Generalized transformation from triple

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) 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 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 lo- cally 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 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 in- variant and thus very easy to handle. However, they do Here, ∇ is the covariant derivative in three dimensions i not include any 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 transfor- mations of spacetime (rather than of the fields) this re- striction is not surprising, though. Indeed, from ∗Electronic address: [email protected] conservation it follows that it is impossible to model a 2 process of absorption by the medium as a local transfor- mation 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 accord- ing 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 , 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 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 ( 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 vec- tors 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 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. A D˜ = s√ √ E˜j − ² H˜l , (15) −g¯00 γ g¯00 transformation material is a material obtained by apply- ij √ ing a transformation T onto a (not necessarily source i g¯ γ¯ g¯0j jil B˜ = s√ √ H˜j + ² E˜l , (16) free) vacuum solution. −g¯00 γ g¯00 There are two types of extensions contained in this with s = ±1 being the sign in (13) and (14). As can definition compared to the previous section: be seen, the media properties are restricted to reciprocal materials (² = ²T , µ = µT , κ = χT ), which, in addi- 1. There exist transformations that leave the equa- tion, obey ² = µ. This result has been obtained in Ref. tions of motion invariant, but change the action by [3] in a slightly different way and encompasses the trans- a constant and thus are not symmetry transforma- formations in Refs. [1, 14]. We do not want to go into tions. A transformation of this type is the so-called further details of this approach but refer to the review electric-magnetic duality. Its effect will briefly be [9], where its geometric optics interpretation is discussed discussed in Section III A. in detail. Indeed, light travels in transformation media 2. We do allow for transformations which leave all of this type along null geodesics of the electromagnetic Maxwell’s equations (1) and (2) invariant, but µ space x¯ , which allows (with some restrictions to be dis- change the media relations (5). This indeed gen- cussed in Section V) a simple and intuitive interpretation eralizes the concept in an important way. of the transformation. To see the origin of the second extension it is impor- tant to realize that the equations of motion of electro- III. TRIPLE SPACETIME METAMATERIALS dynamics separate into two different sets (Eqs. (1) and (2), resp.) with mutually exclusive field content. This Despite the variety of applications of diffeomorphism characteristic is not just an effect of our notation, but transforming metamaterials some results suggest a search as has been shown e.g. in Refs. [15, 16], the equations of for extensions. Indeed, there exist e.g. designs of super- motion of electrodynamics can be derived from first prin- and hyperlenses that make use of indefinite materials ciples without using explicitly the constitutive relation (strong anisotropy) [6–8]. Though both concepts should H = H(F ). As the two sets of equations are separately be perfectly understandable in terms of transformation invariant under diffeomorphisms it should be possible to media, the specific material relations used in these works assign different transformed spaces to H = (D~ , H~ ) and do not fall under the class of diffeomorphism transform- F = (E,~ B~ ). In other words, it must be possible to distort ing metamaterials. the spaces (or the coordinates) of the field strength tensor To understand a possible route to generalize the con- and the excitation tensor separately, whereby the result- cept of diffeomorphism transforming media we have to ing transformation material per constructionem satisfies consider again their basis, namely symmetry transfor- all conditions of the Definition 2. The ensuing constitu- mation. The concept of symmetries is used to identify tive relation as well as the solutions of the equations of different solutions of the equations of motion that effec- motion still follow (almost) as simple as in the case of tively describe the same . By means of the re- Ref. [3]. interpretation in the last step of Figure 1, such symme- To prove the potential of this method we have to ex- try transformations can be used as a simple tool to derive tend the notation compared to the previous section: as within a restricted class of constitutive relations new me- before laboratory space has metric gµν , its fields in vacuo dia properties in a geometrically intuitive and completely are H = (D~ , H~ ) and F = (E,~ B~ ); the fields of the trans- algebraic way. formation material (living in the space with metric gµν ) Nonetheless, within the concept of metamaterials it is are again labeled with a tilde. The transformed space not important that the transformed solution in principle of the field strength tensor has metricg ¯µν and fields describes the same physics as the original one. Still, one F¯ = (E,~¯ B~¯), the one of the excitation tensor g¯ and may want to keep the possibility of mapping source free µν ¯ ~¯ ~¯ solutions onto other source free solutions in a straight- H¯ = (D¯, H¯). This new transformation is illustrated in forward way, as only in this way do we have an effective Figure 2. Applying the two transformations control over passive media and do not risk introducing ex- µ µ ¯µ ¯µ otic sources such as magnetic monopoles. Furthermore a x¯ =x ¯ (x) , x¯ = x¯ (x) (17) geometric interpretation of the transformations is kept, to the constitutive relation (5) with χ being the vacuum which is advantageous in many applications. To weaken relation (6) yields the conditions on transformation materials while keep- ing the advantages of symmetry transformations we thus 1 ∂x¯µ ∂x¯ν ¡ ¢ ∂x¯ρ ∂x¯σ H¯ µν = gλαgτβ − gλβgτα F¯ . propose the following definition: 2 ∂xλ ∂xτ ∂xα ∂xβ ρσ (18) 4

