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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 10 — May. 10, 2010
  • pp: 10168–10176
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Manipulating the field distribution via optical transformation

Xiaofei Zang and Chun Jiang  »View Author Affiliations


Optics Express, Vol. 18, Issue 10, pp. 10168-10176 (2010)
http://dx.doi.org/10.1364/OE.18.010168


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Abstract

Using the coordinate transformation theory, we propose a way to control the field distribution of electromagnetic wave in a waveguide filling with properly designed transformation media. The results show that the field distribution of the electromagnetic wave can be compressed and amplified in both longitudinal and transverse direction. A realizable layered waveguide based on the discrete optical transformation is also designed to manipulate the field distribution of the electromagnetic wave. Potential applications, i.e., turn a narrow (wide) slit into a large (small) window, are investigated in detail.

© 2010 OSA

1. Introduction

Based on the form-invariant of Maxwell’s equations, the exciting issue of achieving invisibility of objects has received much attention in the past few years [1

1. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]

33

33. Y. Luo, J. Zhang, H. Chen, J. Huangfu, and L. Ran, “High-directivity antenna with small antenna aperture,” Appl. Phys. Lett. 95(19), 193506 (2009). [CrossRef]

]. Pendry et al theoretically proposed an invisibility cloak that can hide the objects inside the cloak from detection, the cloak having electrical permittivity and magnetic permeability that are both spatially varying and anisotropic [2

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

]. Therefore, an electromagnetic cloak bends and guides incoming waves smoothly around the cloaked region. Following this theory, several other electromagnetic devices with novel functionalities have been designed both in theory and experiment. For example, field-rotating device rotates the fields in a region and thus makes the information from the outside as it comes from a different angle [34

34. H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]

]; field concentrator focuses the energy to a small region [35

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

]; beam splitter shifts one part of the wave in the one direction and the other part of the wave in the opposite direction [36

36. 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]

], and so on. All of these devices demonstrate that the coordinate transformation theory can be used as an effective way to control the properties of the electromagnetic wave (EMW).

Among the theoretical and experimental studies of the above electromagnetic devices, one of the most important topics is how to manipulate the field distribution of EMW. Recently, Jiang et al, have proposed a layered lens antenna in which the local properties of fields in a small region can be amplified to a large region [37

37. W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. 93(22), 221906 (2008). [CrossRef]

]. Simultaneously, they also used the discrete optical transformation to design multibeam antennas which means that the field distribution of EMW can be amplified in the transverse direction. However, their investigations only focus on the manipulation of the EMW field distribution in traverse direction (perpendicular to the direction of propagation). In this paper, we design transformation media to manipulate the field distribution of EMW both in longitudinal and transverse direction inside a waveguide based on coordinate transformation theory. Such a kind of transformation media in a narrow (wide) slit can achieve the transparency of a much wider (smaller) window.

2. The general modes

We consider a two-dimensional structure in Cartesian coordinate system (shown in Fig. 1
Fig. 1 The schematic illustration the field distribution of EMW is compressed or amplified in the transformation region. In the transformation region: (a) The field distribution of EMW is amplified both in longitudinal and transverse direction. Hollow arrow represents the direction of EMW propagation. (b) The field distribution of EMW is amplified in longitudinal direction but compressed in transverse direction. (c) The field distribution of EMW is amplified in transverse direction but compressed in longitudinal direction. (d) The field distribution of EMW is compressed both in longitudinal and transverse direction.
), where the virtual space is rectangle ABCD and the physical space is ABC’D’. In the transformation region as shown in Fig. 1, the field distribution of EMW can be amplified or compressed both in longitudinal and transverse direction. The geometric coordinate transformation between the new system (x', y', z') and the original system (x, y, z) can be expressed as
x'=x+(ba)xy'Lay'=cyz'=z,
(1)
where (x, y) is the coordinate in virtual space and (x', y') is the coordinate in physical space. From Eq. (1), we can find that the field distribution of EMW will be amplified (compressed) in the transverse direction for b>a (b<a). When c>1(c<1), the field distribution of EMW will be amplified (compressed) in the longitudinal direction.

Using coordinate transformation and mapping a homogeneous isotropic medium with permittivity and permeability ε¯¯ and μ¯¯ onto heterogeneous matrices of permittivity and permeability, they can be described as
ε¯¯=Λε¯ΛTdet(Λ)μ¯¯=Λμ¯ΛTdet(Λ),
(2)
where Λ is the Jacobian matrix with form as Λαα'=xα'/xα and ε¯ = I¯ε0 and μ¯ = I¯μ0 in free space.

