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A general boundary mapping method is proposed to enable the designing of various transformation devices with arbitrary shapes by reducing the traditional space-to-space mapping to boundary-to-boundary mapping. The method also makes the designing of complex-shaped transformation devices more feasible and flexible. Using the boundary mapping method, an arbitrarily shaped perfect electric conductor (PEC) reshaping device, which is called a “PEC reshaper,” is demonstrated to visually reshape a PEC with an arbitrary shape to another arbitrary one. Unlike the previously reported simple PEC reshaping devices, the arbitrarily shaped PEC reshaper designed here does not need to share a common domain. Moreover, the flexibilities of the boundary mapping method are expected to inspire some novel PEC reshapers with attractive new functionalities.
©2011 Optical Society of America
1. Introduction
Since the first design of spherical invisibility cloak by coordinate transformation method [1], various shapes invisibility cloaks have been proposed due to the importance of the shape in practical applications. In principle, coordinate transformation method can be used to design cloaks with arbitrary shapes [2]. However, the process of calculating the material parameters for a given shape is intractable, especially when the given shape is very complex. Up to date, numerous works have been done to derive the material parameters of an arbitrarily shaped invisibility cloak [3–13]. Early focus was on the case when the analytical expressions of the boundaries of the cloak were known prior to the design [3–8]. However, sometimes it was a very tough task to set up the expressions of the cloak boundaries for an actual object. Recently, without analytical expressions of a cloak boundary, the material parameters of arbitrarily shaped invisibility cloaks were calculated by numerically solving the partial differential equations (PDEs) such as Laplace’s [11], Helmholtz’s [12] and Poisson’s [13] equations. Nevertheless, different from the invisibility cloaks whose shapes are almost independent on the cloaking performances, some other transformation devices depend on the shape of themselves much more [14–17]. Therefore, the design of arbitrarily shaped cloaks of them is of more practical significances. Although the method to solve the PDEs has been used to design complex shaped invisibility cloaks, the design of some other forms of transformation media such as an arbitrarily shaped perfect electric conductor (PEC) reshaper [16,17] is still a great challenge due to the highly complicity of the transformation.
A PEC reshaper is a device which visually changes the shape of a PEC to another one [16]. To date, both the designed reshapers and their effective PECs are usually of simple shapes [16–18], though an algorithm of mapping arbitrarily star-shaped regions onto other domains was recently developed to form a series of relatively complex reshapers [19]. Moreover, the existing design methods for the PEC reshapers are all based on the traditional space-to-space mapping and have many constraints or defects, e.g., the physical PEC must share a domain with the effective PEC, and all the boundaries must be smooth and non-convex [16,17]. Above all, the design method for arbitrary-shape PEC reshapers is still lacking.
In this paper, we proposed a general boundary mapping method to design various kinds of transformation devices including PEC reshapers. Compared with the traditional coordinate transformation procedure, the boundary mapping method seeks the transformation equations of the boundaries instead of the areas of the cloaks. As a result, the design process is not only greatly simplified, but also can be applicable to the design of the arbitrarily shaped PEC reshaper, whose effective PEC is also of arbitrary shape. Moreover, by this method, some new forms of the reshapers can be designed and interesting new applications are inspired.
2. The boundary mapping method
2.1 Introduction of the method
For simplicity, we consider the problems in two dimensional cases. According to the traditional coordinate transformation method [1], in order to calculate the material parameters of a cloak, one should know the coordinate transformation equations of the whole transformation region first. If, for example, the transformation equations are expressed as
or equivalently
where (x, y, z) and (x', y', z') are physical and virtual Cartesian spaces, respectively. Suppose that the virtual space is free space, we can then calculate the relative material parameters of the cloak by [1]
where J is the Jacobian transformation tensor with components Jij = ∂xi / ∂xi' (i, j = 1, 2, 3 for the three spatial coordinates), and det(J) represents its determinant.
