A Rigorous Analysis of Plane-Transformed Invisibility Cloaks The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Yu Luo et al. “A Rigorous Analysis of Plane-Transformed Invisibility Cloaks.” Antennas and Propagation, IEEE Transactions on 57.12 (2009): 3926-3933. © 2009 IEEE As Published http://dx.doi.org/10.1109/tap.2009.2027824 Publisher Institute of Electrical and Electronics Engineers Version Final published version Citable link http://hdl.handle.net/1721.1/53728 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. 3926 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 57, NO. 12, DECEMBER 2009 A Rigorous Analysis of Plane-Transformed Invisibility Cloaks Yu Luo, Jingjing Zhang, Hongsheng Chen, Lixin Ran, Bae-Ian Wu, and Jin Au Kong, Fellow, IEEE Abstract—The electromagnetic characteristics of plane-trans- [4]. And a spherical cloak with spatially gradient, anisotropic formed invisibility cloaks are quantitatively studied in this paper. material parameters was proposed, which can bend and guide We take elliptical cylindrical cloak as the example, and use an the incoming wave smoothly around the cloaked region without elliptical cylindrical wave expansion method to obtain the scat- tered field. It is demonstrated that an ideal elliptical cylindrical disturbing the incident field. Leonhardt described a similar op- cloak is inherently visible. Noticeable field scattering and pene- tical conformal mapping method to achieve two-dimensional tration will be induced when the cloak is exposed directly to an (2D) invisibility by assuming the short wavelength geometrical electromagnetic wave. However, as long as the cloak consists of a limit [5]. The effectiveness of the transformation based cloak perfect electric conducting lining at the interior surface, perfect design was first validated by full-wave simulation [6] and ray invisibility can still be achieved along the direction parallel to the major axis of the cloak for transverse magnetic illumination. tracing exercise [7], while a practical 2D cylindrical cloak based Another plane-transformed cloak with a conical geometry is also on the reduced set of material parameters was realized experi- proposed. The advantage of this cloak is that all the permittivity mentally [8]. Following this approach, some authors considered and permeability elements are spatially invariant while none of cloak achieved with a high order transformation [9], aiming to them is singular. Hence, it is easily realizable with artificially minimize the undesired scattering caused by the impedance mis- structured metamaterials. Finally, the applications of this kind of matched at the cloak’s outer boundary. The scattering model cloak are discussed. of spherical cloak was established in [10], [11]. Further theo- Index Terms—Artificially structured metamaterials, invisibility retical treatments dealing with the boundary conditions of the cloak, scattering characteristics, spatial coordinate transforma- point transformed [12] and line transformed [13] cloaks have tion. also been presented. These rapid processes made the transfor- mation a hot topic and triggered more and more study on invis- I. INTRODUCTION ibility cloaks [14]–[22]. We notice that in most of the above discussions, the inner ESIGN of electromagnetic cloak of invisibility has at- boundary of the cloak is transformed from a point or a line, D tracted much attention in the recent years. By exploiting which results in extreme material parameters (zero or infinity) the frequency dispersion of plasmonic materials, Alù and En- at the cloak’s interior surface. Recently, the transformation gheta suggested the possibility of employing plasmonic shell approach was applied to elliptical cylindrical coordinate, and for reducing the scattering from a single or a collection of small an elliptical cylindrical cloak was achieved correspondingly spheres [1], [2]. This concept has also been used to conceal a [23]–[25]. It was shown that none of the parameters of the cloak particle simultaneously at different frequency in optical domain is singular [24], since the inner boundary of the cloak is trans- [3]. Another methodology based on spatial coordinate transfor- formed from a plane in this case. However, the performances mation of Maxwell’s equations was reported by Pendry et al. of this cloak are not so good, especially for transverse electric (TE) illumination [25]. A similar plane-transformed cloak was presented in [26], which was achieved by combing the Manuscript received December 23, 2008; revised July 05, 2009. First segments of cylindrical cloak. And a finite element method was published July 21, 