## Two-dimensional electromagnetic cloaks with arbitrary geometries

Optics Express, Vol. 16, Issue 17, pp. 13414-13420 (2008)

http://dx.doi.org/10.1364/OE.16.013414

Acrobat PDF (1063 KB)

### Abstract

Transformation optics opens an exciting gateway to design electromagnetic “invisibility” cloaks with anisotropic and inhomogeneous medium. In this paper, we establish a generalized transformation procedure to highly improve the flexibilities for the design of two-dimensional (2D) cloaks. The general expressions for the complex medium parameters are developed, which can be readily applied to design 2D cloaks with arbitrary geometries. An example of 2D cloak with irregular cross section is designed and studied by full-wave simulations. The Huygens’ Principle is applied to quantitatively evaluate its unusual electromagnetic behaviors. All the theoretical and numerical results verify the effectiveness of the proposed approach. The generalization in this Paper makes a great step forward for the flexible design of electromagnetic cloaks with arbitrary shapes.

© 2008 Optical Society of America

## 1. Introduction

*et. al.*for the design of invisible cloaking devices based on a forminvariant transformations of Maxwell’s equations [1–4

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

2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science **314**, 977 (2006). [CrossRef] [PubMed]

5. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 224 (2007). [CrossRef]

6. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal Cylindrical Cloak : Perfect but Sensitive to Tiny Perturbations,” Phys. Rev. Lett. **99**, 113903 (2007). [CrossRef] [PubMed]

6. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal Cylindrical Cloak : Perfect but Sensitive to Tiny Perturbations,” Phys. Rev. Lett. **99**, 113903 (2007). [CrossRef] [PubMed]

7. H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. **99**, 063903 (2007). [CrossRef] [PubMed]

8. H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B **76**, 241104 (2007). [CrossRef]

10. 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 Transfromations of Maxwell’s Equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87 (2008). [CrossRef]

11. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. **90**, 241105 (2007). [CrossRef]

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

12. H. Ma, S. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A **77**, 013825 (2007). [CrossRef]

13. D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. **92**, 013505 (2008). [CrossRef]

10. 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 Transfromations of Maxwell’s Equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87 (2008). [CrossRef]

## 2. Medium transformation for 2D electromagnetic cloaks with arbitrary geometries

*θ*, z) in Cylindrical coordinate with relationship,

*r*=R

_{0}(

*θ*), we first introduce a new coordinate variable defined as

*r⃑*in terms of R

_{0}(

*θ*). Then we obtain

*ρ*represent a family of contours with similar shapes as

*r*=R

_{0}(

*θ*). Hence, (

*ρ*,

*θ*,

*z*) forms a conformal coordinate system with the contour

*r*=R

_{0}(

*θ*), as shown in Fig.1(a). To create a cloak, we define a spatial transformation that compresses the cylindrical volume with 0<

*ρ*<1 in the (

*ρ*,

*θ*,

*z*) system into an annular volume with

*τ*<

*ρ*′<1 (

*τ*<1) in the (

*ρ*′,

*θ*′,

*z*′) system via

*ρ*′<

*τ*is completely excluded. Based on above transformation, we can derive the relationship between the transformed Cartesian coordinate (

*x*′,

*y*′,

*z*′) and the original Cartesian coordinate (

*x*,

*y*,

*z*) as

*et al.*, the associated permittivity and permeability tensors of the transformation media become [3

3. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794 (2006). [CrossRef] [PubMed]

*ε*′

_{xz}=

*ε*′

_{yz}=

*ε*′

_{zx}=

*ε*′

_{zy}=0. The permeability tensor

_{0}(

*θ*′) over

*θ*′, and

*τ*represents the linear compressing ratio. Eq.(8) gives the general expressions of the medium parameters for 2D cloaks with outer boundary defined by

*r*′=R

_{0}(

*θ*′) and inner boundary defined by

*r*′=

*τ*R

_{0}(

*θ*′). For the special case R

_{0}(

*θ*′)=

*b*and

*τ*=

*a*/

*b*, the tensors in Eq.(8) are simplified to the medium parameters of the cylindrical-symmetry cloak [1–4

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

*b*and semi-minor axis

*a*, we can insert the contour equation

*τ*into Eq.(8) to get the media parameters. In fact, R

_{0}(

*θ*′) can be chosen as arbitrary continuous functions with period 2π to represent closed contours with arbitrary shapes. It can be generally expressed by a Fourier series as

*θ*, which means the cloaks have sharp corners, the medium parameters will also be discontinuous at the corresponding positions. Fortunately, it has been verified by the square cloak in Ref.10

_{d}10. 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 Transfromations of Maxwell’s Equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87 (2008). [CrossRef]

*s*

_{1}for the inner square and a side length 2

*s*

_{2}for the outer square, we can substitute the contour equation

*τ*=

*s*

_{1}/

*s*

_{2}into Eqs.(8) to obtain the medium parameters. The above discussion means the generalization in this Paper can be specialized to all of the formerly designed cloaks with conformal inner and outer boundaries.

