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We study an invisibility cloak with a twin cavity, simulated by a plane algebraic curve- hippopede. The cloaked region, which looks like eight for some sets of geometric parameters, is expanded from one single point. Using a geometric transformation approach, we demonstrate that the material parameters of cloaking layer can be exactly determined. Numerical simulations show that the incoming rays pass in and out the cloaking region twice, and return to their original trajectory outside the curved cloak. A notable feature is that the cloaking region has two hollow regions in which two objects can be hidden at one time and that they could not perceive each other.
©2009 Optical Society of America
1. Introduction
Pendry et al. [1] and Leonhardt [2] found that by enclosing an object by a layer of tailored material, the electromagnetic field can be controlled, bent around an interior region and return to their original propagation direction without perturbing the exterior field. Numerous theoretical, numerical and experimental developments have been reported in the last few years. Among them, we mention the relevant works: rigorous proofs of form invariance upon transformation [3-5], experimental implementations [6], various physical phenomena [7-11], inhomogeneous media [12-13] and rotated electromagnetic waves [14]. We mention that analogous concepts of invisibility can be dated back to the works [15-16]. Also, earlier references of [17-19] suggested that arbitrary change of coordinates could be useful in electromagnetics, especially in numerical modeling. The latest progress includes that the proposition of anti-cloak [20] which annihilates some effects of invisibility cloak, cloaking under the carpet [21], and cloaking objects at a distance outside the cloaking shell [22]. See also the review [23] for a broader scope.
The key idea of cloaking is to expand one point or a line segment into a circular or spherical cloaking space. Apart from the aforementioned works, another category of studies has been directed toward the feasibility of design of cloaking devices other than circular or spherical shapes. Study of non-circular or non-spherical cloaks is not just of academic curiosity, it provides flexible potentials to fit for the geometries of hidden objects and also for some specific purposes. Relevant findings include slab cloaks [24], elliptic and eccentric elliptic cloaks [25-26], rectangular cloaks with flat surfaces and sharp corners [27], toroidal cloaks [28], and polygonal and arbitrarily shaped cloaks [29-37]. Among the formulations, some utilized analytic approaches, others were simulated by numerical solutions. But, to our best knowledge, all the existing works on cloaking so far are confined to cloaks enclosed with a single hole. In this paper we propose a new type of invisibility cloak that contains two or more cavities. Imagine the situation of a train containing two or more cars. One potential application of a twin or multiple cavities is to hide two or more objects at one time, in which they could not be placed together inside one cavity or would not be detected by each other. We here mention that the twin cavity is indeed expanded from one single point to a simply-connected hollow region with indentations at top and bottom, in which the two indentation points coincide with one single point at the origin. Thus, the derivation of the material parameters of the cloaking layer follows the conventional route for design of cylindrical cloaks.
2. Coordinate transformations for hippopedal cloaks
To begin with, we simulate the cloaked region by a plane algebraic curve- hippopede. A hippopede is a plane curve obeying the equation in polar coordinates [38]
where a and b are positive numbers. Simple algebra suggests that the contour follows the symmetry relations
Fig. 1. The hippopedal curves, described by Eq. (1), with different sets of parameters (a) a > b (dashed lines), (b) a = b (solid line), (c) a < b (dotted lines).
Different values of a and b will correspond to different contours. For example, if a ≥ 2b, it is an oval, and if b < a ≤ 2b, it is an oval with indentations at top and bottom. If a ≤ b, the figure forms a figure eight (see Fig. 1 for an illustration). Here we are particularly interested in the latter situation in which the configuration is composed of two closed contours. Particularly, we note that when θ varies from 0 to 2π, each θ will at most correspond to one single value of r. For example, when b ≥ a, as θ increases from 0 to π/2, the value r will decrease from 2√ab to zero at θ = sin-1√a/b. When sin-1 √a/b ≤ θ ≤ π - sin-1 √a/b , there is no real value of r-that is, r is an imaginary value. For the remaining parts of the locus π/2 ≤ θ ≤ 2π , the contour can be depicted following the symmetry condition Eq. (2). In Cartesian coordinates, one can write Eq. (1) as
The form also suggests that the locus should be symmetric about the x-axis, y-axis as well as the coordinate center O. Particularly, when a = b, one can simplify Eq. (2) as
This reflects that, in this particular case, the locus is composed of two isolated circles with the same radius b, centered respectively at (b, 0) and (−b, 0), and that the two circles intersects at the origin O. It is seen that, in this situation, each θ will correspond to one single value of r. When θ = π/2, we have r = 0. We mention that, for b = 2a , the hippopede corresponds to the lemniscate of Bernoulli [39].