Infinite forms of transformation equations f 1 and f 2 can be used, and they correspond to various forms of coordinate mappings [20]. To ensure that the virtual space is mapped onto the physical space continuously and the material parameters can be derived, the transformation equations should be continuous everywhere (including the boundaries) and differentiable in the transformation regions with respect to x' and y'. As we know, the solutions of many classical PDEs such as Laplace’s equations are harmonic and undoubtedly meet the requirements. Therefore, we can solve the Dirichlet problem [21] to obtain a possible suite of transformation equations once they are determined in advance on the boundaries. Since finding the boundary transformation equations of a device is much easier than finding the transformation equations of the total device directly, such an approach may greatly decrease the complexity of designing a transformation device. Moreover, it also provides a possibility of controlling the material parameter distributions of the designed cloaks if the total transformation is not determined directly [12,22]. After the boundary conditions are specified, the next step is to find the solutions of the PDEs such as Laplace’s [11], Helmholtz’s [12] and Poisson’s [13] equations by solving the Dirichlet problem. In this work, we simply take the PDEs to be solved as Laplace’s equations to illustrate the method. For example, to design an invisibility cloak, we can solve the Laplace’s equations
with specified boundary conditions [11]
where Δ is the Laplace operator, and Γi and Γo represent the inner and outer boundaries of the cloak, respectively. (x 0, y 0) is an arbitrary fixed point in physical space. The inner boundary conditions imply that the cloaked region is mapped as a point in the virtual space, and the outer boundary conditions ensure that the field outside the cloak is not disturbed [11–13,23,24]. As a result, solving of Eq. (3) with boundary conditions (4) leads to a perfect invisibility cloak. For designing other transformation devices such as the field concentrator [25,26], we can also specify the corresponding boundary conditions.
2.2 Application to a cylindrical PEC reshaper
To illustrate the use of the boundary mapping method, in this section we take a very simple cylindrical PEC reshaper design for example. This simple reshaper is designed to reshape a PEC cylinder to a bigger one. Figure 1 shows its cross section and mapping procedure. For convenience, we investigate the transformation in cylindrical coordinate systems. The transformation has a suit of easily derivable transformation equations as the following form [27]
Fig. 1 Scheme of the transformation for the cylindrical PEC reshaper which reshapes a cylinder of diameter of a to another cylinder of diameter of a'. (a) shows the original space and (b) shows the physical space. The outer radius of the reshaper is b.
The mapping can also be described by other forms of equations. Here, we will derive one suit by the boundary mapping method. Suppose that the transformation equations on the annular shell of the cylindrical PEC reshaper are
where u is currently unknown function that we should find out using the boundary mapping method. According to the method, we should firstly find out the exact expression of Eq. (6) on the inner and outer boundaries of the reshaper, and they will be taken as the boundary conditions of the Laplace’s equation to be solved. Since the transformation equations are continuous everywhere including the outer boundary, on the outer boundary it must has the same form of
as on the region outside the reshaper. For the inner boundary, the transformation equation can also be written down directly as follows according to the mapping procedure shown in Fig. 1
Then, on the annular shell of the PEC reshaper, the Laplace’s equation with its boundary conditions is
Thanks to the special shape of the computational region (an annulus) and boundary conditions of Eq. (9), the solution of it is easy to be found as
where
The transformation equations are so far obtained by the boundary mapping method. Using the transformation equations, the material parameters of the reshaper are derived routinely from Eq. (2):
where “diag” represents a diagonal tensor with all its diagonal entries in the brackets. Thus, design of the cylindrical PEC reshaper has been accomplished by the boundary mapping method. Next, we verify the design result with numerical simulation.
Before the simulation start, we first arbitrarily assign the geometry parameters of the reshaper and its effective shape as a = 0.1 m, b = 0.2 m and a' = 0.17 m. In the illumination of a TE (Transverse Electric) plane wave of 2 GHz in frequency, the electric field distributions of the reshaper and its effective PEC are shown in Fig. 2(a) and (b) , respectively. The far fields of them are nearly the same visually. To quantitatively verify this qualitative judgment, we plot their far field scattering width versus the scattering angle for both of them in Fig. 2(c). The highly consistence of the two plots indicates the reshaper performs definitely as designed.
Fig. 2 Electric field distribution of (a) the effective PEC cylinder, (b) the PEC cylindrical reshaper when a TE plane wave is incident from left to right, and (c) scattering width curves correspond to (a) and (b). The dashed line in (b) outlines the effective PEC.
2.3 A boundary mapping principle for arbitrarily shaped PEC reshaper
As illustrated in the above section, for designing regularly shaped PEC reshapers with regular effective PEC object using the boundary mapping method, it would be quite simple as the boundary conditions are evident and easy to set. However, for some irregularly shaped PEC reshapers, their boundary conditions would be hard to determine. Considering this, we propose a boundary mapping principle to determine the boundary conditions. With this principle, designing of arbitrarily shaped PEC reshaper is feasible.