Substituting Eq. (1) into Eq. (2), we obtain

ε¯¯ε=μ¯¯μ=[La+(ba)y'Lac+Lac(ba)2x'2(La+(ba)y')3Lac(ba)x'(La+(ba)y')20Lac(ba)x'(La+(ba)y')2LacLa+(ba)y'000Lac(La+(ba)y')],
(3)

Considering the practical application, we can divide the transformation region into the nth homogeneous pieces where each layer is composed of homogenous and uniaxially anisotropic metamaterial. Therefore, the geometric coordinate transformation between the new system (x', y', z') and the original system (x, y, z) can be expressed as
x'=x+(ba)xk(n)Lay'=cyz'=z,
(4)
where k(n) = L(m-0.5)/n (1mn).

The corresponding permittivity and permeability tensors of the transformation media in Cartesian coordinates are

ε¯¯nε=μ¯¯nμ=[La+(ba)k(n)Lac000LacLa+(ba)k(n)000Lac(La+(ba)k(n))].
(5)

3. Numerical simulations and discussions

In what follows, we perform numerical simulations on manipulating the field distribution in transformation region by using the finite element method (FEM) to demonstrate the designed formulae of Eq. (3) and Eq. (5). TE wave with electric fields polarized in z aixs propagates from the bottom of the normal region to the transformation region.

First, we assume that the transformation region is composed of the continuous and inhomogeneous metamaterials as defined in Eq. (3). The electric field distribution is shown in Fig. 2
Fig. 2 (Color online) Electric field distribution of continuous coordinate transformation [Eq. (3)] with system parameters (a) a = 0.2m, b = 1m, c = 2 (b) a = 0.2m, b = 1m, c = 1/2, (c) a = 0.3m, b = 0.2m, c = 2, (d) a = 0.3m, b = 0.2m, c = 1/2. The frequency of the line source is 2GHz for (a), (b), and 1.4GHz for (c), (d). The inset displays the field distribution of three regions: I-the free space II-the transformation region III-the free space (left) or a homogeneous material (right) with μxx = 0.5, μyy = 2, εzz = 0.5. The parameters in the transformation region are the same as (a)
with different parameters a, b, c, and L. When a = 0.2m, b = L = 1m, and c = 2, the field distribution from the normal region (free space with perfect electrical conductor boundary) is amplified both in longitudinal and transverse direction in the transformation region. When a = 0.2m, b = L = 1m, and c = 1/2, the field distribution is amplified in transverse direction but compressed in longitudinal direction in the transformation region. The filed distribution of the EMW can be compressed in transverse direction and amplified in longitudinal direction when a = 0.2m, b = 0.3m, L = 0.5m, and c = 2. When the incident electromagnetic wave transports into the transformation region, the field distribution can also be compressed both in longitudinal and transverse direction for a = 0.2m, b = 0.3m, L = 0.5m, and c = 1/2. In a word, the field distribution of the EMW in the transformation region is completely decided by the parameters a, b, c. In other words, the field distribution of EMW can be well manipulated by selecting appropriate parameters a, b, c. In addition, we should note that there is reflection at the exit of the transformation region, which is similar to the case of Ref [37

37. W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. 93(22), 221906 (2008). [CrossRef]

] due to the mismatch at the output port, as shown in Fig. 2 of the inset.

Next, we consider the case when the transformation region is composed of layered homogeneous anisotropy metamaterials as defined in Eq. (5). We divide the transformation region into twenty (ten) layers for a<b (a>b). Here, it should be noted that the more layers we divide the transformation region, the better result we’ll get. The field distribution is shown in Fig. 3
Fig. 3 (Color online) Electric field distribution of discrete coordinate transformation [Eqs. (5)] with system parameters (a) a = 0.2m, b = 1m, c = 2 (b) a = 0.2m, b = 1m, c = 1/2, (c) a = 0.3m, b = 0.2m, c = 2, (d) a = 0.3m, b = 0.2m, c = 1/2. The frequency of the line source is 2GHz for (a), (b), and 1.4GHz for (c), (d).
. The field distribution in the layered transformation region (Fig. 3) has almost no difference to in the continuous transformation region (Fig. 2). Therefore, we can manipulate the field distribution of EMW by choosing an appropriate layered homogeneous anisotropy metamaterials. In this case, there is also reflection phenomenon at the exit of the transformation region due to the mismatch at the output port (not shown), which is similar to the continuous coordinate transformation case.

