The approach of designing cloaks with arbitrary shapes was investigated. The coordinate transformation was considered as a homeomorphous topological mapping and the related geometrical description was proposed by introducing position surfaces and tracing lines. Then an approximation approach was presented, which considers a close shape as a polyhedron and performs the spatial compression in a general method. As an example of this approach, we deduced the material parameter equations for two-dimensional polygon cloaks and confirmed the results through numerical simulations. Our approach opens up possibilities of designing practical cloaks with arbitrary shapes through numerical or/and semi-analytical methods.
© 2008 Optical Society of America
The coordinate transformation solutions reported by Pendry et al.  produce a general method for designing electromagnetic cloaks with arbitrary sizes and shapes that renders inner objects perfectly invisible from the outer region. Another approach, which is based on the optical conformal map principle, has been introduced by Leonhardt [2, 3]. The method is used to configure electromagnetic or acoustic cloaks under the short wavelength limit. Using those methods, researchers have designed and fabricated cylindrical meta-material cloaks [4, 5], have proposed transformation media concept to achieve devices such as concentrators, rotation coats, parallel beam shifters and beam splitters [6-8], and have extended electromagnetic cloak concept to acoustic waves [9-12]. Additionally, other researchers have adopted analytical or semi-analytical methods to investigate the field behaviors when electromagnetic waves were put on and propagating in cloaks [13-16]. Some have got the conclusion that perfect invisibility cloaks with dispersive materials cannot be achieved because of the limitation of causality .
Among the above work, designing material parameters is another focus mainly on the cloaks with spherical and circular cylindrical geometries. Recently, square and elliptical cylindrical cloaks have been reported [6, 18-21], which is drawing much attention as it opens up possibilities of designing composite cloaks. More recently, references [22, 23] have made similar discussions and presented the analytical principle in designing arbitrarily shaped cloaks. However, it is difficult to analytically express the associated coordinate transformations for practical objects to be cloaked since their shapes are generally complicated and irregular. What’s more, most theoretical analyses are based on straight-line compression of space and therefore limited to the case that, in the cloak volume, there is at least one inner point from which all points in the cloak surface can be reached along straight lines. In fact, such limitation is not always satisfied in general cases. Practically, the designing of arbitrarily shaped cloaks must rely on approximation approaches and numerical methods.
In this paper, we will report our studies on the approximation approach of designing cloaks with arbitrary shapes. At first, the geometrical means to describe the space deformation based on coordinate transformations is introduced. Then, an approximation approach is investigated and the designing procedures in two-dimensional (2D) spaces are also presented. Finally, to verify this approximation approach, a 2D fish-liked cloak is designed and simulated.
2. Position surfaces and Tracing Lines
Our discussion starts from the coordinate transformation which squeezes space from a volume into a shell surrounding the concealment volume without disturbing the outer fields. Mathematically, such spatial compression is corresponding to a homeomorphous topological mapping (HTM) between two similar topology structures. In order to perform such kind of mapping, we build a shell topology structure by excluding an arbitrary inner point from the continuous volume and treating this point as a mathematical oddity (we use it as the original point). In this way, any continuous volume with an arbitrary inner point being excluded can be compressed as a shell through HTM. As a result, in principle, cloaks with arbitrary close shapes can be designed based on coordinate transformations.
We introduce position surfaces (PSs) (or position contours (PCs) in 2D spaces) and tracing lines (TLs) to describe HTM geometrically. Similar to equal-potential surfaces, on each PS, points that have the “equal position” are located. By assigning all points that fall into a considered region on a series of PSs and seeing the shape changes of them, we can describe the spatial compression geometrically. In short, if the considered region is compressed, the points that on one PS will move to another one, which will result in the shape changes of those PSs. At the same time, we record the positions traced by the points that are moved from one PS to another and connect those positions as TLs. As seen in Fig. 1, the solid lines represent the PSs and the dashed lines represent the TLs. Generally, the HTM and the coordinate transformation can be performed in various methods, which also results in various shapes of the PSs and the TLs. For example, the TLs may be curve, straight, or alternatively curve and straight in different regions (see Fig. 1(a)). The principle of drawing those PSs and TLs are determined by HTM and should be such that, there is no inter-crossing between all the PSs but for each PS, it must intercross with every TL only once, and that for all the TLs, there is no inter-crossing between them but they must intercross with every PS once. This limitation requires every point located on one PS to be uniquely mapped to the exact point on another PS, which therefore guarantees the mapping to be homeomorphous and the coordinate transformation to be unique.
