OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19740–19751
« Show journal navigation

General boundary mapping method and its application in designing an arbitrarily shaped perfect electric conductor reshaper

Jianguo Guan, Wei Li, Wei Wang, and Zhengyi Fu  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19740-19751 (2011)
http://dx.doi.org/10.1364/OE.19.019740


View Full Text Article

Acrobat PDF (2129 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

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 OSA

1. Introduction

Since the first design of spherical invisibility cloak by coordinate transformation method [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

], 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

2. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10(4), 043040 (2008). [CrossRef]

]. 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

3. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16(17), 13414–13420 (2008). [CrossRef] [PubMed]

13

13. J. J. Ma, X. Y. Cao, K. M. Yu, and T. Liu, “Determination the material parameters for arbitrary cloak based on Poisson's equation,” Prog. Electromagn. Res. M 9, 177–184 (2009). [CrossRef]

]. Early focus was on the case when the analytical expressions of the boundaries of the cloak were known prior to the design [3

3. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16(17), 13414–13420 (2008). [CrossRef] [PubMed]

8

8. J. J. Zhang, Y. Luo, H. S. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25(11), 1776–1779 (2008). [CrossRef]

]. 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

11. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]

], Helmholtz’s [12

12. X. Chen, Y. Q. Fu, and N. C. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express 17(5), 3581–3586 (2009). [CrossRef] [PubMed]

] and Poisson’s [13

13. J. J. Ma, X. Y. Cao, K. M. Yu, and T. Liu, “Determination the material parameters for arbitrary cloak based on Poisson's equation,” Prog. Electromagn. Res. M 9, 177–184 (2009). [CrossRef]

] 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

14. W. Li, J. G. Guan, W. Wang, Z. G. Sun, and Z. Y. Fu, “A general cloak to shift the scattering of different objects,” J. Phys. D Appl. Phys. 43(24), 245102 (2010). [CrossRef]

17

17. G. S. Yuan, X. C. Dong, Q. L. Deng, H. T. Gao, C. H. Liu, Y. G. Lu, and C. L. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express 18(6), 6327–6332 (2010). [CrossRef] [PubMed]

]. 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

16. H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008). [CrossRef]

,17

17. G. S. Yuan, X. C. Dong, Q. L. Deng, H. T. Gao, C. H. Liu, Y. G. Lu, and C. L. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express 18(6), 6327–6332 (2010). [CrossRef] [PubMed]

] 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

16. H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008). [CrossRef]

]. To date, both the designed reshapers and their effective PECs are usually of simple shapes [16

16. H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008). [CrossRef]

18

18. A. Diatta, G. Dupont, S. Guenneau, and S. Enoch, “Broadband cloaking and mirages with flying carpets,” Opt. Express 18(11), 11537–11551 (2010). [CrossRef] [PubMed]

], though an algorithm of mapping arbitrarily star-shaped regions onto other domains was recently developed to form a series of relatively complex reshapers [19

19. A. Diatta and S. Guenneau, “Non-singular cloaks allow mimesis,” J. Opt. 13(2), 024012–024022 (2011). [CrossRef]

]. 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

16. H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008). [CrossRef]

,17

17. G. S. Yuan, X. C. Dong, Q. L. Deng, H. T. Gao, C. H. Liu, Y. G. Lu, and C. L. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express 18(6), 6327–6332 (2010). [CrossRef] [PubMed]

]. 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

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

], 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

x'=f1(x,y),y'=f2(x,y),z'=z
(1a)

or equivalently

x=f11(x',y'),y=f21(x',y'),z=z'
(1b)

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

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

]

ε=μ=JJT/det(J)
(2)

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.

Δf1=0,Δf2=0
(3)

with specified boundary conditions [11

11. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]

]

f1|Γi=x0,f1|Γo=x,f2|Γi=y0,f2|Γo=y
(4)

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

11. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]

13

13. J. J. Ma, X. Y. Cao, K. M. Yu, and T. Liu, “Determination the material parameters for arbitrary cloak based on Poisson's equation,” Prog. Electromagn. Res. M 9, 177–184 (2009). [CrossRef]

,23

23. R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24(1), 015016 (2008). [CrossRef]

,24

24. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3), 749–789 (2007). [CrossRef]

]. 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

25. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fund. Appl. 6, 89–95 (2008).

