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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20827–20832
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Hiding in the corner

Weiren Zhu, Ilya Shadrivov, David Powell, and Yuri Kivshar  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20827-20832 (2011)
http://dx.doi.org/10.1364/OE.19.020827


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Abstract

We describe a novel type of electromagnetic cloak designed to conceal an object in a corner, and demonstrate its excellent performance by employing direct numerical simulation. Furthermore, we study the angular dependence and the effect of loss on the invisibility performance and compare ideal and simplified cloaks. The proposed structure has homogeneous constitutive parameters, which greatly simplifies practical realization.

© 2011 OSA

1. Introduction

Up to now, two broad classes of invisibility cloak based on transformation optics have been reported: the free-space cloak [4

4. 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, 977–980 (2006) [CrossRef] [PubMed]

, 8

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

] and the carpet cloak [5

5. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

, 9

9. 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]

11

11. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011) [CrossRef] [PubMed]

]. The free-space cloak can conceal an object within it, and should be invisible to radiation from any incident direction. However, this kind of cloak requires the use of resonant elements, and as a consequence it only operates at a single frequency, thus making it unsuitable for practical applications. The carpet cloak conceals an object located on a flat surface, by mimicking the reflection from the bare surface. Since it can be constructed with non-resonant components, this kind of cloak has attracted more attention from researchers, due to its ease of fabrication and wider operating bandwidth. Moreover, the parameters of the carpet cloak could be homogeneous and rigorously designed [12

12. Y. Luo, J. Zhang, H. Chen, L. Ran, B. Wu, and J. A. Kong, “A rigorous analysis of plane-tansformed invisiblity cloaks,” IEEE Tran. Antenna. Propag. 57, 3926–3933, (2009). [CrossRef]

14

14. J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with unifrom layered structures,” Opt. Express 19, 8625–8631, (2011). [CrossRef] [PubMed]

]. A macroscopic cloak for visible light was also experimentally demonstrated based on the carpet cloak concept, showing great promise for practical applications [10

10. X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nature Commun. 2, 176 (2011) [CrossRef]

, 11

11. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011) [CrossRef] [PubMed]

]. However, carpet cloak designs presented so far have not considered backgrounds other than simple flat surfaces.

In this paper, we design an invisible cloak suitable for non-flat surroundings, the most basic case of which is a corner formed by two walls. We first calculate the constitutive parameters of the proposed cloak, and then demonstrate the cloaking performance with numerical simulations. Moreover, a differential scattering coefficient is introduced to quantitatively characterize and compare the cloaking performance. We show how the performance of the cloak varies with the angle of incidence, and demonstrate the influence of losses.

2. Coordinate transformation for the corner cloak

Fig. 1(a) shows a picture of the proposed cloak, with a schematic giving the important dimensions in Fig. 1(b). The transformation for this cloak is an identity along the z axis. Hence, we only need to consider the two-dimensional cross section, and perform the coordinate transformation in the xoy plane.

Fig. 1 (a) The schematic of the proposed triangular corner cloak, (b) the cross section of the cloak and its geometrical parameters.

For this transformation, we need to compress the space of Quadrilateral ACBD into that of Triangle ABD. Here, a, b, c, and d are the lengths of OA, OB, OC and OD respectively. A spatial transformation can be defined simply in Cartesian coordinates:
x=x,z=z,y={(a+xac+y)dc+d,(RegionI)(bxbc+y)dc+d,(RegionII)
(1)
With the theory presented in Ref. [1

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

], the relative permittivity and permeability tensors can be deduced in the new space: ɛ′ = JɛbJT/|J|, μ′ = JμbJT/|J|. Here J is the Jacobian transformation tensor with components Jij = ∂ri/∂rj.