Seeing Fig. 1, we notice that, the shapes of PSs are determined by the shapes of cloaks, and that the TLs in Fig. 1(a) are more complicated than that in Fig. 1(b), which is due to the styles of cloak shapes. Shown in Fig. 1(a), for a cloak with very complicated shapes, if there is no inner point from which all the points on the cloak surface can be reached along straight lines, the TLs of such cloak must be curves; whereas, as shown in Fig. 1(b), as long as there is one such inner point (for example, point O) in the cloak volume, the TLs can be straight lines. The later case was discussed by other researchers [22, 23], and now, we will consider the general cases, namely, the TLs are curves.
In Cartesian systems, suppose the function of PSs is S(x, y, z)=0, and that the parameter function of TLs is L : x=ϕ(t), y=φ(t), z=ψ(t). Thus the arc length of a TL from the original point (ϕ(t 0), φ(t 0), ψ(t 0) to the considered point (ϕ(t), φ(t), ψ(t)) is:
Then we define a spatial compression, which may be of various forms. But here we give the usual form:
where l a and l b are the arc lengths of a TL from the original point to the point where the TL intercross with the inner and outer shells of the cloak, respectively. Equation (2) guarantees that all the points on a TL are squeezed according to their sequence on the arc between the inner and the outer shells of the cloak. With respect to the functions of PSs and TLs, we obtain the coordinate transformation by substituting equation (1) into (2):
Obviously, equations (1-3) are general in expressing coordinate transformations depending on PSs and TLs. In procedures of designing cloaks, function (1) should be explicitly expressed, which requires to offer the functions of PSs and TLs at first. However, generally, the shapes of cloaks are complicated and irregular, which results in the difficulties in obtaining above functions. In practice, we can use PSs and TLs as the means in describing the space deformation to propose various approximation approaches and to develop efficient numerical methods in designing arbitrarily shaped cloaks.
3. Approximation approach
In this section, depending on PSs and TLs means, we introduce an approximation approach in designing practical cloaks. Firstly, in the general cases, we treat an arbitrarily shaped cloak as a polyhedron since it has a close shape. Under this approximation, we divide the cloak shell as well as the inner space into several pyramids whose top points meet at the original point while the triangular faces may be distorted. This way, the volumes surrounded by PSs can be also treated as the similar polyhedrons especially the interior and outer PSs that are respectively equivalent to the interior and outer shells of the polyhedron cloak. Because the polyhedron surface is connected by planes, the interior and outer PSs of the polyhedron cloak can be expressed by plane functions. As a result, the functions of the PSs in the interlayer of the interior and outer PSs are also of plane forms. Obviously, such approximation brings the feasibility in numerical design of cloaks especially when the cloak shape cannot be described analytically.
Secondly, we consider the simple case shown in Fig. 1(b). In this case, for an arbitrarily shaped cloak which is approximated as a polyhedron, if there is at least one inner point from which straight lines can be drawn to every point on the outer surface, the pyramids that construct such polyhedron have straight triangular faces. Plotting planes being parallel to the base in every pyramid and connecting them, we enclose those planes to a series of polyhedrons that have different volumes, and use the surfaces of those polyhedrons as the PSs. By doing so, we can build plane functions to describe the PSs and use straight line functions to describe the TLs, therefore equations (1-3) can be analytically solved. Comparing with that in the general cases, the polyhedron approximation in this simple case results in the analytical expressions of space deformations and therefore simplifies the design of cloaks.
