Hiding in the corner

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 Optical Society of America OCIS codes: (160.1190) Anisotropic optical materials; (230.0230) Optical devices; (260.2110) Electromagnetic theory. References and links 1. J. B. Pendry, D. Schurig, D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006) 2. U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006) 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) 4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006) 5. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008). 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) 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) 8. W. Cai, U. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics 1, 224–227(2007) 9. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, D. R. Smith, “Broadband Ground-Plane Cloak”, Science 323, 366–369 (2009) 10. X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nature Commun. 2, 176 (2011) 11. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011) 12. Y. Luo, J. Zhang, H. Chen, L. Ran, B. Wu, J. A. Kong, “A rigorous analysis of plane-tansformed invisiblity cloaks,” IEEE Tran. Antenna. Propag. 57, 3926-3933, (2009). 13. W. Li, J. Guan, Z. Sun, W. Wang, Q. Zhang, “A near-perfect invisibility cloak constructed with homogeneous materials,” Opt. Express 17, 23410-23416, (2009). 14. J. Zhang, L. Liu, Y. Luo, S. Zhang, N. A. Mortensen, “Homogeneous optical cloak constructed with unifrom layered structures,” Opt. Express 19, 8625-8631, (2011). 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). #153544 $15.00 USD Received 31 Aug 2011; revised 14 Sep 2011; accepted 17 Sep 2011; published 5 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20827


Introduction
Transformation optics [1,2] has enabled unprecedented flexibility in manipulating electromagnetic waves and producing new functionalities with artificially structured metamaterials [3,4].A series of applications are carried out under the theoretical framework of transformation optics, such as the cloak of invisibility [4,5], field concentrator [6], and illusion devices [7].Among all these application, cloaking is definitely the most exciting innovation, and has been a subject of great scientific and popular interest.
Up to now, two broad classes of invisibility cloak based on transformation optics have been reported: the free-space cloak [4,8] and the carpet cloak [5,[9][10][11].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][13][14].A macroscopic cloak for visible light was also experimentally demonstrated based on the carpet cloak concept, showing great promise for practical applications [10,11].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.

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.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: With the theory presented in Ref. [1], the relative permittivity and permeability tensors can be deduced in the new space: Here J is the Jacobian transformation tensor with components We consider here a free-space background medium with ε b = μ b = 1.Therefore, the permittivity and permeability tensors can be expressed as 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 where sgn(x) = 1 for x > 0 and −1 for x < 0. If the incident wave is TE polarized, only the μ x , μ y and ε z components are relevant.Then, the parameters in Eq. ( 3) can be reduced as 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 Here ε = 1 is only an example of the simplification.We can choose any value of ε to satisfy the practical requirements of fabrication.

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 E z distribution between the ideal cloak with parameters shown in Eq. ( 3) and the simplified cloak with parameters shown in Eq. ( 5).In Fig. 2 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.To better understand the performance of the cloak, we plot the distribution of the real part of E z 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 .
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].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 fi-nite 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: where E e and E c are the real parts of the E z field for the cases when the PEC corner is empty and covered with the cloak, respectively, and E i is the incident field.The integration path for E c and E e is taken along the line y = 20λ , for E i it is taken across the source aperture.If the cloak is perfect, E e should be equal to E c , and the differential scattering should be zero.Obviously, a smaller value of S d 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 E z 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 S d 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 S d remains low when the incident angle is However, the scattering increase when the incident wave deviates from y axis.For θ from 5°to 35°, S d remains below 0.1, and steeply climbs above 0.4 when θ = 40 • .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: ε = ε (1 + iσ ) and μ = μ (1 + iσ ).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 S d 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, S d 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] or natural uniaxial crystals [10,11].In the latter case, low values of σ and wide band of invisibility may be achievable.

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.

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

Fig. 2 .
Fig. 2. E z 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.

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