The refractive index of reciprocal electromagnetic media

We study the electromagnetics of media described by identical inhomogeneous relative dielectric and magnetic tensors, ϵ = &mgr; . ?> Such media occur generically as spatial transformation media, i.e. electromagnetic media that are defined by a deformation of space. We show that such media are completely described by a refractive index n ( r , s ˆ ) ?> that depends on position r ?> and direction s ˆ , ?> but is independent of polarization. The phase surface is always ellipsoidal, and n ( r , s ˆ ) ?> is therefore represented by the radius vector to the surface of the ellipsoid. We apply our method to calculate the angular dependence of the refractive index in the well-studied cylindrical cloak and to a new kind of structurally chiral medium induced by a twist deformation. By way of a simple example we also show that media for which ϵ = &mgr; ?> do not in general preserve the impedance properties of vacuum. The implications of this somewhat surprising conclusion for the field of transformation optics are discussed.


Introduction
Spatial transformation optics relates deformations of Euclidean space to an electromagnetic medium, so that essentially any desired continuous mapping of the electromagnetic field can be achieved. The technique has been employed in a variety of contexts, most notably to propose [1][2][3], and then implement [4] an electromagnetic cloak, in which linear rays are distorted in such a way as to avoid a certain region of space, rather like water flowing around an obstruction in a river.
The electromagnetic medium arising from applying the spatial transformation optics algorithm is one in which the relative permittivity and permeability tensors are equal, i.e.
.  m = This can be understood by noting that transformation optics mimics the deformation of vacuum, so that in order for the transformed medium to respond to electric and magnetic fields in the same proportion as for vacuum, inevitably we have that , 1 ( )  m k = º say. We refer to such a medium as being electromagnetically reciprocal or a 'k' medium. The spatial deformation in general changes k from , 0 k its value in flat vacuum, to , k a value that is both inhomogeneous and anisotropic. In the ideal casek will be real-valued, just like 0 k is real for vacuum, though a perfectly lossless medium can only be approximated in practice. The required complexity can in principle be accessed through metamaterials' technology, which precisely seeks to engineer the appropriate anisotropy and inhomogeneity via effective medium parameter values that are not found in nature. However, for most demonstrations of transformation optics to date some level of approximation has been invoked to bring the desired functionality within reach of current technology. A common approach is the socalled reduced parameter scheme [5], in which the precise medium values are replaced with ones which at least yield the desired refractive index distribution. Rays are then refracted appropriately, although light is also scattered due to impedance changes. In the first demonstration of the electromagnetic cloak, for example, the reduced parameter scheme was used to eliminate the spatial dependence of one of the permeability components, and the geometry restricted to TE polarization [4].
Despite these compromises, the transformation medium defined byk is of intrinsic interest since, from a technological viewpoint, the rapid advance of metamaterials technology brings 'perfect' transformation optics media closer to reality. More generally, a k-medium can mimic gravitational curvature without the need for the enormous mass densities required to actually distort space [6]. A curious feature of electromagnetic media described by the single tensor k is that even if its principal values are all distinct, it is not birefringent. A detailed analysis of the geometric origin of such nonbirefringence was made by Favaro et al [7] which embraced, for example, the well studied example of anisotropic unirefringence encountered in moving media [8,9].
The purpose of this paper is to study the electromagnetic characteristics of media described by , k and thereby to increase our understanding of generic electromagnetic media formulated through spatial transformation optics. After some mathematical preliminaries in section 2 we will show in section 3 that media described by k are completely characterized by a refractive index function n r s , (ˆ) which depends on position r and propagation direction ŝ, but is independent of polarization. The index is obtained from a simple ellipsoid construction, distinct from the usual constant energy index ellipsoid associated with birefringent media [10]. Thereby, we develop a geometrical optics of spatial transformation optics. To illustrate our approach, in section 4 we calculate the index along the rays of the traditional electromagnetic cloak, and then, in section 5, for a novel structurally chiral medium induced by a twist deformation. For this medium it is shown that it is integral lines of the Poynting vector that are modified by the deformation, the integral lines of the wave vector being unaffected.
Before concluding in section 7, we address the problem of impedance matching in spatial transformation optics in section 6. We show by a simple example that although spatial dilations of vacuum can be successfully mimicked by the polarization-independent index n r s , , (ˆ) in general it is not possible to preserve the polarization independence of the impedance. This has significant implications for the future of transformation optics, as it shows that perfect cloaking cannot be achieved, even in principle.

