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Generalized transformation optics of linear materials

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Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF OPTICS J. Opt. 13 (2011) 055105 (12pp) doi:10.1088/2040-8978/13/5/055105 Generalized transformation optics of linear materials

Robert T Thompson1, Steven A Cummer2 and Jorg¨ Frauendiener1,3

1 Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand 2 Department of Electrical and Computer Engineering and Center for Metamaterials and Integrated Plasmonics, Duke University, Durham, NC 27708, USA 3 Centre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway

E-mail: [email protected]

Received 4 February 2011, accepted for publication 8 March 2011 Published 7 April 2011 Online at stacks.iop.org/JOpt/13/055105

Abstract We continue the development of a manifestly four-dimensional, completely covariant, approach to transformation optics in linear dielectric materials begun in a previous paper. This approach, which generalizes the Plebanski based approach, is systematically applicable for all transformations and all general linear materials. Importantly, it enables useful applications such as arbitrary relative motion, transformations from arbitrary non-vacuum initial dielectric media, and arbitrary space–times. This approach is demonstrated for a resulting material that moves with uniform linear velocity, and in particular for a moving cloak. The inverse problem of this covariant approach is shown to generalize Gordon’s ‘optical metric’. Keywords: transformation optics, optical metric, electrodynamics in materials, metamaterials

1. Introduction properties, and demonstrated how it could be used to create novel devices. Closely related work by Greenleaf The emerging field of transformation optics, where useful et al [9] developed a similar concept for electric current arrangements of man-made ‘metamaterials’ [1–3] are designed flow and applied it to impedance tomography. The initial via transformations of electromagnetic fields, has theoretical approaches to transformation optics relied on purely spatial roots stretching back nearly a century, to the early days of transformations [8, 10, 11]. Leonhardt and Philbin generalized general relativity. The idea that the behavior of light in a these to transformations involving both space and time by gravitational field can be replicated by a suitable distribution using the explicit equivalence given by De Felice, thus linking of refractive media appears to have been first postulated transformation optics to differential geometry [12]. Another by Eddington [4]. Subsequently, Gordon [5] studied the approach [13], based on field-transforming metamaterials, inverse problem, that of representing a refractive medium considered more general transformations in the Fourier as a vacuum space–time described by an ‘optical metric’. domain, but at the expense of the intuitive and appealing Later, Plebanski [6] found effective constitutive relations geometric interpretation. For more recent reviews see [14, 15]. for electromagnetic waves propagating in vacuum space– In examining the details of the Plebanski–De Felice times, but, while recognizing the formal equivalence of these approach, we find some limitations that are addressed here. equations to those in a macroscopic medium in flat space–time, As pointed out by Plebanski himself [6], the equation now did not exploit this equivalence to actually describe such a bearing his name is not strictly covariant, because its derivation medium. This was first done by De Felice [7], who used the requires a matrix inversion that is not a true tensor operation. Plebanski equations to describe the equivalent medium of both One consequence of this, in the context of transformation a spherically symmetric gravitational system and Friedmann– optics, is that magneto-electric coupling terms cannot always Robertson–Walker space–time. be simply interpreted as a material velocity, as is frequently More recently, Pendry [8] pointed out the specific done. This is because the Plebanski equations can only relationship between spatial transformations and material be identified with stationary media or with slowly moving

2040-8978/11/055105+12$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA J. Opt. 13 (2011) 055105 R T Thompson et al (i.e. nonrelativistic), isotropic media [16]. This prompts the a relatively self-contained and more detailed description of question of whether the approach of [12] may be generalized covariant electrodynamics in both vacuum and linear materials. to clearly distinguish magneto-electric couplings from material This development relies on the geometric language and tools velocity, and allow for specially relativistic corrections. Just of differential geometry, such as exterior derivative, wedge such an approach was outlined in [16], and was demonstrated product, and the pullback of a tensor. These aspects of to recover several results obtained through other means. differential geometry are described in a myriad of excellent Here we provide a complete derivation of the approach sources, such as [20–23]. The most important of these for our outlined in [16], further generalizing the result obtained purpose, the pullback map, is described in appendix B.The there. A physically realistic scenario for transformation optics development and notation, in particular the sign convention, designed devices is that the device moves with arbitrary follow that of [24], while more information, particularly for velocity. We demonstrate that the approach described here may electrodynamics in materials, can be found in [25, 26]. We be applied to find the material properties of a transformation use the Einstein summation convention, indices are lowered αβ when the resultant material is constrained to move with (raised) by the metric tensor gαβ (its inverse g ), and the speed arbitrary uniform velocity with respect to the frame in which of light and Newton’s constant are set to c = G = 1. the transformations are given. This represents a departure from most previous examples in transformation optics, where either 2.1. Field strength tensor the resulting material is stationary or the velocity is dictated by the transformation itself. While a few examples exist of In free space, classical electrodynamics is modeled as a nonrelativistic moving dielectrics in transformation optics for principal U(1) fiber bundle over a space–time manifold special cases [13, 17], we provide a systematic approach that M (which we assume to be equipped with a metric) with is widely applicable for any velocity. connection one-form A = Aμ (frequently called a ‘gauge Another limitation of the Plebanski–De Felice approach is field’, A is the covariant version of the four-vector potential). that the resulting material must reside in vacuum Minkowski The field strength F = Fμν is the curvature two-form of the space–time. The approach described here relaxes this U(1) fiber bundle, equal to the exterior derivative of A, condition, allowing for physically realistic scenarios such as transformations in arbitrary non-vacuum initial dielectric F = dA ⇒ Fμν = Aν,μ − Aμ,ν, (1) media [18], or in arbitrary space–times—thus providing general relativistic corrections for transformations in arbitrary where the comma indicates a derivative. The components of space–times, such as the weakly curved space–time near F can be represented as a matrix, that in a local orthonormal Earth [19]. Lastly, we show that this covariant approach is frame (or Minkowski space–time with Cartesian coordinates) consistent with, and generalizes, Gordon’s optical metric as have values essentially the inverse problem of transformation optics. ⎛ ⎞ 0 −E −E −E The paper is organized as follows: in section 2 we review x y z ⎜ Ex 0 Bz −By ⎟ the completely covariant theory of vacuum electrodynamics Fμν = ⎝ ⎠ . (2) E −B 0 B using modern language, for which the necessary ideas y z x E B −B 0 and notation from differential geometry may be found in z y x appendix B. This review is presented in some detail, because With this choice of values for the components of F, the four- in section 3 the covariant theory of section 2 is extended force vector on a particle moving with four-velocity uν and to describe electrodynamics in macroscopic linear dielectric charge q is α αμ ν materials. This section presents a slight departure from the f = qg Fμνu , (3) usual description of electrodynamics in dielectric materials in order to clearly distinguish material effects from space–time which is just the Lorentz force. For example, a particle at rest ν = ( , , , ) effects. Section 4 describes the concept of transformation with respect to this system has four-velocity u 1 0 0 0 , μ = ( , , , ) optics and presents an interpretation consistent with the for which f q 0 Ex Ey Ez , recovering the usual notion geometric picture of the preceding sections. The main result that a charged particle at rest feels only the electric part of the ν x y z for applications in transformation optics is equation (30). field. For a particle moving with u = γ(1,v ,v ,v ),the μ Section 4.1 examines a particular transformation both when the spatial components of f are f = qγ(E +v × B), while the resulting material is at rest and when it is in motion relative time component is the change in energy per unit time, or the to the frame in which the fields have been measured. As power. expected, it is found that the results for the material in motion The E and B fields are now tightly intertwined, simply smoothly recover, in the limit v → 0, the results for the representing different components of a single object, F. material at rest. In section 5 we study the inverse problem Because the second exterior derivative of any form vanishes of transformation optics, that of finding an equivalent vacuum and the fact that F = dA, it immediately follows that space–time starting from an initial dielectric, thus generalizing = . Gordon’s optical metric idea. We conclude with section 6. dF 0 (4)

2. Classical electrodynamics in vacuum This is nothing more than the covariant form of the homogeneous Maxwell equations, and shows that the The basic elements of covariant electrodynamics needed for homogeneous equations are simply geometric conditions transformation optics were presented in [16]; here we present imposed on the fields.

