## Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation

Optics Express, Vol. 17, Issue 3, pp. 1308-1320 (2009)

http://dx.doi.org/10.1364/OE.17.001308

Acrobat PDF (1683 KB)

### Abstract

In transformation optics, the space transformation is viewed as the deformation of a material. The permittivity and permeability tensors in the transformed space are found to correlate with the deformation field of the material. By solving the Laplace’s equation, which describes how the material will deform during a transformation, we can design electromagnetic cloaks with arbitrary shapes if the boundary conditions of the cloak are considered. As examples, the material parameters of the spherical and elliptical cylindrical cloaks are derived based on the analytical solutions of the Laplace’s equation. For cloaks with irregular shapes, the material parameters of the transformation medium are determined numerically by solving the Laplace’s equation. Full-wave simulations based on the Maxwell’s equations validate the designed cloaks. The proposed method can be easily extended to design other transformation materials for electromagnetic and acoustic wave phenomena.

© 2009 Optical Society of America

## 1. Introduction

*et al*[2

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

4. 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**, 977–980 (2006). [CrossRef] [PubMed]

6. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science **305**, 788–792 (2004). [CrossRef] [PubMed]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

7. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. **8**, 118 (2006). [CrossRef]

8. U. Leonhardt, “General relativity in electrical engineering,” New J. Phys. **8**, 247 (2006). [CrossRef]

9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations,” Photon. Nanostruct. Fundam. Appl. **6**, 87–95 (2008). [CrossRef]

10. H. Ma, S. B. 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 (2008). [CrossRef]

14. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. **10**, 043040 (2006). [CrossRef]

*et al*. [14

14. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. **10**, 043040 (2006). [CrossRef]

*et al*. [15

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

*et al*. [16

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

*et al*. [17

17. H. Ma, S. Qu, Z. Xu, and J. Wang, “Numerical method for designing approximate cloaks with arbitrary shapes,” Phys. Rev. E **78**, 036608(2008). [CrossRef]

**A**for an arbitrary cloak. So a general and flexible method for designing an arbitrary cloak is still necessary. Especially, the answer to the following question needs to be clarified: is there a governing equation that allows us to evaluate the transformed material parameters for cloaks with arbitrary shapes in a systematic way?

## 2. Design method

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

8. U. Leonhardt, “General relativity in electrical engineering,” New J. Phys. **8**, 247 (2006). [CrossRef]

18. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. **8**, 248(2006). [CrossRef]

**x**to a distorted space

**x**′(

**x**), the permittivity

**ε′**and permeability

**μ′**in the transformed space are given by [18

18. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. **8**, 248(2006). [CrossRef]

**ε**and

**μ**are the permittivity and permeability of the original space, respectively.

**A**is the Jacobian transformation tensor with the components

*A*=

_{ij}*∂x*′

_{i}/

*∂x*, which characterizes the geometrical variations between the original space Ω and the transformed space Ω′. The determination of the matrix

_{j}**A**is the crucial point for designing transformation mediums.

**A**is called the deformation gradient tensor for an infinitesimal element

*d*

**x**deformed to

*d*

**x**′ under the space transformation. The transformation can be decomposed into a pure stretch deformation (described by a positive definite symmetric tensor

**V**) and a rigid body rotation (described by a proper orthogonal tensor

**R**), so the tensor

**A**can be expressed as

**A**=

**VR**[19]. Suppose that material parameters in the original space are homogeneous and isotropic, and they are expressed by scalar parameters

*ε*

_{0}and

*μ*

_{0}. Consider the left Cauchy-Green deformation tensor

**B**=

**V**=

^{2}**AA**

^{T}, Eq. (1) can be rewritten as

**B**can be expressed in the diagonal form

*λ*(

_{i}*i*=1,2,3) are the eigenvalues of the tensor

**V**or the principal stretches for an infinitesimal element. Using det

**A**=

*λ*

_{1}

*λ*

_{2}

*λ*

_{3}and Eq. (3), we can rewrite Eq. (2) as

*b*, inside of this region, we define a point denoted by

*a*. An arbitrary cloak can be constructed by enlarging the point

*a*to the inner boundary

*a*′, while keeping the outer boundary of the region fixed (

*b*=

*b*′). This condition can be expressed by

*U*′(

*a*) =

*a*′ and

*U*′(

*b*) =

*b*′, where the operator

*U*′ is the new coordinates for a given point during the transformation. Now the problem is how to determine the deformation field

*∂U*′ /

**∂**

**x**within the cloak layer enclosed by the inner and outer boundary

*a*′ and

*b*′ for a specific transformation. The commonly used operator

*U*′ for designing a cloak is a linear transformation [2

**312**, 1780–1782 (2006). [CrossRef] [PubMed]

9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations,” Photon. Nanostruct. Fundam. Appl. **6**, 87–95 (2008). [CrossRef]

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

*r*′ = (

*b*′−

*a*′)

*r*/

*b*′ +

*a*′. However, for an arbitrary cloak, it is very difficult to express analytically the boundaries

*a*′ and

*b*′, the calculation of deformation field is usually very complicated.

*U*′ = 0 with the boundary conditions

*U*′(

*a*) =

*a*′ and

*U*′(

*b*) =

*b*′. In order to keep from the singular solution of the Laplace’s equations, we can use the inverse form of the Laplace’s equations as

*U*denotes the original coordinates in the original space. The corresponding Dirichlet boundary conditions then become

_{i}*U*(

*a*′) =

*a*and

*U*(

*b*′) =

*b*. After solving Eq. (5) with proper boundary conditions, we can get the deformation field inside of the cloak layer that characterizes the distortion of the element. The transformed material parameters can then be calculated from Eq. (4). This method is flexible whatever the shape of cloaks. In the next two sections, we will explain in detail how to obtain analytical expressions of material parameters for regular cloaks and how to design irregular cloaks by solving Eq. (5) numerically.

## 3. Application to regular cloaks

### 3.1 Spherical cloak

*θ*′ =

*θ*and

*φ*′ =

*φ*. Then the radial coordinate satisfies the following equation

*r*(

*r*′ =

*a*′) = 0 and

*r*(

*r*′ =

*b*′) =

*b*′ (

*b*′ =

*b*), the solution of Eq. (7) is given by

*a*′ and

*b*′ are respectively the radii of the inner and outer boundaries of the cloak. Equation (9) indicates a nonlinear transformation, which is different from the linear one

*r*′ = (

*b*′−

*a*′)

*r*/

*b*′+

*a*′ commonly used for a spherical cloak [2

**312**, 1780–1782 (2006). [CrossRef] [PubMed]

### 3.2 Elliptical cylindrical cloak

*x*,

*y*,

*z*) can be expressed by the elliptic cylindrical coordinates (

*ξ*,

*η*,

*z*) as

*c*is the focal length of the elliptic cylinder. The scale factors of the elliptical cylindrical coordinate system are [21]

*η*′=

*η*and

*z*′ =

*z*, then Eq. (14) becomes

*∂*

^{2}

*ξ*/

*∂*

*ξ*′

^{2}= 0 . With the boundary conditions

*ξ*(

*ξ*′ =

*a*′) = 1 and

*ξ*(

*ξ*′ =

*b*′)=

*b*(

*b*′ =

*b*), we can obtain a linear transformation relation

*ξ*′ = (

*b*′ −

*a*′)(

*ξ*-1)/(

*b*′−1)+

*a*′, where

*a*′ and

*b*′ are respectively the coordinates of the inner and outer boundaries of the elliptic cloak. The principal stretches for this linear transformation are given by

*ε*

_{0}=

*μ*

_{0}= 1. It can be found that Eq. (16) agrees with the results given by Ma

*et al*[10

10. H. Ma, S. B. 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 (2008). [CrossRef]

## 4. Application to irregular cloaks

*a*′ and

*b*′ for an arbitrary cloak and set the boundary conditions

*U*(

*a*′) = 0 and

*U*(

*b*′)=

*b*(

*b*′=

*b*), then solve the Laplace’s equations to obtain the deformation field

*∂U*/

*∂*

**x**′ inside of the cloak layer. The material parameters of the cloak can be calculated numerically by the principle stretches

*λ*

_{1},

*λ*

_{2}, and

*λ*

_{3}(evaluated from the deformation field) according to Eq. (4). This can be achieved with the PDE solver provided by the commercial software COMSOL Multiphysics. In order to check the performance of the designed cloak, the cloak device will be illuminated by a plane electromagnetic wave with help of the RF module in the same software. So the design and validation of the cloak can be well integrated in a two-step modeling.

*U*(

*a*′) = 0 and

*U*(

*b*′) =

*b*(

*b*′ =

*b*) for the inner and outer boundaries, then solve the equation ∆′

*U*= 0 to get the coordinates

**x**=

*U*(

**x**′) [

**x**′ ∊ (

*a*′,

*b*′) ], from which we obtain

**x**′ =

*U*′(

**x**) [

**x**∊(0,

*b*)]. Figure 4 gives the coordinate lines of

**x**′ transformed from a flat space

**x**. The line path implies that electromagnetic waves will propagate around the area bounded by

*a*′ and any objects put inside of this area will be invisible. For the verification, we let a plane electromagnetic wave of frequency 20 GHz with the electric field polarized along the

*z*direction illuminate the cloak. The full fields of the system are solved with help of the TE waves mode of COMSOL Multiphysics. Fig. 5 (a) and 5(b) shows the contour plots of the electric field

*E*for the waves incident on the cloak horizontally and at an angle of 45° from the

_{z}*x*direction, respectively. It can be seen from the figures that the constructed arbitrary cloak doesn’t disturb the incident waves and can shield an irregular obstacle from detection. For the physical realization, one can easily retrieve the material parameters of the arbitrary cloak from the deformation field according to Eq. (4). The values of some components of the permittivity and permeability tensors,

*ε*,

_{zz}*μ*,

_{xx}*μ*, and

_{xy}*μ*of the cloak are shown in Figs. 6(a), 6(b), 6(c), and 6(d), respectively. It can be found that the cloak are highly anisotropic and must be realized with the metamaterial technology.

_{yy}*O*

_{1}(0,1),

*O*

_{2}(1,0),

*O*

_{3}(0,-1), and

*O*

_{4}(-1,0). The boundary conditions for numerically solving the Laplace’s equation then become

*U*(

*a*′) =

_{i}*O*(

_{i}*i*=1, 2,3,4) and

*U*(

*b*′) =

*b*. Figures 8 (a) and 8(b) show the contour plots of the electric fields

*E*for the waves incident on this cloak horizontally and at an angle of 45° from the x direction, respectively. We can see from the figures that the perfect invisibility is still achieved even if the embedded obstacles are separated apart.

_{z}*E*for TE electromagnetic waves impinging the cloak horizontally and vertically respectively. It can be seen that the designed cloak does not disturb the outside fields and shields the inner random object from detection.

_{z}## 5. Discussion and conclusion

## Acknowledgments

## References and links

1. | A. Greenleaf, M. Lassas, and G. Uhlmann, “On non-uniqueness for Calderón’s inverse problem,” Math. Res.Lett. |

2. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

3. | U. Leonhardt, “Optical conformal mapping,” Science |

4. | 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 |

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

6. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science |

7. | U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. |

8. | U. Leonhardt, “General relativity in electrical engineering,” New J. Phys. |

9. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations,” Photon. Nanostruct. Fundam. Appl. |

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

11. | Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express |

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

13. | W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys. |

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

15. | W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E |

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

17. | H. Ma, S. Qu, Z. Xu, and J. Wang, “Numerical method for designing approximate cloaks with arbitrary shapes,” Phys. Rev. E |

18. | G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. |

19. | W. M. Lai, D. Rubin, and E. Krempl, |

20. | R. Courant and D. Hilbert, |

21. | G. Arfken, |

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

23. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. |

24. | M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders”, Opt. Express |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(220.2740) Optical design and fabrication : Geometric optical design

(260.1180) Physical optics : Crystal optics

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 20, 2008

Revised Manuscript: January 16, 2009

Manuscript Accepted: January 16, 2009

Published: January 22, 2009

**Citation**

Jin Hu, Xiaoming Zhou, and Gengkai Hu, "Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation," Opt. Express **17**, 1308-1320 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1308

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### References

- A. Greenleaf, M. Lassas, and G. Uhlmann, "On non-uniqueness for Calderón’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (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, 977-980 (2006). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- U. Leonhardt, "Notes on conformal invisibility devices," New J. Phys. 8, 118 (2006). [CrossRef]
- U. Leonhardt, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006). [CrossRef]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, and D. R. Smith, "Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell’s Equations," Photon. Nanostruct. Fundam. Appl. 6, 87-95 (2008). [CrossRef]
- H. Ma, S. B. 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 (2008). [CrossRef]
- Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, "Invisibility cloaks for irregular particles using coordinate transformations," Opt. Express 16, 6134-6145 (2008). [CrossRef] [PubMed]
- D. Kwon and D. H. Werner, "Two-dimensional eccentric elliptic electromagnetic cloaks," Appl. Phys. Lett. 92, 013505 (2008). [CrossRef]
- W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, "Arbitrarily elliptical-cylindrical invisible cloaking," J. Phys. D: Appl. Phys. 41, 085504 (2008). [CrossRef]
- W. Yan, M. Yan, Z. Ruan, and M. Qiu, "Coordinate transformations make perfect invisibility cloaks with arbitrary shape," New J. Phys. 10, 043040 (2006). [CrossRef]
- W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Phys. Rev. E 77, 066607 (2008). [CrossRef]
- A. Nicolet, F. Zolla, and S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett. 33, 1584-1586 (2008). [CrossRef] [PubMed]
- H. Ma, S. Qu, Z. Xu, and J. Wang, "Numerical method for designing approximate cloaks with arbitrary shapes," Phys. Rev. E 78, 036608 (2008). [CrossRef]
- G. W. Milton, M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006). [CrossRef]
- W. M. Lai, D. Rubin, and E. Krempl, Introduction to Continuum Mechanics, 3 edition, (Butterworth-Heinemann, Burlington, 1995).
- R. Courant and D. Hilbert, Methods of Mathematical Physics, (Wiley-Interscience, New York, 1989) Vol. 2, 1st Edition.
- G. Arfken, Mathematical Methods for Physicists (Academic Press, Orlando, 1970).
- H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]
- M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, D. R. Smith, "Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations," Phys. Rev. Lett. 100, 063903 (2008) [CrossRef] [PubMed]
- M. Rahm, D. A. Roberts, J. B. Pendry and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008). [CrossRef] [PubMed]

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