Preliminaries
It is common to treat the design of a cloak or other device as a coordinate transformation [1]. However, this conflicts with the notion of the coordinate invariance of any physical process. Consequently, it is more accurate mathematically to replace the notion of design by coordinate transformation with that of design by deformation, in which one physical solution on a manifold is taken into another, and coordinates are relegated to the local labelling of points.
Consider figure 1, where the deformation j of a manifold  may be described as the mapping of  onto itself, i.e. as : The coordinate representation of j is therefore given by the composite map   of the dielectric tensor 0  will be taken under j to [11], where the summation convention is assumed and we denote morphed quantities with a tilde. By contrast, a coordinate transformation relates distinct charts at ,  and is described by the composition figure 1(b). The transformation rule for the dielectric tensor 0  between coordinate systems (i.e. x x is given by [12] L L L det , 5 with an analogous equation for transforming the permeability tensor components . ij 0 m equation (5) and its permeability analogue can be used to transform the dielectric/permeability tensors from Cartesian coordinates to polar coordinates, for example. The L j i ¢ are prescribed by the nature of the basis j The coordinate representation of the morphism is : .  .  We will assume the usual Euclidean metric given in Cartesian coordinates by g .
ij ij d = Transformation optics is concerned with maps of the form of (2), represented by the functions x x . m ĩ ( ) An electromagnetic medium that produces the same deformations as if flat vacuum were distorted by j is one whose dielectric tensor is given by equation (3) and whose permeability tensor is analogously given by In Cartesian coordinates the vacuum constitutive parameters are just ij Since the permittivity and permeability induced by a spatial deformation are the same up to a constant, we may characterize the electromagnetic medium by a single tensork whose components are We note that provided j is a diffeomorphism,k is invertible. We also note that a lossy transformed k-medium results if 0 k is complex-valued.
In the following, the distinctions between deformations and coordinate transformations, and between coordinate and orthonormal bases will all be important in calculating the generalized refractive index associated with spatial deformations.
It is known that the eigen-indices associated with the Fresnel equation for media formed from equation (7) are always degenerate [7], i.e. for light whose wave vector lies parallel to the unit vector ŝ there is just one refractive index n r s , (ˆ) which depends on position r and direction s, but is independent of the field polarization. In the following section we calculate n r s , (ˆ) explicitly for a general morphism j.
3. The refractive index n r; b s À Á When calculating the index n r s , (ˆ) it is most convenient to use Cartesian coordinates. Consider a plane wave propagating in a medium characterized by .
k The idea of a plane wave propagating through an inhomogeneous medium is an approximation that is useful when the wavelength is much greater than the scale of the inhomogeneity of . k Since anyk k = induced by deformation is inevitably inhomogeneous (see equation (7)) our results are only strictly valid in the geometric optics limit. If the local wave-vector of such a wave is k, then from the frequency domain Maxwell curl relations Momentarily, let us set the propagation direction to be the z-axis i.e. k k z.