,26

26. C. F. Yang, J. J. Yang, M. Huang, J. H. Peng, and W. W. Niu, “Electromagnetic concentrators with arbitrary geometries based on Laplace’s equation,” J. Opt. Soc. Am. A 27(9), 1994–1998 (2010). [CrossRef] [PubMed]

], 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
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.
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

27. A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys. 10(11), 115022 (2008). [CrossRef]

]

r'=ba'bar+a'abab,θ'=θ,z'=z
(5)

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

r'=u(r,θ),θ'=θ,z'=z
(6)

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

u(b,θ)=b
(7)

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

u(a,θ)=a'
(8)

Then, on the annular shell of the PEC reshaper, the Laplace’s equation with its boundary conditions is

{2ur2+1rur+1r22uθ2=0u(a,θ)=a',u(b,θ)=b
(9)

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

u(r,θ)=c1+c2lnr
(10)

where

c1=b(ba')lnbln(b/a),   and  c2=ba'ln(b/a)
(11)

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):

μ=ε=diag(r'c2,c2r',c2r'r2)
(12)

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)
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.
, 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.

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

x'=u(x,y),y'=v(x,y),z'=z
(13)

The idea of the boundary mapping for an arbitrarily shaped PEC reshaper is illustrated in Fig. 3
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).
. 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

u|Γo=x,v|Γo=y
(14)

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(l1=Γids,l2=Γ'ids'), and AB and A'B' (AB=ABds,A'B'=A'B'ds') 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

u|Γi=h1,v|Γi=h2
(15)

By using the boundary conditions described in Eqs. (13) and (14), and solving the Dirichlet problem of the following Laplace’s equations,

2ux2+2uy2=0
(16)

and

2vx2+2vy2=0
(17)

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
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.
, 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.

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.,

x'(0.006)0(0.006)=y'(0.057)0(0.057)=y(0.1)0(0.1)
(18)

We can therefore derive the expressions of the transformation equations on AB:

x'=0.06(y+0.1)0.006,y'=0.57(y+0.1)0.057
(19)

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 BC
    x'=1.18y,y'=0.46y
    (20)
  • for CD
    x'=0.0266(x+0.1)0.118,y'=0.018(x+0.1)+0.046
    (21)
  • for DE
    x'=0.0234(y0.1)+0.015,y'=0.028(y0.1)0.136
    (22)
  • for EA
    x'=0.0276(x+0.1)0.006,y'=0.0106(y+0.1)0.057
    (23)

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

28. M. Machura and R. A. Sweet, “A survey of software for partial differential equations,” ACM Trans. Math. Softw. 6(4), 461–488 (1980). [CrossRef]

] 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
Fig. 5 Numerical solutions of the Laplace equations of (16) (left panel) and (17) (right panel).
.

Using the numerically computational results and combining Eq. (2), we can further calculate the material parameters of the device as shown in Fig. 6
Fig. 6 Material parameter distribution of the square PEC reshaper. (a) ηxx, (b) ηxyyx), (c) ηyy, (d) ηzz. η represents μ or ε. Other components of the material tensor all equal to zero everywhere.
. 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)
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.
. 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.

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)
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.
[and 8(b)], respectively.

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
Fig. 9 Numerical solutions of the Laplace equations for (a) u and (b) v in the domain of the arbitrarily shaped PEC reshaper.
. The material parameters of the PEC reshaper are then calculated as shown in Fig. 10
Fig. 10 Material parameter distribution of the PEC reshaper. (a) ηxx, (b) ηxyyx), (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

29. J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41(2), 345–351 (1994). [CrossRef]

]), it is possible to optimize the distribution of material parameters by changing the PDEs when solving the Dirichlet problem [12

12. X. Chen, Y. Q. Fu, and N. C. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express 17(5), 3581–3586 (2009). [CrossRef] [PubMed]

]. 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)
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.
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.

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

14. W. Li, J. G. Guan, W. Wang, Z. G. Sun, and Z. Y. Fu, “A general cloak to shift the scattering of different objects,” J. Phys. D Appl. Phys. 43(24), 245102 (2010). [CrossRef]

,15

15. W. Li, J. G. Guan, Z. G. Sun, and W. Wang, “Shifting cloaks constructed with homogeneous materials,” Comput. Mater. Sci. 50(2), 607–611 (2010). [CrossRef]

] and the PEC reshaper [16

16. H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008). [CrossRef]

,17

17. G. S. Yuan, X. C. Dong, Q. L. Deng, H. T. Gao, C. H. Liu, Y. G. Lu, and C. L. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express 18(6), 6327–6332 (2010). [CrossRef] [PubMed]

]. The design procedure is very similar to that described in section 4 and here we do not repeat it. Figures 12(a) and (b)
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.
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

14. W. Li, J. G. Guan, W. Wang, Z. G. Sun, and Z. Y. Fu, “A general cloak to shift the scattering of different objects,” J. Phys. D Appl. Phys. 43(24), 245102 (2010). [CrossRef]

,15

15. W. Li, J. G. Guan, Z. G. Sun, and W. Wang, “Shifting cloaks constructed with homogeneous materials,” Comput. Mater. Sci. 50(2), 607–611 (2010). [CrossRef]

].

