Generalized Transformation Optics from Triple Spacetime Metamaterials

Generalized transformation optics from triple spacetime metamaterials Luzi Bergamin∗ European Space Agency, The Advanced Concepts Team (DG-PI), Keplerlaan 1, 2201 AZ Noordijk, The Netherlands (Dated: July 25, 2008) In this paper, various extensions of the design strategy for transformation media are proposed. We show that it is possible to assign different transformed spaces to the field strength tensor (electric field and magnetic induction) and to the excitation tensor (displacement field and magnetic field), resp. In this way, several limitations of standard transformation media can be overcome. In particular, it is possible to provide a geometric interpretation of non-reciprocal as well as indefinite materials. We show that these transformations can be complemented by a continuous version of electric- magnetic duality and comment on the relation to the complementary approach of field-transforming metamaterials. I. INTRODUCTION with the space metric γij and its determinant γ. For many manipulations it will be advantageous to use In the field of metamaterials, artificial electromagnetic relativistically covariant quantities. Therefore, Eqs. (1) materials, the use of spacetime transformations as a de- and (2) are rewritten in terms of the field strength tensor µν µ sign tool for new materials has been proved very success- Fµν , the excitation tensor H and a four current J (cf. ful recently [1–3]. As basic idea of this concept a meta- Appendix): material mimics a transformed, but empty space. The ²µνρσ∂ F = 0 ,D Hµν = −J µ ,D J µ = 0 . (4) light rays follow the trajectories according to Fermat’s ν ρσ ν µ principle in this transformed (electromagnetic) space in- The four dimensional covariant derivative Dµ is defined stead of laboratory space. This allows one to design in an analogously to (3), whereby the space metric is replaced efficient way materials with various characteristics such √ by the spacetime metric gµν and its volume element −g. as invisibility cloaks [1, 2, 4], perfect lenses [3], magnifi- We wish to analyze these equations of motion from cation devices [5], an optical analogue of the Aharonov- the point of view of transformation media. All transfor- Bohm effect or even artificial black holes [3]. Still the mation materials have in common that they follow as a media relations accessible in this way are rather limited, transformation from a (not necessarily source-free) vac- in particular non-reciprocal or indefinite media (materi- uum solution of the equations of motion, which maps als exhibiting strong anisotropy) are not covered. But this solution onto a solution of the equations of motion these types of materials also have been linked to some of of the transformation material [25]. The crucial ingredi- the mentioned concepts, in particular perfect lenses [6, 7] ent in the definition of transformation media then is the and hyperlenses [8]. This raises the question whether class of transformations to be considered. As space of all there exists an extension of the concept of transforma- transformations we restrict ourselves to all linear trans- tion media such as to cover those materials as well and formations in four-dimensional spacetime. Consequently, to provide a geometric interpretation thereof. all media exhibit linear constitutive relations, which may In this paper we propose an extension of this type. As be written within the covariant formulation as [12] in Refs. [1–3] our concept is based on diffeomorphisms locally represented as coordinate transformations. There- 1 Hµν = χµνρσF . (5) fore many of our result allow a geometric interpretation 2 ρσ similar to the one of Refs. [3, 9] as opposed to another recently suggested route to overcome the restrictions of In vacuum one obtains [26] diffeomorphism transforming media [10, 11]. The start- 1 ing point of our considerations are Maxwell’s equations χµνρσ = (gµρgνσ − gµσgνρ) , (6) 2 in possibly curved, but vacuous space [24]: such that the standard result E~ = D~ and B~ = H~ emerges. ∇ Bi = 0 , ∇ Bi + ²ijk∂ E = 0 , (1) i 0 j k These transformations and the ensuing media proper- i ijk i i ∇iD = ρ , ² ∂jHk − ∇0D = j . (2) ties (5) have the advantage of being relativistically invariant and thus very easy to handle. However, they do Here, ∇ is the covariant derivative in three dimensions i not include any frequency dependence and remain strictly 1 √ real, which perhaps is the most severe restriction that fol- ∇ Ai = (∂ + Γi )Aj = √ ∂ ( γAi) , (3) i i ij γ i lows from the coordinate transformation approach. As long as the linear transformations are seen as transformations of spacetime (rather than of the fields) this restriction is not surprising, though. Indeed, from energy ∗Electronic address: [email protected] conservation it follows that it is impossible