=
Denoting ⊥ as the projection of vectors and operators onto the transverse x-y plane we find, since k is invertible, that An eigen-equation can then be formed in E^as: and k form an orthogonal set, and (d) k^and H^are in the plane perpendicular to k.
where n k here  is the 2×2 identity. Hence the refractive index n associated with the wave vector k is given by Although this key result has been calculated in Cartesian coordinates, it is valid in any orthonormal basis. It relates the refractive index for light propagating in a particular direction in a transformation medium to the deformation j (see equations (2), (7) and (17)). Equation (17) is also valid for a complex k-medium, in which case the imaginary part of n corresponds to absorption.
Aligning k along the z-axis requires generally two rotations (say f followed by θ-see figure 3), so that Now since c k is symmetric it will be diagonal in an appropriately oriented Cartesian system. If we choose the local Cartesian axes to be along the principal axes of diag , , , From equation (17) it is then found that Setting n nS C n nS S n nC , , where n n n , , .
For the lossless case where all the principal values are real, equation (22) becomes an ellipsoid as illustrated in figure 4. It represents the phase surface, and the refractive index, for arbitrary polarization, is equal to the radius vector to the surface of the ellipsoid. It represents a quite distinct index ellipsoid construction to the standard one associated with the constant energy surface W E E 1 2 · ·  = [10,13]. There, the two polarizations and their associated indices for a given wave vector are given by the orientation and length of the sectional ellipse orthogonal to k. The phase surface in standard birefringent media, where m is proportional to the identity and  is tensorial, can self-intersect [10], supporting conical refraction along the direction of such 'diabolical' points [14]. By contrast, we have shown that in electromagnetically reciprocal media (k-media) the phase surface is always The length of the radius vector extending from the origin to the surface of the ellipsoid represents the refractive index experienced by light travelling in that direction. ellipsoidal and consequently conical refraction cannot be supported in transformation media. Note that electromagnetically n r s , (ˆ) completely describes the medium. Given this function 1 the electromagnetic medium can be reconstructed by finding its three maximal radii (n n n , , 3 ), each of which occurs along a principal axis, and then calculating the principal constitutive parameters as n n n n n n n n n , , .
( ( ) ( ) ( )) k k k together with the three parameters that specify the orientation at r of the local principal axes ( r r r , , , Since k is symmetric, the specification of equation (7) results in six differential equations in terms of the six known quantities r r r r r r , , ,

Cylindrical cloak
It is instructive to apply the previous results to the cylindrical cloak [4], which expands the origin to a circle of radius a, compressing the disc of radius b into an annulus of inner radius a and outer radius b. In cylindrical polar coordinates the deformation is described via Transforming the Cartesian representation of k (i.e. ij ij k d = ) to cylindrical polar coordinates (see equation (5)) we obtain r r r diag , , . The only refractive index component that varies in space inside the cloak is n θ , which depends only on r. Choosing a=1 and b=2 and restricting to the x-y plane, the index ellipses at different points inside the cloak are illustrated in  figure 5. In the figure, the refractive index ellipses at distinct points along a ray curving around the central region are highlighted. Each light ray is a geodesic that minimises the optical path length. However, it does not follow that the refractive index seen by the ray at a given point is the smallest possible at that point. Instead, each ray sees a single refractive index given by the radius to the ellipse surface in the ray direction. The refractive index experienced along the highlighted ray of figure 5 is shown in figure 6. As expected, we find that the optical path length along the ray is equal to the undeformed (vacuum) path length: The other point to note is that the inclusion of isotropic absorption simply results in an exponential decay in the field intensity that is the same along any given geodesic.

