the cloaked region. For simplicity, in this study we limit ourselves to toroidal cloaks with a conformal toroidal cloaked region. The conformal toroidal cloaks can be continuously deformed into cloaks with any genus one geometry in the same manner as spherical cloaks are generalized to arbitrarily-shaped genus zero cloaks, which has been extensively discussed in the literature.
3. Material properties for toroidal cloaks
To obtain the cloaking material property tensors ε and μ via the coordinate transformation approach, we follow the scheme used in Ref.  by working in the toroidal coordinates but use slightly different notations. Consider a coordinate transformation from coordinates (q 1,q 2,q 3) to coordinates (q′1,q′2,q′3), the tensor representation of the transformation formula is given by 
A point in the original space is indicated as (x, y, z) in Cartesian coordinates and (r,u,v) in the toroidal coordinates, and the corresponding point in the transformed space is indicated as (x′,y′, z′) in Cartesian coordinates and (r′,u′,v′) in the toroidal coordinates. Furthermore, we consider the sequential coordinate transformations as follows:
For toroidal cloaks, the transformations (x,y,z) → (r,u,v) and (r,u,v) → (r′,u′,v′) have been determined by Eqs. (1) and (2), respectively. The transformation (r′,u′,v′)→(x′,y′,z′) can be determined by Eq. (1) with (x,y, z) and (r,u,v) replaced by (x′,y′,z′) and (r′,u′,v′), respectively.
The Jacobian matrices involved in these coordinate transformations are
Since we have Eq. (1), it is more straightforward to calculate Λ-1 1 = ∂(x, y, z)/∂ (r,u,v) rather than Λ1. According to the chain rule, the Jacobian matrix associated with the coordinate transformation (x, y, z)→(x′,y′,z′) can be written as
where Q 1 = (Λ-1 1)TΛ-1 1. Eq. (8) is the matrix representation of the permittivity and permeability tensors in Cartesian coordinates, e.g., ε′ = ∑ij ε′ij exi exj, where exi is the ith Cartesian unit vector. To write Eq. (8) as a matrix representation in the toroidal coordinates, we use the following relation
where D = diag[h 1,h 2,h 3] with hi = (Λ2 3,1i + Λ2 3,2i + Λ2 3,3i)1/2 the scale factors of the toroidal coordinates and DΛ-1 3 is an orthogonal matrix (DΛ-1 3)T = (DΛ-1 3)-1. Eq. (8) now becomes
where Q 3 = ΛT 3Λ3.
Therefore, the non-vanishing diagonal components of the permittivity and permeability tensors in the toroidal cloaking region are given by
where we have eliminated primes on ε′, μ′ and r′. All off-diagonal components vanish.
As predicted by Ref. , the tangential components εuu and μuu approach infinity at the inner boundary r→a. This implies that, similar to perfect 2D cylindrical cloaks, perfect toroidal cloaks are difficult to realize using metamaterials. To fabricate toroidal cloaks, simplified material properties [23, 24, 25, 26, 27] may have to be used. It is interesting to compare Eqs. (12a)-(12c) with the material properties of 2D cylindrical cloaks given by, for example, Eqs. (1) and (2) in Ref. 
where R 2 and R 1 are used to denote radii of the outer and inner boundaries, respectively. The correspondences between the r-, u- and v-components in the toroidal coordinates and the r-, ϕ - and z-components in the cylindrical coordinates are obvious. For toroidal cloaks, each diagonal component of the material property tensors has an extra term that depends on both r and u coordinates. This is reasonable since the geometry is no longer invariant with respect to the u coordinate. Specifically, Eqs. (12a)-(12c) reduce to Eqs. (13a)-(13c) when b/R approaches 0, or when sinu = 0, r = a or r = b if b/R is fixed.
Figure 3 shows the diagonal components of the permittivity and permeability tensors for a toroidal cloak with b = R/2 and a = b/2. Since the components depend on the u coordinate, we show them for various u values. The material properties for a toroidal cloak are the same as those for a 2D cylindrical cloak when u = 0. Additionally, the percentage changes (with respect to the 2D cylindrical results) introduced by the u-dependence are on the order of 10%, with the r- and u-components decreased and the v-component increased. The u-dependence seems negligible for angles where u > 60°.