Suppose that the transformation equations of the PEC reshaper are
The idea of the boundary mapping for an arbitrarily shaped PEC reshaper is illustrated in Fig. 3 . The region in virtual space as shown in Fig. 3(a) is reshaped to another region in physical space as shown in Fig. 3(b). The original outer boundary (Γ'o) and the domain outside it do not deform during the transformation. Therefore we get the boundary condition for Γo, i.e., the mapping between Γo and Γ'o
Fig. 3 Scheme illustration of the transformation of the arbitrarily shaped PEC reshaper. (a) The original domain in virtual space, and (b) the transformed domain in physical space. Starting from the reference points (A and A'), the points on Γ'i are mapped onto Γi one-to-one. For an arbitrary pair of points (B and B'), we have A'B'/l 1 = AB/l 2 (where l 1 and l 2 are total length of Γ'i and Γi, respectively).
In order to obtain the material parameters of the device, we also need to establish a relationship between the original inner boundary (Γ'i) and the deformed inner boundary (Γi). To do this, we first arbitrarily select a point A' (a 1', a 2') on Γ'i and arbitrarily map it onto Γi as A (a 1, a 2). Then, for any point B on Γi and its corresponding mirror point B' on Γ'i we construct the relationship of A'B'/l 1 = AB/l 2. Here l 1 and l 2 are the lengths of the two boundaries, and AB and A'B' () are the path lengths as shown in Fig. 3. By this way, all the points on Γi can be mapped onto Γ'i one-to-one. This series of one-to-one relationships can be expressed as two functions: x' = h 1 (x, y) and y' = h 2 (x, y), where h 1 and h 2 are determined according to the process described above. Then we get the boundary conditions for Γ'i
By using the boundary conditions described in Eqs. (13) and (14), and solving the Dirichlet problem of the following Laplace’s equations,
and
we can get the transformation equations of the PEC reshaper and then calculate the material parameters.
3. An example of designing a square PEC reshaper
As an illustration, in this section, a square PEC reshaper with a relatively complex effective PEC is designed using the boundary mapping method, and its boundary conditions are determined by the principle described in section 2.3. As shown in Fig. 4 , an arbitrary polygon in the original space is mapped onto a square in the physical space, while the room between the polygon and the outer square border is twisted as well to form a square frame. The transformation occurred on the square frame is what we are interested.
Fig. 4 Scheme illustration of the transformation of the rectangular PEC reshaper. (a) The original domain in virtual space, and (b) the transformed domain in physical space. Coordinates of the vertexes are A' (−0.006, −0.057), B' (0, 0), C' (−0.118, 0.046), D' (0.015, 0.136), E' (0.132, −0.004) in original space, and A (−0.1, −0.1), B (−0.1, 0), C (−0.1, 0.1), D (0.1, 0.1), E (0.1, −0.1) in physical space. The outer boundaries in (a) and (b) are both squares with side lengths of 0.2 m and center at origin.
Suppose that the transformation equations of it are represented by Eq. (13). Using the traditional coordinate transformation method, it would be intractable to find the exact form of Eq. (13) because of the complexity of the transformation. However, we can use the boundary mapping method to do it. The key step of the method is to find the boundary conditions of the transformation equations, and the boundary mapping principle as described in section 2.3 can be used to achieve this. However, for convenience, here we make a little improvement to the process to fit the current problem. That is, we apply the process separately for every segment but not for the whole boundary once. For example, in Fig. 4, for segment AB which is mapped from segment A'B' in the original space, we select A (−0.1, −0.1) in physical space and A' (-0.006, −0.057) in original space as start points, and they move toward B (−0.1, 0) and B' (0, 0), respectively. According to the boundary mapping principle in section 2.3, for an arbitrary point P' (x', y') on A'B' and its corresponding mirror point P (x, y) on AB, we have AP/AB = A'P'/A'B', i.e.,
We can therefore derive the expressions of the transformation equations on AB:
For other segments on the inner boundary of the square reshaper, the transformation expressions are easily obtained by similar ways. Here we leave out the details and list the results directly as follows:
For the outer boundary of the reshaper, its transformation equations follow the general principle of most of the transformation device designs, i.e., the transformation should keep the space outside the device unchanged. Therefore, the transformation equations on the outer boundary are the same as Eq. (14).
Next, we solve the Laplace’s equations of Eqs. (16) and (17) along with the above obtained boundary conditions (14) and (19)-(23) to accomplish a suit of transformation equations for the design. However, due to the complexity of the boundary conditions as well as the shape of the computational region, analytically solving the equations is quite an intractable task. On the other hand, numerically solving the PDEs including the Laplace’s equations by FEM (Finite Element Method) or FDTD (Finite-Difference Time-Domain) method has been fully investigated decades ago and implemented in many commercial or free mathematic tools [28] such as Matlab, Maple, Comsol and Meep. In the assistance of one of the computational tool, the only thing we need to do is to input the boundary conditions for every boundary, and the results are plotted in Fig. 5 .