## 3. Electromagntic properties of a 2D cloak with irregular geometry

*x*=-1m,

*y*=-1m to generate the cylindrical wave. The results are given in Fig.3. It’s seen that the cylindrical wave is perfectly guided around the cloaked object without any obvious scattering.

*σ*(the 2D equivalent of a radar cross section) is calculated based on the Huygens’ Principle. To determine

*σ*, the scattered electric field in far field region is calculated by the integration of the simulated near field along the outer boundary of the scattering object or any other contours which enclosing the scattering object. The integration expression for

*σ*in terms of the near field is

*E⃑*and

_{c}*H→*is the EM fields on the integration contour

_{s}*C*,

*r̂*

_{0}is the unit vector of the scattering direction,

*r→*′ is the position vector on the contour

*C*, and

*η*

_{0}is the free space wave impedance. Considering the non-symmetry of the cloak structure introduced above, the scatter widths for four different incident directions are calculated. The cases with and without cloak are both investigated. The scattering patterns are plotted in Fig.4. Table1 lists some parameters to describe and compare the scattering properties, including the averaged and the maximum scatter widths, and the ratios between the cases with and without cloaks. It’s seen that the cloak greatly reduces the scatter width in different scattering angles. The total scatter power (equivalent to the averaged scatter width) of the irregular PEC cylinder is reduced more than 20 times and the maximum scatter width is reduced more than 90 times. No doubt the ratios could be pushed even larger with finer meshes in simulation. A more interesting phenomenon is that the scattering power of the cloaked structure is almost isotropy over all the angles, which is very different from conventional scattering from objects with irregular shapes.

## 4. Conclusion

## Acknowledgements

## References and links

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

2. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science |

3. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

4. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys |

5. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

6. | Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal Cylindrical Cloak : Perfect but Sensitive to Tiny Perturbations,” Phys. Rev. Lett. |

7. | H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. |

8. | H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. Ma, and C. T. Chan, “Extending the bandwidth of electromagnetic cloaks,” Phys. Rev. B |

9. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

10. | 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 Transfromations of Maxwell’s Equations,” Photon. Nanostruct.: Fundam. Applic. |

11. | H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. |

12. | H. Ma, S. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A |

13. | D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.0230) Optical devices : Optical devices

(260.2110) Physical optics : Electromagnetic optics

(160.2710) Materials : Inhomogeneous optical media

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 4, 2008

Revised Manuscript: August 5, 2008

Manuscript Accepted: August 5, 2008

Published: August 15, 2008

**Citation**

Chao Li and Fang Li, "Two-dimensional electromagnetic cloaks with arbitrary geometries," Opt. Express **16**, 13414-13420 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13414

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### References

- J. B. Pendry, D. Schurig, D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977 (2006). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006). [CrossRef] [PubMed]
- U. Leonhardt, T. G. Philbin, "General relativity in electrical engineering," New J. Phys 8, 247 (2006). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224 (2007). [CrossRef]
- Z. Ruan, M. Yan. C. W. Neff, and M. Qiu, "Ideal Cylindrical Cloak:Perfect but Sensitive to Tiny Perturbations," Phys. Rev. Lett. 99, 113903 (2007). [CrossRef] [PubMed]
- H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, "Electromagnetic wave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]
- H. Y. Chen, Z. X. Liang, P. J. Yao, X. Y. Jiang, H. Ma, and C. T. Chan, "Extending the bandwidth of electromagnetic cloaks," Phys. Rev. B 76, 241104 (2007). [CrossRef]
- F. Zolla, S. Guenneau, A. Nicolet, J. B. Pendry, "Cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069 (2007). [CrossRef] [PubMed]
- 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 Transfromations of Maxwell??s Equations," Photon. Nanostruct.: Fundam. Applic. 6, 87 (2008). [CrossRef]
- H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]
- H. Ma, S. Qu, Z. Xu, J. Q. Zhang, B. W. Chen, and J. F. Wang, "Material parameter equation for elliptical cylindrical cloaks," Phys. Rev. A 77, 013825 (2007). [CrossRef]
- D. H. Kwon, D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]

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