The sketches for PSs and TLs under polyhedron (or polygon in 2D spaces) approximation are shown in Fig. 2, which provides a comparison with that in Fig. 1. In Fig. 2, the PSs (or PCs in 2D spaces) are constructed by planes(or straight lines in 2D spaces), and the TLs, especially in (a), their shapes in some regions are changed and can be optimized as straight lines, which of course brings the facility in designing cloaks. In addition, we notice that, such an approximation may produce approximation errors, but in practical procedures, we can reduce them by increasing the number of partitions and improving the modeling methods, that is to say, the approximation errors can be controlled under a small range.
4. Design of 2D cloaks
For the sake of simplicity in explaining the above approximation approach, we present our procedures in 2D spaces, and note that the conclusions can be easily generalized to three-dimensional cases.
As shown in Fig. 2(b), in 2D cases, we divide the region Σ into several triangles with their hemlines approximating to the arc length of the outer shell. We plot straight lines being parallel to the hemline in each triangle and connect those lines to form the PCs. At the same time, we draw straight lines from the original point O to the points that on the outer contours and treat these lines as the TLs. It can be geometrically proved that the PC functions in every triangle are determined by the gradient k and the end position (x b, y b) of the triangle hemline. Let spatial compression being performed as equation (2), we then obtain the coordinate transformation in a selected triangle as follows:
and (x a, y a) is the end position of the inner shell cut by the selected triangle. By using the method prescribed by Rahm et al. , we can retrieve the value of the permittivity and permeability via α′(x′,y′)=Aα(x,y)A T/det A, where α(x,y) (before transformation) and α′(x′,y′) (after transformation), respectively, denote the permittivity or permeability tensors, and A is the matrix whose elements are as follows:
Note that the above equations, though deduced in a sampled triangle, is valid for all the triangles by assigning k, x b and y b the right values determined by their respective hemline positions. Obviously, these equations are useful to design an arbitrarily shaped cloak approximated by a polygon and are convenient for numerical simulations without any further coordinate transformation since they are already expressed in Cartesian systems.
Full-wave finite-element simulations by using COMSOL Multiphysics software were performed to verify our conclusions. Fig. 3 displays a cut in the x-y plane of the simulation results using the TE-mode harmonic waves. The computation domain is such that the inner shell of the cloak is perfect electric conducting (PEC) and the boundaries being paralleled to the wave propagation are perfect magnetic conducting (PMC). In the simulations, we designed a 2D fish-liked cloak that can be approximated by a polygon. The electromagnetic waves are put on the cloak horizontally or perpendicularly. The results show that, outside the cloaking shell, the wave fronts remain planar and the scattering is quite small. However, inside the cloaking shell, the phase fronts are warped, which causes power-flow to bend around the inner shell. In short, the electromagnetic fields are smoothly excluded from the interior region with minimal scattering, and the cloaking effect is quite obvious.
In conclusion, PSs and TLs are adopted to geometrically describe the coordinate transformations in designing arbitrarily shaped cloaks. Based on these geometrical means, an approximation approach which treats arbitrary close shapes as polyhedrons (or polygons in 2D spaces) is proposed. Under such an approximation, PSs and TLs can be simplified and can be easily expressed by plane and straight-line functions, which therefore provides an efficient approach to overcome the difficulties in analytically expressing coordinate transformations of arbitrary shapes. The approach should be given great attention, because it opens up possibilities of designing arbitrary practical cloaks through numerical or/and semi-analytical methods. This approach is also useful in designing acoustic cloaks with arbitrary shapes. What should be mentioned here, the errors of the approximation that cover other issues related to tectonics, mechanics, and so on, should be studied in those related fields specially.
This work was supported by the National Natural Science Foundation of China (Grant Nos.50632030 and 10474077) and the 973-project of the Ministry of Science and Technology of China (Grant No.2002CB613307). The Innovation Funds of the College of Science, Air Force University of Engineering, also supported this work.
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