Twist deformation-a new kind of structurally chiral medium
The above formalism can also be used to design a new kind of structurally chiral medium where rays parallel to the z-axis starting at a point in the plane z=0 are twisted so that they form helices about the z-axis for z 0.
> Consider the following transformation, expressed in Cartesian coordinates as Note that all indices throughout the medium are independent of z. However, the principal axes of the ellipsoid vary, both within the x-y plane, and with z. The fact that n n n 1 1 2 3 = indicates that the map of equation (29) is volume preserving. Figure 7 shows the evolution of the eigen-directions ofk at x y , 1,1 ( ) ( ) = for K=0.3. The principal directions associated withk are calculated, and to each principal direction an arrow of length equal to the corresponding refractive index is plotted. The direction associated with the index n 1 1 = is always radial in the x-y plane. The other two directions are associated with an index that is 1 < (respectively 1 > ), which points in the same (contra-) direction to the twist, though out of the x-y plane. Together the three directions are mutually orthogonal. The triad at other points the same distance from the axis can be adduced by rotation. Figure 8 shows the evolution of light propagating along an eigenvector, in this case that associated with the smallest refractive index. The trajectories shown result from the local propagation direction in the plane z=0 lying along the eigen-direction associated with n=0.8. The resulting trajectories are helical and the figure shows the tumbling evolution of the index ellipsoid as it travels along one of these trajectories.
As well as the radial direction in the x-y plane, there is another direction in which n=1. From equation (30) we have that For propagation along the axis of the twist (i.e. along the zaxis) we see that rom which n=1 follows from equation (17). An axial monochromatic plane wave incident from vacuum to the medium occupying the half-space z 0 > has an electric and magnetic field for z 0 < given by We emphasise the distinction between the medium discussed here and previously studied structurally chiral media (SCM). The most commonly studied SCM consists of a medium whose dielectric tensor is expressed as Kz Kz Such a medium has been widely studied both in the context of sculptured thin films [15][16][17][18][19][20][21], and in cholesteric liquid crystals [22]. The progressive rotation of the principal axes of a birefringent medium produces a Bragg grating that for axial propagation reflects one circular polarization while transmitting the other [23]. Plane waves, whether transmitted or reflected, follow linear paths.
However, for the case of the structurally chiral medium considered here, the pre-transformed medium is vacuum and there is no birefringence. Anisotropy is induced as a result of the helical transformation. As shown above, the energy flow associated with axial propagation follows a helical path as shown in figure 8, while the integral lines of the wave-vector are straight lines parallel to the z-axis. The medium described here is also distinct from the 'field rotator' implemented by Chen et al [24], and from an optically active medium.
Although difficult to manufacture, the medium discussed here would have interesting properties. Since the transformation transports the electromagnetic field, an image input to a slab of the proposed medium would emerge rotated, but with the polarization preserved. This is in contrast to standard methods of optically rotating an image (e.g. a dove prism [10]), which do not preserve polarization in general.

Impedance anisotropy
In this section we consider whether the constraint  m = implies that all plane waves propagating in the k medium experience the same impedance as vacuum. We will show by way of a simple example that this is not the case. Consider the trivial morphism that dilates one spatial direction This index behaviour is largely unremarkable; along the direction of dilation ( 0 q = ) it is just λ, and orthogonal to this direction ( 2 q p = ) the vacuum value n=1 is maintained. However, despite the fact that ,  m = the impedance behaviour of the medium is in general polarization dependent. For a field propagating along the z-direction, the impedance is independent of polarization and equal to its vacuum value , but for a field propagating along the x-direction we have according to equation (12) that So if the electric field points along y the impedance is λ −1 times its vacuum value, while if the field points along z, the impedance is l times its vacuum value. It is worth noting that this polarization dependence of the impedance is intrinsic to any transformation optics design, and cannot be fixed by reduced parameter scaling. Since every transformation optics device is built out of similar spatial compressions/expansions that vary in degree at each point, no transformation device can preserve the impedance properties of vacuum.

Conclusion
In this paper we have analysed electromagnetically reciprocal media for which  m k = º that arise canonically as the electromagnetic media that replicate a morphing of flat space. By clearly distinguishing between a morphing of space and a coordinate transformation, we emphasise the intrinsic, coordinate-free nature of transformation optics, and avoid the possible error of associating a transformation medium with a coordinate transformation such as a switch between Cartesian and polar coordinates.
We showed that the electromagnetics of a k-medium can be described in terms of a refractive index function n r s , , (ˆ) which can be represented by an ellipsoidal phase surface. A plane wave propagating in the ŝ direction sees just one refractive index, independent of polarization. If the k-medium is inhomogeneous, then the index at r is interpreted as that seen by a local plane wave under a geometrical optics approximation.
When applied to the original cylindrical cloak, the index along the morphed rays was calculated. A new kind of structurally chiral medium resulted when a k-medium was designed by morphing space with a progressive twist along the z-axis. It was found that the wave vector of an axially incident plane wave to such a medium is undeviated, while the Poynting vector morphs exactly as prescribed by the deformation.
Finally, we showed that whatever the deformation, preserving the impedance properties of vacuum is not generally possible. Propagation transverse to the deformation will inevitably result in a dependence of the impedance on polarization. In other words, perfect cloaking is impossible.
The components of the electromagnetic field tensor are, in Cartesian coordinates  wherek is given by equation (30).