For toroidal cloaks, the permittivity and permeability tensors also depend on the ratio b/R, as shown in Eqs. (12a). Figure 4 shows the diagonal components for a toroidal cloak with b = R/10. This is a toroid with a long thin tube. In this case, the changes introduced by the u-dependence are even smaller, with percentage changes of no more than 4%.
4. Numerical simulations
To confirm the cloaking effect of toroidal invisibility cloaks, we simulated the distribution of the electric field in the vicinity of toroidal cloaks using the generalized discrete-dipole approximation (DDA) method [28, 29, 30, 31]. Since the u-components of the required permittivity and permeability tensors vary with a large gradient and become infinite at the inner boundary of the cloaking region, it is not likely possible to perfectly mimic the required u-components in a discretized numerical calculation. In contrast, the non-singular r- and v-components can be represented by the discrete DDA grids to the extent that the simulated electric field distributions show the cloaking effect [15, 31].
Figure 5 shows the simulated electric field distribution in the vicinity of two toroidal cloaks. Shown in the figure are the field contours observed in the x-y plane. The scales are in units of the wavelength (λ) of the incident plane wave propagating from left to right. The cross sections of the inner and outer parts of the cloaking region are separated (the thin annuli confined by black circles). The cross sections of the cloaked region are the thicker annuli between the two parts of the cloaking region. The cloaked regions are filled with isotropic and homogeneous impedance-matched material with ε =μ = 1.2. Figure 5(a) shows a toroidal cloak with a = 3/4b, b = R/2 and R = 0.45λ. The cloaked region is a relatively small toroid with a fat tube and a thin layer of cloaking material compared with the cloaked toroid. Figure 5(b) shows a cloak with a = 3/4b, b = 3/4R, and R = 1.196λ. The cloaked region is a larger and thinner toroid. The first phenomenon noticed from these figures is that, the electric field is highly compressed into the cloaking region, leading to a radiation-free cloaked region. As predicted by the coordinate transformation approach shown in Fig. 2, the electric field is compressed into both sides of the cloaked region. Outside of the cloaking region, the plane wave structure is well kept, as if no scatterer is present. There are slight deviations from the plane wave patterns at the far end of the cloaks, as the infinitely large u-components at the inner boundary of the cloaking region were not perfectly represented. The implementation of the material property specifications in this numerical simulation is practical for an experiment using metamaterials, yet the resultant cloaking effect is impressive.
The invisibility cloak for the simplest genus-one geometry in three-dimensional space, the toroid, was defined on the basis of the coordinate transformation approach. This design broadens the extent of arbitrarily-shaped 3D cloaks. The permittivity and permeability tensors for toroidal cloaks are similar to those for 2D cylindrical cloaks rather than those for spherical cloaks. A tangential component is required to be infinite at the inner boundary of the cloaking region. Numerical calculations based on the generalized DDA method were carried out to simulate the electric field distributions in the vicinity of toroidal cloaks subject to the incident plane wave. The results suggest that the electric field outside of a cloak is almost unaltered as if neither the cloaking material nor the cloaked toroids were present.
Similar to 2D cylindrical cloaks, the toroidal cloaks may be difficult to realize due to the singularity of the material properties at the inner boundary. The present numerical simulations suggest that the cloaking effect can be impressive even if non-perfect material properties are used. Since the required material properties for a toroidal cloak are similar to those for an infinitely long cylindrical cloak, experimental realizations of toroidal cloaks are possible by using the metamaterial structures consisting of unit cells reported in the literature for cylindrical cloaks [23, 24, 25] and by bending the cylindrical structure into a toroid, except that the extra u-dependence needs to be included by varying the three components of the permittivity and permeability tensors of each individual unit cell along the u direction. However, this u-dependence may be neglected if both the cloaked toroid and the cloaking layer are extremely thin such that R≫b.
This research was partially supported by the Office of Naval Research under contract N00014-06-1-0069. Ping Yang acknowledges support from the National Science Foundation (ATM-0803779).
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