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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 22 — Oct. 26, 2009
  • pp: 19656–19661
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Metamaterial electromagnetic concentrators with arbitrary geometries

Jingjing Yang, Ming Huang, Chengfu Yang, Zhe Xiao, and Jinhui Peng  »View Author Affiliations


Optics Express, Vol. 17, Issue 22, pp. 19656-19661 (2009)
http://dx.doi.org/10.1364/OE.17.019656


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Abstract

The electromagnetic concentrators play an important role in the harnessing of light in solar cells or similar devices, where high field intensities are required. The material parameters for two-dimensional (2D) metamaterial-assisted electromagnetic concentrators with arbitrary geometries are derived based on transformation-optical approach. Enhancements in field intensities of the 2D concentrator have been shown by full-wave simulation. All theoretical and numerical results validate the material parameters for the 2D concentrator with irregular cross section we developed.

© 2009 OSA

1. Introduction

Control of electromagnetic wave with metamaterials is of great topical interest, and is fuelled by rapid progress in electromagnetic cloaks [1

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

6

6. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

]. Cloaking techniques rely on the transformation of coordinates, e.g., a point in the electromagnetic space is transformed into a special volume in the physical space, thus leading to the creation of the volume where electromagnetic fields do not exist, but are instead guided around this volume [7

7. P. Alitalo and S. Tretyakov, “Electromagnetic cloaking with metamaterials,” Mater. Today 12(3), 22–29 (2009). [CrossRef]

]. Based on the coordinate transformation idea, some interesting optical and microwave applications, such as illusion device [8

8. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

], concentrators [9

9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics and Nanostructures-Fundamentals and Applications 6, 89 (2008). [CrossRef]

], field shifters [10

10. 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. 100(6), 063903 (2008). [CrossRef] [PubMed]

], anti-cloaks [11

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

], super-scatterers [12

12. 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]

], superabsorbers [13

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

], remote cloaks [14

14. Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

] and cloaking sensor [15

15. A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102(23), 233901 (2009). [CrossRef] [PubMed]

] have been proposed. Among various novel applications, the phenomenon of near-field concentration of light plays an important role in the harnessing of light in solar cells or similar devices, where high field intensities are required [9

9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics and Nanostructures-Fundamentals and Applications 6, 89 (2008). [CrossRef]

,16

16. K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008). [CrossRef] [PubMed]

]. Recently, Rahm et al [9

9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics and Nanostructures-Fundamentals and Applications 6, 89 (2008). [CrossRef]

] derived the material parameters for circular cylindrical metamaterial electromagnetic concentrator, and confirmed by numerical simulation. However, to the best of our knowledge, there is no report about the metamaterial electromagnetic concentrators with any other geometry.

Inspired by the work of Li et al [17

17. C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express 16(23), 19366–19374 (2008). [CrossRef]

], we develop the material parameters for 2D concentrators with arbitrary geometries, and validate them by numerical simulation. This work has greatly improved the designing flexibility of concentrators, since material parameters for the concentrator with arbitrary geometries can be easily obtained for the given contour equations. We show that the material parameters developed in this paper can be also specialized to the 2D concentrator with conformal inner and outer boundaries or the other regular shapes, such as circular, elliptical and square, which represents an important progress towards the realization of arbitrary shaped concentrator.

2. Theoretical model

The schematic diagram of the space transformation is shown in Fig. 1
Fig. 1 Schematic diagram of the space transformation for the design of arbitrary shaped 2D concentrator.
, where three cylinders with arbitrary cross section enclosed by contours R1(θ), R2(θ)and R3(θ)divide the space into three regions S1, S2 and S3. The space is compressed into S1 at the expense of an expansion of space in regions S2 and S3. Here,Rn(θ) (n=1,2,3)is an arbitrary continuous function with period 2π [17

17. C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express 16(23), 19366–19374 (2008). [CrossRef]

]. According to the coordinate transformation method, the permittivity εij and permeability μij tensors of the transformation media can be written as [18

18. 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]

]
εij=ΛiiΛjj|det(Λii)|1εij,μij=ΛiiΛjj|det(Λii)|1μij.
(1)
where Λiiis the Jacobian transformation matrix. It is just the derivative of the transformed coordinates with respect to the original coordinates. |det(Λii)| is the determinant of the matrix. εijandμijare the permittivity and permeability of the original space, respectively.