Although the transformation region can manipulate (amplification or compression) the field distribution of EMW, how can we return to the original shape of the field distribution when the electromagnetic wave move away from the transformation region. We design a “mirror” transformation region (the region II in Fig. 4
Fig. 4 (Color online) Electric field distribution demonstration the shape of the origin EMW can be restored when a “mirror” transformation region (region II) add to the system. The system parameters are (a) aI = 0.2m, bI = 1m, cI = 1/2 LI = 1m, aII = 0.2m, bII = 1m, cII = 2, LII = 1m, with continuous coordinate transformation, (b) the parameters are the same as (a) but with discrete coordinate transformation, (c) aI = 0.2m, bI = 0.3m, cI = 1/2 LI = 0.4m,, aII = 0.3m, bII = 0.2m, cII = 2, LII = 0.4m,with continuous coordinate transformation, (d) the parameters are the same as (c) but with discrete coordinate transformation. The frequency of the line source is 2GHz for (a), (b), and 1.4GHz for (c), (d).
) to restore the field distribution. As can be seen in Fig. 4(a), when the incident electromagnetic wave propagates from the bottom normal region to transformation region I with parameters aI = 0.2m, bI = 1m, cI = 1/2, LI = 1m, the shape of the field distribution is amplified in transverse direction and compressed in longitudinal direction. When the EMW transports into the “mirror” transformation region II with parameters aII = 1m, bII = 0.2m, cII = 2, LII = 1m, the field distribution is amplified in longitudinal direction and compressed in transverse direction. After coming out from the region II, the Electromagnetic field distribution in the top normal region is the same as in the bottom region. In Fig. 4(b)4(d), the situation is the same as in Fig. 4(a) except for different system parameters.

Finally, we propose the potential applications of the transformation media discussed above. First, we design a small slit [the left figure of Fig. 5(f)
Fig. 5 (Color online) (a) The electric field distribution for an electric line source near the slit without transformation media in it. (b) The electric field distribution for an electric line source near the designed transformation media slit with c1 = c2 = 1. (c) The electric field distribution for an electric line source near a big “window”. (d) The electric field distribution for an electric line source near the designed transformation media slit with 3C1 = C2 = 1. (e) The electric field distribution for an electric line source near a much bigger “window”. (f) The schematic diagram represents the function of the small slit of (b). a1 = a2 = 0.3m, b = 0.1m, c1 = c2 = 1, L1 = L2 = 0.4m, a’ = 0.3m, b’ = 0.8m. (g) The schematic diagram represents the function of the small slit of (d). a1 = a2 = 0.3m, b = 0.1m, c1 = c2 = 1, L1 = L2 = 0.4m, a” = 0.3m, b” = 1.6m.The frequency of the line source is 1.4GHz.
, 5(g)] composed of layered homogeneous anisotropy metamaterials with parameters a1 = a2 = 0.3m, L1 = L2 = 0.4m and b = 0.1m. For the different parameters of c1 and c2, the small slit can replace different large windows [the right figure of Fig. 5(c) 5(f)] . When the small slit in Fig. 5(a) without transformation media in it, there is not any electromagnetic field distribution in the bottom of the free space as shown in Fig. 5(a). When the small slit in Fig. 5(b) embedded with layered transformation media and c1 = c2 = 1, the electromagnetic field distribution in the bottom of the free space as shown in Fig. 5(b) is the same as in Fig. 5(c). Compared with Fig. 5(b) and 5(c), the large window in Fig. 5(c) with a’ = 0.3m and b’ = 0.8m [see the right figure in Fig. 5(f)] can be replaced by a small slit of Fig. 5(b). In Fig. 5(d) and 5(e), the electromagnetic field distribution in the bottom of the free space is also identical with 3c1 = c2 = 1. But, the small slit replace with a much bigger window with a” = 0.3m b” = 1.6m [see the right figure in Fig. 5(g)]. It should be noted that the big window in Fig. 5(e) is much higher than that in Fig. 5(c). But the small slit in Fig. 5(b) and 5(d) is identical. Second, we design a large slit [the left figure of Fig. 6(f)
Fig. 6 (Color online) (a) The electric field distribution for an electric line source near the slit without transformation media in it. (b) The electric field distribution for an electric line source near the designed transformation media slit with c1 = c2 = 1. (c) The electric field distribution for an electric line source near a small “window”. (d) The electric field distribution for an electric line source near the designed transformation media slit with (1/2)c1 = c2 = 1. (e) The electric field distribution for an electric line source near a much smaller “window”. (f) The schematic diagram represents the function of the large slit of (a). a1 = a2 = 0.2m, b = 1m, c1 = c2 = 1, L1 = L2 = 0.5m, a’ = 0.2m, b’ = 1m. (g) The schematic diagram represents the function of the large slit of (c). a1 = a2 = 0.2m, b = 1m, (1/2)c1 = c2 = 1,L1 = L2 = 0.5m, a” = 0.2m, b” = 0.75m.The frequency of the line source is 2GHz.
, 6(g)] composed of layered homogeneous anisotropy metamaterials with parameters a1 = a2 = 0.1m, L1 = L2 = 0.5m and b = 0.5m. In this case, the large slits can replace a small window. The electric field distribution near the slit without transformation media in it is shown in Fig. 6(a). In Fig. 6(b) and 6(c), it is found that the flied distribution of the line source in the bottom of the free space is identical which demonstrates that the small window can be replaced with a large slit. The case of Fig. 6(d) and 6(e) is similar to Fig. 6(b) and 6(c), but the small window in Fig. 6(e) is much shorter than that in Fig. 6(b). These potential applications demonstrate that our device can gather (expand) information from different angles of light sources like an arbitrarily wide (small) window.