We consider here a free-space background medium with ɛb = μb = 1. Therefore, the permittivity and permeability tensors can be expressed as
ɛ=μ={[d+cdca0cad(a2+c2)a2(c+d)000d+cd](RegionI)[d+cdcb0cbd(b2+c2)b2(c+d)000d+cd](RegionII)
(2)
Equations (2) give full ideal design parameters for the permittivity and permeability tensors in the triangular cloak layer. One advantage of this cloak is that the constitutive parameters have constant value within each region, determined by the geometrical dimensions a, b, c, and d. If we fix the geometry of the cloak, the permittivity and permeability tensors are both constant in each region of the cloak. This property will greatly reduce the difficulty of practical design and fabrication as well as avoiding the singularity of the parameters which occurs in the cylindrical cloak.

To illustrate the performance of the proposed homogeneous cloak with the constitutive parameters corresponding to Eq. (2), the geometric dimensions of a = b = c = d = 5λ are chosen for simplicity. In this case
ɛ=μ=[2sgn(x)0sgn(x)10002],
(3)
where sgn(x) = 1 for x > 0 and −1 for x < 0. If the incident wave is TE polarized, only the μx, μy and ɛz components are relevant. Then, the parameters in Eq. (3) can be reduced as
ɛ=2μ=[2sgn(x)sgn(x)1].
(4)

Furthermore, the constitutive parameters of the proposed cloak can be further simplified if we wish to primarily demonstrate the wave trajectory of the cloak, which can eliminate the need for materials with both an electric and magnetic response. Thus the parameters in Eq. (4) are simplified as
ɛ=1μ=[42sgn(x)2sgn(x)2].
(5)
Here ɛ′ = 1 is only an example of the simplification. We can choose any value of ɛ′ to satisfy the practical requirements of fabrication.

3. Simulation results

The cloaking behavior is verified using the commercial finite element package COMSOL Multiphysics. The incident wave is excited by an electric field source of constant amplitude and finite width. We first compare the Ez distribution between the ideal cloak with parameters shown in Eq. (3) and the simplified cloak with parameters shown in Eq. (5). In Fig. 2(a), we show the reference field distribution when neither the cloak nor the obstacle is present. The incident wave propagating in the −y direction is directly reflected from the two perfect electric conductor (PEC) boundaries, and returns in +y direction. When the full cloak [Fig. 2(b)] or simplified cloak [Fig. 2(c)] is placed in the corner the incident wave will be compressed into the cloak region, and reflected back along the same direction as shown in Fig. 2(a). Both the ideal cloak and simplified cloak show excellent performance for hiding the obstacle, and it is difficult to observe the difference between the two cases in the field distributions outside the cloaks. In addition, we also investigate the performances with different angles of incidence. In Fig. 2(d) the case of simplified cloak is shown when incident angle is 45° from the y axis. The field distribution in this case is similar to the case that the wave is normally incident to the right PEC boundary, the reflected wave together with the incident wave forming a standing wave, which also shows good cloaking performance.

Fig. 2 Ez distributions in the PEC corner. (a) ( Media 1) Empty, (b) ( Media 2) covered with the ideal cloak, (c) ( Media 3) covered with the simplified cloak, and (d) ( Media 4) at oblique incidence to the simplified cloak. The obstacles here are modeled by PEC boundaries.

To better understand the performance of the cloak, we plot the distribution of the real part of Ez along the line y = 20λ, which clearly shows the tiny difference among the direct reflection from the PEC corner, ideal cloak, and simplified cloak. As shown in Fig. 3(a), for the ideal cloak, the field distribution has very small error, of purely numerical origin. For the simplified cloak shown in Fig. 3(b), some differences appear, especially in the region x = λ to 4λ.

Fig. 3 The distribution of Ez along the line y = 20λ with and without the cloak, for different incident angles θ. (a) θ = 0° for the ideal cloak, (b) θ = 0° for the simplified cloak, (c) θ = 40° for the ideal cloak, (d) θ = 40° for the simplified cloak.