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.

References and links

1.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

2.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10(4), 043040 (2008). [CrossRef]

3.

C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16(17), 13414–13420 (2008). [CrossRef] [PubMed]

4.

G. Dupont, S. Guenneau, S. Enoch, G. Demesy, A. Nicolet, F. Zolla, and A. Diatta, “Revolution analysis of three-dimensional arbitrary cloaks,” Opt. Express 17(25), 22603–22608 (2009). [CrossRef] [PubMed]

5.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. P. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(6 ), 066607 (2008). [CrossRef] [PubMed]

6.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33(14), 1584–1586 (2008). [CrossRef] [PubMed]

7.

A. Veltri, “Designs for electromagnetic cloaking a three-dimensional arbitrary shaped star-domain,” Opt. Express 17(22), 20494–20501 (2009). [CrossRef] [PubMed]

8.

J. J. Zhang, Y. Luo, H. S. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25(11), 1776–1779 (2008). [CrossRef]

9.

H. Ma, S. B. Qu, Z. Xu, and J. F. Wang, “Numerical method for designing approximate cloaks with arbitrary shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(3), 036608 (2008). [CrossRef] [PubMed]

10.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Investigations of the electromagnetic properties of three-dimensional arbitrarily-shaped cloaks,” Prog. Electromagn. Res. 94, 105–117 (2009). [CrossRef]

11.

J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]

12.

X. Chen, Y. Q. Fu, and N. C. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express 17(5), 3581–3586 (2009). [CrossRef] [PubMed]

13.

J. J. Ma, X. Y. Cao, K. M. Yu, and T. Liu, “Determination the material parameters for arbitrary cloak based on Poisson's equation,” Prog. Electromagn. Res. M 9, 177–184 (2009). [CrossRef]

14.

W. Li, J. G. Guan, W. Wang, Z. G. Sun, and Z. Y. Fu, “A general cloak to shift the scattering of different objects,” J. Phys. D Appl. Phys. 43(24), 245102 (2010). [CrossRef]

15.

W. Li, J. G. Guan, Z. G. Sun, and W. Wang, “Shifting cloaks constructed with homogeneous materials,” Comput. Mater. Sci. 50(2), 607–611 (2010). [CrossRef]

16.

H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008). [CrossRef]

17.

G. S. Yuan, X. C. Dong, Q. L. Deng, H. T. Gao, C. H. Liu, Y. G. Lu, and C. L. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express 18(6), 6327–6332 (2010). [CrossRef] [PubMed]

18.

A. Diatta, G. Dupont, S. Guenneau, and S. Enoch, “Broadband cloaking and mirages with flying carpets,” Opt. Express 18(11), 11537–11551 (2010). [CrossRef] [PubMed]

19.

A. Diatta and S. Guenneau, “Non-singular cloaks allow mimesis,” J. Opt. 13(2), 024012–024022 (2011). [CrossRef]

20.

C.-W. Qiu, A. Novitsky, and L. Gao, “Inverse design mechanism of cylindrical cloaks without knowledge of the required coordinate transformation,” J. Opt. Soc. Am. A 27(5), 1079–1082 (2010). [CrossRef] [PubMed]

21.

R. Courant, The Dirichlet Principle, Conformal Mapping and Minimal Surfaces (Interscience, 1950).

22.

A. Novitsky, C. W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys. 11(11), 113001 (2009). [CrossRef]

23.

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24(1), 015016 (2008). [CrossRef]

24.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3), 749–789 (2007). [CrossRef]

25.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fund. Appl. 6, 89–95 (2008).

26.

C. F. Yang, J. J. Yang, M. Huang, J. H. Peng, and W. W. Niu, “Electromagnetic concentrators with arbitrary geometries based on Laplace’s equation,” J. Opt. Soc. Am. A 27(9), 1994–1998 (2010). [CrossRef] [PubMed]

27.

A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys. 10(11), 115022 (2008). [CrossRef]

28.

M. Machura and R. A. Sweet, “A survey of software for partial differential equations,” ACM Trans. Math. Softw. 6(4), 461–488 (1980). [CrossRef]

29.

