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

Optics Express, Vol. 19, Issue 20, pp. 19740-19751 (2011)

http://dx.doi.org/10.1364/OE.19.019740

Acrobat PDF (2129 KB)

### 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

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]

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]

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

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]

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]

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]

## 2. The boundary mapping method

### 2.1 Introduction of the method

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

*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]

*J*is the Jacobian transformation tensor with components

*J*= ∂

_{ij}*x*/ ∂

_{i}*x*' (

_{i}*i*,

*j*= 1, 2, 3 for the three spatial coordinates), and det(

*J*) represents its determinant.

*f*

_{1}and

*f*

_{2}can be used, and they correspond to various forms of coordinate mappings [20

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]

*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

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]

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]

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]

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]

*and Γ*

_{i}*represent the inner and outer boundaries of the cloak, respectively. (*

_{o}*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

**17**(3), 1308–1320 (2009). [CrossRef] [PubMed]

**9**, 177–184 (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]

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]

## 2.2 Application to a cylindrical PEC reshaper

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

*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

*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.

*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.

## 2.3 A boundary mapping principle for arbitrarily shaped PEC reshaper

*) and the domain outside it do not deform during the transformation. Therefore we get the boundary condition for Γ*

_{o}*, i.e., the mapping between Γ*

_{o}*and Γ*

_{o}*'*

_{o}*) and the deformed inner boundary (Γ*

_{i}*). To do this, we first arbitrarily select a point A' (*

_{i}*a*

_{1}',

*a*

_{2}') on Γ'

*and arbitrarily map it onto Γ*

_{i}*as A (*

_{i}*a*

_{1},

*a*

_{2}). Then, for any point B on Γ

*and its corresponding mirror point B' on Γ'*

_{i}*we construct the relationship of A'B'/*

_{i}*l*

_{1}= AB/

*l*

_{2}. Here

*l*

_{1}and

*l*

_{2}are the lengths of the two boundaries

*can be mapped onto Γ*

_{i}*'*one-to-one. This series of one-to-one relationships can be expressed as two functions:

_{i}*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}## 3. An example of designing a square PEC reshaper

*-*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.,

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]

## 4. An example of an arbitrarily shaped PEC reshaper

*in Fig. 3(a). To use the boundary mapping method, we first assume that the transformation equations are also represented by Eq. (13).*

_{i}*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 Γ

*and A' (−0.106, −0.01) on Γ*

_{i}*'*. Then, for every point B (

_{i}*x*,

*y*) on Γ

*and its mirror point B' (*

_{i}*x*',

*y*') on Γ

*'*we have A'B'/

_{i}*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.

*'*and Γ

_{i}*are*

_{i}*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 Γ

*, is determined by the two plots in Fig. 8 associated with the equation A'B'/*

_{i}*l*

_{1}= AB/

*l*

_{2}.

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]

**17**(5), 3581–3586 (2009). [CrossRef] [PubMed]

*'*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.

_{i}## 5. Development of two kinds of new reshapers

## 6. Conclusions

## Acknowledgments

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

2. | W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. |

3. | C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express |

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 |

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. |

6. | A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. |

7. | A. Veltri, “Designs for electromagnetic cloaking a three-dimensional arbitrary shaped star-domain,” Opt. Express |

8. | J. J. Zhang, Y. Luo, H. S. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B |

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. |

10. | Q. Cheng, W. X. Jiang, and T. J. Cui, “Investigations of the electromagnetic properties of three-dimensional arbitrarily-shaped cloaks,” Prog. Electromagn. Res. |

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 |

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 |

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 |

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. |

15. | W. Li, J. G. Guan, Z. G. Sun, and W. Wang, “Shifting cloaks constructed with homogeneous materials,” Comput. Mater. Sci. |

16. | H. Y. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. |

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. | A. Diatta, G. Dupont, S. Guenneau, and S. Enoch, “Broadband cloaking and mirages with flying carpets,” Opt. Express |

19. | A. Diatta and S. Guenneau, “Non-singular cloaks allow mimesis,” J. Opt. |

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 |

21. | R. Courant, |

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. |

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. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. |

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. |

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. | A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys. |

28. | M. Machura and R. A. Sweet, “A survey of software for partial differential equations,” ACM Trans. Math. Softw. |

29. | J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. |

**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: Year | Journal | Reset

### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- 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]
- C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express16(17), 13414–13420 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- A. Veltri, “Designs for electromagnetic cloaking a three-dimensional arbitrary shaped star-domain,” Opt. Express17(22), 20494–20501 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. Diatta, G. Dupont, S. Guenneau, and S. Enoch, “Broadband cloaking and mirages with flying carpets,” Opt. Express18(11), 11537–11551 (2010). [CrossRef] [PubMed]
- A. Diatta and S. Guenneau, “Non-singular cloaks allow mimesis,” J. Opt.13(2), 024012–024022 (2011). [CrossRef]
- 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]
- R. Courant, The Dirichlet Principle, Conformal Mapping and Minimal Surfaces (Interscience, 1950).
- 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]
- 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]
- 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]
- 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).
- 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]
- A. D. Yaghjian and S. Maci, “Alternative derivation of electromagnetic cloaks and concentrators,” New J. Phys.10(11), 115022 (2008). [CrossRef]
- M. Machura and R. A. Sweet, “A survey of software for partial differential equations,” ACM Trans. Math. Softw.6(4), 461–488 (1980). [CrossRef]
- 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.