To characterize the behavior of the proposed cloak, it is necessary to have a quantitative measure of its performance. For a free-space cloak subject to uniform illumination, the radar cross section averaged over all scattering angles is the most appropriate measure [15

15. D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” New J. Phys. 10, 115039, (2008). [CrossRef]

]. However, for carpet and corner cloaks, there is strong background scattering which occurs even when the cloak is operating perfectly. Therefore, we consider the differential scattering - the difference between the observed scattered field and that of an empty corner. In addition, due to the finite size of the exciting beam, we require a dimensionless scattering coefficient instead of a scattering cross section. We call this the differential scattering coefficient and define it as:
Sd=|EeEc|2dx|Ei|2dx
(6)
where Ee and Ec are the real parts of the Ez field for the cases when the PEC corner is empty and covered with the cloak, respectively, and Ei is the incident field. The integration path for Ec and Ee is taken along the line y = 20λ, for Ei it is taken across the source aperture. If the cloak is perfect, Ee should be equal to Ec, and the differential scattering should be zero. Obviously, a smaller value of Sd reflects better performance of the cloak. With the data in Fig. 3(a) and (b), the calculated scattering coefficients of the ideal and simplified cloak are 0.005 and 0.02 respectively, which shows that the performance of the ideal cloak is better than that of the simplified cloak. However, the performance in these two cases is still excellent as the differential scattering is very small.

Furthermore, we investigate the performance of the proposed cloak for several angles of incidence for both ideal and simplified parameters. In Fig. 3(c) and (d), we plot the real component of Ez for the ideal and simplified cloak, for an incident angle of 40°. For the ideal cloak, the field distribution is almost the same as the empty case, however for the simplified cloak, there is a significant difference for x from 2λ to 20λ. We calculate the scattering coefficient over a range of angles as shown in Fig. 4(a). The values of Sd for the ideal cloak are near zero for the whole angle region from 0° to 45°, which shows the excellent performance. For the simplified cloak, the value of Sd remains low when the incident angle is zero. However, the scattering increase when the incident wave deviates from y axis. For θ from 5° to 35°, Sd remains below 0.1, and steeply climbs above 0.4 when θ = 40°.

Fig. 4 (a) The scattering coefficient at different angles of incidence, comparing the ideal cloak and simplified cloaks. (b) The calculated scattering coefficient versus loss.

Up to this point, we have considered the materials comprising the cloak to be lossless. However, losses may be the most critical factor that degrades the performance of the cloak, as they are unavoidable in a practical structure. Here, we investigate the effect of loss on our triangular corner cloak. A loss parameter σ is introduced into the constitutive parameters of the proposed cloak: ɛ″ = ɛ′(1+ ) and μ″ = μ′(1+ ). We modify both the permittivity and the permeability tensors in order to ensure no additional impedance mismatch is introduced.

In Fig. 4(b) the differential scattering Sd is shown as a function of σ for both ideal and simplified cloaks when the wave is incident along the y axis. It can be seen that the scattering increases monotonically and rapidly when the loss increases in both curves, which means the performance of cloak sharply degrades as expected. If σ is larger than 0.005, Sd will exceed 0.1, in which case the reflected wave is highly suppressed by losses. The cloak we discussed here is for TE incidence, but similar results could also be obtained for the TM case, which can be realized by employing non-resonant elements [14

14. J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with unifrom layered structures,” Opt. Express 19, 8625–8631, (2011). [CrossRef] [PubMed]

] or natural uniaxial crystals [10

10. X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nature Commun. 2, 176 (2011) [CrossRef]

, 11

11. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011) [CrossRef] [PubMed]

]. In the latter case, low values of σ and wide band of invisibility may be achievable.

