## Isotropic non-ideal cloaks providing improved invisibility by adaptive segmentation and optimal refractive index profile from ordering isotropic materials |

Optics Express, Vol. 18, Issue 14, pp. 14950-14959 (2010)

http://dx.doi.org/10.1364/OE.18.014950

Acrobat PDF (1469 KB)

### Abstract

Mimicking the ideal cloak, which is anisotropic and inhomogeneous, can be achieved by alternating homogeneous isotropic materials, whose permittivity and permeability of each isotropic coating can be determined from effective medium theory. An improved two-fold method is proposed by optimally discretizing the cloak and re-ordering the combination of the effective parameters of each layer to form a smooth step-index profile. The roles of impedance matching and index matching are investigated for cloaking effects. Smoothing the index profile leads to better invisibility than that obtained by smoothing the impedance profile, since the forward scattering can be further diminished. Nonlinear-transformation-based spherical ideal cloaks are studied, and improved design method is explored together with different segmentation schemes. Significant improvement in invisibility is always observed for the optimal segmentation in virtual space with the proposed two-fold design method no matter how nonlinear the coordinate transformation is.

© 2010 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. 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]

5. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E **72**, 016623 (2005). [CrossRef]

6. A. Alu and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express **15**, 3318–3332 (2007). [CrossRef] [PubMed]

7. L. Gao, T. H. Fung, K. W. Yu, and C.-W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E **78**, 046609 (2008). [CrossRef]

8. X. Cai, Q. Deng, and G. Hu, “Experimental study on electromagnetic wave transparency for coated metallic cylinders,” J. Appl. Phys. **105**, 103112 (2009). [CrossRef]

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

10. 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]

11. C.W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloak,” Phys. Rev. E **80**, 016604 (2009). [CrossRef]

12. 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]

13. H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express **16**, 15449–15454 (2008). [CrossRef] [PubMed]

14. 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]

15. 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]

16. 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**, 113001 (2009). [CrossRef]

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

17. 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]

18. 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]

19. 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]

11. C.W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloak,” Phys. Rev. E **80**, 016604 (2009). [CrossRef]

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

16. 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**, 113001 (2009). [CrossRef]

16. 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**, 113001 (2009). [CrossRef]

11. C.W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloak,” Phys. Rev. E **80**, 016604 (2009). [CrossRef]

17. 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]

**80**, 016604 (2009). [CrossRef]

22. C.W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E **79**, 047602 (2009). [CrossRef]

23. M. Zedler, C. Caloz, and P. Russer, “A 3-D isotropic left-handed metamaterial based on the rotated transmission-line matrix (TLM) scheme,” IEEE Trans. Microw. Theory Tech. **55**, 2930–2941 (2007). [CrossRef]

24. P. Alitalo, O. Luukkonen, L. Jylha, J. Venermo, and S. A. Tretyakov, “Transmission-line networks cloaking objects from electromagnetic fields,” IEEE Trans. Antennas Propagat. **56**, 416–424 (2008). [CrossRef]

22. C.W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E **79**, 047602 (2009). [CrossRef]

22. C.W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E **79**, 047602 (2009). [CrossRef]

**79**, 047602 (2009). [CrossRef]

**79**, 047602 (2009). [CrossRef]

## 2. NTB spherical cloaks and design method for improved invisibility

*z*axis with the electric field polarized along

*x*axis. The core region (0 <

*r*<

*R*

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

*R*

_{1}<

*r*<

*R*

_{2}) is filled by the NTB cloak, characterized by the relative parameters

*(*

**ε**̅*r*) =

*ε*(

_{r}*r*)

*r̂r̂*+

*ε*(

_{t}*r*)(

*+*θ ^ θ ^

*) and*ϕ ^ ϕ ^

*=*

_{μ}̅*μ*(

_{r}*r*)

*r̂r̂*+

*μ*(

_{t}*r*)(

*+*θ ^ θ ^

*) (*ϕ ^ ϕ ^

*ε*(

_{r}*r*) =

*μ*(

_{r}*r*) and

*ε*(

_{t}*r*) =

*μ*(

_{t}*r*)). The prescribed function

*f*(

*r*) to transform the original virtual space

*r*′ to the compressed physical space

*r*is designed to be of a nonlinear class

*x*denotes the nonlinearity of the spatial compression. Based on the well-established TO method [1