2 J. Opt. 13 (2011) 055105 R T Thompson et al Essentially, this term is a total four-divergence and contributes nothing to the integral. Writing out the wedge product shows that F ∧ F =−F ∧ F, so the second possibility also contributes nothing to the integral. The last possibility, however, gives a contribution Figure 1. The metric generates a bijection g between k and 0 0 k ( ∧ ) = 4 | | ( , ) = 4 | |( μν ). alternating tensors, while the volume form ω provides a bijection F F d x g g F F d x g F Fμν between k and 0 alternating tensors (for any k < m). The M M M 0 m−k (9)  = ω ◦ g m−k k composition provides a bijection between 0 and 0 The component form of this expression—the right-hand side 0 0 alternating tensors and between m−k and k alternating tensors. of equation (9)—is the version most commonly encountered in μν the literature. That this appears in component form as F Fμν 2.2. Field strength dual should be regarded as a happy coincidence that is an artifact of the classical vacuum; do not lose sight of the fact that the Naturally associated to each point of the space-time manifold fundamental tensors making up the Lagrangian density are F are four m-dimensional vector spaces, where m = dim(M). and F, as this will be generalized shortly. The metric generates a bijection g between the space of k- ∧k ∗( ) ∧k ∗( ) forms Tp M and the space of k-vectors Tp M .The volume form provides a bijection ω between the space of 2.4. Excitation tensor (m−k) k-forms and the space of (m − k)-vectors ∧ Tp(M). The composition of maps, called the Hodge dual, makes the The field strength tensor F encodes some information about diagram of figure 1 commutative. The Hodge dual provides a the fields, namely the electric field strength and the magnetic flux. Consider that on the other side of this coin, the magnetic natural two-form dual to the field strength Fμν . In particular, we define the map field strength and the electric flux are encoded in another tensor G, called the excitation tensor. The components of G can  :∧2 ∗( ) →∧2 ∗( ) be represented as a matrix, that in a local orthonormal frame Tp M Tp M (5) (or Minkowski space–time with Cartesian coordinates) have as the composition  = ω ◦ g applied to two-forms (where values ∧2T ∗(M) is the space of two-forms), or in component form ⎛ ⎞ p 0 H H H x y z 1 αγ βδ ⎜ −Hx 0 Dz −Dy ⎟ ( )μν = | |μναβ γδ. = . F 2 g g g F (6) Gμν ⎝ ⎠ (10) −Hy −Dz 0 Dx − − The components of F can also be represented as a matrix, that Hz Dy Dx 0 in a local orthonormal frame (or Minkowski space–time with Then the identification Cartesian coordinates) have values ⎛ ⎞ G = F (11) 0 Bx By Bz ⎜ −Bx 0 Ez −Ey ⎟ (F)μν = ⎝ ⎠ . (7) is a linear map that takes F to G and provides a set of −By −Ez 0 Ex constitutive relations for the components of G in terms of those −Bz Ey −Ex 0 of F. Comparing equation (10) with (2), it is clear that G = F The dual nature is now explicit; where we had the reduces to the trivial, vacuum, constitutive relations Ha = Ba = a ∧ + ( a ∧ b) decomposition F Eadx dt Bab dx dx we now have and Da = Ea (where ε0 = μ0 = 1whenc = 1). Thus we a a b the dual decomposition F =−Badx ∧ dt + Eab(dx ∧ dx ). find that a trivial linear map recovers the correct constitutive relations in vacuum, but this will be extended to something 2.3. Vacuum action non-trivial in section 3. To describe electrodynamics by means of a variational We would like to stress that we take G to be a tensor, not a principle requires√ an action, and to construct an action tensor density. Expressing the excitation as a tensor density S = d4x |g|L requires a suitable Lagrangian density L. is common in the literature, but using G = F makes use Differential geometry tells us that we can only integrate a k- of the metric structure and explicitly shows the space–time form over a k-dimensional (sub)manifold. There are three contributions encoded by , which will be very useful when ways to construct a four-form from F and F that can be discussing materials and, in particular, transformation optics. integrated over the four-dimensional space–time manifold: F∧ Including an interaction term, the action is generalized to F, F ∧ F,andF ∧ F (we neglect a possibility such as A ∧ A, which results in a massive photon described by the S = 1 F ∧ G + J ∧ A. (12) Proca equations [27]). Consider the first possibility. Since 2 F = dA, In the interaction term, A is the connection one-form, and J = Jαβγ is the charge–current three-form. This action is (F ∧ F) = d(A ∧ F) = (A ∧ F). (8) M M ∂ M invariant under a gauge transformation A → A + d f for some