Using the numerically computational results and combining Eq. (2), we can further calculate the material parameters of the device as shown in Fig. 6 . With the material parameters, numerical verifications of the performance of the reshaper can be conducted. Using the same wave source as in Fig. 3, the electric field distributions of the PEC reshaper and its designed effective PEC are presented as Fig. 7(a) and (b) . The two patterns are very close. Their far fields are nearly the same in all angles [see Fig. 7(c)]. Therefore, the design of the square PEC reshaper is successful with the boundary mapping method.
Fig. 6 Material parameter distribution of the square PEC reshaper. (a) ηxx, (b) ηxy (ηyx), (c) ηyy, (d) ηzz. η represents μ or ε. Other components of the material tensor all equal to zero everywhere.
Fig. 7 Electric field distribution of (a) the effective polygon PEC, (b) the square PEC reshaper when a TE plane wave is incident from left to right, and (c) scattering width curves for the effective PEC in (a), and the PEC reshaper in (b). The dashed line in (b) outlines the effective polygon PEC.
4. An example of an arbitrarily shaped PEC reshaper
In the above case of the square PEC reshaper design, the boundary conditions can be expressed analytically as the shapes of the effective PEC and the reshaper are not very complicated. However, in practice, both of the two shapes are mostly much more complicated, and their boundaries may not be able to be expressed analytically. In this case, the boundary conditions must be determined and given numerically. As an example, we consider designing a PEC reshaper whose boundaries are depicted as in Fig. 3(b) and its effective PEC is outlined by Γ'i in Fig. 3(a). To use the boundary mapping method, we first assume that the transformation equations are also represented by Eq. (13). u and v are continuous functions defined on the total range of the PEC reshaper including the two boundaries of it. On the outer boundary, it has the common form of Eq. (14). However, on the inner boundary, we cannot give explicit analytical expressions of them. According to the boundary mapping principle in section 2.3, we can build up the boundary conditions numerically. Firstly, the start points are arbitrarily selected as A (−0.155, 0.046) on Γi and A' (−0.106, −0.01) on Γ'i. Then, for every point B (x, y) on Γi and its mirror point B' (x', y') on Γ'i we have A'B'/l 1 = AB/l 2. The path lengths and total lengths of the two boundaries can be derived numerically, and we can plot the path length A'B' (and AB) versus coordinates of B' (and B) as in Fig. 8(a) [and 8(b)], respectively.
Fig. 8 The plots of path lengths versus coordinates for the (a) effective PEC boundary in original space (x', y') and (b) inner boundary in physical space (x, y) of the PEC reshaper.
From Fig. 8 we know that the total lengths of Γ'i and Γi are l 1 = 0.698 m and l 2 = 1.337 m. According to the relationship of A'B'/l 1 = AB/l 2, one can find out the coordinates of the mirror point B' for every specified point B. Therefore, the relationships between B and B', i.e., the transformation equations on Γi, is determined by the two plots in Fig. 8 associated with the equation A'B'/l 1 = AB/l 2.
After the establishment of the boundary conditions on inner and outer boundaries, we solve the Laplace’s equations numerically and the results are shown in Fig. 9 . The material parameters of the PEC reshaper are then calculated as shown in Fig. 10 .
Fig. 9 Numerical solutions of the Laplace equations for (a) u and (b) v in the domain of the arbitrarily shaped PEC reshaper.
Fig. 10 Material parameter distribution of the PEC reshaper. (a) ηxx, (b) ηxy (ηyx), (c) ηyy, (d) ηzz. η represents μ or ε. The reshaper can be simplified by replacing the abnormal parameters (values exceeding the range of the color scale bars or with absolute value less than 0.2) with their nearest appropriate values. Other components of the material tensor all equal to zero everywhere.
As the transformation does not map a point onto a region like the perfect invisibility cloaks, the material parameters of the cloak should be non-singular. Although we know from Fig. 10 that there are a few places where the parameters are non-ideal (generally we hope the refractive index to be relatively small values to facilitate the realization [29]), it is possible to optimize the distribution of material parameters by changing the PDEs when solving the Dirichlet problem [12]. The selection of the reference points as well as the shapes of the physical and effective PECs also affects the material parameter distributions. It is believed that the investigation of these influencing factors in the future can adjust the material parameters in a desirable range. Moreover, we will see that even for the current design we can still greatly lower the material requirements by simplification with little influence on the performance.