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

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

17. 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]

**80**, 016604 (2009). [CrossRef]

18. 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]

**79**, 047602 (2009). [CrossRef]

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

*f*(

*r*) is a linear straight line with respect to

*r*(i.e.,

*x*= 1 in Eq. (1)), and thus it does not make difference whether one equally discretizes the cloak in either physical (i.e., Ω(

*r*),

*R*

_{1}<

*r*<

*R*

_{2}) or virtual (i.e., Ω′(

*r*′), 0 <

*r*′ <

*R*

_{2}) spaces. On the contrary, given a nonlinear prescribed function

*r*′ =

*f*(

*r*) between the virtual space and the physical space, the self-adaptive stepwise segmentation in physical space

*r*—finer segmentation is needed where

*f*(

*r*) turns to be more steep—is desired to mimic the transformation function. Hence it is obvious to find that the most economic way is to equally discretize the virtual space

*r*′ into

*M*layers and project them onto physical space

*r*via the transformation curve. Then we can obtain a set of

*r*which automatically control the steps according to the slope of

_{n}*f*(

*r*), e.g., in the region where

*f*(

*r*) is changing dramatically, more discretized anisotropic layers are assigned, and less for the region where

*f*(

*r*) seems flat. In this paper, this self-adaptive scheme is adopted throughout unless stated otherwise (e.g., for comparison purposes in Figs. 7 and 8), and its advantage is also justified by comparing with the cloaking performance obtained from directly dividing the physical space

*r*into

*M*layers of identical thickness. By projecting the equal segmentation in

*r*′

_{n}=

*R*

_{2}·

*n*/

*M*onto the physical

*r*, one has

*ε*(

_{r}*r*) =

*μ*(

_{r}*r*) and

*ε*(

_{t}*r*) =

*μ*(

_{t}*r*), the

*set of material parameters of isotropic medium-A and medium-B dielectrics can be obtained [22*

**old****79**, 047602 (2009). [CrossRef]

25. 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]

*set*

**new***M*cannot be too small (we choose

*M*= 40 herein). Nevertheless, based on the optimization algorithm, a much smaller

*M*can be used as shown in [26

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

*r*and

_{n−1}*r*to determine the anisotropic parameters in Eq. (2), and thus we assume the thin shell in the upper-left illustration in Fig. 1 has uniform anisotropic parameters. Now, one can apply either Eq. (4) or Eq. (5) to derive the isotropic parameters needed in the lower-left illustration in Fig. 1. However, it is found that this pre-chosen position within a thin shell cannot be randomly put even if each discretized shell itself is thin enough to be treated as a homogeneous anisotropic shell. In Fig. 2, we consider three simple choices of this pre-chosen position for the

_{n}*n*-th anisotropic shell, i.e., the left (

*r*=

*r*

_{n−1}); the middle (

*r*= (

*r*

_{n−1}+

*r*)/2); and the right (

_{n}*r*=

*r*). It can be seen that, those three choices in Fig. 2 will lead to significantly distinct performances in scattering reduction as shown in Fig. 3.

_{n}*set of effective isotropic materials, selecting the parameters corresponding to the middle point*

**new****M**of each discretized ansotropic layer as a uniform anisotropic layer will dramatically outperform the other two choices (i.e., the left

**L**and right

**R**) especially the coordinate transformation becomes more nonlinear, and this invisibility improvement is quite stable along with the increase in the nonlinearity factor

*x*. On the contrary, for the

**set, selecting the right point**

*old***R**in Fig. 2 gives rise to lowest total scattering cross section. Nevertheless, for nonlinear transformation based cloaks, the choice of the middle point

**M**under the

**set leads to the most improved invisibility for isotropic non-ideal cloaks. Therefore, the choice of middle point**

*new**M*is adopted in the following.