4. Conclusion

In summary, we have proposed a way to manipulate the field distribution of EMW based on the coordinate transformation theory. Our numerical simulations demonstrated that electromagnetic field distribution of EMW could be well manipulated in both longitudinal and transverse directions. A kind of layered transformation media in a small (large) slit was also designed to achieve the transparency of a much wider (smaller) window.

Acknowledgements

This work was supported by Ministry of Education Foundation of China (Grant No. 708038).

References and links

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A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]

2.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (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, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

5.

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]

6.

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

7.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32(9), 1069–1071 (2007). [CrossRef] [PubMed]

8.

H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 113903 (2007).

9.

B. L. 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(6), 063904 (2008). [CrossRef] [PubMed]

10.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007). [CrossRef] [PubMed]

11.

M. Yan, Z. C. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. 99(23), 233901 (2007). [CrossRef]

12.

X. Zhou and G. K. Hu, “Acoustic wave transparency for a multilayered sphere with acoustic metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046606 (2007). [CrossRef] [PubMed]

13.

H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]

14.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett. 33(14), 1584 (2008). [CrossRef] [PubMed]

15.

D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloak,” Appl. Phys. Lett. 92(1), 013505 (2008). [CrossRef]

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D.-H. Kwon and D. H. Werner, “Two-dimensional electromagnetic cloak having a uniform thickness for elliptic cylindrical regions,” Appl. Phys. Lett. 92(11), 113502 (2008). [CrossRef]

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P. Zhang, Y. Jin, and S. He, “Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak,” Appl. Phys. Lett. 93(24), 243502 (2008). [CrossRef]

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S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100(12), 123002 (2008). [CrossRef] [PubMed]

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P. Yao, Z. Liang, and X. Jiang, “Limitations of the electromagnetic cloak with dispersive material,” Appl. Phys. Lett. 92(3), 031111 (2008). [CrossRef]

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32.

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33.

Y. Luo, J. Zhang, H. Chen, J. Huangfu, and L. Ran, “High-directivity antenna with small antenna aperture,” Appl. Phys. Lett. 95(19), 193506 (2009). [CrossRef]

34.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]

35.

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

36.

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]

37.

W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. 93(22), 221906 (2008). [CrossRef]

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

ToC Category:
Physical Optics

History
Original Manuscript: January 8, 2010
Revised Manuscript: April 15, 2010
Manuscript Accepted: April 15, 2010
Published: April 30, 2010

Citation
Xiaofei Zang and Chun Jiang, "Manipulating the field distribution via optical transformation," Opt. Express 18, 10168-10176 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-10168


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References

  1. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (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, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  5. 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]
  6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterial,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
  7. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32(9), 1069–1071 (2007). [CrossRef] [PubMed]
  8. H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 113903 (2007).
  9. B. L. 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(6), 063904 (2008). [CrossRef] [PubMed]
  10. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007). [CrossRef] [PubMed]
  11. M. Yan, Z. C. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Phys. Rev. Lett. 99(23), 233901 (2007). [CrossRef]
  12. X. Zhou and G. K. Hu, “Acoustic wave transparency for a multilayered sphere with acoustic metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046606 (2007). [CrossRef] [PubMed]
  13. H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]
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