From the design process we know that only rectifiability of the boundaries is required. But the boundaries do not need to be smooth and of non-convex, and the physical PEC do not need to share a domain with the effective PEC, either. These advantages may bring much flexibility in designing the arbitrarily shaped PEC reshapers, and even the designing of other new forms of PEC reshapers is possible.
In the property simulation, we assume that a TE plane wave with unit amplitude and a frequency of 2 GHz travels from left to right. Figure 11(a) shows the snapshot of the total electric field caused by the effective PEC object with its outer boundary depicted by Γ'i in Fig. 3(a). Figure 11(b) shows the total electric field induced by the designed PEC reshaper. Comparing the two patterns, we can conclude that they are almost the same in the far-field. This conclusion can be further confirmed by Fig. 11(c), which shows the far-field scattering width curves of the two cases as well as the simplified PEC reshaper by restricting the parameter values within the range of the scale bars in Fig. 10 [and excluding the range of (−0.2, 0.2)]. It is obvious that the three curves in Fig. 11(c) agree well with each other. Thus the design method is proved to be successful and the parameters can also be controlled in a relatively small range.
Fig. 11 Electric field distribution of (a) the effective PEC, (b) the PEC reshaper when a TE plane wave is incident from left to right, and (c) scattering width curves for the effective PEC in (a), the PEC reshaper in (b) and the simplified PEC reshaper. The dashed line in (b) outlines the effective PEC.
Obviously, using the boundary mapping method proposed here can design an arbitrarily shaped PEC reshaper successfully. The effective PEC is also in an arbitrary shape, and the design process is facile with the assistance of modern numerical calculation packages.
5. Development of two kinds of new reshapers
As we discussed, the boundary mapping method breaks the constraint that the physical PEC must share a domain with the effective PEC. This would bring more flexibility in designing the PEC reshapers and inspire more interesting applications. For example, one can design a device which visually reshapes the cloaked PEC and simultaneously shifts it to a distance. Such a device combines the concepts of the shifting cloak [14,15] and the PEC reshaper [16,17]. The design procedure is very similar to that described in section 4 and here we do not repeat it. Figures 12(a) and (b) show the scattering patterns of the effective PEC and the PEC reshaper respectively when the plane wave is incident from right to left. As predicted, the two patterns are coincident in far-field and thus the effectiveness of the PEC reshaper is confirmed. Another interesting application is that we can design a special device to visually reshape and shift multiple physical PECs to one effective PEC. To design such a device, we just need to apply the boundary mapping twice to map the two boundaries of the physical PECs onto the boundary of the common effective PEC. The scattering pattern of such a device whose effective PEC is the one in Fig. 12(a) is shown in Fig. 12(c). It is obvious that the two patterns are almost the same and the design is proved to be effective. To further show that the three cases in Figs. 12(a), (b) and (c) have equivalent images, we also plot the far-field scattering of them in Fig. 12(d). It is obvious that the curves are nearly the same, and the functions of the devices are confirmed. Notice that the position of the effective PEC can be arbitrarily designed in theory (e.g. totally out of the reshaper) like in the design of the shifting cloak [14,15].
Fig. 12 Electric field distribution of (a) the effective PEC, (b) the PEC reshaper with one physical PEC, (c) the PEC reshaper with two physical PEC, under the illumination of a TE plane wave from right to left, and (d) scattering width curves corresponding to (a), (b) and (c). The dashed line in (b, c) outlines the effective PEC. The physical PECs of the reshapers do not share a domain with their effective PEC.
6. Conclusions
In conclusion, we have proposed a facile boundary mapping method to design various transformation devices with arbitrary shapes. Arbitrarily shaped PEC reshapers with simultaneously arbitrary-shaped effective PECs were designed using the method. The PEC reshaper designed by the boundary mapping method does not require the cloaked PEC to share a domain with the effective PEC, and thus more forms of PEC reshapers with new functionalities are expected to be designed. As examples, we designed a device which visually shifted and reshaped the cloaked PECs, and another device which visually turned two objects into one. The design method can be applied to more complex transformation devices with arbitrary shapes.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities (2010-11-008), the Young Teacher Grant from Fok Ying Tung Education Foundation under Grant No. 101049 and the Ministry of education of China under Grant No. PCSIRT0644.
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