## 3. Numerical results and verifications of improved cloaking effects

*x*= 0.1,

*x*= 4, and

*x*= 20 in Eq. (1) corresponding to small, medium, and large nonlinear transformations, respectively. In Fig. 4, the bistatic radar cross section (RCS) is plotted to demonstrate the invisibility as well as the improvement of the

**isotropic cloak compared with the**

*new***one. It can be observed that the non-ideal cloak made of either**

*old***set of effective isotropic layers in Eq. (4) or**

*old***set in Eq. (5) are able to reduce the scattering induced by the original bare PEC sphere. However the forward scattering of the**

*new***set becomes even larger than that of a bare PEC sphere, which is not desired (**

*old**see*blue lines in Fig. 4). It can be overcome by considering the

**set which further lowers down the forward scattering by more than 25 dBsm, and this huge improvement is stable even when the spatial compression is quite nonlinear (i.e., very large values of**

*new**x*), thanks to the proposed self-adaptive discretization scheme. There is a small sacrifice in backscattering of the improved

**set compared with the backscattering of the**

*new***set, but it is evidently worthwhile because the backscattering of the**

*old***set is still negligible.**

*new**f*(

*r*). It is clear to see that the

**set leads to much lower scattering particularly in forward direction, which in turn brings down the total SCS. It can be concluded that the reduction in forward scattering significantly contributes to the total scattering suppression, and the proposed self-adaptive scheme in Eq.(3) makes the improved invisibility pronounced continuously for NTB spherical cloaks at arbitrary nonlinearity**

*new**x*in spatial transformation.

*x*is very small and

*x*is very large. Fig. 6 reveals that even though the far-field diagrams for

*x*= 0.1 and

*x*= 20 are quite similar as Fig. 4 shows, their corresponding near fields, especially in the cloak shell region, are distinct. In Fig. 6, it is shown that when

*x*increases, the field intensity inside the cloak shell will be increased significantly while the nearly undisturbed wave fronts still hold outside the cloak (

*r*>

*R*

_{2}) as shown in the inset of Fig. 6(b). In addition, those high-intensity areas will be more squeezed into the region towards the outer radius

*R*

_{2}when

*x*is getting larger. The highly nonlinear transformation based cloak can be realized as an isotropic non-ideal cloak, restore the invisibility after the self-adaptive discretization, and produce improved invisibility by using the

**set of effective mediums.**

*new***and**

*new***sets of isotropic parameters in respective cases as shown in Fig. 7.**

*old***set of parameters designed in Eq. (5) while the difference in choosing “virtual” or “physical” discretization scheme is negligible. It is due to the fact that when**

*new**x*= 0.1 the transformation function

*f*(

*r*) is nearly linear against

*r*. Under such circumstance, if one equally discretizes the ideal NTB cloak in virtual space and then projects onto physical space, the stepping intervals will be almost the same as those obtained from equally dividing the physical space directly. However, if

*x*becomes large enough, namely,

*f*(

*r*) is quite nonlinear, one has to adopt the current two-fold method to achieve improved invisibility: i) “virtual” discretization scheme is used; ii) the new

**of effective parameters is used. One can observe that the “virtual” scheme is even more important than adopting the**

*set***set of parameters by inspecting blue solid line and green dashed line in Fig. 7(b): even though the former (blue solid line in Fig. 7(b)) uses**

*new***set in Eq. (4), both the total and forward scattering are much smaller than the latter (green dashed line in Fig. 7(b)) based on the**

*old***set in Eq. (5). Of course, it is evident that our current two-fold method (“virtual, new” corresponding to the green solid line) clearly outperforms all other cases in terms of scattering reduction.**

*new**x*= 20 as shown in Fig. 7(b). We compare the impedance and index profiles for the worst (“physical, old”) and the best candidates (“virtual, new”) in Fig. 8. It can be seen in Fig. 8(a) that the impedance is always matched with that of free space for “physical, old” case designed by equally dividing the physical space and using Eq. (4) while the relative impedance of isotropic cloak for “virtual, new” based on our two-fold method has only two values, i.e., 0.025 and 40 which changes in an alternating manner in the cloak region. In Fig. 8(b), our two-fold method leads to a much smoother index profile which matches that of the free space at the outer radius

*R*

_{2}. It can be seen that, for these isotropic non-ideal cloaks realized from ideal NTB anisotropic cloaks, a smooth index profile would be more crucial in achieving improved invisibility than a smooth impedance profile. The index matching will diminish the forward scattering further down though a negligible increase in backscattering may occur, while the impedance match is mainly efficient in removing the backscattering. Therefore, the effective medium of each interlayer has to be properly determined to take into account the slope of

*f*(

*r*) and as the index profile.