that √ −g¯ ¯¯ ¯¯ ¯¯ ¯¯ Aij = −s¯ √ (g00gij − g0jgi0) , (24) γ √ γ B = −s¯√ (g¯ g¯ − g¯ g¯ ) , (25) ij −g¯ 0¯0¯ ¯i¯j 0¯¯j ¯i0¯ √ s¯ −g¯ ¯¯ ¯¯ ¯¯ ¯¯ C j = − √ ² (gk0glj − gkjgl0) , (26) i 2 γ ikl √ i s¯ −g¯ 0¯k¯ ¯i¯l 0¯¯l ¯ik¯ D j = √ ²jkl(g g − g g ) , (27) FIG. 2: Illustration and notation of the generalized “triple 2 γ spacetime metamaterials”. Notice that the diffeomorphism I acts only on the fields E~ and B~ , while diffeomorphism II acts which are the defining tensors of the Boys-Post relation. on D~ and H~ . After some algebra the Tellegen relation √ √ √ i −g¯ ¯i¯j −g¯ −g¯ ¯ik¯ 0¯¯l m¯ ¯j D˜ = −s¯√ g E˜j − s¯s¯ g g ²klmg H˜j , Introducing the notation γg0¯0¯ γg0¯0¯ (28) µ ν ν µ √ √ √ ¯ ∂x¯ ∂x¯ ∂x¯ ∂x¯ −g¯ ¯ −g¯ −g¯ ¯ ¯ gµ¯ν¯ = gρσ = g¯µρ = g¯ρν (19) B˜i = −s¯ g¯i¯jH˜ +s ¯s¯ g¯ik² gl0¯gm¯ ¯jE˜ ∂xρ ∂xσ ∂x¯ρ ∂x¯ρ √ j klm j γg0¯0¯ γg0¯0¯ (29) the relation may be written as is found, which in the limit of g¯µν =g ¯µν is equivalent to 1 ¡ ¯ ¯ ¯ ¯ ¢ Eqs. (15) and (16). An important comment is in order: H¯ µν = gµ¯ρ¯gν¯σ¯ − gµ¯σ¯ gν¯ρ¯ F¯ . (20) µν 2 ρσ due to the different transformations applied to H and Fµν , resp., the constitutive relation (22), or (28) and (29), µ¯ν¯ relates fields from different spacetime points in the orig- It should be noted that g in Eq. (19) is no longer a met- inal space, e.g. E˜ (˜x =x ¯(x)) refers the field E (x) at a ric, in particular it need not be symmetric in its indices i i different point xµ in the original space than D˜i (˜x = x¯(x)) and it need not have signature (3, 1). does. To derive the new constitutive relations in the original Let us comment on the more technical parts of this (laboratory) space we proceed analogously to the previ- result. In Section II we saw that transformation materials ous section. All fields have to be rescaled in order to obey derived from symmetry transformations are restricted to the equations of motion in the original space with metric reciprocal materials with ² = µ. These restrictions can gµν , which implies be overcome partially with the above result:

√ √ ¯i¯j ¯j¯i T −g¯ −g¯ • As g = (g ) it follows that permittivity and F˜ = ±F¯ , H˜ µν = √ H¯ µν , J˜µ = √ J¯µ . µν µν −g −g permeability are related as (21) p √ s¯ −gµ¯ ij =s ¯ −g²¯ ji . (30)

Thus the constitutive relation becomes It should not come as a surprise that permittiv- √ ity and permeability cannot be independent, as by µν µνρσ 1 −g¯ ¡ µ¯ρ¯ ν¯σ¯ µ¯σ¯ ν¯ρ¯¢ virtue of the definition of the relativistically covari- H˜ =χ ˜ F˜ρσ = ± √ g g − g g F˜ρσ , 2 −g ant tensors Fµν and Hµν such transformations can- (22) not act independently on E~ and B~ or D~ and H~ , where the sign refers to the possible change of orienta- resp. A possible route to relax this restriction is tion in the transformation xµ → x¯µ. For the equivalent discussed in Section VI. relation in terms of space vectors the notation • Permittivity and permeability need no longer be √ √ symmetric. Therefore it is possible to describe −g −g ²µνρσ =s ¯√ ²¯µνρσ = s¯√ ²¯µνρσ (23) non-reciprocal materials, or, in the language of Eq. −g¯ −g¯ (A.23), the skewon part need not vanish. This hap- pens if the mapping between the two electromag- is used, where s¯ and s¯ are the respective signs due to netic spaces, ∂x¯µ/∂x¯ν , is not symmetric in µ and the change of orientation in the transformations to lab- ν, e.g. for a material with mappingx ¯ = x − z, oratory space. Now it easily follows from (A.17)–(A.20) x¯ = x + z. 5