J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41(2), 345–351 (1994). [CrossRef]

OCIS Codes
(230.0230) Optical devices : Optical devices
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: July 22, 2011
Manuscript Accepted: September 14, 2011
Published: September 23, 2011

Citation
Jianguo Guan, Wei Li, Wei Wang, and Zhengyi Fu, "General boundary mapping method and its application in designing an arbitrarily shaped perfect electric conductor reshaper," Opt. Express 19, 19740-19751 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19740


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  2. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(4), 043040 (2008). [CrossRef]
  3. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express16(17), 13414–13420 (2008). [CrossRef] [PubMed]
  4. G. Dupont, S. Guenneau, S. Enoch, G. Demesy, A. Nicolet, F. Zolla, and A. Diatta, “Revolution analysis of three-dimensional arbitrary cloaks,” Opt. Express17(25), 22603–22608 (2009). [CrossRef] [PubMed]
  5. W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. P. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.77(6 ), 066607 (2008). [CrossRef] [PubMed]
  6. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett.33(14), 1584–1586 (2008). [CrossRef] [PubMed]
  7. A. Veltri, “Designs for electromagnetic cloaking a three-dimensional arbitrary shaped star-domain,” Opt. Express17(22), 20494–20501 (2009). [CrossRef] [PubMed]
  8. J. J. Zhang, Y. Luo, H. S. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B25(11), 1776–1779 (2008). [CrossRef]
  9. H. Ma, S. B. Qu, Z. Xu, and J. F. Wang, “Numerical method for designing approximate cloaks with arbitrary shapes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(3), 036608 (2008). [CrossRef] [PubMed]
  10. Q. Cheng, W. X. Jiang, and T. J. Cui, “Investigations of the electromagnetic properties of three-dimensional arbitrarily-shaped cloaks,” Prog. Electromagn. Res.94, 105–117 (2009). [CrossRef]
  11. J. Hu, X. M. Zhou, and G. K. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express17(3), 1308–1320 (2009). [CrossRef] [PubMed]
  12. X. Chen, Y. Q. Fu, and N. C. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express17(5), 3581–3586 (2009). [CrossRef] [PubMed]
  13. J. J. Ma, X. Y. Cao, K. M. Yu, and T. Liu, “Determination the material parameters for arbitrary cloak based on Poisson's equation,” Prog. Electromagn. Res. M9, 177–184 (2009). [CrossRef]
  14. W. Li, J. G. Guan, W. Wang, Z. G. Sun, and Z. Y. Fu, “A general cloak to shift the scattering of different objects,” J. Phys. D Appl. Phys.43(24), 245102 (2010). [CrossRef]
  15. W. Li, J. G. Guan, Z. G. Sun, and W. Wang, “Shifting cloaks constructed with homogeneous materials,” Comput. Mater. Sci.50(2), 607–611 (2010). [CrossRef]
  16. H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys.10(11), 113016 (2008). [CrossRef]
  17. G. S. Yuan, X. C. Dong, Q. L. Deng, H. T. Gao, C. H. Liu, Y. G. Lu, and C. L. Du, “A design method to change the effective shape of scattering cross section for PEC objects based on transformation optics,” Opt. Express18(6), 6327–6332 (2010). [CrossRef] [PubMed]
  18. A. Diatta, G. Dupont, S. Guenneau, and S. Enoch, “Broadband cloaking and mirages with flying carpets,” Opt. Express18(11), 11537–11551 (2010). [CrossRef] [PubMed]
  19. A. Diatta and S. Guenneau, “Non-singular cloaks allow mimesis,” J. Opt.13(2), 024012–024022 (2011). [CrossRef]
  20. C.-W. Qiu, A. Novitsky, and L. Gao, “Inverse design mechanism of cylindrical cloaks without knowledge of the required coordinate transformation,” J. Opt. Soc. Am. A27(5), 1079–1082 (2010). [CrossRef] [PubMed]
  21. R. Courant, The Dirichlet Principle, Conformal Mapping and Minimal Surfaces (Interscience, 1950).
  22. A. Novitsky, C. W. Qiu, and S. Zouhdi, “Transformation-based spherical cloaks designed by an implicit transformation-independent method: theory and optimization,” New J. Phys.11(11), 113001 (2009). [CrossRef]
  23. R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl.24(1), 015016 (2008). [CrossRef]
  24. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys.275(3), 749–789 (2007). [CrossRef]
  25. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fund. Appl.6, 89–95 (2008).
  26. C. F. Yang, J. J. Yang, M. Huang, J. H. Peng, and W. W. Niu, “Electromagnetic concentrators with arbitrary geometries based on Laplace’s equation,” J. Opt. Soc. Am. A27(9), 1994–1998 (2010). [CrossRef] [PubMed]
  27. A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys.10(11), 115022 (2008). [CrossRef]
  28. M. Machura and R. A. Sweet, “A survey of software for partial differential equations,” ACM Trans. Math. Softw.6(4), 461–488 (1980). [CrossRef]
  29. J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt.41(2), 345–351 (1994). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited