## Spherical cloaking using nonlinear transformations for improved segmentation into concentric isotropic coatings

Optics Express, Vol. 17, Issue 16, pp. 13467-13478 (2009)

http://dx.doi.org/10.1364/OE.17.013467

Acrobat PDF (678 KB)

### Abstract

Two novel classes of spherical invisibility cloaks based on nonlinear transformation have been studied. The cloaking characteristics are presented by segmenting the nonlinear transformation based spherical cloak into concentric isotropic homogeneous coatings. Detailed investigations of the optimal discretization (e.g., thickness control of each layer, nonlinear factor, etc.) are presented for both linear and nonlinear spherical cloaks and their effects on invisibility performance are also discussed. The cloaking properties and our choice of optimal segmentation are verified by the numerical simulation of not only near-field electric-field distribution but also the far-field radar cross section (RCS).

© 2009 Optical Society of America

## 1. Introduction

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

2. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

3. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express **15**, 14772–14782 (2007). [CrossRef] [PubMed]

5. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. **24**, 413–419 (2003). [CrossRef] [PubMed]

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

6. H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. **91**, 183518 (2007). [CrossRef]

7. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. **101**, 1345011 (2008). [CrossRef]

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

9. B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary Surface Voltage Effect in the Invisibility Cloak with an Active Device Inside,” Phys. Rev. Lett. **100**, 063904 (2008). [CrossRef] [PubMed]

10. 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]

11. M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B **79**, 033102 (2009). [CrossRef]

12. M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. **94**, 061903 (2009). [CrossRef]

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

13. 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,” Photonics Nanostruc. Fundam. Appl. **6**, 87 (2008). [CrossRef]

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

15. W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily ellipticalCcylindrical invisible cloaking,” J. Phys. D: Appl. Phys. **41**, 085504 (2008). [CrossRef]

16. M. Yan, Z. Chao, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. **99**, 233901 (2007). [CrossRef]

17. M. Yan, Z. Chao, and M. Qiu, “Scattering characteristics of simplified cylindrical invisibility cloaks,” Opt. Express **15**, 17772C17782 (2007). [CrossRef]

18. 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]

19. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 063904 (2007). [CrossRef]

21. 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]

22. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science **323**, 366–369 (2009). [CrossRef] [PubMed]

23. 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]

24. 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]

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

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

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

*anisotropy ratio*[26

26. C.W. Qiu, L.W. Li, T. S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E **75**, 026609 (2007). [CrossRef]

*ε*=

_{t}*ε*=

_{r}*r*

^{2}=(

*r*−

*R*

_{1})

^{2}, where

*R*

_{1}is the radius of the cloaked region. The subtle point of the singularity in the coordinate transformation at the innner surface of the cloak was analytically shown to correpond to surface voltages [9

9. B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary Surface Voltage Effect in the Invisibility Cloak with an Active Device Inside,” Phys. Rev. Lett. **100**, 063904 (2008). [CrossRef] [PubMed]

28. R. Weder, “A rigorous analysis of high-order electromagnetic invisibility cloaks,” J. Phys. A: Math. Theor. **41**, 065207 (2008). [CrossRef]

*r*

^{2}=(

*r*−

*R*

_{1})

^{2}, it will lead to quite complicated formulations for the field expressions, which cannot be treated by the previous method for a position-dependent anisotropy ratio [8

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

29. C. W. Qiu, L. W. Li, Q. Wu, and T. S. Yeo, “Field representations in general gyrotropic media in spherical coordinates,” IEEE Antennas Wirel. Propagat. Lett. **4**, 467–470 (2007). [CrossRef]

30. C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: Application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propagat. **55**, 3515–3523 (2007). [CrossRef]

## 2. Preliminaries

*r*<

*R*

_{1}) is a perfect electric conductor (PEC) and the intermediate region (

*R*

_{1}<

*r*<

*R*

_{2}) is filled by the nonlinearly transformed spherical cloak, characterized by

*(*ε ¯

*r*) and

*µ*̄(

*r*), which will be discussed below. The electric field is polarized along the

*x*axis and propagating along the

*z*axis.

26. C.W. Qiu, L.W. Li, T. S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E **75**, 026609 (2007). [CrossRef]

26. C.W. Qiu, L.W. Li, T. S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E **75**, 026609 (2007). [CrossRef]

*̄ and*

**ε***̄ of perfect linear spherical cloaks [1*

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

*A*=

_{e}*A*=

_{m}*r*

^{2}=(

*r*−

*R*

_{1})

^{2}, Eq. (1) is reduced to

*f*(

*r*) can be solved in a way similar to that for isotropic materials, except for the change in the argument of resultant Bessel/Hankel functions. However, given a set of

*̄ and*

**ε***̄ derived from a certain transform, the anisotropy ratio may not be*

**µ***r*

^{2}=(

*r*−

*R*

_{1})

^{2}anymore, and then the radial component cannot be solved explicitly in the same way. In this situation, we could approximate the original inhomogeneous anisotropic cloak by many thin, homogeneous anisotropic coatings, and the diffraction problem can thus be solved in terms of analytical Bessel/Hankel functions satisfying boundary conditions at each interface. Nevertheless, the requirement of anisotropic materials still remains. Alternatively, to further alleviate the restriction in material complexity, the original inhomogeneous anisotropic cloaking materials can be approximated by the limit of many thin, concentric, homogeneous,

*isotropic*coatings, forming an effective anisotropic medium, which will be discussed in this section.

*̄=*

**ε***ε*(

_{r}*r*)

*r*̂

*r*̂+

*ε*(

_{t}*r*)(

*θ*̂

*θ*̂+

*ϕ*̂

*ϕ*̂) and

*̄=*

**µ***µ*(

_{r}*r*)

*r*

*̂*̂+

_{r}*µ*(

_{t}*r*)(

*θ*̂

*θ*̂+

*ϕ*̂

*ϕ*̂), in which

*ε*(

_{r}*r*)=

*µ*(

_{r}*r*) and

*ε*(

_{t}*r*)=

*µ*(

_{t}*r*) are position-dependent in general. It is then divided into

*M*initial-layers (anisotropic) but the thickness of individual initial layers may or may not be identical, depending on the transformation. Given a coordinate transformation function

*r*′=

*f*(

*r*) between the virtual space (i.e.,

*Ω*′(

*r*′), 0<

*r*′<

*R*

_{2}) and the compressed space (i.e.,

*Ω*(

*r*),

*R*

_{1}<

*r*<

*R*

_{2}), the stepwise segmentation in physical space (

*r*′) is desired to mimic the transformation function as well as possible and we also desire that the segmentation not be too complicated. We find that equally dividing the

*virtual*space (

*r*0) into

*M*initial-layers will make the segmentation on the physical space (

*r*) “self-adaptive” in a simple way—it automatically uses a finer discretization in regions of physical space where the anisotropic materials are varying more rapidly—which will result in better invisibility performance. This conversion is illustrated in Fig. 2. Throughout this paper, we apply the same condition, i.e., the segmented layers in

*r*′ are of equal thickness, which represents a good choice of segmentation in the initial-layers in the cloak shell

*R*

_{1}<

*r*<

*R*

_{2}. By projecting the segmentation in

*r*′

*=*

_{n}*R*

_{2}·

*n*=

*M*onto the physical

*r*, one has

*r*=

_{n}*f*

^{−1}(

*r*′

*); (n=1, 2,…, M). Thus, the geometry of every initial-layer in Fig. 2(a) is determined.*

_{n}*r*−

_{n}*r*

_{n}_{−1})=

^{2}, as shown in Fig. 2(b). The material parameters of type-A and type-B isotropic dielectrics can be inferred and derived from the result for radial conductivity by Sten [31

31. J. C. E. Sten, “DC fields and analytical image solutions for a radially anisotropic spherical conductor,” IEEE Trans. Diel. Elec. Insul. **2**, 360–367 (1995). [CrossRef]

*n*-th initial layer into its pair of effective isotropic sub-layers, we need to pick a specific radial position for those

*ε*and

_{r}*ε*on the right-hand side of Eq. (3) within the initial layer in order to determine corresponding parameters of the two isotropic dielectrics (

_{t}*ε*,

_{A}*µ*) and (

_{A}*ε*,

_{B}*µ*) on the left-hand side of Eq. (3) and in Fig. 2(b). According to [33], such a discretization mechanism using

_{B}*r*=

*r*in Eq. (3) for the

_{n}*n*-th initial layer will give a good compromise for both forward and backward scatterings, while retaining good invisibility performance.

## 3. Effects of Nonlinear Transformation in Nonlinear Spherical Cloaks

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

*r*′ against the physical radius

*r*. In what follows, we introduce two classes of nonlinear-transformation spherical cloaks, and we discuss how to restore and improve the invisibility performance after discretization by choosing a proper nonlinear transformation and by choosing a suitable compensation scheme while discretizing the original spherical cloak into multilayer isotropic structures. The two types of spherical cloaks considered here are classified in terms of the negative (i.e., concave-down) or positive (i.e., concave-up) sign of the second derivative of the transformation function. It is important to reiterate that all three designs—linear, concave-up, and concave-down—are perfect cloaks for the exact inhomogeous design, and we are only considering the breakdown of invisibility when the design is discretized into homogeneous layers.

### 3.1. Concave-Down Nonlinear Transformation

*R*

_{2}in Ω′ (original) space into a shell at the region

*R*1<

*r*<

*R*

_{2}in W (compressed) space, we propose a class of prescribed functions

*x*denotes the degree of the nonlinearity in the transformation. When

*x*is very small in Eq. (4), the curves are difficult to distinguish and all approach to the same limiting case when

*x*→0:

*̄,*

**ε***̄ in the transformed coordinates can be written in term of*

**µ***̄′,*

**ε***̄′ in the original space by*

**µ***A*is the Jacobian matrix with elements defined as

*A*=

_{ki}*∂*=

_{k}*∂*′

*.*

_{i}*r*, and we term this class of transformations the

*concave-down*nonlinear transformation. The nonlinear transformation function in Eq. (4) only depends on the radial component

*r*in the spherical coordinate system (

*r*,

*θ*,

*ϕ*). Thus it is easy to find that the Jacobian matrix

*A*is diagonal. Considering that the original space is filled with air (

*R*

_{1}<

*r*<

*R*

_{2}) are shown to be

*A*=

_{e,m}*ε*/

_{t}*ε*=

_{r}*x*

^{2}

*R*

^{2x}

_{1}=(

*r*−

^{x}*R*

^{x}

_{1})

^{2}in our current case. In addition, one can also find that when the virtual space is equally discretized using a concave-down nonlinear transformation, the segmentation of the initial layers in Fig. 2 in the physical space

*r*is

*x*in the near-field and far-field of the discretized nonlinear-transformation spherical cloaks. We fix M=40, and consider the cases of x=0.1, x=1, x=4, and x=10 in concave-down nonlinear cloaks, shown in Fig. 4. It can be seen from Figs. 4(a–d) that when

*x*increases, the magnitude of electric field increases significantly inside the cloak. This is because more energy is guided towards the inner boundary of the PEC core, which in turn makes the cloaked PEC more

*visible*to external incidences. To prove that the large electric fields only occur in the region

*R*

_{1}<

*r*<

*R*

_{2}and to study the effect of

*x*on individual near-field perturbations more explicitly, only the fields outside the cloak are presented in Figs. 4(e–f). One can again confirm that, for discretized concave-down nonlinear spherical cloaks, smaller

*x*leads to smaller disturbance in electric fields in the outer space. In particular, the degradation in the invisibility in the forward direction is proportional to the value of

*x*(see Fig. 4(g) and Fig. 4(h)).

*M*initial-layers in the physical space (see Fig. 3). From Fig. 5, it can be seen that for concave-down nonlinear transformations, the cloaking property is better retained when

*x*is small. Except for the forward direction, concave-down nonlinear transformation with small

*x*can achieve even lower RCS over a wide range of observation angles. As

*x*keeps increasing, their RCSs are increasing dramatically and can be larger than that of a uncloaked PEC core (e.g.,

*x*=10). No matter how large

*x*becomes, the suppressed backward scattering is still maintained [8

**99**, 063903 (2007). [CrossRef] [PubMed]

### 3.2. Concave-Up Nonlinear Transformation

*x*→0, it also approaches to the same limit as Eq. (5). In the same manner, we can obtain the desired parameters in the compressed space (

*R*

_{1}<

*r*<

*R*

_{2})

*x*=1 is exactly Pendry’s linear spherical cloak.

*concave-up*nonlinear transformations. Also, it is found that the limiting case of Eq. (5), i.e,

*x*→0, is the dividing line between the convacedown and concave-up classes. Since

*x*≪1 in Eq. (11) is actually very close to the situation of

*x*≪1 for concave-down nonlinear transformations, we will not revisit that case here. Instead, we only consider

*x*=1,

*x*=4, and

*x*=10 concave-up transformations [Eq. (12)], and the segmentation for concave-up nonlinear spherical cloaks in physical space

*r*turns out to be

*x*, concave-up transformation cloaks have lower peak amplitudes than concave-down transformation cloaks, i.e., the perturbation inside the cloak region is smaller; (3) the perturbation outside the cloak of concave-up transformation cloaks is also lower than that of concave-down transformation cloaks in all cases of

*x*; (4) the invisibility performance is better maintained for concave-up transformations even when

*x*=10 or even larger. Hence, spherical cloaks based on concave-up nonlinear transformations exhibit better invisibility after discretization.

*x*. More importantly, when

*x*increases (e.g.,

*x*=4 or

*x*=10), though the RCS will be a bit larger than Pendry’s linear cloak at most angles, the RCS near the forward direction will be reduced greatly compared with that of classic linear cloak. Such far-field phenomena are also connected with the near-field patterns. It is important to point out that the increase of x in concave-up transformations [see Figs. 7(a–c)] will push the “hot” areas (where the electric field is very high) further and further away from the spherical PEC core, so the induced shadow and the far-field pattern become stable with increasing

*x*. However, the increase of

*x*in the concave-down transformations has little effect in shifting away the “hot” positions from the core [see Fig. 4(a–d)], which results in larger interaction with the PEC core. Hence, the RCS reduction in the far field is degraded with the increase of

*x*in the concave-down class as shown in Fig. 5.

## 4. Justification of Improved Segmentation

*M*during the discretization process, otherwise the conversion scheme in Fig. 2 will lose its accuracy. We also find that a nonlin- ear transformation with a given

*x*of its concave-up class will yield a better invisibility by taking the segmentation as Eq. (13) in the physical space (this is also the only way to determine sublayers of type-A and type-B dielectrics in practice). Here the justification is given. The concave-up class with

*x*=4, which is characterized by the mapping curve

*r*(1) for concave-up cloaks corresponds to the case of equally dividing the physical space.

_{n}*r*(

_{n}*x*) sets (see Eq. 13) for real radius

*r*are applied to

*r*′(4), e.g., the sets of

*x*=1 (

*r*(1)),

_{n}*x*=2 (

*r*(2)),

_{n}*x*=6 (

*r*(6)) and

_{n}*x*=10 (

*r*(10)). The total cross sections corresponding to those four sets are compared with the best segmentation set of its own

_{n}*r*(4), i.e., equally dividing the virtual space of

_{n}*x*=4 into 40 initial-layers and then projecting onto physical space. We find that, for concave-up class

*x*=4,

*r*(4) is indeed its optimal segmentation among these possibilities.

_{n}## 5. Conclusion

32. B. I. Popa and S. A. Cummer, “Cloaking with optimized homogeneous anisotropic layers,” Phys. Rev. A **79**, 023806 (2009). [CrossRef]

## 6. Acknowledgement

## References and links

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

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

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

4. | D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express |

5. | A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. |

6. | H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. |

7. | M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. |

8. | H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. |

9. | B. Zhang, H. Chen, B. I. Wu, and J. A. Kong, “Extraordinary Surface Voltage Effect in the Invisibility Cloak with an Active Device Inside,” Phys. Rev. Lett. |

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

11. | M. Farhat, S. Guenneau, S. Enoch, and A. Movchan, “Cloaking bending waves propagating in thin elastic plates,” Phys. Rev. B |

12. | M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. |

13. | 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,” Photonics Nanostruc. Fundam. Appl. |

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

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

16. | M. Yan, Z. Chao, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. |

17. | M. Yan, Z. Chao, and M. Qiu, “Scattering characteristics of simplified cylindrical invisibility cloaks,” Opt. Express |

18. | 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 |

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

20. | B. Zhang, H. Chen, and B. I. Wu, “Limitations of high-order transformation and incident angle on simplified invisibility cloaks,” Opt. Express |

21. | 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 |

22. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science |

23. | 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 |

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

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

26. | C.W. Qiu, L.W. Li, T. S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E |

27. | W. Cai, U. K. Chettiar, A.K. Kildishev, G. W. Milton, and V. M. Shalaev, “Non-magnetic cloak without reflection,” arXiv:0707.3641v1. |

28. | R. Weder, “A rigorous analysis of high-order electromagnetic invisibility cloaks,” J. Phys. A: Math. Theor. |

29. | C. W. Qiu, L. W. Li, Q. Wu, and T. S. Yeo, “Field representations in general gyrotropic media in spherical coordinates,” IEEE Antennas Wirel. Propagat. Lett. |

30. | C. W. Qiu, S. Zouhdi, and A. Razek, “Modified spherical wave functions with anisotropy ratio: Application to the analysis of scattering by multilayered anisotropic shells,” IEEE Trans. Antennas Propagat. |

31. | J. C. E. Sten, “DC fields and analytical image solutions for a radially anisotropic spherical conductor,” IEEE Trans. Diel. Elec. Insul. |

32. | B. I. Popa and S. A. Cummer, “Cloaking with optimized homogeneous anisotropic layers,” Phys. Rev. A |

33. | C. W. Qiu, L. Hu, and L. Gao, “Trade-off between forward and backward scatterings of Linear and Nonlinear Spherical Invisibility Cloaks,” Progress In Electromagnetics Research, to be submitted (2009). |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(230.3205) Optical devices : Invisibility cloaks

(290.5839) Scattering : Scattering, invisibility

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 12, 2009

Revised Manuscript: July 7, 2009

Manuscript Accepted: July 10, 2009

Published: July 21, 2009

**Citation**

Chengwei Qiu, Li Hu, Baile Zhang, Bae-Ian Wu, Steven G. Johnson, and John D. Joannopoulos, "Spherical cloaking using nonlinear
transformations for improved
segmentation into concentric isotropic
coatings," Opt. Express **17**, 13467-13478 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13467

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (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, 9794-9804 (2006). [CrossRef] [PubMed]
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