Let us now consider two similar hippopedes Ωi and Ωo represented by
in which m > 1. Here the subscripts i and o are used to designate the inner and outer hippopedes. According to Pendry [1], we introduce a coordinate transformation that maps the physical space x onto a new space x′. The process is to expand a point from the origin to a hippopedal cavity Ωi , while for points outside Ωo they remain unchanged. This will compress space inside the region Ωo to a hippopedal layer Ωo\Ωi. Such a transformation will lead to two cavities Ωi, which can hide two objects inside two separate holes at the same time. The two hidden objects are invisible to the observers outside and that they could not perceive each other. Previous cloaking geometries reported so far are confined to one single cavity only. Here the cloak region is a twin cavity and is expanded from one single point at the origin. In this way, one can now write the transformation in the form r′ = (m − 1)r/m + ri, θ′ = θ, and z′ = z where r = |x i + y j| and r′ = |x′i + y′j|. In terms of Cartesian components, one can rewrite the transformation relation explicitly in terms of Cartesian coordinates as
where α = (m - 1)/m. The new relative permittivity and permeability follow [4]
where ε and μ are the original properties of background material, which can be represented by an identity matrix in free space with rectangular coordinates, and A is the Jacobian matrix defined by
A referee brought to our attention that the exact connection of Eq. (7) was also stated in Nicolet et al. [40]. Now working out the algebra in Eq. (7), we could express each component of transformed material properties as
In terms of polar coordinate coordinates, the transformed permittivity and permeability follow the form
Equation (10) provides full design parameters for the hippopedal cloaks in the polar coordinates based on the center O. Clearly, the cloak is composed of inhomogeneous and anisotropic metamaterials. We note that, when we set θ = 0 in Eq. (1), the contours characterized in Eq. (5) become two concentric circles and, in this situation, the transformed material properties in Eq. (10) recover to those of the circular cloak [6]. It is known that, for circular cloaks, singular material parameters are distributed at the inner boundary. Here, in Fig. 2, we plot the values of εxx′, εyy′, εxy′ and εzz′ along the portion of cloaking layer for the case of a = 1 and b = 1.
Fig. 2. The magnitudes of material properties εxx′, εyy′, εzz′, and εxy′ inside the hippopedal cloaking layer (a = 1, b = 1) .
We find that extreme material properties also prevail along the inner boundary of the cloak as in the circular cloaks. In particular, we note that at the origin, the material property is necessarily singular as, at that point, the inner and outer cloaking boundaries shrink to one point and the thickness of the cloaking layer becomes effectively zero. The presence of a singularity arises from the fact that the origin point O is mapped onto the origin in the new space, but its neighborhood is tearing apart to four distinct directions in the transformed (new) space.
3. Numerical simulations
To demonstrate the effectiveness of the designed cloak, we perform numerical simulations based on finite element calculations. In the following, the transverse-electric (TE) polarized electromagnetic field is considered. In this case, only εzz′, μrr′, μrθ′ and μθθ′ are required in the simulations. The operating frequency is 3GHz. The background material is assumed to be air and a TE plane wave is incident from left to right. The size and shape of the cloak are controlled by the positive factors a, b and m. Three types of cloak characterized by different sets of parameters are considered in this paper. Parameters associated with the inner boundary, for the three different curves, are taken respectively as (a) a = 0.12, b = 0.1, (b) a = 0.15, b = 0.15, (c) a = 0.11, b = 0.16. The outer boundary of the cloak follows a similar shape with that of the inner one, and a factor of m = 2 is selected. The cloaking layer between the inner and outer boundaries of the cloak is filled with a tailored material as specified in Eq. (9) or Eq. (10). The simulation results are illustrated in Fig. 3. In Figs. 3(a)-3(c), we plot the electric field distribution with three types of cloak. It is seen that the fields are bent around the cloaking region and return to its original trajectory outside the curved cloak. It can be seen that, in cases (b) and (c), the cloaking region looks like a doubly connected cavity. In fact, it is a simply connected bounded oval with indentations at top and bottom, in which the two indentation points coincide with one single point at the origin. Thus the incoming rays pass in and out the cloaking region twice. Physically, one can imagine that two objects could be hidden at the same time under the same cloak, and yet they could not perceive each other. Next an incident wave, with an angle of 30° with respect to the x-axis, is considered (Fig. 4). In this configuration, the propagation direction is not parallel, nor perpendicular, to the principal axes of the cloak region. The incident wave is again bent smoothly around the twin cavity and returns to its original trajectory without any distortion.
Fig. 3. Snapshot of the electric field distribution. The plane waves are incident in the horizontal direction. (a) a = 0.12, b = 0.1, (b) a = 0.15, b = 0.15, (c) a = 0.11, b = 0.16.
Fig. 4. Snapshot of the electric field distribution (a), and stream lines (b). The plane waves are incident at an oblique angle of 30° with a = 0.15, b = 0.15.
Fig. 5. A schematic illustration of a few plane curves. (a) double folium, r = 4acosθsin2 θ (dashed line), (b) sinusoidal spirals, r = (acos3θ)1/3 (dotted line).
4. Conclusions
In summary, we have demonstrated that, by using the concept of coordinate transformation, it is possible to expand one point into a twin cavity so that two objects can be hidden at one time. Numerical simulations show that the wave can be bent around the cloaking region in and out twice with two hidden objects inside. Creating the desired properties for the cloaking shell with singular properties could be a challenging task, but reports of new findings of design of cloaking devices may encourage further developments and inspire potential applications of cloaking with a twin cavity. We mention that recent studies [41-42] have proposed some schemes that may allow an invisibility cloak to be designed without singularities. Also, it should be mentioned that hippopedes in plane algebraic geometry are not the only plane curve that may lead to multiply connected domains. There are other kinds of geometries, such as double folium or sinusoidal spirals (Fig. 5) [38], which may also lead to a configuration with multiple cavities. The design of cloaks of the latter geometries is indeed parallel to the formulation presented in this work.
Acknowledgment
This work was supported by the National Science Council, Taiwan, under contract NSC97-2211-E-006-117-MY3.
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