3 J. Opt. 13 (2011) 055105 R T Thompson et al scalar function f . The three-form J is related to the usual four- The tensor χ contains information on the dielectric material’s vector current j = j μ by the volume dual, J = ω(j),orin properties, and can be thought of as representing an averaging component form over all the material contributions to an action that describes μ a more fundamental quantum field theory. The motivation Jαβγ = |g|αβγ μ j . (13) for using this constitutive relation is to explicitly separate the space–time effects (i.e. the Plebanski relations) from the What advantage is there to thinking in terms of the charge– material effects, which will be useful for transformation optics. current three-form J rather than the four-vector j? Again, the To retain the symmetry properties and usual notions of G integral is only defined for forms of the same dimensionality and F, χ must be independently antisymmetric on its first two as the manifold on which the integration is performed. The and last two indices, and in vacuum χ(F) = F. Thus the three-form J may be integrated over a three-dimensional classical vacuum is a perfect dielectric for which a trivial χ hypersurface of the space–time manifold. Choosing a constant- describes the electrodynamics, and we extend this idea to a time hypersurface is equivalent to a spatial three-volume. non-trivial χ describing electrodynamics in arbitrary, linear, Integrating J over this spatial three-volume gives the charge dielectric media. These conditions reduce the number of enclosed. Integrating J over a 1+2 hypersurface corresponding free parameters of χ to 36. One may further decompose χ to time and a spatial two-surface gives the current flowing through the spatial surface. into principal, skewon, and axion parts [26, 33], but we do Equipped with the constraint equation dF = 0 not consider this here. Additional symmetry conditions may (the homogeneous Maxwell equations), and the constitutive be imposed based on thermodynamic or energy conservation equation (15), varying the action of equation (12) with respect arguments, or by the lack of an observed directive effect to A gives in naturally occurring stationary materials [25, 34]. Having dG = J. (14) recently entered an era of engineered materials which may incorporate active elements [35, 36], however, we leave open This comprises the inhomogeneous Maxwell equations, the discussion of additional symmetries and consider the although the reader may be more familiar with the expression three conditions above to be the minimal requirements. The obtained by taking the dual of both sides of equation (14). χ ( ) =  condition vac F F is sufficient to uniquely specify all components of χ for the vacuum, they are [16] 3. Electrodynamics in linear media χ = (χ ) σρ = 1 vac vac γδ Electrodynamics in materials is somewhat more complicated ⎛ ⎞ ⎛ 2 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 0000 0 100 0010 0 001 ⎞ than electrodynamics in vacuum. Here we expand on the brief ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0000⎟ ⎜ −1000⎟ ⎜ 0000⎟ ⎜ 0 000⎟ introduction given in [16], whose main points are embedded ⎜ ⎝ 0000⎠ ⎝ 0 000⎠ ⎝ −1000⎠ ⎝ 0 000⎠ ⎟ ⎜ 0000 0 000 0000 −1000 ⎟ here for a complete and self-contained discussion. The ⎜ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ ⎜ 0 −100 0000 0000 0000⎟ microscopic theory of electrodynamics would be a quantum ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ 1000⎟ ⎜ 0000⎟ ⎜ 0010⎟ ⎜ 0001⎟ ⎟ field theory described by some complicated action that includes ⎜ ⎝ 0000⎠ ⎝ 0000⎠ ⎝ 0 −100⎠ ⎝ 0000⎠ ⎟ ⎜ 0000 0000 0000 0 −100 ⎟ not only the electromagnetic fields, but also the various matter ⎜ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎟ . ⎜ 00−10 0000 0000 0000⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ fields making up the material, along with their interactions ⎜ ⎜ 00 0 0⎟ ⎜ 00−10⎟ ⎜ 0000⎟ ⎜ 0000⎟ ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎟ and associated gauge fields [28, 29]. Considering the vast ⎜ 10 0 0 0100 0000 0001⎟ ⎜ 00 0 0 0000 0000 00−10 ⎟ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ number of fields contained in a sample of ordinary material, ⎜ 000−1 000 0 000 0 0000 ⎟ ⎝ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎠ it would be an impossible task to examine the full exact theory. ⎜ 000 0⎟ ⎜ 000−1 ⎟ ⎜ 000 0⎟ ⎜ 0000⎟ ⎝ 000 0⎠ ⎝ 000 0⎠ ⎝ 000−1 ⎠ ⎝ 0000⎠ Fortunately, in the thermodynamic limit, electrodynamics in 100 0 010 0 001 0 0000 media can be described by an effective theory. This generally (17) comes in the form of a material dependent set of constitutive relations. Equation (17) expresses χ as a matrix of matrices, the μν The standard vector relations D = εE and H = first two indices of χ αβ give the αβ component of the large −1 −1 μ B (where  and μ may be matrix-valued) are matrix, which is itself a matrix described by the second set of μν frequently combined into an expression such as G = indices. The component values of χ for the vacuum are unique μναβ ζ Fαβ [25, 26]. Such an expression can be useful, and independent of coordinate system. For a more general particularly when exploring electrodynamics in the absence of material, the component values can easily be determined by ametric[30–32]. However, since we assume the existence simply matching the results of the constitutive equation G = of a metric, we instead choose to retain the usual space–time χ(F) with the usual flat-space constitutive relations in a notions of metric and Hodge dual , and take the minimal particular coordinate system, as shown in appendix A.The approach of extending the trivial constitutive equation G = F components of the constitutive equation provide a set of six in vacuum to a more general linear constitutive equation [16] independent equations that can locally be collected in the form

G = χ(F) (15) −1 b ∗ b Ha = (μˇ )a Bb + (γˇ1 )a Eb, (18) ∗ b ∗ b that in component form reads Da = (εˇ )a Eb + (γˇ2 )a Bb, αβ Gμν = χ μν (F)αβ . (16) where we use the notation aˇ to denote a 3 × 3matrix.

4 J. Opt. 13 (2011) 055105 R T Thompson et al Rearranging these to

b b Ba = (μ)ˇ a Hb + (γˇ1)a Eb, (19) b b Da = (ε)ˇ a Eb + (γˇ2)a Hb gives the more familiar representation for the constitutive relations. These three-dimensional representations of the Figure 2. Under the map T the points of M are mapped to the image completely covariant equation (15) are essentially equivalent, M˜ . The electromagnetic fields are transformed by the pullback of T , and it is a simple matter to switch between them using T ∗. The metric, on the other hand, is transformed by the pullback of T ˜ ⊆ −1 −1 ∗ ∗ ∗ , which for present purposes is the identity map. Thus M M,and μˇ = (μˇ ) , εˇ =ˇε −ˇγ2 μˇ γˇ1 , ˜ (20) the pulled back fields, which exist on M, may be excluded from ∗ ∗ some region of M. γˇ1 =−ˇμγ1 , γˇ2 =ˇγ2 μ.ˇ

One should be aware that these 3 × 3 matrices are not tensors, but simply components of χ that have been collected into how the image M˜ is mapped to the original manifold. In matrices. They could be made into tensors by incorporating certain situations it may be possible to set T = T −1,but the appropriate three-dimensional Hodge dual of a space- in general T −1 may not even exist. Furthermore, while the like hypersurface. As they stand, equations (18)and(19) transformation T acts to transform the fields, the space–time are somewhat misleading, because while Ea and Ha are metric is transformed by a different map T, which here we take components of a one-form, Da and Ba are really selected to be the identity. We will have more to say about this below. a b a b components of the two-forms Dab dx ∧dx and Bab dx ∧dx . To make this more concrete, consider a pair of maps Thus a more appropriate version of equations (19) would be (T, T ). We demand that T acts only on the metric via its something like pullback T : M → M (22) b b ( B) = (μ)ˇ H + (γˇ ) E , (21a) a a b 1 a b T∗(g) =ˆg. (23) b b ( D)a = (ε)ˇ a Eb + (γˇ2)a Hb, (21b) On the other hand, we demand that the map T acts on the electromagnetic fields via its pullback where  is the Hodge dual on the three-dimensional space- like hypersurface. But this requires that we resolve the space– T : M˜ ⊆ M → M (24) time into space and time components, selecting an observer ∗ ˜ to define a direction of time. The spatial hypersurface is then T (F) = F (25) orthogonal to the selected direction of time. Transforming as depicted in figure 2. Under (T, T ), the metric and fields are to the local frame of the selected observer we can make the transformed from the configuration (g,,χ, F, G, J),where identifications of equation (A.2) and then give the constitutive dF = 0, dG = J,andG = χ(F), to a new configuration equations a three-dimensional representation. However, this (gˆ, ,ˆ χ˜ , F˜ , G˜ , J˜),wheredF˜ = 0, dG˜ = J˜,andG˜ =˜χ(ˆF˜ ). requires greater care, and the inclusion of time transformations The way to physically achieve such a transformation is is not immediately evident. We will therefore continue with a to introduce some kind of material, just like introducing a manifestly four-dimensional approach. dielectric between the plates of a parallel plate capacitor. Thus we know that the new configuration of electromagnetic 4. Transformation optics fields must arise from a material distribution χ˜ and must obey G˜ =˜χ(ˆF˜ ). The question of transformation optics is: given The completely covariant approach to transformation optics a transformation, is it possible to determine the χ˜ that will was outlined in [16]; here we derive this method in greater support the transformed fields? detail. Start with an initial space–time M and field Consider equations (23)and(25) in more detail. At a point configuration (g,,χ, F, G, J),wheredF = 0, dG = J,and x ∈ M˜ we have G = χ(F). The usual approach to transformation optics ˜ = T ∗( ) = T ∗(χ ◦  ◦ ), begins by imagining a coordinate transformation T that in Gx GT (x) T (x) T (x) FT (x) (26) some way ‘deforms’ the manifold. For example, in the case where the second line follows from the constitutive relation of an electromagnetic cloak [8, 37], we imagine a coordinate G = χ(F). But from the constitutive relation G˜ =˜χ(ˆF˜ ) transformation that stretches out a hole in Minkowski space– we also have time (with a point removed). ˜ =˜χ ◦ ˆ ◦ T ∗( ). However, certain subtleties involved in this picture should Gx x x FT (x) (27) be clarified. The idea of stretching open a hole or otherwise The right-hand sides of equations (26)and(27) must be equal, deforming the space–time is non-physical; therefore it is more but to clearly see what is going on we must consider the action appropriate to imagine a map which takes M to an image of G˜ on a bi-vector V ∈ T 2(M˜ ),so M˜ ⊆ M, as in figure 2. While it may be intuitive to think of the x x x χ ◦  ◦ ◦ T ( ) =[˜χ ◦ ˆ ◦ T ∗( )]( ). map T as given, the transformation of two-forms (representing T (x) T (x) FT (x) d Vx x x FT (x) Vx the fields of interest) requires instead a map T that describes (28)