In the compressive region (r<R1(θ)), with the transformationr=k3r,θ=θ, z=z, where k3=R1(θ)/R2(θ),we obtain the Jacobi matrix Λii=[C1,C2,0;D1,D2,0;0,0,1]and its determinant det(Λii)=C1D2C2D1, whereC1=k3k4sinθcosθ, C2=k4cos2θ, D1= k4sin2θ,D2=k3+k4sinθcosθ, k4=[R2(θ)dR1(θ)/dθR1(θ)dR2(θ)/dθ]/R22(θ). Substituting Λii and det(Λii)into Eq. (1), we can obtain the relative permittivity and permeability tensors for compressive region as

εij=μij=[(C12+C22)/(C1D2C2D1)(C1D1+C2D2)/(C1D2C2D1)0(C1D1+C2D2)/(C1D2C2D1)(D12+D22)/(C1D2C2D1)0001/(C1D2C2D1)]
(2)

In the stretching region (R1(θ)<r<R3(θ)), with the transformation r=k1r+k2, θ=θ, z=z, where k1=[R3(θ)R1(θ)]/[R3(θ)R2(θ)],k2=R3(θ)(1k1), the coordinate transformation equations are expressed by x=rcos(θ)=k1x+k2x/x2+y2, y=rsin(θ)=k1y+k2y/x2+y2,z=zwith the Jacobi matrix Λii=[A1,A2,0;B1,B2, 0;0,0,1]and its determinant det(Λii)= A1B2A2B1, where

A1=k1asinθcosθ+(k2/r)sin2θ(b/r)sinθcosθ,
A2=acos2θ(k2/r)sinθcosθ+(b/r)cos2θ,
B1=asin2θ(k2/r)sinθcosθ(b/r)sin2θ,
B2=k1+asinθcosθ+(k2/r)cos2θ+(b/r)sinθcosθ,  with
a={[R3(θ)R2(θ)][R3(θ)R1(θ)][R3(θ)R1(θ)][R3(θ)R2(θ)]}[R3(θ)R2(θ)]2,
b={R3(θ)[R1(θ)R2(θ)]+R3(θ)[R1(θ)R2(θ)]}[R3(θ)R2(θ)]1
R3(θ)[R1(θ)R2(θ)][R3(θ)R2(θ)][R3(θ)R2(θ)]2.

Substituting Λii and det(Λii) into Eq. (1), we can obtain the relative permittivity and permeability tensors for the stretching region as

εij=μij=[(A12+A22)/(A1B2A2B1)(A1B1+A2B2)/(A1B2A2B1)0(A1B1+A2B2)/(A1B2A2B1)(B12+B22)/(A1B2A2B1)0001/(A1B2A2B1)]
(3)

3. Simulation results and discussion

Figure 2(a)
Fig. 2 (a) Electric field distribution of the concentrator with non-conformal boundaries. (b) The corresponding power flow distribution of (a). (c) Electric field distribution of the concentrator with conformal boundaries. (d) The corresponding power flow distribution of (d).
displays the electric field distribution in the vicinity of the concentrator with non-conformal boundaries under TE wave irradiation. The frequency of the TE wave is 4GHz. The contour equations used in the simulation are as follows

R1(θ)=(12+2cos(θ)+sin(2θ)2sin(3θ))/320,R2(θ)=(20+2sin(2θ)3sin(5θ)+5cos(7θ))/288,R3(θ)=0.175(0.8+0.1cos(θ)+0.2cos(5θ)+0.1sin(5θ)),R4(θ)=(10+sin(θ)sin(2θ)+2cos(5θ))/48.

As can be seen in Fig. 2(a), the waves are focused into the compressive region with slight distortion. Figure 2(b) is the corresponding power flow distribution. It is calculated according to the Poynting vector S=E×H, where E andHare the electric field and magnetic field in the computational domain. Red lines indicate the direction of the power flow. It can be seen that although the power flow is enhanced in the compressive region, it exhibits strong spatial fluctuations as a consequence of the non-conformal inner and outer boundaries. Figure 2(c) and 2(d) show the simulation results for the concentrator with conformal boundaries, of which the geometry parameters are chosen asR(θ)= R1(θ),t1=0.3,t2=0.8, t3=1. Form Fig. 2(c), it can be clearly seen that the waves are focused by the concentrator into the compressive region without any distortion compared with Fig. 2(a). Comparing Fig. 2(d) with 2(b), we can observe that the power flow distribution in the compressive region is spatially uniform and the intensity of the power flow is strongly enhanced. Significantly stronger enhancements can be achieved by increasing the ratio ofR2/R1.

Since the proposed concentrator has no symmetry in any direction, it’s necessary to study its interaction with electromagnetic waves from different orientations. Figure 3
Fig. 3 Power flow distribution in the vicinity of the concentrator with conformal boundaries under cylindrical wave irradiation. The line source is located at (−0.45, 0), (0, 0.45) and (−0.32, −0.32) for (a), (b) and (c).
shows the power flow distribution for the concentrator with conformal boundaries under cylindrical wave irradiation. The line source with a current of 0.001A/m is located at (−0.45, 0), (0, 0.45) and (−0.32, −0.32) for panels (a), (b) and (c), respectively. It can be clearly seen that the power flow is strongly enhanced in the compressive region, and the focusing effects are independent on the location of the line source.

Since metamaterials are always lossy in real applications, it does make sense to investigate the effects of loss on the performance of the concentrator. Figure 4(a)
Fig. 4 (a) Power flow distribution in the computational domain for the concentrator with loss tangent of 0.01. (b) The power flow distributions along x axis for the concentrators with different loss tangents.
shows the power flow distribution in the vicinity of the concentrator with electric and magnetic-loss tangents of 0.01. It can be seen that the power flow distribution in the compressive region is greatly fluctuated with the influence of loss. The electric field distributions along the x axis of the concentrator with different electric and magnetic-loss are shown in Fig. 4(b). It can be seen that the power flow distribution is basically undisturbed when loss tangent of 0.001 is added to both permittivity and permeability tensors of the anisotropic and inhomogeneous materials. For the back-scattering region, performance of the concentrator is independent on the loss of the metamaterials. The increase of loss mainly deteriorates the performance of the concentrator in the transformation region and the forward-scattering region of the near field.

