Invisibility Cloaks in Electromagnetic Theory Marco A. Ribeiro †, Carlos R. Paiva ¥ † Instituto de Telecomunicações, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Phone: +351-218418457, Fax: +351-218418472, e-mail: [email protected] ¥ Instituto de Telecomunicações, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Phone: +351-218418479, Fax: +351-218418472, e-mail: [email protected] Abstract 1— In this communication, using a previously derived for electromagnetics and show how it can be used to design framework that we have called the equivalence principle for invisibility devices. electromagnetics, we analyze invisibility cloaking devices. These In section II we address the technical details regarding the devices compress and drag the electromagnetic field, without design of cylindrical invisibility devices. We present two disturbing their functionality, to render an object invisible examples cloakings: first, a cloak in which we can control within the concealment region. To assess the accuracy of our how the electromagnetic field is compressed inside it; second, geometric modeling full-wave numerical simulations are also a cloaking which is capable of dragging the electromagnetic provided. field inside it. Finally, in section III we provide full-wave numerical simulations, using the FDTD method, for the invisibility I. INTRODUCTION devices introduced in section II. The unprecedented control over the material properties that metamaterials can offer [1] has prompted a new and powerful II. THEORETICAL FORMULATION strategy for designing electromagnetic devices. This strategy stems from an equivalence principle for electromagnetics [2]: A. Electrodynamics in general relativity a given geometry creates an effective medium in the same way as a given electromagnetic medium creates an effective In general relativity, and using tensor analysis, the free- geometry [3, 4]. This interplay between geometry and space Maxwell equations are written as [8] F Fαβ j α , (1) electromagnetic media can be explored for designing []αβ; γ =0, ;β = invisibility devices. In fact, cloaking devices that aim the where the semi-colon means covariant differentiation and the concealment of a given object have been the subject of some square brackets indicate anti-symmetrization. It is possible to recent research in electromagnetics [5, 6]. They belong to a cast the information in these equations in the form of two family of extraordinary new devices – called transformation generally covariant Maxwell equations (homogeneous and media – which have many promising applications, e.g., inhomogeneous) and address all the geometric aspects of the shielding from mobile phone and radar radiation, new optical underlying pseudo-Riemannian manifold (M,g ) into a devices and medical imaging. The physical origin for this equivalence principle can be separate spacetime relation. By performing this separation tracked down to Fermat’s principle: light rays follow the between the two Maxwell equations and the metric structure stationary optical path in media. That is why especially of spacetime, we get a framework called pre-metric electrodynamics [9]. Using the coordinate-free approach tailored inhomogeneous metamaterials – the up mentioned 2 transformation media – can achieve active transformations by presented in [2] these pre-metric equations are written as dF = 0,( homogeneous equation ) deviating electromagnetic radiation along a prescribed (2) trajectory, thereby obnubilating the object covered with a δ G= J,() inhomogeneous equation cloaking device. 0 123 where F= E ∧ e + ce B is the EM (electromagnetic In this communication we show how the metric-free (and coordinate-free) mathematical setup introduced in [2] - which field) strength untwisted 2-form, G=e123 Hc − D ∧ e 0 is the is based simultaneously on the grammar of Clifford’s EM field excitation bivector density, J is the charge-current geometric algebras and the language of differential forms - vector density whereas d and δ are the exterior and can be used to design several cylindrical invisibility devices contraction derivatives respectively (see [2] for details). in light of the equivalence principle for electromagnetics. The spacetime relation corresponding to free-space, without This communication is organized as follows. In section I we which the set (2) would not be complete, is given by present a brief description of the theoretical formulation * G= y0 − gg ( F ) , (3) introduced in [2] which was employed - as a unifying approach - to design invisibility devices and study moving media in [7]. We also introduce the principle of equivalence 2 The notation is the same as in [2]: we use italic fonts for forms defined over the cotangent bundle (e.g. F ) and regular fonts for vectors defined over the tangent bundle (e.g. G and J ). where g