• The generalized transformations yield many more similarly to Eq. (15) that all electric-magnetic cou- possibilities considering the signs of the eigenval- plings vanish if the transformation does not mix ues of permittivity and permeability. Within the space and time. In this case the crucial compo- ¯ ¯ method of Ref. [3], µ and ² are essentially deter- nents g0¯¯l and g0¯¯l may be written as mined by the spatial metric of the electromagnetic 0 l 0 l space (cf. Eqs. (15) and (16) and recall the relation ¯ ∂x¯ ∂x¯ ∂x¯ ∂x¯ g0¯l = g00 + gij , (34) gij = γij.) However, a spatial metric by definition ∂x0 ∂x0 ∂xi ∂xj must have three positive eigenvalues, a characteris- 0 l 0 l ¯¯ ∂x¯ ∂x¯ ∂x¯ ∂x¯ tic that cannot be changed by any diffeomorphism. g0l = g00 + gij . (35) ∂x0 ∂x0 ∂xi ∂xj Thus it follows that in any medium of this type the eigenvalues of ² and µ are all of the same sign. Most importantly it is found from these expres- sions that one of the two bi-anisotropic couplings – Within the generalized setup of “triple space- may vanish while the other one is non-vanishing, time metamaterials”, however, the signs of the which is impossible within the context of diffeomor- eigenvalues in ² can be chosen freely, as the phism transforming media. Moreover, in the latter metric is multiplied by a transformation ma- case the bi-anisotropic couplings must be symmet- trix, ric matrices, which need no longer be the case in j the present context. ¯ ∂x¯ g¯i¯j = g¯iµ , (31) ∂x¯µ • Finally, the result (28) and (29) reduces to the rela- tions (15), (16) if gµ¯ν¯ is a symmetric√ matrix√ of sig- and no restrictions on the signs of the eigen- nature (3, 1) and, in addition, −g¯ = −g¯. This values of the transformation matrix exist. In does not necessarily implyx ¯µ = x¯µ but rather that this way indefinite materials [6, 7] can be de- there exists yet a different space which describes signed as a result of different space inversions the same media properties in terms of the transfor- in the two different mappings. As an ex- mations of Section (II). ample the mapping z¯ = −z, z¯ = z (with all other directions mapped trivially) yields ij ij ² = diag(−1, −1, 1), µ = diag(1, 1, −1). A. Electric-magnetic duality and rotation – Furthermore the relative sign between the eigenvalues of ² and those of µ can be chosen Finally, we should ask whether Eqs. (28), (29) indeed as a consequence of the factor s¯ in Eq. (30). describe the most general media fulfilling Definition 2. This change in the relative sign may be inter- Taken separately, the two sets of equations in (1) and (2) preted as a partial reversal of time as can be do not exhibit more symmetries than diffeomorphisms. seen in the following list (space maps trivially However, there exists the possibility of transformations µν here and all media are assumed to be homo- that mix Fµν and H . Indeed a transformation of this geneous): type is known as electric-magnetic duality, which has im- portant implications in modern theoretical high-energy t¯ t¯ ² µ physics [17]. It represents the fact that under the ex- change I t t 1 1 II t −t −1 1 Fµν ↔ ∗Hµν (36) III −t t 1 −1 or in terms of space vectors IV −t −t −1 −1 i i B → −D , Hi → −Ei , (37) We note that all eight classes of materials discussed E → H , Di → Bi , (38) in Ref. [6] allow a geometric interpretation within i i the setup of “triple spacetime metamaterials.” the source-free equations of motion do not change (the • More complicated than permittivity and perme- action changes by an overall sign.) Of course, this duality ability are the bi-anisotropic couplings. With the transformation is problematic when applied to a solution standard assumption of g0i = 0 in laboratory space with sources, as it transforms electric charges and cur- it follows from rents into magnetic charges and currents and vice versa. √ √ In the remainder of this section we thus restrict to source- −g¯ −g¯ ¯ ¯ ¯ free solutions or should allow the possibility of artificial ij ¯ ¯ik¯ 0¯¯l m¯ ¯j ξ = −s¯s¯ g g ²klmg , (32) magnetic monopoles. Then it can be checked straight- γg0¯0¯ √ √ forwardly that electric-magnetic duality applied to the ij −g¯ −g¯ ¯ik¯ ¯l0¯ m¯ ¯j κ =s ¯s¯ g ²klmg g , (33) result (20), or (28) and (29), does not yield media rela- γg0¯0¯ tions not yet covered by diffeomorphisms alone. 6

IV. PERFECT LENS FROM INDEFINITE MATERIAL: AN EXAMPLE

To provide a better understanding of the formalism developed in the previous section a concrete example is demonstrated. To keep things simple we show how a pro- posal taken from the literature can be given a geometric interpretation. In Ref. [6] it has been pointed out that two slabs of in- definite material (media with strong anisotropy) can form FIG. 3: Illustration and notation of the generalized “triple a perfect lens. Since, in constrast to standard diffeomor- spacetime metamaterials” complemented by electric-magnetic phism transforming media, strong anisotropy is available rotation. The electric-magnetic rotation must act after the in triple spacetime metamaterials the question appears transformation of spacetime as these two steps do not com- whether a geometric interpretation of the lens proposed mute. in Ref. [6] can be given (cf. Ref. [18] for a related discus- sion.) We consider the lens to be an infinite slab in the x-y plane with a certain thickness in the z direction. In However, as far as the equations of motion (1) and (2) its simplest form the lens consists of two slabs of equal are concerned, electric-magnetic duality can be promoted thickness d, where the media properties of the first slab to a continuous U(1) symmetry with transformation [28] are

B˜i = cos αBi − sin αDi , D˜i = cos αDi + sin αBi , ²ij = µij = diag(1, 1, −1) , (43) (39) while in the second slab E˜i = cos αEi + sin αHi , H˜i = cos αHi − sin αEi . (40) ²ij = µij = diag(−1, −1, 1) . (44)