5 J. Opt. 13 (2011) 055105 R T Thompson et al μ μ Letting  ν be the Jacobian matrix of T ( ν is the matrix a material. Thus there is no symmetry that should be preserved. representation of dT ), this may be written in component form The point of transformation optics is that the transformed fields as do not constitute a solution to Maxwell’s equations in the μν σρ α β λκ original space–time but do constitute a solution to Maxwell’s χ αβ |T (x) μν |T (x) Fσρ|T (x)( λ κ )|x V x equations in the appropriate material. Thus the mappings = (χ˜ ξζˆ γδ)| | (σ ρ )| λκ , λκ ξζ x Fσρ T (x) γ δ x Vx (29) involved are not mappings between equivalent solutions, but where we have indicated explicitly where each tensor or rather a method of generating new solutions from the original Jacobian matrix is evaluated. Eliminating F and Vx from both (untransformed) solution. sides and solving for χ˜ gives the final result Furthermore, one might be concerned that the pullback τη α β μν σρ of the fields under T might not be contained in the image of χ˜ λκ (x) =− λ κ χ αβ |T (x) μν |T (x) the pullback of the metric under T. Ultimately, this map may × (−1)π (−1)θ ˆ τη| . σ ρ πθ x (30) only be defined locally. But since the final material parameters In equation (30) Λ−1 is the matrix inverse of Λ, both Λ and must ultimately be measured by a local observer with a local Λ−1 are evaluated at x, and in solving for χ˜ we have made metric, the depiction in figure 2 can always be defined locally use of the fact that on a four-dimensional Lorentzian manifold, (neglecting irrelevant situations like transformation optics near acting twice with  returns the negative, F =−F. singularities). What is ˆ at the point x? The Hodge dual  is not directly Finally, we note that because of the symmetries of transformed by the pullback of T. Rather, the metric is pulled F and G, which have only six independent components back with T∗ and then ˆ is computed from the pulled back each, and of χ and , which have at most 36 independent metric. But notice that components, we could re-express these results in terms of two six-dimensional vectors encoding the information of F ˆ = T∗( ) = ◦ T gx gT(x) gT(x) d (31) and G and two 6 × 6 matrices encoding the information of χ and . This six-dimensional representation is commonly depends on g at the point T(x), which is not, in general, the encountered in the literature, and could be advantageous if same point as T (x).Soχ˜ (x) depends not only on the point performing calculations by hand, because of the large number T (x), as explicitly shown in equation (30), but also depends of terms that must be computed in equation (30). However, on the point T(x) through the evaluation of ˆ at x. a modern computer algebra package can handle the full Equation (30) is the core of transformation optics. Start calculation with ease, and the manifestly four-dimensional with a given space–time with metric g and associated dual , nature of the expressions is useful when further manipulations and with given dielectric material properties described by the are performed. tensor χ. The initial space–time may be Minkowski and the initial dielectric may be the vacuum, but this is not necessary. We imagine a transformation that changes the fields in some 4.1. Examples χ˜ way. We ask what is required to physically achieve such a The completely covariant approach to transformation optics transformation. The answer is given by equation (30). developed above provides a concrete and powerful framework A few remarks are in order concerning equation (30). For for analyzing any desired configuration of fields and T applications in transformation optics, is generally taken to linear dielectric materials in any space–time and with any T( ) = ˆ =  be the identity map, meaning that x x and . relative velocities. Several previous examples have been T However, this is not strictly necessary, and allowing to be a given [16] which demonstrate that the results obtained with more general map may be useful in the study of analog space– the completely covariant approach agree with results obtained T times [38]. Furthermore, if in addition to being the identity through other means. This section presents some additional χ  =  the initial space–time is vacuum, then since vac , examples to illustrate the applicability of the completely τη α β σρ −1 π −1 θ τη covariant approach to arbitrary dielectric velocities. χ˜ λκ (x) =− λ κ  αβ |T (x)( ) σ ( ) ρ  πθ , (32) where the first  is evaluated at T (x) and everything else 4.1.1. Mixed transformation. Starting from vacuum is evaluated at x [16]. Notice that the prescriptions of Minkowski space–time, consider the transformation T defined equations (30)or(32) are meaningful only for points x ∈ by M˜ . So for transformations such as the electromagnetic cloak, T (t , x , y , z ) = (t, x, y, z) = (ax t , x , bx + cy , z ). where there is a hole in M˜ , the material parameters inside the (33) hole are unspecified and completely arbitrary. In this way, any The temporal part of this transformation has previously been uncharged material may be hidden inside the cloak without considered in more detail [39], but the spatial transformation affecting the behavior of the fields outside the cloak. provides some additional complexity. As discussed above, the One might be concerned by the step of assigning two fields are actually transformed by T rather than T ,whichwe different transformations, one to the metric and one to the take to be fields, and argue that such a step is not allowed because it t y − bx does not constitute a symmetry of the system. However, these T (t, x, y, z) = (t , x , y , z ) = , x, , z , are not maps between physically equivalent systems, clearly ax c evidenced by the fact that one is vacuum and the other contains x = 0 (34)

