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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 9422–9427
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Design of invisibility anti-cloak for two-dimensional arbitrary geometries

Long Li, Feifei Huo, Yuanming Zhang, Yang Chen, and Changhong Liang  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 9422-9427 (2013)
http://dx.doi.org/10.1364/OE.21.009422


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Abstract

In this paper, we derive the material parameter formulae for designing an electromagnetic invisibility anti-cloak of two-dimensional arbitrary geometry, which is conformal with the cloaked object. Different shapes of electromagnetic invisibility anti-cloaks are proposed to verify the correctness and effectiveness of the proposed formulae. The simulation results show that the invisibility anti-cloak can break cloak shielding and make the external electromagnetic waves into the cloak. This is not only to realize the transfer of information, but will not affect the role of cloak of stealth.

© 2013 OSA

1. Introduction

As electronic devices occupy an increasingly important position in modern warfare, how to make an object invisible from the enemy radar has become a hot research spot in recent years. In 2006, Pendry et al. proposed a new vision-electromagnetic invisible cloak, which is based on the optical transformation [1

1. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

]. The coordinate transformation is used to squeeze space to control the electromagnetic waves propagation in metamaterials, so that the target in the cloak is invisible to the outside radiation [2

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

]. The method for controlling electromagnetic fields is based on Maxwell’s equations form invariance principle in different spatial coordinates [3

3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006). [CrossRef]

5

5. 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(10), 248 (2006). [CrossRef]

]. The theory of the electromagnetic cloak of stealth was first confirmed in the microwave regime by experiment [6

6. 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(5801), 977–980 (2006). [CrossRef] [PubMed]

]. In [7

7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

], the working principle was demonstrated in optical frequencies by using non-magnetic materials and tiny metal segments to constitute the invisibility cloak. The transformation approach has been extended for many electromagnetic applications [8

8. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]

11

11. J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009). [CrossRef] [PubMed]

]. Subsequently, a lot of analyses and research of electromagnetic cloaks have been done, such as full-wave simulations employing the finite element methods and the finite difference time domain methods, and the analysis of Fourier-Bessel functions [12

12. D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010). [CrossRef]

, 13

13. X. Chen, Y. Fu, and N. 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]

].

However, most of aforementioned examples verified the perfect effect of cloak of stealth, but did not consider embedded objects with anisotropic material inside the cloak. In [14

14. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008). [CrossRef] [PubMed]

], Chen et al. proposed a new concept-electromagnetic invisibility anti-cloak. An anti-cloak with a specific form of negative index anisotropic material is embedded inside the cloak, makes the external electromagnetic wave penetrate into the interior of the invisibility cloak and partially defeats the cloaking effect. This concept has been elaborated in [15

15. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009). [CrossRef] [PubMed]

18

18. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(1), 016603 (2011). [CrossRef] [PubMed]

]. In [15

15. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009). [CrossRef] [PubMed]

], the objects in the cloaked domain can receive the external energy via a “tunneling” between cloak and anti-cloak, and the waveform of electromagnetic waves is kept unchanged. In other words, the objects are still invisible for the outside.

In this paper, according to the anti-cloak theory, we derive the transformation formulae of the permittivity and permeability for designing the electromagnetic invisibility anti-cloak of two-dimensional arbitrary geometries. Different shapes of electromagnetic invisibility anti-cloaks are proposed to verify the correctness and effectiveness of the proposed formulae.

2. Invisibility anti-cloak for two-dimensional arbitrary geometries

Maxwell’s equations are form-invariant under the coordinate transformation. According to the coordinate transformation, a volume in the virtual space can be compressed into a shell surrounding the concealment volume in the real space [19

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

]. This shell is the electromagnetic invisibility cloak which has the specific permittivity and permeability. The cloak can control the electromagnetic waves bypassing the concealed region and restore the waveform outside the cloak, so that the observer is insensible of the objects inside the cloak.