These transformations comply with Definition 2 and thus To provide a geometric interpretation we start with the their action onto a medium with general constitutive re- observation that a standard perfect lens with ² = µ = −1 lation (A.13) should be studied. The result may be produced by two different transformations, either a space inversionz ¯ = −z, or a time reversal t¯ = −t. ¡ ¢ij From Eqs. (15) and (16) it follows straightforwardly that D˜i = cos2 α² + sin2 αµ + sin α cos α(κ + ξ) E˜ j these two transformations yield the same media proper- ¡ 2 2 ¢ij + cos ακ − sin αξ + sin α cos α(µ − ²) H˜j , ties. Within triple spacetime metamaterials we now may (41) ask the question of what happens if space inversion is ¡ ¢ applied to one set of the fields, while time reversal is ap- ˜i 2 2 ij ˜ B = cos αµ + sin α² − sin α cos α(κ + ξ) Hj plied to the other set. For concreteness, space inversion ¡ ¢ 2 2 ij is applied to the fields E~ and B~ and thus + cos αξ − sin ακ + sin α cos α(µ − ²) E˜j (42) z¯ = −z + Z1 , (45) shows that the transformation acts trivially if ² = µ and where Z1 is an unimportant constant necessary to meet ξ = −κ, in particular in vacuo and consequently for all the boundary conditions. All other fieldsx ¯µ are mapped diffeomorphism transforming media (15). However, they trivially. The second set of fields, D~ and H~ , transform yield new media relations when acting on a solution of according to the type (28) and (29). Therefore these new relations ¯ are part of the materials covered by Definition 2. They t¯= −t + T1 (46) are derived here for completeness, though their geomet- ric interpretation is not immediate. The coordinate lines with all other fields transformed trivially. Consider now x¯µ(x) and x¯µ(x) could be understood as the electromag- these two transformations in Eqs. (28) and (29). From (19) one finds netic spaces of the linear combinations (E,~˜ B~˜) and (D~˜, H~˜) as given in (39) and (40), resp. Still, one should be careful ¯¯ ¯¯ gij = gij = diag(1, 1, −1) . (47) with this interpretation: as the transformation of space- time does not commute with the electric-magnetic rota- Furthermore, s¯ = s¯ = −1 as both tranformations are tion one cannot modify the situation in Figure 3 in such a orientation changing. Furthermore, g¯¯ = 1 (remember way that the two electromagnetic spaces,x ¯µ and x¯µ, are 00 our convention g00 = −1), such that indeed the media identified with certain linear combinations of (E,~ B~ ) and properties (43) are found in this slab. (D~ , H~ ), resp.; rather the electric-magnetic rotation acts This single slab of indefinite material does not establish upon the fields after the transformation of spacetime. a perfect lens, as can be seen easily when studying how a 7

(SEM tensor.) While in the generic situation of electro- dynamics in media, the definition of the “SEM tensor of electrodynamics” is not unique [19, 20], we do not have to deal with these subtleties in the present situation as our (idealized) media are lossless and dispersion free and thus allow for a definition of a complete action (cf. Eq. (7)) without any reference to “.” Therefrom we imme- diately derive the covariant SEM tensor µ ¶ 1 1 T µν = − F gσ{µHν}ρ + gµν F Hρσ , (49) 4π ρσ 4 ρσ µν FIG. 4: Mapping of the world-line s(t) = (t, z(t)) in the origi- DµT = 0 . (50) nal space onto the deformed spaces by means of the two differ- ent transformations. lines indicate the trajectory outside The advantage of this tensor over the canonical SEM ten- of the lens (trivial mapping of all xµ), the line represents sor is the simple behavior under diffeomorphisms: being the first slab, the line the second slab. The parametriza- a real tensor field, Tµν transforms exactly in the same tion of the world-line in the original space is assumed to obey way as the metric. z(t) = t. Let us now look at the materials as described in Sec- tion II. Thanks to its transformation properties the SEM tensor in the electromagnetic space follows immediately world-line s(τ) = (t(τ), x(τ), y(τ), z(τ)) is mapped onto as the two deformed spaces. For simplicity time may be µ ¶ interpreted with the parametrization variable t(τ) = τ µν 1 σ{µ ν}ρ 1 µν ρσ T¯ = − F¯ρσg¯ H¯ + g¯ F¯ρσH¯ . (51) and furthermore we can assume without loss of generality 4π 4 z(τ) = τ = t [29]. The situation is illustrated in Figure 4. As can be seen the mappings do not agree after the first But how about T˜µν ? Of course one could define an “in- slab, both trajectories are at the same point z = 0, but duced SEM tensor” from the electromagnetic space as they differ in time. This must be corrected in the second (cf. Eqs. (12) and (13)) slab. In our example we have chosen a completely trivial √ µ ¶ mapping for D~ and H~ , so these fields propagate in the ˜µν −g 1 ˜ σ{µ ˜ ν}ρ 1 µν ˜ ˜ ρσ TI = ∓√ Fρσg¯ H + g¯ FρσH , second slab as in free space. E~ and B~ , however, are −g¯ 4π 4 transformed as (52) but obviously this tensor is not conserved in laboratory ˜µν t¯= −t + T2 , z¯ = −z + Z2 , (48) space, DµTI 6= 0, since it depends explicitly on the met- ricg ¯µν . In other words, the crucial trick to re-interpret in which actually reverts the transformation (45) and at the the dynamical equations the coordinates in electromag- same time applies (46). Not surprisingly, the two trajec- netic space,x ¯µ, as those in laboratory space, xµ, works tories now meet at the same point again and the perfect in the equations of motion (1) and (2), but does not work lens is established. Again it is immediate that this trans- for the SEM tensor and its conservation. formation establishes the media relations (44). There- Of course, the correct SEM tensor in laboratory space fore, triple spacetime metamaterials indeed can provide immediately follows from (49) as a geometric interpretation of the lens of Ref. [6]. It should µ ¶ be noted, that this specific lens has focal length zero, it µν 1 σ{µ ν}ρ 1 µν ρσ T˜ = − F˜ρσg H˜ + g F˜ρσH˜ . (53) shrinks the effective width of the device from 2D to zero, 4π 4 but not to a negative value as is necessary for a real lens. Clearly, requiring equivalence of the two tensors would not even allow for conformal transformations. But even when looking at integrated quantities (total energy and V. ENERGY, MOMENTUM AND WAVE VECTOR momentum flux in the material), Z µ 3 √ 0µ So far we studied solutions of Maxwell’s equations P = d x γ T , (54) which—up to rescalings—are equivalent to certain vac- uum solutions. Still we did not ask up to what point these the induced tensor does not yield the correct quantity in transformation materials really are “media that look like laboratory space. Of course, the situation is even more empty space.” To do so it is not sufficient to consider the complicated for triple spacetime metamaterials: since the transformation of the fields and sources, but equally well transformation of the explicit metrics appearing in Eq. we should look at the conservation laws, summarized in (49) is not defined, an “induced SEM tensor” cannot even the conservation of the stress-energy-momentum tensor be defined. 8