is a Lorentz metric, g * its inverse bundle map, The parameters ε , ξ , µ and ζ in (4) relate to the metric of spacetime in the following way: g= det (g ) and y0= ε 0 µ 0 . −g The particularly distinctive feature of this approach is that ∗ ε ()E= − g⊥ () E , the two Maxwell equations belong to different bundles of M g00 (see Fig. 1): the homogeneous equation belongs to forms 1 ∗ ξ ()H=e123() H ∧ g ⊥ () e 0 , bundle ( ∧T M ) whereas the in-homogeneous equation g 00 (5) M belongs to exterior algebra bundle ( ∧T ). By using this −g µ ()H= − g∗ () H , strategy, we are able to write our spacetime relation (3) g ⊥ completely free of indices. It seams that this is not easily done 00 with an approach based solely on differential forms [9]. 1 ζ ()E= −e123() E ∧ g ⊥ () e 0 . g00 III. PERMEABILITY AND PERMITIVITY FUNCTIONS OF CLOAKING DEVICES Cloaking devices do not exhibit magneto-electric coupling ( g⊥ (e0 ) = 0 ). Therefore, in (5) we make ξ= ζ = 0 . On the other hand, there is some freedom associated with g00 and Fig. 1. The generally covariant Maxwell equations are written in we chose g00 = − 1 for commodity. Hence from (5) we find different bundles of M . the compact expression − 1 2 B. The equivalence principle for electromagnetics ε= µ = (g∗ ) g* . (6) According to the equivalence principle [2], there are two A. Cylindrical cloak – compression transformation alternative interpretations for (3): the topological and the materials interpretation. The equivalence principle states that Cylindrical cloaks are devices mapping the space in a 3- dimensional cylindrical volume (radius r ) into a cylindrical the topological interpretation (M,g ) is, from the 2 shell of inner radius and outer radius (Fig. 2). electromagnetic point of view, entirely equivalent to the r1 r2 materials interpretation: it corresponds to a trivial manifold (M, I ) filled with a medium described by a CR (constitutive relation) that caries the information contained in g : M M ()()(),g ⇔,I + CR g վ վ վ topological interpretation equivalence principle materials interpretation The strength of these interpretations relies in the fact that the generally covariant Maxwell equations are preserved by any diffeomorphism over the four-dimensional spacetime M : we can pull and stretch spacetime, thereby modifying its metric, without any effect on the form of Maxwell equations. Actually, only the spacetime relation, which characterizes free-space from the electromagnetic point of view, will change. One should stress that the information about the flexible metric in (M,g ) cannot be transferred into the realm of (M, I ) : it is now ingrained in the texture of the CR Fig. 2. A light ray passing through a cloak with inner radius characterizing the medium that permeates the trivial and rigid r= 0.1 m and outer radius r= 0.2 m for two values of the M 1 2 ( , I ) ; the flexible metric g gives place, in the materials compression factor n . a) materials interpretation; b) topological interpretation, to a medium – that we call a simple medium - interpretation for n = 1 ; c) topological interpretation for n = 0.4 ; characterized by a CR that corresponds in three-dimensional d) profile of the transformation. space to an inhomogeneous bianisotropic medium [2] D=εε()E + εµξ () H , We consider a transformation r֏ f( r ) which is 0 0 0 (4) characterized by the compression function B=µµ0()H + εµζ 0 0 () E . r2− r 1 n r r r− r 1 fr() =n rr +1, 0 << rr 2 , (7) ε()()e= µ en = er + φ () re θ , r2 r whose profile is illustrated in Fig. 2a for compression factors 1+φ 2 ()r r εeereθ= µ θ = φ + e , (10) n = 1 and n = 0.4 . The effect of the compression factor is ()() () r θ n r− r 1 revealed in Fig. 2b: as we decrease n light rays will pass 2 2 n−1 closer to the outer shell. The transformed space, z z 1 r2 ()r− r 1 ε()()e= µ e = 2 e z . corresponding to the topological interpretation, is also shown n()r− r n r in Fig. 2c and Fig. 2d for n = 1 and n = 0.4 respectively. 2 1 where, Here we can see clearly the intrinsic relation between the ray rθ 1 trajectory and the structure of space which is revealed in the 0 n φ ()r= −1 () r − r 1 (11) topological interpretation. ()r− r n Using the methodology described in [2,7] we can easily 2 1 derive the medium parameters for this cloaking: IV. NUMERICAL SIMULATIONS r r r− r 1 ε()()e= µ en = e r , r Finally, we present full-wave numerical simulations of the devices that have been analyzed in this communication. All θ θ 1 r ε()()e= µ e = e θ , (8) the simulations are preformed at the working frequency of n r− r 1 2GHz using the FDTD method. The visualization window is a 2 −1 1 r 2 ()r− r n rectangular grid of 1.2m (8 wave-lengths) by 0.8m (2.67 εez= µ e z = 2 1 e . ()() 2 z wave-lengths) and we have used material average nr− r n r ()2 1 optimization (nine subcells) in all devices in order to reduce B.