6 J. Opt. 13 (2011) 055105 R T Thompson et al and the Jacobian matrix of T is can be arbitrarily tuned, it is not always possible to tune the ⎛ ⎞ 1 − t velocity such that for a given transformation the material does 2 00 ⎜ ax ax ⎟ not require inherent magneto-electric couplings. In the totally μ ⎜ 0100⎟ covariant method presented here, the material velocity is an  ν = ⎜ ⎟ . (35) ⎝ − b 1 ⎠ 0 c c 0 arbitrary parameter compatible with any value of magneto- 0001 electric coupling. This section presents some examples of a transformation Turning the crank on equation (30), and comparing with applied to a uniformly moving material in vacuum Minkowski equation (A.2), the components of χ can be extracted and space–time. For simplicity, suppose the resulting material collected as moves in the x-direction with some speed β with respect to ⎛ ⎞ the laboratory frame in which the electromagnetic fields are 1 b 0 μ a 2 2 v εˇ∗ = ⎝ 2 + 2 − c t ⎠ , known. A prime will denote an object in the material x bb c a2 x4 0 c 2 frame, while unprimed objects reside in the laboratory frame. 001− t a2 x4 A transformation to the material frame from the laboratory 2 2 μ μ ν b + c −b 0 frame is done with a Lorentz boost such that for v = L ν v , − 1 μˇ 1 = −b 10, (36) ⎛ ⎞ acx γ −γβ 00 00c2 μ ⎜ −γβ γ 00⎟ 00 b L ν = ⎝ ⎠ , (38) ∗ ∗ T t 0010 γˇ =−(γˇ ) = 00−1 . 1 2 acx2 0001 0 c2 0 where γ = (1 − β2)−1/2. The inverse boost, from the material Changing to the representation of equation (19)byusing frame to the laboratory frame, is obtained by setting β →−β, ν ¯ ν μ equation (20) results in so that v = L μ v ,where ⎛ ⎞ a 1 b 0 γγβ00 εˇ =ˇμ = x bb2 + c2 , 0 ν ⎜ γβ γ 00⎟ c ¯ 001 L μ = ⎝ ⎠ . (39) (37) 0010 000 0001 γˇ =ˇγ T = t . 1 2 001 ¯ ν x − An object with lowered indices boosts as ωμ = ων L μ or 0 10 μ ων = ωμ L ν . The time transformation is responsible for the appearance The calculation now follows more or less straightfor- of the non-zero magneto-electric coupling terms [16, 39]. It wardly from equation (32), but includes a boost. Let χ was shown in [16]thatifεˇ =ˇμ is proportional to the be the material parameters in the rest frame of the material; identity matrix, then the magneto-electric coupling terms can the material parameters described in the laboratory frame are, be simply identified as a velocity, i.e. the magneto-electric schematically χ = −1(χ ). coupling may arise solely from a material velocity. However, lab L (40) if εˇ =ˇμ is not proportional to the identity matrix, as in this But the material parameters in the laboratory frame are example, then the magneto-electric coupling cannot always be precisely those obtained from equation (32), χ =˜χ.It interpreted as a simple material velocity. Thus in this example lab follows that in the frame of the material, where the material we must interpret these results as an anisotropic material with parameters are to be measured, magneto-electric coupling, at rest with respect to the frame ψ ζ ¯ λ ¯ κ ψ ζ in which the fields are given. Indeed, it is not possible to χ ϕ ξ =−L ϕ L ξ L τ L η α β σρ −1 π −1 θ τη boost to a uniformly translating frame in which the magneto- × ( λ κ  αβ ( ) σ ( ) ρ  πθ ,). (41) electric coupling vanishes, as we show below. Note that the Let the transformation again be that of equation (33). time-dependent behavior of these material parameters makes Performing the calculation, extracting the components of εˇ∗, a validating calculation quite difficult. Such a calculation −1 ∗ ∗ μˇ , γˇ1 and γˇ2 , and converting to the representation of requires an analysis for both the initial and transformed media equation (19), results in that is beyond the scope of this paper [39]. 3 εˇ =ˇμ = acx a2x 4(b2 + c2)β2 − c2(x + tβ)2 4.1.2. Moving materials. In a real world scenario, it is likely 2 2 β2 − ( + t β)2 − b ( + t β) that the design parameters require the desired material to move a x 1 x γ 1 x 0 − b ( + t β) − b2+c2 with respect to the frame in which the field transformations × γ 1 x γ 2 0 , 00− 1 are defined. In the Plebanski based approach outlined in [12], γ 2 (42) the magneto-electric coupling term is interpreted as a velocity γˇ =ˇγ T = 1 dictated by the transformation, rather than being a free 1 2 a2x 4(b2 + c2)β2 − c2(x + tβ)2 parameter. As discussed in [16], a magneto-electric coupling 00∗ can only be independently identified with a velocity if the × 00∗ , ba2x4 2 2 4 2 2 material is isotropic. Therefore even if the material velocity − γ β c (x + tβ)(t + xβ) − a x β(b + c ) 0

7 J. Opt. 13 (2011) 055105 R T Thompson et al where ∗ indicates components that are obtained by the 4.1.3. Moving cloak. Finally, consider a cloak moving in antisymmetry of the matrix. the x-direction with constant speed β. What velocity-induced While these results are fairly complicated and perhaps corrections are required to achieve cloaking with respect to do not offer much intuition regarding moving dielectrics, this fields described in the laboratory frame? A square cloak is example demonstrates the method by which material velocities obtained from the transformation [37] may be easily incorporated into the theory, even for non- T ( , , , ) = ( , , , ) t x y z t x y z trivial transformations. Essentially, the calculation consists of s (x − s ) s y(x − s ) χ˜ = t, 2 1 , 2 1 , z , calculating in the frame in which the desired transformation − ( − ) (47) is specified, and then transforming that result to the rest frame s2 s1 x s2 s1   − <  | | < | | of the corresponding moving material. It can be readily seen for s1 x s2, s2 y s2,and y x . that the limit β → 0 recovers the results of equation (37). From equation (32), the well-known results for the equivalent material parameters of a stationary cloak are As a somewhat simpler example, consider the transforma- ⎛ ⎞ − s1 − y tion 1 s1 2 0 ⎜ x x ⎟ y − cx x4 +s2 y2 εˇ =ˇμ = ⎜ − y 1 ⎟ T (t, x, y, z) = (t , x , y , z ) = at + bx, x, , z , ⎝ s1 2 3 ( − ) 0 ⎠ (48) d x x x s1 s2(x−s ) (43) 2 1 , 00x(s −s )2 which is similar to that of equation (33), but avoids 1 2 with γ = γ = 0. Now assume that the cloak is moving. From the complication of time-dependent parameters found in 1 2 equation (41), it follows that in the frame of the cloak and to equations (37)and(42). For a stationary material, the first order in the speed β, the material parameters now have the equivalent material parameters are additional magneto-electric couplings 1 c 0 β 1 T s1 εˇ =ˇμ = 2 + 2 , γˇ1 =ˇγ2 = cc d 0 (44a) x 3(s − s )2 ad 1 2 001 ∗ 00 000 × 00∗ , (49) T b 2 γˇ =ˇγ = 001. (44b) −y(x − s )s2 x 3(2s − s ) + y s s2 0 1 2 a 1 2 2 1 x 1 2 0 −10 where * indicates components that are obtained by the It is tempting to associate the magneto-electric coupling with anisymmetry of the matrix. a simple material velocity in the x-direction, proportional to The moving cloak therefore requires inherent magneto- b/a. To check whether this is the case, suppose we take a electric couplings designed to compensate for these velocity- material described by equation (44a)forεˇ =ˇμ, but with induced magneto-electric couplings, which are more compli- γ1 = γ2 = 0,andmovinginthex-direction with speed β. cated than might be expected from previous experience with In general, εˇ =ˇμ,andγˇ1 =ˇγ2 will all now depend on β. isotropic materials. These corrections are likely to be small For low speeds, expanding to leading order in β finds εˇ =ˇμ for ordinary nonrelativistic velocities, but depend specifically unchanged to first order, but with on the other cloak parameters. As before, the corrections to β εˇ =ˇμ are second order in β. γˇ =ˇγ T = 1 2 2 2 a d 00 −c 5. The Gordon formalism × 00(a2d2 − c2 − d2) . (45) c −(a2d2 − c2 − d2) 0 In a problem closely related to the methods of transformation Thus, interpreting the magneto-electric coupling of optics, Gordon, in 1923, studied the possibility of representing equation (44b) as a simple low-speed motion in the x- electrodynamics in a material by an equivalent vacuum space– direction is incorrect, as this generates additional magneto- time [5]. One motivation for such a representation is that electric coupling components of the same order. However, it allows the material to be described by the action of γˇ = β/ 2 2 equation (12), with the constitutive relation G = F,where 1xz c a d indicates that including a y-component for the velocity might allow us to recover the desired results. Indeed  is the Hodge dual for a vacuum space–time described by this is the case, and it can be shown that a material with εˇ =ˇμ an effective, or ‘optical’ metric. This is, in some sense, described by equation (44a)andγ1 = γ2 = 0, moving with the ‘inverse’ problem of transformation optics. In particular, velocity components suppose we are given a dielectric material described by χ residing in a space–time with metric g and corresponding ( 2 2 − ) ab a f 1 dual , then from equation (28) it may be possible to find ˆ βx = , 1 − a2(1 − c2) − f 2(1 − a2) corresponding to an effective vacuum space–time, (46) abc χ = χ ˆ = .ˆ (50) βy = , vac 1 − a2(1 − c2) − f 2(1 − a2) To be more explicit, the initial setup consists of a material β generates the results of equations (44) to leading order in x residing in a space–time, for which the relevant part of the β εˇ =ˇμ and y. Velocity-induced corrections to are of second action is β β order, i.e. proportional to x y, and of course we are restricted = ∧ = ∧ χ( ). (β2 + β2)1/2 =|β| S F G F F (51) to x y 1. M M