Instead of the correct, directly evaluated SEM tensor spatial transformations as a covector [1], while ni from (53) a slightly different tensor is considered in the follow- Eq. (60) behaves as a vector: ing. To see its advantage we make the standard assump- √ 0 i 0 i −g¯ ∂x ∂x¯ j tion that our laboratory space metric has g00 = −1 (x S˜ =s ¯√ √ S , (64) −g −g ∂x¯0 ∂xj is our laboratory time) and g0i = 0 (the measure of dis- 00 tances is time independent.) Then it is straightforward √ ∂x¯0 ∂xj n˜ =s ¯ −g¯ n . (65) that the quantity i ∂x0 ∂x¯i j µν µρ σν Of course, the relative orientation of Si and ni is pre- M = −g FρσH (55) served under the diffeomorphisms, but this is no longer ˜i i contains the and the direction of the true for S andn ˜ , since indices are raised/lowered by the wave vector, space metric γij in laboratory space as opposed toγ ¯ij in electromagnetic space. i 0i ijk S = M = ² EjHk , (56) For triple spacetime metamaterials no linear transfor- ˜i i j i i0 ij k l i mation S = T jS exists. This makes the interpretation n = M = γ ²jklD B k k . (57) a little bit more complicated, but at the same time is the source of the numerous additional possibilities within this We recall that the original fields obey the constitutive generalized setup. In general, the value of the element relations of vacuous space and thus trivially M 0i = M i0. T ijk defines the component of the Poynting vector in di- From the transformation rules (21) the transformed ten- rection xi as generated by electric and magnetic fields sor is found as that point in the original space in the directions xj and √ k −g¯ x , resp. In this way it is easy to engineer the direction M˜ µν = −s¯√ gµρF¯ H¯ σν −g ρσ of the Poynting vector in the medium for a given polar- √ (58) ization of the incoming wave in vacuum. Similar conclu- −g¯ ∂xλ ∂xτ ∂x¯σ ∂x¯ν µρ αβ sions apply for the transformation matrix Uijk, with the = −s¯√ g Fλτ H . −g ∂x¯ρ ∂x¯σ ∂xα ∂xβ notable restriction thatn ˜i can be parallel or anti-parallel to ki. Whether (D˜i, B˜j, k˜l) form a right- or left-handed µν The transformation law of M encodes in a geometric triple can be deduced from language how energy flux and phase velocity behave in a ˜ √ ˜ medium. For simplicity let us now concentrate on media j k ki jk −g¯ ki ¯jk¯ ²ijkD˜ B˜ = E˜j² E˜k = −s¯√ E˜jγ E˜k . (66) without bi-anisotropic couplings, in other words we allow ω˜ γg0¯0¯ ω˜ for general spatial transformations as well as stretchings and reversal of time, but keepg ¯0i = g¯0i = 0. Then we find for the transformed space vectors (cf. Eqs. (A.9)– A. Wave vector and dispersion relations (A.12)): √ While the above relations correctly reproduce the di- −g¯ ∂x0 ∂xm ∂xn S˜i = M˜ 0i = −s¯s¯σ¯ 00 ²ijk E H (59) rection of the Poynting and the wave vector, they cannot g ∂x¯0 ∂x¯j m ∂x¯k n distinguish between propagating and evanescent modes. √ 00 √ 0 k l Consider as an example the following transformation: i i0 −g¯ γ¯ ∂x¯ ij ∂x¯ m ∂x¯ n n˜ = M˜ =σ ¯ √ γ ²jkl D B (60) γ ∂x0 ∂xm ∂xn x¯0 = x0 , x¯i = xi , x¯0 = −x0 , x¯i = xi . (67)