Assume the coordinate for the virtual space is (u,v,w), and the coordinate for the real space is (u',v',w'), so the coordinate transformation can be described by
{u'=u'(u,v,w)v'=v'(u,v,w)w'=w'(u,v,w).
(1)
Furthermore, the material parameters permittivity tensor ε'¯¯ and permeability tensor μ'¯¯ are obtained via
ε¯¯'=Aε¯¯ATdetA,μ¯¯'=Aμ¯¯ATdetA,
(2)
where A is the Jacobi matrix of the transformation from the virtual space to the real space defined by

A=[u'/uu'/vu'/wv'/uv'/vv'/ww'/uw'/vw'/w].
(3)

Figure 1(a)
Fig. 1 (a) Schematic diagram of stealth cloak with anti-cloak. (b) Coordinate transformation of the anti-cloak.
shows the structure of a two-dimensional stealth cloak with an anti-cloak. Using a cylindrical coordinate system, the anti-cloak shell is embedded between the cloak shell and the masked object which permits the electromagnetic waves propagate into the cloak and the anti-cloak. The relative permittivity and permeability of the inner cylinder are εr and μr, and the outside of the cloak is vacuum (εr=1,μr=1).

To create a cloak shown in Fig. 1, we use a spatial transformation that compresses the cylindrical region with 0rR3(θ) into an annular region with R2(θ')r'R3(θ'),
r'=R3(θ)R2(θ)R3(θ)r+R2(θ),θ'=θ,z'=z,0rR3(θ).
(4)
The coordinate transformation of the anti-cloak is given as follows
r'=R1(θ)R2(θ)R1(θ)r+R2(θ),θ'=θ,z'=z,0rR1(θ).
(5)
The region with 0rR1(θ) in the virtual space is folded into the region with R1(θ')r'R2(θ') in the real space. The whole transformation is illustrated in Fig. 1(b). If we set that the stealth cloak and the anti-cloak have the same shapes of inner and outer boundaries with the object, i.e. conformal cloak and anti-cloak, the coefficient of the transformation r'=f(r) is a constant. Let (R3(θ)R2(θ))/R3(θ)=t2 and (R1(θ)R2(θ))/R1(θ)=t1, according to Eqs. (4) and (5), we get the uniform coordinate transformation form of the cloak and the anti-cloak
r'=tir+R2(θ),i=1,2.
(6)
Substituting Eq. (6) into Eq. (2), we can obtain the permittivity and the permeability of the cloak and the anti-cloak. Because of the uniform of ε'¯¯ and μ'¯¯, the components of the relative permittivity tensor can be given in cylindrical coordinates as follows
{εrr={r'R2(θ')r'+1r'[r'R2(θ')][R2(θ')θ]2}εrεrθ=εθr=1r'R2(θ')R2(θ')θεrεθθ=r'r'R2(θ')εrεzz=r'R2(θ')ti2r'εr.
(7)
From the above formulae, it is found that in the region of the anti-cloak (r'<R2(θ')), the corresponding constitutive parameters are opposite in sign to the inner cylinder [15

15. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009). [CrossRef] [PubMed]

]. The electromagnetic parameters are negative, i.e. double negative (DNG) materials which could be realized by using metamaterials. In addition, the parameters of the cloak and the anti-cloak have the same forms in r-direction and in θ-direction, and the only difference is the components with different constant factors in z-direction.

3. Full-wave simulation and analysis

Firstly, a two-dimensional cylindrical anti-cloak is established to verify the above formulae. As shown in Fig. 2(b)
Fig. 2 Total electric field distribution near the anti-cloak under an incident TE plane-wave from left to right for (a) a dieletic cylindrical object (εr=2,μr=1) with radius 100mm in a cloak; (b) a cylindrical object (εr=2,μr=1) surrounded by an ideal anti-cloak embedded in the cloak; (c) a cylindrical object (εr=2,μr=1) surrounded by a simplified anti-cloak embedded in the cloak.
, the cylinder is put along z-direction. The inner radius of the anti-cloak is R1(θ')=a=100mm, the outer radius is R2(θ')=b=200mm, and R3(θ')=c=400mm, the anti-cloak is filled with an inner cylinder (εr=2,μr=1). In a two-dimensional space, the electromagnetic wave can be divided into a transverse electric (TE) polarization and a transverse magnetic (TM) polarization. If the source is a TE-polarized plane-wave spread vertical z direction, the effective electromagnetic parameters are εzz, μrr and μθθ.