8 J. Opt. 13 (2011) 055105 R T Thompson et al Similarly to equation (9), this is for the case of an isotropic material with vanishing magneto- electric coupling, it is relatively easy to check that Gordon’s F ∧ χ(F) = d4x |g|g(F, -(χ  F)). (52) optical metric M M μν μν μ ν γμν = gμν +(1−εμ)uμuν ,γ= g −(εμ−1)u u , If this is to be regarded as a vacuum space–time with metric γ , (56) then μ where u is the four-velocity of the material, agrees with our μ = ∧ χ( ) = ∧ ˆ = 4 |γ |γ( , ). method. For an observer at rest with the material, u F F F F d x F F (53) ( , , , ) M M M 1 0 0 0 , while for a material in relative motion to the observer, the χ in equation (50) is first transformed with the χ The most general has 36 independent components, but appropriate boost. the metric only has ten components, meaning that not every dielectric material can be represented by an equivalent space– time. To find the conditions that a material must satisfy 6. Conclusions to be representable as a vacuum space–time, return first to equation (50). Calculating ˆ for an arbitrary metric γ ,and We have continued the development of a fully covariant comparing with χ for arbitrary χ, it is readily observed that and manifestly four-dimensional formalism for transformation ∗ −1 ∗ ∗ T optics in general linear materials that began in [16]. The both εˇ and μˇ must be symmetric, while γˇ2 =−(γˇ1 ) is traceless. This reduces the number of apparently independent main result of this construction, equation (30), represents a equations to 21, plus one constraint. However, the equations sort of master equation for transformation optics of general, are not linear and are given in a particular representation. linear, nondispersive media. The benefit of this approach Converting to the representation of equation (19) one finds is that it is valid for arbitrary background space–times [19], T arbitrary initial material [18], and for arbitrary transformations. that εˇ =ˇμ,andthatγˇ2 =ˇγ1 is not just traceless but antisymmetric. These, of course, are precisely the constraints The motivation for this construction was the observation that expected from the Plebanski–De Felice equations. the Plebanski–De Felice equations that are frequently used in For a dielectric material to be representable by a curved transformation optics and identified with material velocities are T only valid in the low velocity limit for isotropic materials. vacuum space–time it must have εˇ =ˇμ,andγˇ2 =ˇγ1 , which reduces the number of free parameters of χ to nine. To arrive at the completely covariant method it was first Unfortunately, this is insufficient to uniquely determine the necessary to find a modest generalization of electrodynamics ten free parameters of the metric. On a four-dimensional in linear dielectric media. This came by way of a modification = χ( ) Lorentzian manifold, every metric g is uniquely associated to the constitutive relations to the form G F . Within with a map  from two-forms to two-forms by these relations the vacuum is identified as being a perfect linear dielectric with relative permittivity and permeability equal to μν 1 σμ ρν  αβ = | |αβσρ . χ 2 g g g (54) 1—corresponding to a trivial —and is therefore treated just like any other dielectric media.  → Notice that is invariant to a scaling of the metric g ag, Next, the interpretation of transformation optics was  so given it is only possible to determine the metric up clarified by realizing that the system is being transformed to an overall scale factor. On the other hand this means by two distinct maps, one applied to the metric and a γ that when trying to determine a particular from a given different one applied to the electromagnetic fields, because a ˆ , one is free to choose a convenient representative from the transformation of the fields is physically realized by changing equivalence class of conformally related space–times, such the dielectric material properties (e.g. by introducing a non- γ =− as the representative with 00 1. The situation may vacuum dielectric material to a vacuum region) but generally = improve somewhat if there are sources present, J 0, since the not the metric. Hodge dual of one-forms and three-forms is not invariant under As an illustration of the completely covariant approach conformal transformations of the metric. This may impose we have examined the results of a transformation when some additional constrains on the metric, but remains to be the resulting material is constrained to move with arbitrary fully investigated. uniform velocity—a physically realistic scenario that has There exists another condition from which to obtain not, as far as we know, been previously considered. This constraints on the material parameters. Recall that one of is a departure from previous results, where the material the defining characteristics of the Hodge dual on a four- velocity has been dictated by the transformation or has been  =− dimensional Lorentzian manifold is that F F. It follows constrained to the nonrelativistic limit. It is furthermore that shown that the magneto-electric coupling does not arise as (χ)(χ) =− , id2 (55) straightforwardly from a material velocity as expected from wherewewriteid2 as the identity of the map from two- previous interpretations of such couplings. For the particular forms to two-forms. This does not provide any independent transformation considered it was found that it is in fact constraints on the material, but may serve as a useful check possible to approximately get magneto-electric couplings from when performing calculations. a material velocity, but only in the low velocity limit where It is not our purpose to find solutions for a particular corrections to the permeability and permittivity are neglected, distribution of materials, and it is sufficient to remark that, and even then the velocity was in an unexpected direction. It