The transformation of the Poynting vector may be ab- From Eqs. (15) and (16) it is found that this is a homo- breviated as geneous material with ² = −1 and µ = 1. As fields in vacuo, E~ = D~ and B~ = H~ , we consider a monochromatic i ijk S˜ = T EjHk , (61) wave ~ E~ = ~eei(k~x−ωt) + c.c. , ~k · ~e = 0 , (68) and it is then easily seen that ni transforms as ~ 1 √ B~ = ~bei(k~x−ωt) + c.c. , ~b = ~k × ~e. (69) γ¯γ¯ ∂x¯0 ∂x¯0 ω n˜ =σ ¯s¯σ¯s¯ g U DjBk , (62) i 2γ 00 ∂x0 ∂x0 ikj After the transformation the fields E~˜, B~˜ and D~˜, H~˜ refer ijk to the original fields at different time instances: where Uijk is the inverse of T in the sense of ~ µ µ ljk l E˜ (˜x =x ¯ (x)) = E~ (~x,t) , (70) UijkT = δi . (63) B~˜ (˜xµ =x ¯µ(x)) = B~ (~x,t) , (71) While these formulae might look cumbersome, their geo- metric interpretation actually is quite straightforward. D~˜ (˜xµ = x¯µ(x)) = −D~ (~x, −t) , (72)

In the case of diffeomorphism transforming materials, ~ µ µ i H˜ (˜x = x¯ (x)) = −H~ (~x, −t) . (73) g¯µν = g¯µν , Eq. (59) states that S behaves under purely 9

ρσ Of course, Maxwell’s equations are satisfied by the new mapped on a solution of the new equations (Ψˆ µν is the fields by construction. Still, the partial exchange of pos- ˆ λτ ρσ ρ σ σ ρ inverse matrix Ψµν Ψλτ = (δµδν − δµδν )/2) itive and negative angular has important im- plications in the dispersion relation as any propagating µν ν µν ρσ wave in vacuo becomes evanescent in the medium and DµH¯ = J¯ = Dµ(Ω ρσH ) , (78) vice versa. ²µνρσ∂ F¯ = J¯µ = ²µνρσ∂ (Ψˆ τλF ) , (79) Though this behavior may not appear immediate when ν ρσ M ν ρσ τλ transforming the monochromatic wave (68) and (69) with (70)–(73), it can be made explicit from geometric quan- provided appropriate currents are introduced. These tities as well. Indeed, from the relativistic wave equation transformations in general are not symmetry transforma- [12] tions and accordingly a source free solution is no longer mapped on another source free solution. The transfor- µνρσ ν Dν χ DρAσ = −J (74) mations (76) and the ensuing equations of motion (78) and (79) are the relativistic form of the transformations it follows straightforwardly that “triple-space metamate- proposed in Refs. [10, 11]. rials” in the absence of charges and currents and in the limit of approximate obey the dispersion re- We recover the invariant transformations of Section lation (III) by introducing the restrictions