to model a 2 process of absorption by the medium as a local transformation of spacetime (notice that the spacetime itself is not dynamical and thus cannot contribute to the energy). II. DIFFEOMORPHISM TRANSFORMING METAMATERIALS FIG. 1: Diffeomorphism transforming metamaterials according to Ref. [3]. Obviously, the concept of transformation materials as sketched above is related to symmetry transformations, Now, as the basic idea of Ref. [3], if empty space can as those are by definition linear transformations that map appear like a medium, a medium should also be able to a solution of the equations of motion onto another one. appear as empty space. One starts with electrodynam- Therefore it is worth working out this relation in some ics in vacuo, we call these fields F and Hµν with flat more detail. µν metric gµν . Now we apply a diffeomorphism, locally rep- A symmetry is a transformation which leaves the resented as a coordinate transformation xµ → x¯µ(x). As source-free [27] action of the theory, here the equations of motion by definition are invariant un- Z √ der diffeomorphisms, all relations remain the same with S = d4x −gF Hµν , (7) µν the fields F , H and the metric gµν replaced by the new barred quantities. As a last step one re-interprets in the invariant, whereby surface terms are dropped. It dynamical equations (1) and (2) the coordinatesx ¯µ as straightforwardly follows that a symmetry transforma- µ the original ones x , while keepingg ¯µν in the constitu- tion applied on a solution of the equations of motion still tive relation. To make this possible some fields must be solves the latter. In the above action a general, not nec- rescaled in order to transform barred covariant deriva- essarily flat, spacetime is considered. The symmetries of tives (containingg ¯) into unbarred ones (containing g.) this action are well known: these are the U(1) gauge The situation of diffeomorphism transforming metama- symmetry of electromagnetism and the symmetries of terials is illustrated in Figure 1, which also summarizes spacetime (diffeomorphisms). The gauge symmetry can- our notation. As a more technical remark it should be not help in designing materials as the media relations noted that this manipulation is possible as we consider are formulated exclusively in terms of gauge invariant just Maxwell’s theory on a curved background rather quantities. However, diffeomorphisms change the media than Einstein-Maxwell theory (general relativity coupled relations, as is pointed out e.g. in Ref. [13] and as it has to electrodynamics.) In the former case the metric is an been applied to metamaterials in Ref. [3]. Thus one way external parameter and thus this manipulation is possi- to define transformation media is: ble as long as none of the involved quantities depends Definition 1. A transformation material follows from explicitly on the metric. a symmetry transformation applied to a vacuum solution To keep the whole discussion fully covariant the fields of Maxwell’s equations. This vacuum solution need not are transformed at the level of the field strength and ex- be source free. citation tensor (rather than at the level of space vectors as was done in Ref. [3]). To transform the covariant The space of all possible transformation materials of derivatives D¯ µ into the original Dµ we have to apply the this kind has been derived in Ref. [3]; here we briefly rescalings want to summarize the result of that paper. The starting √ √ point is the observation that a curved space in Maxwell’s −g¯ −g¯ equations looks like a medium. Indeed, in empty but H˜ µν = √ H¯ µν J˜µ = √ J¯µ . (12) −g −g possibly curved space the constitutive relation among the electromagnetic fields is found by exploiting In addition the transformation xµ → x¯µ(x) may not pre- 0j kl F0i = (g00gij − g0jgi0)H + g0kgilH , (8) serve the orientation of the manifold, which technically ij i0 jk ik jl means that the Levi-Civita tensor changes sign [3]. This H = 2g g F0k + g g Fkl , (9) is corrected by introducing the sign ambiguity which in terms of the space vectors reads F˜µν = ±F¯µν (13) ij i g g0j jil D = √ Ej − ² Hl , (10) −g00 g00 with the plus sign for orientation preserving, the mi- ij nus for non-preserving transformations. These new fields i g g0j jil B = √ Hj + ² El . (11) again live in the original space with metric g , but now −g g µν 00 00 the space is filled with a medium with Thus empty space can appear like a medium with per- ij ij ij √ √ meability and permittivity ² = µ = g / −g00 and µνρσ 1 −g¯ µρ νσ µσ νρ ij ij lij χ˜ = ± √ (¯g g¯ − g¯ g¯ ) , (14) with bi-anisotropic couplings ξ = −κ = ² g0l/g00. 2 −g 3 or, in terms of space vectors, Definition 2. Consider the set of all transformations T which map a source free solution of the equations of ij √ i g¯ γ¯ g¯0j jil motion (1) and (2) onto another source free solution.

Load more