9 J. Opt. 13 (2011) 055105 R T Thompson et al ⎛ ⎞ was shown that for a moving cloak, magneto-electric couplings 0 −μ−1 −μ−1 −μ−1 of the order of the velocity have to be designed into the cloak zx zy zz ⎜ μ−1 0 −γ γ ⎟ to compensate for the velocity-induced couplings. D = ⎜ zx 1zz 1zy ⎟ , ⎝ μ−1 γ 0 −γ ⎠ It was shown in section 5 that our formalism generalizes, zy 1zz 1zx μ−1 −γ γ in a covariant way, Gordon’s optical metric [5], which turns out zz 1zy 1zx 0 to be essentially the inverse problem of transformation optics. ⎛ ⎞ Namely, given a dielectric, can we find an equivalent vacuum 0 γ2yx γ2yy γ2yz ⎜ −γ  − ⎟ space–time? The answer appears to be yes, but only if the E = ⎝ 2yx 0 yz yy ⎠ , dielectric satisfies certain properties, and even then only up to −γ2yy −yz 0 yx a conformal factor of the metric. −γ2yz yy −yx 0 It is worth reiterating that the component forms of the ⎛ ⎞ constitutive relations in the covariant formalism are of the 0 −γ2xx −γ2xy −γ2xz form of equation (18) rather than equation (19). This leads ⎜ γ2xx 0 −xz xy ⎟ × εˇ∗ F = ⎝ ⎠ to a slightly different meaning for the 3 3 matrices , γ2xy xz 0 −xx μ−1 γˇ ∗ γˇ ∗ , 1 ,and 2 than what would be expected from the γ2xz −xy xx 0 relations (19). One may readily switch back and forth where the ∗ indicates entries that are antisymmetric on either between the two representations for the constitutive relations σρ the first or second set of indices on χ γδ . × via equation (20). Keep in mind, however, that the 3 3 In cylindrical coordinates (t, r,θ,z), the Minkowski matrices of equation (18) are merely selected components of 2 metric tensor is gμν = diag(−1, 1, r , 1). Because we have the main quantity of interest, the covariant 2 -tensor χ. 2 c = 1, time and space are measured in the same units, meaning that Eα and Bα are both measured in units of V m−1 and the Acknowledgments units for F are V m−1. Changing to cylindrical coordinates does not change the units of F,so The Duke University component of this work was partly F = Er dr ∧ dt + rEθ dθ ∧ dt + Ez dz ∧ dt + rBr dθ ∧ dz supported by a Lockheed Martin University Research Initiative − Bθ dr ∧ dz + rB dr ∧ dθ. (A.3) award and by a Multiple University Research Initiative from z the Army Research Office (Contract No. W911NF-09-1-0539). This definition for the components of F in cylindrical coordinates assumes that all Eα and Bα are still measured −1 θ Appendix A. Cartesian coordinates in units of V m , despite the dimensionless coordinate . Similar arguments hold for G. These are written in matrix form as In Cartesian coordinates, the field strength tensor is defined as ⎛ ⎞ 0 −E −rEθ , −E F = Ex dx ∧ dt + Ey dy ∧ dt + Ez dz ∧ dt r z ⎜ Er 0 rBz −Bθ ⎟ + Bx dy ∧ dz − By dx ∧ dz + Bz dx ∧ dy. (A.1) Fμν = ⎝ ⎠ , and rEθ −rBz 0 rBr This may be represented in matrix form by equation (2). To Ez Bθ −rBr 0 recover the usual component relations of equation (18), the ⎛ ⎞ (A.4) , constitutive equation G = χ(F) allows us to make the 0 Hr rHθ Hz ⎜ −Hr 0 rDz −Dθ ⎟ identification [16] Gμν = ⎝ ⎠ . ⎛ ⎞ −rHθ −rDz 0 rDr 0 ∗∗∗ −Hz Dθ −rDr 0 σρ 1 ⎜ A 0 ∗∗⎟ χ γδ = ⎝ ⎠ , (A.2) = χ( ) 2 BC0 ∗ Matching the constitutive equation G F for DEF0 Minkowski space–time in cylindrical coordinates results in the identification where ⎛ ⎞ ⎛ ⎞ ∗∗∗ −μ−1 −μ−1 −μ−1 0 0 xx xy xz σρ 1 ⎜ G 0 ∗∗⎟ ⎜ μ−1 −γ γ ⎟ χ γδ = ⎝ ⎠ , (A.5) A = ⎜ xx 0 1xz 1xy ⎟ , 2 HI 0 ∗ ⎝ μ−1 γ −γ ⎠ xy 1xz 0 1xx JKM0 μ−1 −γ γ 0 xz 1xy 1xx where − ⎛ ⎞ ⎛ − μ 1 − ⎞ 0 −μ−1 −μ−1 −μ−1 0 −μ 1 − rθ −μ 1 yx yy yz rr γr rz − ⎜ −1 1rz ⎟ ⎜ μ 1 −γ γ ⎟ μ 0 − γ1rθ ⎜ yx 0 1yz 1yy ⎟ G = ⎜ rr r ⎟ , B = , ⎝ μ−1 γ γ ⎠ ⎝ μ−1 γ −γ ⎠ rθ 1rz − 1rr yy 1yz 0 1yx r r 0 r − −1 γ1rr μ 1 −γ γ μ −γ θ yz 1yy 1yx 0 rz 1r r 0

⎛ ⎞ ⎛ − μ−1 −μ−1 − μ−1 ⎞ 0 −γ2zx −γ2zy −γ2zz 0 r θr θθ r θz ⎜ γ −  ⎟ ⎜ μ−1 −γ γ ⎟ 2zx 0 zz zy H = r θr 0 1θz r 1θθ , C = ⎝ ⎠ , ⎝ −1 ⎠ γ2zy zz 0 −zx μθθ γ1θz 0 −γ1θr γ −  μ−1 − γ γ 2zz zy zx 0 r θz r 1θθ 1θr 0

10 J. Opt. 13 (2011) 055105 R T Thompson et al

⎛ ⎞ ⎛ −γ ⎞ − γ −γ − γ − γ −γ 2ϕϕ 0 r 2zr 2zθ r 2zz 0 r 2ϕr 2ϕθ s ϕθ ⎜ rγ 0 − r θ ⎟ ⎜ rγ ϕ 0 −ϕϕ ⎟ I = ⎝ 2zr zz z ⎠ , ⎜ 2 r s ⎟ P = −ϕ , γ zθ zz 0 −zr ⎝ γ  ⎠ 2 2ϕθ ϕϕ 0 rs γ −  rγ2zz −rzθ zr 0 2ϕϕ ϕθ ϕ s s rs 0

⎛ μ−1 ⎞ ⎛ ⎞ −μ−1 − zθ −μ−1 − μ−1 − μ−1 −μ−1 0 zr r zz 0 rs ϕr s ϕθ ϕϕ −1 γ1zz ⎜ μ 0 − γ θ ⎟ ⎜ μ−1 − γ γ ⎟ ⎜ zr r 1z ⎟ ⎜ rs ϕr 0 s 1ϕϕ 1ϕθ ⎟ J = −1 , Q = , ⎝ μ θ γ γ ⎠ ⎝ −1 −γ1ϕr ⎠ z 1zz − 1zr μ γ ϕϕ r r 0 r s ϕθ s 1 0 r −1 γ1zr − γ ϕ μ −γ θ 0 μ 1 −γ 1 r zz 1z r ϕϕ 1ϕθ r 0

⎛ γ ⎞ ⎛ ⎞ γ 2θθ γ γ γ γ 0 2θr r 2θz 0 rs 2θr s 2θθ 2θϕ θ ⎜ −γ θ 0 z −θθ ⎟ ⎜ − γ  − ⎟ K = ⎝ 2 r r ⎠ , R = rs 2θr 0 s θϕ θθ , γ θθ θz θ ⎝ θ ⎠ − 2 − 0 r − γ θθ − θϕ r r r r s 2 s 0 r θ −θ −γ θ θθ − r 0 −γ θϕ θθ r 2 z r 2 r 0

⎛ ⎞ ⎛ 2 ⎞ 0 −rγ2rr −γ2rθ −rγ2rz 0 −r sγ2rr −rsγ2rθ −rγ2rϕ 2 ⎜ rγ − r θ ⎟ ⎜ γ −  ϕ  θ ⎟ M = ⎝ 2rr 0 rz r ⎠ . S = ⎝ r s 2rr 0 rs r r r ⎠ . γ2rθ rz 0 −rr rsγ2rθ rsrϕ 0 −rr rγ2rz −rrθ rr 0 rγ2rϕ −rrθ rr 0 The form of the components in both of equations (A.4) Like the cylindrical result, the spherical result could and (A.5) could equally well be determined by transforming be obtained equally well by transforming the Cartesian the Cartesian versions, equations (2), (10), and (A.2), with the equations (2), (10), and (A.2), with the appropriate Cartesian– appropriate Cartesian–cylindrical transformation matrix. spherical transformation matrix. In spherical coordinates (t, r,θ,ϕ), the Minkowski metric 2 2 2 tensor has components gμν = diag(−1, 1, r , r sin θ).This Appendix B. Pullback and pushforward time there are two dimensionless coordinates, so Suppose two manifolds are related by a smooth map ϕ: M → F = Er dr ∧ dt + rEθ dθ ∧ dt + r sin θ Eϕ dϕ ∧ dt N. Thus ϕ maps the points of M to an image in N, p ∈ M → + 2 θ θ ∧ ϕ − θ ∧ ϕ + ∧ θ r sin Br d d r sin Bθ dr d rBϕ dr d ϕ(p) ∈ N. Given a function f on N,thepullback of f by ϕ, (A.6) denoted ϕ∗ f , is the composite function f ◦ ϕ. So the pullback ϕ∗ ϕ ϕ∗ (every dθ gets a factor of r,everydϕ gets a factor of r sin θ), takes a function f on N to a -related function f on M. ϕ ∈ which is written in matrix form as The differential of at a point p M defines a map ϕ: → ⎛ ⎞ of tangent spaces d Tp M Tϕ(p) N. Given a vector − − , − θ 0 Er rEθ r sin Eϕ Vp ∈ Tp M,thepushforward of Vp assigns a ϕ-related vector ⎜ ϕ − θ θ ⎟ ϕ( ) ∈ = Er 0 rB r sin B . d Vp Tϕ(p) N, such that Fμν ⎝ 2 ⎠ rEθ −rBϕ 0 r sin θ Br ∗ 2 ϕ( ) = (ϕ ) r sin θ Eϕ r sin θ Bθ −r sin θ Br 0 d V ϕ(p) f Vp f (B.1) (A.7) μ Letting s = sin θ, the same identification procedure as before for some f on N. In particular, given coordinate charts {x } { μ} leads to ⎛ ⎞ on M and y on N this is 0 ∗∗∗ ∂ ∂ σρ 1 ⎜ N 0 ∗∗⎟ ( ϕ( ))α = μ (ϕ∗ ) χ γδ = ⎝ ⎠ , (A.8) d Vp ϕ(p) α f Vp μ f (B.2) 2 OP0 ∗ ∂y ∂x QRS 0 ∂ = V μ ( f ◦ ϕ) (B.3) where p ∂ μ x ⎛ μ−1 μ−1 ⎞ α −μ−1 − rθ − rϕ μ ∂ϕ ∂ 0 rr = . r rs Vp μ α f (B.4) ⎜ − −γ1rϕ γ θ ⎟ ∂ ∂ μ 1 1r x p y ⎜ rr 0 r rs ⎟ N = ⎜ −1 ⎟ , μ γ ϕ −γ ⎝ rθ 1r 1rr ⎠ Thus the pushforward of Vp is r r 0 r 2s μ−1 −γ γ rϕ 1rθ 1rr 0 ( ϕ( ))α = α μ, rs rs r 2s d Vp ϕ(p) μVp (B.5)