µ¯ν¯ ~ g kµkν = 0 , kµ = (ω, k) . (75) µν µ ν ρσ ρ σ Ω ρσ = S ρS σ , Ψµν = Tµ Tν . (80) ¯ In our example the partial reversal of time yields g0¯0¯ = 1 and thus ω2 + ~k2 = 0. Let us first count the degrees of freedom in the transfor- mations. χµνρσ is a rank four tensor, anti-symmetric in (µ, ν) and (ρ, σ) and thus has 36 independent components VI. NON-INVARIANT TRANSFORMATIONS (20 components of the principal part, 15 of the skewon part and one axion coupling). The same applies to the Within the approaches to transformation media dis- transformation matrices Ω and Ψ. Restriction to diffeo- cussed so far invariant transformations of the equations morphisms according to Eq. (80) reduces this number to of motions were used exclusively. This means that the 6 parameters for each, Ω and Ψ; the electric-magnetic transformations “do not introduce charges or currents”, rotation of Section III A adds another parameter in form in other words the transformation medium based on a of a rotation angle. Here, another important difference source free vacuum solution will be source free as well. between a transformation material according to Defini- What happens if this restriction is abandoned? Still in- tion 2 and 3 emerges. In both cases the transforma- sisting on a constitutive relation of the form (5) this sug- tion yielding certain media properties is not unique. In gests the following definition: the former case, however, different transformations are physically equivalent as they are connected by symme- Definition 3. A transformation medium is defined by try transformations (isometries of the laboratory metric an arbitrary linear transformation applied to a (not nec- gµν ). In the more general case of Eq. (76) the different essarily source free) vacuum solution of Maxwell’s equa- transformations need not be physically equivalent. As tions. The linear transformation constitutes the media is immediate from Eq. (76) a certain medium exhibit- properties as well as charges and currents of the trans- ing sources due to non-invariant transformations can be formation medium. designed using electric charges and currents, magnetic charges and currents or both, which clearly character- Though not in its most general form, this approach was izes physically different situations with the same media proposed in [10, 11]. Starting from the vacuum relation properties χ. (6) the most general linear relation can be achieved by the field transformations [30] Does there exist the possibility of a geometric interpre- tation of Eq. (76)? If this shall be possible space must be µν µν ρσ ρσ transformed differently for different components of F µν H¯ = Ω ρσH ,Fµν = Ψµν F¯ρσ , (76) and Hµν . In fact, the most general linear transforma- with the transformed χ¯, tion can be interpreted as a separate transformation of spacetime for each component of the two tensors. Let us χ¯µνρσ = (ΩχΨ)µνρσ . (77) provide a simplified example, where independent spatial transformations are applied to E~ , B~ , D~ and H~ . Labora- µν i The original fields Fµν and H by assumption are solu- tory space is denoted by x , the electromagnetic spaces i i i i tions to the equations of motion. If we allow besides the by xE, xB, xD and xH , resp. Under time-independent standard electric four-current J µ also a magnetic four- spatial transformations all four fields transform as (co- µ current JM , any transformation of the type (76) can be )vectors and thus Eqs. (A.9)–(A.12) suggest the inter- 10 pretations light rays travel along trajectories as if the medium was √ a transformed, empty space. Still, the transformation ∂xj γ ∂xi ˜ ˜i B B j medium in general is quite different from transformed Ei = sE i Ej , B = √ j B , (81) ∂xE γ ∂x empty space, if the conservation laws from the stress- √ γ ∂xi ∂xj energy-momentum tensor are considered. This aspect is D˜i = D D Dj , H˜ = s H , (82) √ j i H i j even more important within the extension proposed here, γ ∂x ∂xH as there exist two different transformed (electromagnetic) yielding for permittivity and permeability spaces and light rays don’t follow the geodesics of any of √ i j them. We have shown that one can make a virtue out ij γD ∂xD kl ∂xE ² = sE √ γ , (83) of necessity: the geometric approach does not just pro- γ ∂xk ∂xl vide a tool to design the path of light in a medium, but √ i j γB ∂x ∂x equally well it may be used to design the behavior of µij = s √ B γkl H . (84) H γ ∂xk ∂xl (parts of) the stress-energy-momentum tensor, e.g. the direction of the Poynting vector, and/or the behavior of As is seen from (1) and (2) Gauss’ law for B˜i and D˜i the wave vector. Here the proposed generalization of- remain unchanged, while Faraday’s and Amp`ere’slaws fers many more possibilities compared to the known dif- are changed according to feomorphism transforming media. In particular we have √ γ i derived the geometric relations that describe the trans- E ∂xE ˜j ijk ˜ √ j ∇0B + ² ∂jEk = 0 , (85) formation of the Poynting vector, of the direction of the γB ∂x B √ wave vector as well as the dispersion relation. γ i ijk ˜ H ∂xH ˜j Finally we have commented on a different route to ² ∂jHk − √ j ∇0D = 0 , (86) γD ∂x generalize the notion of transformation media [10, 11]. D These field-transforming media are not based on invari- making the electric and magnetic currents ant transformations of the equations of motion and con- à ! √ i sequently source free solutions of the original configura- γH ∂x ji = − δi − √ H ∇ D˜j , (87) tion are not mapped onto source free solutions of the j γ j 0 D ∂xD new medium. We have shown that also this approach à ! √ i may be covered by a generalized concept of coordinate γE ∂x ji = δi − √ E ∇ B˜j (88) transformations. Still, there remains a fundamental dif- M j γ j 0 B ∂xB ference between the approach of Refs. [10, 11] and the necessary. For completeness it should be mentioned that one discussed here: While in the former case the trans- the transformations (81) and (82) allow a straightforward formations are ultra-local (the transformed fields at the µ interpretation since each of the four Maxwell’s equations point x are defined in terms of the original fields at this still can be transformed as a whole. Taking even more point), in the latter they are essentially non-local, as the µ general transformations, e.g. transforming each compo- transformed fields atx ˜ are related to the original fields µ µ nent of the electric and magnetic fields separately, does at some x 6=x ˜ . The preferable approach depends on no longer allow this manipulation in a simple way and the specific problem at hand, also a combination of the thus will make the derivation of the necessary media pa- two is conceivable. rameters more complicated.

