This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. Response of a cylindrical invisibility cloak to electromagnetic waves Zhang, Baile; Chen, Hongsheng; Wu, Bae‑Ian; Luo, Yu; Ran, Lixin; Kong, Jin Au 2007 Zhang, B., Chen, H., Wu, B. I., Luo, Y., Ran, L., & Kong, J. A. (2007). Response of a cylindrical invisibility cloak to electromagnetic waves. Physical Review B, 76(12). https://hdl.handle.net/10356/95353 https://doi.org/10.1103/PhysRevB.76.121101 © 2007 The American Physical Society. This paper was published in Physical Review B and is made available as an electronic reprint (preprint) with permission of The American Physical Society. The paper can be found at the following official DOI: http://dx.doi.org/10.1103/PhysRevB.76.121101. One print or electronic copy may be made for personal use only. 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Downloaded on 28 Sep 2021 12:32:39 SGT RAPID COMMUNICATIONS PHYSICAL REVIEW B 76, 121101͑R͒͑2007͒ Response of a cylindrical invisibility cloak to electromagnetic waves Baile Zhang, Hongsheng Chen,* Bae-Ian Wu, Yu Luo, Lixin Ran, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA and The Electromagnetics Academy at Zhejiang University, Zhejiang University, Hangzhou 310058, China ͑Received 17 July 2007; revised manuscript received 13 August 2007; published 4 September 2007͒ Based on electromagnetic wave scattering theory, we demonstrate that magnetic and electric surface currents are induced at the inner boundary of a cylindrical cloak by incoming waves. These surface currents have no counterparts in the coordinate transform theory. In addition, electromagnetic fields can penetrate into the concealed region under specified conditions, but carry no power. The field distribution can be unsymmetrical when a circularly polarized wave is obliquely incident onto the cloak. The far-field scattering for a nonideal cylindrical cloak is also theoretically addressed. DOI: 10.1103/PhysRevB.76.121101 PACS number͑s͒: 41.20.Jb, 42.25.Fx Recently, invisibility cloaking based on coordinate trans- discussed. All of these aspects provide us with deeper in- forms has been discussed, where empty space is transformed sights into the cloaking phenomenon. into a shell surrounding the object to be concealed.1 Another Figure 1 shows the configuration of the scattering from a ͑ ͒ method was reported where the two-dimensional 2D Helm- cylindrical cloak with outer radius R2 and inner radius R1 on holtz equation is transformed to produce similar effects in which a time-harmonic plane wave E¯ is obliquely incident. the geometric limit.2 Some related work on invisibility cloak- i ¯ ͑ ˆ ͒ ik¯·¯r ¯ 2 ␻2␮ ⑀ ˆ ing utilizing plasmonic resonance has also been published.3,4 Let Ei = vˆ iEvi +hiEhi e , where k=xˆk␳ +zˆkz, k = 0 0, hi ϫ ˆ ͉ ϫ ˆ͉ ˆ ϫ ˆ 11 The effectiveness of a transformation-based cloak design =zˆ k/ zˆ k , and vˆ i =hi k. Without loss of generality, we was first confirmed computationally in the geometric optics assume that the concealed region is filled with an isotropic 1,5 6,7 ⑀ ␮ limit, and then in full-wave, finite-element simulations, material with permittivity 2 and permeability 2. The as well as in a full-wave spherical scattering model.8 A prac- cloak layer between R1 and R2 is a specific anisotropic tical 2D cylindrical cloak, which has been realized in experi- and inhomogeneous medium with permittivity tensor ment, showed that the cloak reduces both backscattering ͑re- ⑀ញ =⑀␳␳ˆ␳ˆ +⑀␾␾ˆ ␾ˆ +⑀ zˆzˆ and permeability tensor ␮ញ =␮␳␳ˆ␳ˆ flection͒ and forward scattering ͑shadow͒.9 An attempt to use z ␮ ␾ˆ ␾ˆ ␮ scattering theory to demonstrate the perfect invisibility of the + ␾ + zzˆzˆ. According to Refs. 6 and 9, we choose ⑀ ⑀ ␮ ␮ ͑␳ ͒ ␳ ⑀ ⑀ ␮ ␮ ␳ ͑␳ ͒ cylindrical cloak in a normally incident ͑or 2D͒ case was also ␳ / 1 = ␳ / 1 = −R1 / , ␾ / 1 = ␾ / 1 = / −R1 , and ⑀ ⑀ ␮ ␮ ͓ ͑ ͒2͔͑␳ ͒ ␳ ⑀ ⑀ made in Ref. 10. However, the basic physics behind the z / 1 = z / 1 = R2 / R2 −R1 −R1 / . When 1 = 0 and ␮ ␮ cloaking phenomenon of a cylindrical cloak, for example, 1 = 0, the cloak corresponds to the ideal cloak. the peculiar behavior of the inner boundary of the cloak in In a source-free medium, the fields can be decomposed response to incoming waves, the field distribution, and the into TEz and TMz modes with respect to zˆ,12 corresponding ␺z ␺z far-field scattering behavior due to an obliquely incident to scalar potentials TM and TE, respectively, wave