⎛ −μ−1 ⎞ − μ−1 −μ−1 θϕ where α 0 r θr θθ s α ∂ϕ ⎜ − γ ⎟  = ⎜ μ 1 −γ 1θθ ⎟ μ μ r θr 0 1θϕ s ∂x p O = ⎜ − −γ ⎟ , ⎝ μ 1 γ 1θr ⎠ θθ 1θϕ 0 rs ϕ μ−1 is the Jacobian matrix of the map evaluated at p.Note θϕ −γ1θθ γ1θr s s rs 0 that the pushforward of a vector field is not always defined.

11 J. Opt. 13 (2011) 055105 R T Thompson et al

For example, if ϕ is not injective, then for two distinct points [12] Leonhardt U and Philbin T G 2006 New J. Phys. p, q ∈ M such that ϕ(p) = ϕ(q) ∈ N, the vectors Vp and 8 247 arXiv:cond-mat/0607418 [13] Tretyakov S, Nefedov I and Alitalo P 2008 New J. Phys. Vq would be pushed to the same point in N,butthereisno ϕ( ) ϕ( ) ϕ 10 115028 reason why d Vp should equal d Vq .However,if is a [14] Greenleaf A, Kurylev Y, Lassas M and Uhlmann G 2009 SIAM diffeomorphism, then the pushforward of a vector field is well Rev. 51 3 http://link.aip.org/link/?SIR/51/3/1 defined. [15] Leonhardt U and Philbin T G 2009 Prog. Opt. 53 Like a function (which is a 0-form), a one-form on N can 69 arXiv:0805.4778 be pulled back to a one-form on M such that [16] Thompson R T, Cummer S A and Frauendiener J 2011 J. Opt. 13 024008 arXiv:1006.3118 (ϕ∗ω)( ) = ω( ϕ( )). [17] Cheng X, Chen H, Wu B-I and Kong J A 2009 Prog. Electron. Vp d Vp (B.6) Res. 89 199 [18] Thompson R T 2010 Phys. Rev. A 82 053801 arXiv:1011.0753 In particular, expressing this in coordinate charts as above, we [19] Thompson R T 2011 arXiv:1102.1512 find [20] Tu L W 2008 An Introduction to Manifolds (Berlin: Springer) [21] Frankel T 2004 The Geometry of Physics: An Introduction 2nd ∗ α β ∂ α [(ϕ ωϕ( ))| ]α dx V = (ωϕ( ))α dy edn (Cambridge: Cambridge University Press) p p p ∂x β p [22] Bleecker D 2005 Gauge Theory and Variational Principles ∂ϕβ ∂ (New York: Dover) ISBN 0-486-44546-1 × V μ , (B.7) ∂ μ p ∂ β [23] Baez J and Muniain J P 1994 Gauge Fields, Knots and Gravity x p y (Singapore: World Scientific) ISBN 981-02-1729-3 which simplifies to [24] Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco, CA: Freeman) ∂ϕβ [25] Post E J 1962 Formal Structure of Electromagnetics ∗ α α [(ϕ ωϕ( ))| ]α V = (ωϕ( ))α V , (B.8) (Amsterdam: North-Holland) p p ∂ μ p x p [26] Hehl F W and Obukhov Y N 2003 Foundations of Classical Electrodynamics: Charge, Flux, and Metric (Boston, MA: or Birkhauser) ISBN 0-8176-4222-6 ∗ β (ϕ ω)α = ωβ  α, (B.9) [27] Jackson J D 1998 Classical Electrodynamics 3rd edn β (New York: Wiley) p 808 ISBN 0-471-30932-X where  α is again the Jacobian matrix of ϕ evaluated at p. [28] Hopfield J J 1958 Phys. Rev. 112 1555 [29] Huttner B and Barnett S M 1992 Phys. Rev. A 46 4306 [30] Obukhov Y N and Hehl F W 1999 Phys. Lett. B References 458 466 arXiv:gr-qc/9904067 [31] Hehl F W and Obukhov Y N 2005 Found. Phys. 35 2007 arXiv:physics/0404101 [1] Sievenpiper D F, Sickmiller M E and Yablonovitch E 1996 [32] Hehl F W and Obukhov Y N 2006 Lect. Notes Phys. 702 Phys. Rev. Lett. 76 2480 163 arXiv:gr-qc/0508024 [2] Pendry J B, Holden A J, Robbins D J and Stewart W J 1999 [33] Hehl F W, Obukhov Y N, Rivera J-P and Schmid H 2008 Phys. IEEE Trans. Microwave Theory Technol. 47 2075 Rev. A 77 022106 [3] Smith D R, Pendry J B and Wiltshire M C K 2004 Science [34] Landau L D and Lifshitz E M 1960 Electrodynamics of 305 788 Continuous Media (Oxford: Pergamon) (translated from the [4] Eddington A S 1920 Space, Time, and Gravitation (Cambridge: Russian by J B Sykes and J S Bell) Cambridge University Press) [35] Yuan Y, Popa B-I and Cummer S A 2009 Opt. Express [5] Gordon W 1923 Ann. Phys. 72 421 17 16135 [6] Plebanski J 1959 Phys. Rev. 118 1396 [36] Popa B-I and Cummer S A 2007 Microwave Opt. Technol. Lett. [7] Felice F D 1971 Gen. Relativ. Gravit. 2 347 49 2574 [8] Pendry J, Schurig D and Smith D R 2006 Science 312 1780 [37] Rahm M, Schurig D, Roberts D A, Cummer S A, Smith D R [9] Greenleaf A, Lassas M and Uhlmann G 2003 Math. Res. Lett. and Pendry J B 2008 Photon. Nanostruct. Fundam. Appl. 10 685 6 87 [10] Schurig D, Pendry J B and Smith D R 2006 Opt. Express [38] Thompson R T and Frauendiener J 2010 Phys. Rev. D 14 9794 82 124021 arXiv:1010.1587 [11] Milton G W, Briane M and Willis J R 2006 New J. Phys. 8 248 [39] Cummer S A and Thompson R T 2011 J. Opt. 13 024007

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