4. Conclusions

We presented an approach to design a cloak of invisibility suitable for a non-flat surface, rather than just a perfectly flat mirror. As an example, the proposed triangular cloak shows the ability to conceal an object in a PEC corner. By introducing a differential scattering coefficient, the performance of ideal and simplified cloaks is quantitatively characterized and compared. For various angles of incidence, the ideal cloak shows consistently excellent performance, while the performance of the simplified cloak varies greatly. Further simulation indicates the loss in the cloak will degrade the cloaking performance. The presented design extends the repertoire of available cloaking geometries to include objects located in a corner.

Acknowledgments

We acknowledge funding from the Australian Research Council.

References and links

1.

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

2.

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006) [CrossRef] [PubMed]

3.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000) [CrossRef] [PubMed]

4.

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, 977–980 (2006) [CrossRef] [PubMed]

5.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008). [CrossRef] [PubMed]

6.

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations”, Phys. Rev. B 77, 125127 (2008) [CrossRef]

7.

Y. Lai, J. Ng, H.Y. Chen, D. Han, J. Xiao, Z. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object”, Phys. Rev. Lett. 102, 253902 (2009) [CrossRef] [PubMed]

8.

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

9.

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]

10.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nature Commun. 2, 176 (2011) [CrossRef]

11.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011) [CrossRef] [PubMed]

12.

Y. Luo, J. Zhang, H. Chen, L. Ran, B. Wu, and J. A. Kong, “A rigorous analysis of plane-tansformed invisiblity cloaks,” IEEE Tran. Antenna. Propag. 57, 3926–3933, (2009). [CrossRef]

13.

W. Li, J. Guan, Z. Sun, W. Wang, and Q. Zhang, “A near-perfect invisibility cloak constructed with homogeneous materials,” Opt. Express 17, 23410–23416, (2009). [CrossRef]

14.

J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with unifrom layered structures,” Opt. Express 19, 8625–8631, (2011). [CrossRef] [PubMed]

15.

D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” New J. Phys. 10, 115039, (2008). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(230.0230) Optical devices : Optical devices
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

History
Original Manuscript: August 31, 2011
Revised Manuscript: September 14, 2011
Manuscript Accepted: September 17, 2011
Published: October 5, 2011

Citation
Weiren Zhu, Ilya Shadrivov, David Powell, and Yuri Kivshar, "Hiding in the corner," Opt. Express 19, 20827-20832 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20827


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References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006) [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006) [CrossRef] [PubMed]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000) [CrossRef] [PubMed]
  4. 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, 977–980 (2006) [CrossRef] [PubMed]
  5. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101, 203901 (2008). [CrossRef] [PubMed]
  6. Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations”, Phys. Rev. B77, 125127 (2008) [CrossRef]
  7. Y. Lai, J. Ng, H.Y. Chen, D. Han, J. Xiao, Z. Zhang, and C. T. Chan, “Illusion Optics: The Optical Transformation of an Object into Another Object”, Phys. Rev. Lett.102, 253902 (2009) [CrossRef] [PubMed]
  8. W. Cai, U. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics1, 224–227(2007) [CrossRef]
  9. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak”, Science323, 366–369 (2009) [CrossRef] [PubMed]
  10. X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nature Commun.2, 176 (2011) [CrossRef]
  11. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106, 033901 (2011) [CrossRef] [PubMed]
  12. Y. Luo, J. Zhang, H. Chen, L. Ran, B. Wu, and J. A. Kong, “A rigorous analysis of plane-tansformed invisiblity cloaks,” IEEE Tran. Antenna. Propag.57, 3926–3933, (2009). [CrossRef]
  13. W. Li, J. Guan, Z. Sun, W. Wang, and Q. Zhang, “A near-perfect invisibility cloak constructed with homogeneous materials,” Opt. Express17, 23410–23416, (2009). [CrossRef]
  14. J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with unifrom layered structures,” Opt. Express19, 8625–8631, (2011). [CrossRef] [PubMed]
  15. D. P. Gaillot, C. Croënne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal–dielectric planar hyperlens,” New J. Phys.10, 115039, (2008). [CrossRef]

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