2009; current version published December 01, 2009. This performed to validate the cloaking performance, showing that work was sponsored in part by the National Science Foundation of China under Grants 60531020, 60701007, and 60801005, in part by the NCET-07-0750, this cloak can only achieve perfect invisibility for transverse the Zhejiang National Science Foundation under Grant R1080320, the Ph.D. magnetic (TM) plane wave propagating in a definite direction Programs Foundation of MEC under Grant 200803351025, the ONR under [26]. Since all these facts indicate that the scattering property Contract N00014-06-1-0001, the Department of the Air Force under Air Force Contract F19628-00-C-0002, and the Foundation for the Author of National of the plane transformed cloak is unclear, a rigorous analysis is Excellent Doctoral Dissertation of PR China under Grant 200950. quite necessary. Y. Luo, J. Zhang, H. Chen and L. Ran are with the The Electromag- In this paper, we study the electromagnetic characteristics of netics Academy, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]; [email protected]; hansomchen@zju. the elliptical cylindrical cloak by rigorously solving Maxwell’s edu.cn; [email protected]). equations in elliptical cylindrical coordinate. The general B.-I. Wu is with the Research Laboratory of Electronics, Massachusetts In- boundary equations are derived. It is demonstrated that an ideal stitute of Technology, Cambridge, MA 02139 USA. elliptical cylindrical cloak cannot achieve perfect invisibility J. A. Kong (deceased) was with the The Electromagnetics Academy, Zhejiang University, Hangzhou 310027, China and also with the Research Laboratory when it is exposed to an EM wave directly. However, when of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 the inner boundary of the cloak is a perfect electric conducting USA. lining, the odd scattering coefficients are exactly zero for TE Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. illumination, while the even scattering coefficients are always Digital Object Identifier 10.1109/TAP.2009.2027824 zero for TM illumination. This result indicates that the elliptical 0018-926X/$26.00 © 2009 IEEE Authorized licensed use limited to: MIT Libraries. Downloaded on February 11, 2010 at 14:49 from IEEE Xplore. Restrictions apply. LUO et al.: A RIGOROUS ANALYSIS OF PLANE-TRANSFORMED INVISIBILITY CLOAKS 3927 into an annular region in the new co- ordinate via (2) where and are the inner and outer shell coordinate param- eters of the elliptical cylindrical cloak, and is an arbitrary monotonic function which satisfies the condition and . Under this transformation, a plane is transformed to the cloak’s interior surface [24], [25]. And the corresponding relative permittivity and permeability el- ements of the cloak shell ( ) can be expressed in the transformed physical coordinate system as follows: Fig. 1. Geometry of an elliptical cylindrical cloak. The transformation media (3) that comprises the cloak lies between the two ellipses. and all the off-diagonal terms are zero. For simplicity, we select as a linear function . Thus, cylindrical cloak with a perfect electric conducting lining at (3) is reduced to the interior surface can still achieve perfect invisibility along the direction parallel to the cloak’s major axis when it is illuminated by a TM wave. We have also analyzed how the non-ideal parameters affect the cloaking performance quanti- tatively. It is found that adding loss to the material parameters (4) can dramatically suppress the backward scattering. All these scattering characteristics can be found in any other plane-trans- where is a definite constant. It can be seen formed cloaks. As an example, we propose another type of from (4) that both (or ) and (or ) are spatially plane-transformed cloak which has a cone shaped geometry. uniform, while (or ) is dependent on both and .In Full-wave finite-element method is used to verify the cloaking Fig. 2, we plot the three components of the material parameters performance. It is found that apart from achieving one-dimen- in terms of for different angular coordinate parameters ( sional invisibility, this kind of cloak has the similar capability , , ). Here the semi-focal distance is selected as with the one proposed by Li and Pendry [27] in that it can give , while the inner and outer shell coordinate parameters are any cloaked obstacle the appearance of a flat conducting sheet. set to be and , respectively. The results show However, the cloak presented here has the advantage that all that the permittivity and permeability elements are singular only the material parameters are