Power flow distributions for the concentrator with circular, elliptical and square cross sections under TE wave irradiation are shown in Fig. 5
Fig. 5 Power flow distribution of the concentrator with circular (a), elliptical (b), and square (c) cross section under TE wave irradiation.
. The TE wave is irradiated along the x axis. In the simulation, the linear compression ratios between the inner and outer boundaries are chosen as t1=0.3,t2=0.8, t3=1. The radius for the out boundary of the circular concentrator is 0.2m; the semi-major axis and the semi-minor axis for the out boundary of the elliptical concentrator is 0.22m and 0.11m, respectively; the side length for the outer boundary of the square concentrator is 0.3m. From Fig. 5, we can observe that the power flow is strongly enhanced in the compressive region. Besides, under the irradiation of cylindrical wave, the concentration phenomenon can also be observed, as shown in Fig. 6
Fig. 6 Power flow distribution of the concentrator with circular (a), elliptical (b), and square (c) cross section under cylindrical wave irradiation.
.The simulation results for the circular concentrator are in good agreement with Ref [9

9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics and Nanostructures-Fundamentals and Applications 6, 89 (2008). [CrossRef]

], which further confirms the effectiveness and the generality of the material parameters we developed.

6. Conclusion

The material parameters for 2D concentrator with arbitrary geometries are developed, which can be specialized to the formally designed concentrators with conformal inner and outer boundaries. All theoretical and numerical results validate the effectiveness and the generality of the material parameter for the 2D concentrator with arbitrary cross section we deduced. Besides, we have investigated the influence of metamaterial loss on the performance of the device, and found that the power flow distribution of the concentrator is basically undisturbed when loss tangent of metamaterials is less than 0.001. The phenomenon of field concentration of light plays an important role in the harnessing of light in solar cells similar devices. It is expected that our works are helpful for designing concentrators for electromagnetic and optical fields, where high field intensities are required.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. 60861002), Training Program of Yunnan Province for Middle-aged and Young Leaders of Disciplines in Science and Technology (Grant No. 2008PY031), the Research Foundation from Ministry of Education of China (grant no. 208133), the Natural Science Foundation of Yunnan Province (grant no.2007F005M), Research Foundation of Education Bureau of Yunnan Province (grant no. 07Z10875), and the National Basic Research Program of China (973 Program) (grant no. 2007CB613606).

References and links

1.

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

2.

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

3.

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]

4.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]

5.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]

6.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

7.

P. Alitalo and S. Tretyakov, “Electromagnetic cloaking with metamaterials,” Mater. Today 12(3), 22–29 (2009). [CrossRef]

8.

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]

9.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics and Nanostructures-Fundamentals and Applications 6, 89 (2008). [CrossRef]

10.

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. 100(6), 063903 (2008). [CrossRef] [PubMed]

11.

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

12.

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]

13.

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

14.

Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]

15.

A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102(23), 233901 (2009). [CrossRef] [PubMed]

16.

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008). [CrossRef] [PubMed]

17.

C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express 16(23), 19366–19374 (2008). [CrossRef]

18.

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]

19.

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

OCIS Codes
(220.1770) Optical design and fabrication : Concentrators
(230.0230) Optical devices : Optical devices
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: August 10, 2009
Revised Manuscript: September 20, 2009
Manuscript Accepted: October 5, 2009
Published: October 15, 2009

Citation
Jingjing Yang, Ming Huang, Chengfu Yang, Zhe Xiao, and Jinhui Peng, "Metamaterial electromagnetic concentrators with arbitrary geometries," Opt. Express 17, 19656-19661 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19656


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References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  3. 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]
  4. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]
  5. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
  6. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
  7. P. Alitalo and S. Tretyakov, “Electromagnetic cloaking with metamaterials,” Mater. Today 12(3), 22–29 (2009). [CrossRef]
  8. Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]
  9. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics and Nanostructures-Fundamentals and Applications 6, 89 (2008). [CrossRef]
  10. 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. 100(6), 063903 (2008). [CrossRef] [PubMed]
  11. H. Y. Chen, X. D. Luo, H. R. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008). [CrossRef] [PubMed]
  12. 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]
  13. J. Ng, H. Y. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009). [CrossRef] [PubMed]
  14. Y. Lai, H. Y. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009). [CrossRef] [PubMed]
  15. A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102(23), 233901 (2009). [CrossRef] [PubMed]
  16. K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16(26), 21793–21800 (2008). [CrossRef] [PubMed]
  17. C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express 16(23), 19366–19374 (2008). [CrossRef]
  18. 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]
  19. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16(17), 13414 (2008). [CrossRef] [PubMed]

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