To demonstrate the electromagnetic properties of the anti-cloak, a commercial simulation tool based on the finite-element methods, Comsol Multiphysics [20

20. Comsol Multiphysics” (Comsol AB), <http://www.comsol.com>.

] was used in this paper. A TE-polarized plane-wave illuminates from the left to the right, and the anti-cloak works at the frequency of 700MHz.

Figure 2(a) shows the total electric field when the cylindrical object with εr=2and μr=1 is enclosed by an annular cloak. It can be seen that the electromagnetic wave is smoothly bent around the cloaked region and the waveform is restored after the wave depart from the cloak. The object is invisible to the observer in this case. Simultaneously, the object cannot receive any incoming wave outside the cloak, and be “blindness” for the external situation. When a conformal anti-cloak with DNG materials is laid directly between the cloak and the object, it is capable of receiving the outer fields and restoring the plane waveform. Shown in Fig. 2(b) is an ideal anti-cloak with anisotropic parameters which derived from Eq. (7), the cloaking performance is perfect. Note that μθθ is infinite at the inner boundary of the cloak and the outer boundary of the anti-cloak, which is difficult to realize in practice, so we can use the simplified parameters as shown in [21

21. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006). [CrossRef] [PubMed]

]. The cloak will have constants εzz=4 and μθθ=1 with μrr varying radially from 0 to 0.25, while the anti-cloak have constants εzz=2 and μθθ=1 with μrr varying radially from −1 to 0. The result shown in Fig. 2(c) indicates that the reduced cloaking material degrades cloaking performance but makes the anti-cloak easier to realize in practice with metamaterials.

Furthermore, several anti-cloaks with different special geometries, such as an elliptic anti-cloak with a plane curve obeying the equation in polar coordinate R2(θ)=b1+3cos2(θ)/(3sin2(θ)+1), a -contour anti-cloak with a plane curve obeying the equation in polar coordinate R2(θ)=b1+3cos2(θ), and an anti-cloak with arbitrary geometry expressed by the equation in polar coordinate R2(θ)=b(4+sin(θ)+sin(3θ)+1.5sin(5θ)) are established and simulated. They also obey the phenomenon that mentioned above. The full-wave simulation resuls are shown in Figs. 3(a)
Fig. 3 Total electric fields distribution near the anti-cloak filled with dielectric (εr=2,μr=1) under an incident TE plane-wave from left to right for (a) an elliptical anti-cloak; (b) a -contour anti-cloak; (c) an arbitrary geometry anti-cloak.
-3(c), respectively, which verify the effectiveness and correctness of the proposed anti-cloak design.

In addition, the formulae also can be applied to a polygonal anti-cloak. A hexagon anti-cload shown in Fig. 4(a)
Fig. 4 Total electric field distribution near the anti-cloak filled with dielectric (εr=2,μr=1) under an incident TE plane-wave from left to right for (a) a hexagon anti-cloak; (b) a discretization anti-cloak for unknown boundary object.
is divided into six regions, and each region has a different contour equation. Substituting the contour equation into the formula we get the permittivity and permeability of each region.

As illustrated in Eq. (7), we observe that if the contour equation and electromagnetic parameters of an object are known, the parameters of the anti-cloak for this object can be deduced. However, at most time, the contour of a random object is difficult to be expressed with an equation. In this case, we can make the formulae discretization by sampling some points on the boundary of the object. Because the distances from the sampling points to the origin are easily measured, more sampling points can be used to restore the boundary. Moreover, the distances from the anti-cloak boundary to the origin are known for the reason that the anti-cloak is conformal with the stealth object. In order to verify this method is effective, we check the stealth effect of an anti-cloak with a random shape. The discretization permittivity and permeability can be determined with Eq. (7) by sampling the points every one degree. The result is shown in Fig. 4(b).

The fact that the electric fields exist in the cloaked area demonstrates that an observer in the anti-cloak can recieve the information from the outside, but remain invisibility for the outside. That is to say, an anti-cloak is equivalent to an invisibility cloak with a “window”, and through the “window”, information transmission between the outside and the inside can be implemented.