Acknowledgments VII. CONCLUSIONS The author wishes to thank J. Llorens Montolio for In this paper we have introduced a generalization of the helpful discussions. This work profited a lot from fruitful concept of diffeomorphism transforming media, the ba- discussion with C. Simovski, S.A. Tretyakov, I.S. Neve- sis of transformation optics [1–3]. As basic idea we have dov, P. Alitalo, M. Qiu and M. Yan during a cooperation found that spacetime can be transformed differently for of the Advanced Concepts Team of the European Space the field strength tensor (containing E~ and B~ ) and the Agency with the Helsinki University of Technology and excitation tensor (encompassing D~ and H~ ). This exten- the Royal Institute of Technology (KTH). The coopera- sion allows design of non-reciprocal media, in particular tion was funded under the Ariadna program of ESA. permittivity and permeability need not longer be sym- metric. Furthermore, this approach permits a geometric interpretation of indefinite media [6, 7]. APPENDIX: COVARIANT FORMULATION Diffeomorphism transforming media are motivated by the wish to produce a medium that looks like a trans- In this Appendix we present our notations and conven- formed but empty space. The basis of this interpreta- tions regarding the covariant formulation of Maxwell’s tion is Fermat’s principle applied to these media [9]: in- equations on a possibly curved manifold. For a detailed deed it is found that in a transformation medium the introduction to the topic we refer to the relevant litera- 11 ture, e.g. [12, 13]. Throughout the whole paper natural tory space units with ²0 = µ0 = c = 1 are used. µµ ¶ ∂x0 ∂xj ∂x0 ∂xj Greek indices µ, ν, ρ, . . . are spacetime indices and run E˜ =s ¯ − E i ∂x¯0 ∂x¯i ∂x¯i ∂x¯0 j from 0 to 3, Latin indices i, j, k, . . . space indices with (A.9) j k ¶ values from 1 to 3. For the metric we use the “mostly ∂x ∂x l − ²jklB , plus” convention, so the standard flat metric is gµν = ∂x¯0 ∂x¯i diag(−1, 1, 1, 1). If we interpret x0 = t with (laboratory) √ γ¯ ∂x¯i ∂x0 ∂xl time the space metric can be obtained as [13] B˜i =σ ¯ √ Bj − s²¯ ijk E , (A.10) γ ∂xj ∂x¯j ∂x¯k l √ µµ ¶ ij ij g0ig0j ij i −g¯ ∂x¯0 ∂x¯i ∂x¯0 ∂x¯i γ = g , γij = glk − , γ γjk = δk . D˜i = √ − Dj g00 −g ∂x0 ∂xj ∂xj ∂x0 (A.1) ¶ (A.11) ¯0 ¯i This implies as relation between the determinant of the ∂x¯ ∂x¯ jkl + ² Hl , spacetime metric, g, and the one of the space metric, γ, ∂xj ∂xk √ √ −g¯ ∂xj −g¯ ∂x¯j ∂x¯k ˜ ¯¯ √ 00 √ l Hi = s¯σ¯ i Hj + ²ijk 0 l D . (A.12) −g = −g00γ (A.2) −g00 ∂x¯ −g ∂x ∂x

The four dimensional Levi-Civita tensor is defined as We characterize the general linear, lossless media usu- ally by means of the Tellegen relations √ 1 ² = −g[µνρσ] , ²µνρσ = −√ [µνρσ] , (A.3) Di = ²ij E + κijH ,Bi = µijH + ξijE . (A.13) µνρσ −g j j j j In terms of field strength and excitation tensor the media with [0123] = 1. Therefore the reduction of the four relations become the Boys-Post relation dimensional to the three dimensional tensor reads µν 1 µνρσ H = χ Fρσ , (A.14) √ 0ijk 1 ijk 2 ²0ijk = −g00²ijk , ² = −√ ² . (A.4) −g00 where χ must be invertible with inverse

λτρσ ρ σ σ ρ An additional complication arises in the definition of ²¯ijk χˆµνλτ χ = (δµδν − δµδν ) . (A.15) and ²¯ijk, since the orientation of the spacetime manifold may change without changing the orientation of space By virtue of Eqs. (A.6) and (A.7) Eq. (A.14) may be (e.g. by a mapping t¯= −t.) Therefore the corresponding written as relations should be written as à ! à !à ! H C j B E i = i ij j , (A.16) √ p i ij i j D A D j B ²¯0ijk =σ ¯ −g¯00²¯ijk , ²¯0ijk = σ¯ −g¯00²¯ijk , (A.5) with where σ¯ = +1 if space and spacetime have the same ij √ 0i0j orientation andσ ¯ = −1 otherwise. A = − −g00χ , (A.17) The field strength tensor F encompasses the electric µν 1√ klmn field and the magnetic induction, the excitation tensor Bij = −g00²ikl²jmnχ , (A.18) µν 8 H the displacement vector and the magnetic field with 1√ C j = − −g ² χkl0j , (A.19) the identification: i 2 00 ikl i 1√ 0ikl i 1 ijk D j = −g00²jklχ . (A.20) Ei = F0i ,B = − ² Fjk , (A.6) 2 √2 i √ 0i −g00 jk The Tellegen and Boys-Post formulations are related by D = − −g00H , Hi = − ²ijkH . (A.7) 2 ¡ ¢ij ¡ ¢ij ²ij = A − DB−1C κij = DB−1 (A.21) Finally, and current are combined into a ij −1 ij ij ¡ −1 ¢ij µ √ i √ µν µ = (B ) ξ = − B C (A.22) four-current J = (ρ/ −g00, j / −g00). Fµν and H are tensors, thus under the transformations of Section III Finally, we mention that the rank 4 tensor χµνρσ may be they behave as decomposed as [15, 16]

∂xρ ∂xσ ∂x¯µ ∂x¯ν χµνρσ = (1)χµνρσ + ²µνλ[ρS σ] − ²ρσλ[µS ν] + α²µνρσ , F¯ = F , H¯ µν = Hρσ . (A.8) λ λ µν ∂x¯µ ρσ ∂x¯ν ∂xρ ∂xσ (A.23) where the principal part (1)χµνρσ has no part completely This implies for the transformed space vectors in labora- anti-symmetric in its indices and is symmetric under the 12 exchange (µ, ν) ↔ (ρ, σ). The principal part has been ties of the material [16, 21]), while α represents the well- ν discussed extensively in [12]. Sµ was introduced in known axion coupling [22]. Refs. [15, 16] as skewon part (related to chiral proper-

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