with arbitrary polarization, and the question of whether fields can penetrate into the concealed region, is still un- ͒ ͑ ͒ ␮ញ −1 ٌ ϫ ͑ ␺z ¯ solved. HTM = · zˆ TM , 1a In this paper, a general derivation based on the full-wave scattering model for a cylindrical cloak under oblique inci- dence is presented. The scattering field from an ideal cylin- E =−⑀ញ−1 · ٌ ϫ ͑zˆ␺z ͒, ͑1b͒¯ drical cloak with parameters obtained in Refs. 1 and 6 is TE TE shown to be indeed zero, not only in the normal incidence case,10 but also in the oblique incidence case, corroborating 1 the effectiveness of the cloak design process specified in Ref. E¯ = ⑀ញ−1 · ͓␮ញ −1 · ٌ ϫ ͑zˆ␺z ͔͒, ͑1c͒ 1. Meanwhile, the electromagnetic fields across the inner TM − i␻ TM boundary of such a cloak are discontinuous, and at the inner boundary of the cloak, electric and magnetic surface currents z along the ␾ direction, which are not predicted by the coor- dinate transform theory, will be induced. This indicates that a y θ i k full-wave analytical scattering calculation is very necessary for further investigation of the response of the cloak with a x ε o ε ε μ 2 peculiar inner boundary. Extending the inner boundary in- o μ μ2 ward will cause the fields, but not the power, to penetrate R1 into the concealed region. In addition, the field distribution R2 due to a circularly polarized incident wave can be unsym- metrical while the power is still distributed symmetrically. FIG. 1. Geometry of a cylindrical cloak. The incident wave The sensitivity of the cloak to nonideal parameters is also propagates in the direction of ¯k. 1098-0121/2007/76͑12͒/121101͑4͒ 121101-1 ©2007 The American Physical Society RAPID COMMUNICATIONS ZHANG et al. PHYSICAL REVIEW B 76, 121101͑R͒͑2007͒ ikz ikz 1 e Ae ik z H¯ = ␮ញ −1 · ͓⑀ញ−1 · ٌ ϫ ͑zˆ␺z ͔͒. ͑1d͒ Te 2 TE − i␻ TE Re-ikz -ikz Hereafter the superscript z is suppressed. After substituting Ce ͑ ͒ Eqs. 1 into the Maxwell equations, we can get the wave μooε μ11ε μ22ε equation for the scalar potentials as follows: z=0 z=d 2␺ FIG. 2. Three-layer model with a normally incident wave forץ ␺ 1ץ ץ 1 ͑␳ − R ͒ + -␾2 demonstrating the surface currents at the inner boundary of a cylinץ ␳ ͑␳ − R ͒2ץ ␳ 1ץ ␳ − R 1 1 ␮ drical cloak. The slab’s permeability 1 goes to infinity while the . ⑀ 2␺ permittivity goes to zero, but their product is kept equal to ␮ץ R2 + 2 ͩ␻2␮ ⑀ ␺ + ͪ =0. ͑2͒ 0 0 2 ץ 1 1 2͒ ͑ R2 − R1 z the Neumann function, and the Hankel function of the first ͑M͒ ͑M͒ ͑M͒ ͑M͒ Using the method of separation of variables, the general ex- kind, respectively; an , gn , dn , and fn are unknown pression for the two scalar potentials is obtained as coefficients corresponding to the scattering field, the internal field, and the field inside the cloak, respectively. After solv- ͑␳ ͒ ͑M͒ ͑M͒ ͑M͒ − R1 ␾ ␺ ͩ ͪ in +ikzz ͑ ͒ ing all the equations, it is seen that an = fn =gn =0, indi- = B R k␳ e , 3 n 2 1 ͑ ͒ R2 − R1 cating that both the internal field and the scattering field are zero. The most interesting thing that is not shown in Ref. 10 ͑ ͒ where Bn represents the solutions of the Bessel equation of M is that, when n=0, though f0 =0, the product ͱ␻2⑀ ␮ 2 ͱ 2 2 ͑M͒ nth order, and k␳1 = 1 1 −kz = k1 −kz . As in the spherical f N ͕͓R /͑R −R ͔͒k␦͖ in the limit ␦→0 is equal to E /i␻. 8 ␺i ␺s ␺c ␺int 0 0 2 2 1 vi cloak, the scalar potentials , , , and for the inci- Obviously, the value of this product in Eq. ͑6͒ is nonzero ͑␳Ͼ ͒ ͑␳Ͼ ͒ dent fields R2 , the scattered fields R2 , the fields only at the inner boundary, which can be characterized as a ͑ Ͻ␳Ͻ ͒ inside the cloak layer R1 R2 and the internal fields magnetic surface current raised by the infinite ␮␾ at the inner ͑␳Ͻ ͒ R1 , respectively, for both TE and TM waves, can be boundary of the cloak. Since the tangential electric field on represented. By matching the boundary conditions, all the the cloak side of the inner boundary is Evi, the magnetic expanding coefficients of these scalar potentials can be de- surface current with an amplitude Evi shields the concealed duced. object, making the field inside exactly zero. Reference 10 For convenience of demonstration, we first consider the mentioned the “slow convergence” of the scattering coeffi- ͑ ͒ simple TM case where a vertically polarized Ez plane wave cients but did not note that this is actually due to the surface is normally incident onto an ideal cylindrical cloak, i.e., current at the inner boundary.