4. Conclusion

The design formulae for the electromagnetic invisibility anti-cloak of two-dimensional arbitrary shape objects are given in this paper. By setting the anti-cloak that is conformal with the stealth cloak and the masked object, we can get the uniform transformation for the cloak and anti-cloak of two-dimensional arbitrary geometries. In the region of the anti-cloak, the transformation media is anisotropic inhomogeneous, which can be realized by using electromagnetic metamaterials. Different two-dimensional anti-cloak models were given to verify the accuracy and effectiveness of the proposed design formulae. The anti-cloak plays an important role in the transmission of information from outside to inside of the cloak, meanwhile the object inside the cloak preserves invisibility to the outside. Relative to the invisibility cloak, the anti-cloak could be better to meet the practical application requirements.

Acknowledgments

This work is supported partly by the Program for New Century Excellent Talents in University of China, and partly supported by the National Natural Science Foundation of China under Contract No. 61072017, Fundamental Research Funds for the Central Universities (K5051202051), National Key Laboratory Foundation, and Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and Shaanxi Province.

References and links

1.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

2.

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

3.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006). [CrossRef]

4.

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]

5.

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(10), 248 (2006). [CrossRef]

6.

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(5801), 977–980 (2006). [CrossRef] [PubMed]

7.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

8.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]

9.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]

10.

T. Yang, H. Y. Chen, X. D. Luo, and H. R. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]

11.

J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009). [CrossRef] [PubMed]

12.

D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010). [CrossRef]

13.

X. Chen, Y. Fu, and N. 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]

14.

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008). [CrossRef] [PubMed]

15.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009). [CrossRef] [PubMed]

16.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48(6), 455–467 (2011). [CrossRef]

17.

I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B 81(12), 125124 (2010). [CrossRef]

18.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(1), 016603 (2011). [CrossRef] [PubMed]

19.

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

20.

Comsol Multiphysics” (Comsol AB), <http://www.comsol.com>.

21.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036621 (2006). [CrossRef] [PubMed]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(230.0230) Optical devices : Optical devices
(160.3918) Materials : Metamaterials
(260.2710) Physical optics : Inhomogeneous optical media
(230.3205) Optical devices : Invisibility cloaks

ToC Category:
Physical Optics

History
Original Manuscript: February 1, 2013
Revised Manuscript: March 20, 2013
Manuscript Accepted: April 2, 2013
Published: April 9, 2013

Citation
Long Li, Feifei Huo, Yuanming Zhang, Yang Chen, and Changhong Liang, "Design of invisibility anti-cloak for two-dimensional arbitrary geometries," Opt. Express 21, 9422-9427 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9422


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References

  1. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006). [CrossRef]
  4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006). [CrossRef] [PubMed]
  5. 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(10), 248 (2006). [CrossRef]
  6. 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,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007). [CrossRef]
  8. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90(24), 241105 (2007). [CrossRef]
  9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Desigh of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008). [CrossRef]
  10. T. Yang, H. Y. Chen, X. D. Luo, and H. R. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express16(22), 18545–18550 (2008). [CrossRef] [PubMed]
  11. J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett.34(5), 644–646 (2009). [CrossRef] [PubMed]
  12. D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Trans. Antennas Propag. 52(1), 24–46 (2010). [CrossRef]
  13. X. Chen, Y. Fu, and N. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express17(5), 3581–3586 (2009). [CrossRef] [PubMed]
  14. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express16(19), 14603–14608 (2008). [CrossRef] [PubMed]
  15. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express17(5), 3101–3114 (2009). [CrossRef] [PubMed]
  16. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion48(6), 455–467 (2011). [CrossRef]
  17. I. Gallina, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “General class of metamaterial transformation slabs,” Phys. Rev. B81(12), 125124 (2010). [CrossRef]
  18. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.83(1), 016603 (2011). [CrossRef] [PubMed]
  19. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express16(17), 13414–13420 (2008). [CrossRef] [PubMed]
  20. Comsol Multiphysics” (Comsol AB), < http://www.comsol.com >.
  21. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036621 (2006). [CrossRef] [PubMed]

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