## 4. Conclusion

**set of effective medium with oscillating impedance profile but smooth index profile further lowers down the scattering in forward scattering with a rewarding sacrifice in backscattering which is still weak enough to be ignored.**

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

5. | A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E |

6. | A. Alu and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express |

7. | L. Gao, T. H. Fung, K. W. Yu, and C.-W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E |

8. | X. Cai, Q. Deng, and G. Hu, “Experimental study on electromagnetic wave transparency for coated metallic cylinders,” J. Appl. Phys. |

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

10. | 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. |

11. | C.W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloak,” Phys. Rev. E |

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

13. | H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express |

14. | 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 |

15. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

16. | 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. |

17. | H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. |

18. | 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 |

19. | 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. |

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

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

22. | C.W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E |

23. | M. Zedler, C. Caloz, and P. Russer, “A 3-D isotropic left-handed metamaterial based on the rotated transmission-line matrix (TLM) scheme,” IEEE Trans. Microw. Theory Tech. |

24. | P. Alitalo, O. Luukkonen, L. Jylha, J. Venermo, and S. A. Tretyakov, “Transmission-line networks cloaking objects from electromagnetic fields,” IEEE Trans. Antennas Propagat. |

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

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

**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: December 1, 2009

Revised Manuscript: December 25, 2009

Manuscript Accepted: December 25, 2009

Published: June 29, 2010

**Citation**

C.W. Qiu, L. Hu, and S. Zouhdi, "Isotropic non-ideal cloaks providing improved invisibility by adaptive
segmentation and optimal refractive index profile from ordering isotropic
materials," Opt. Express **18**, 14950-14959 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14950

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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]
- U. Leonhardt,“Optical conformal mapping,” Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- 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]
- A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005). [CrossRef]
- A. Alu and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318 – 3332 (2007). [CrossRef] [PubMed]
- L. Gao, T. H. Fung, K. W. Yu, and C.-W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008). [CrossRef]
- X. Cai, Q. Deng, and G. Hu, “Experimental study on electromagnetic wave transparency for coated metallic cylinders,” J. Appl. Phys. 105, 103112 (2009). [CrossRef]
- 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 ellipticalCcylindrical invisible cloaking,” J. Phys. D: Appl. Phys. 41, 085504 (2008). [CrossRef]
- C. W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloak,” Phys. Rev. E 80, 016604 (2009). [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]
- H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449 - 15454 (2008). [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]
- 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]
- 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, 113001 (2009). [CrossRef]
- 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]
- 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]
- 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]
- W. Cai, U. K. Chettiar, A. K. Kildishev, G. W. Milton, and V. M. Shalaev, “Non-magnetic cloak without reflection,” arXiv:0707.3641v1.
- R. Weder, “A rigorous analysis of high-order electromagnetic invisibility cloaks,” J. Phys. A: Math. Theor. 41, 065207 (2008). [CrossRef]
- C. W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E 79, 047602 (2009). [CrossRef]
- M. Zedler, C. Caloz, and P. Russer, “A 3-D isotropic left-handed metamaterial based on the rotated transmissionline matrix (TLM) scheme,” IEEE Trans. Microw. Theory Tech. 55, 2930-2941 (2007). [CrossRef]
- P. Alitalo, O. Luukkonen, L. Jylha, J. Venermo, S. A. Tretyakov, “Transmission-line networks cloaking objects from electromagnetic fields,” IEEE Trans. Antennas Propagat. 56, 416-424 (2008). [CrossRef]
- 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]
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