Negative refraction in 2D checkerboards related by mirror anti-symmetry and 3D corner lenses

We investigate the electromagnetic response of a periodic checker- board consisting of alternating rectangular cells of positive refractive index (ε = +1, μ = +1) and negative refractive index (ε = −1, μ = −1). We show that the system has peculiar imaging properties in that it reproduces images of a source in one cell in every other cell. Using coordinate transformations, we map this system into a class of imaging systems in three dimensions consisting of three orthogonal planes delimiting eight alternating cubical regions of positive and negative index media sharing the same vertex. We also generalize these results to more general checkerboards that are inhomogeneous and anisotropic that can then be used to generate a class of three-dimensional (3D) corner imaging systems.


Introduction
In 1967, Veselago [1] speculated that negative values for the dielectric permittivity (ε) and the magnetic permeability (µ) would result in a negative refractive index (NRI). Negative refractive index materials (NRMs) consisting of structured metamaterials have now been proposed [2,3] and demonstrated [4,5] at microwave and radio frequencies creating excitement in the scientific community due to the intriguing possibilities these metamaterials offer (see [6] for a recent review). Perhaps one of the most interesting applications that these metamaterials appear to offer is the possibility of a new class of perfect lenses which do not suffer the diffraction limitation of conventional lenses [7]. A slab of NRM, with n = −1, can image both the far-field propagating modes and the near-field evanescent modes of a source, and thus in principle can act as a perfect lens. The resolution of this system is not limited by wavelength, but only by the extent of dissipation [8]- [10], dispersion [11,12] and imperfections in the constituent materials [6]. In any physical realization, the materials are always dissipative, spatially dispersive at short lengthscales [13], and at very short lengthscales even the assumption of an effective medium breaks down. Thus there is always a cut-off for the smallest (but sub-wavelength) lengthscale that can be imaged by such a lens. In most physical realizations, this is primarily set by the levels of dissipation and next by the intrinsic lengthscale of the structure of the metamaterial. Hence, although a perfect image would not be attainable, substantial sub-wavelength image resolution is possible [6,14,19].
In fact, the perfect slab lens is only one member of a whole category of systems which satisfy a generalized lens theorem [15]. A negatively refracting slab is in some sense complementary to an equal thickness of vacuum, and cancels its presence for both propagating and evanescent nearfield modes. In fact, the materials involved do not even have to be homogeneous and could have an arbitrary variation (within the approximation of a metamaterial) in the directions transverse to the imaging axis. Now consider the more general situation where the dielectric permittivity and the magnetic permeability are arbitrary functions of the transverse spatial co-ordinates (see figure 1): 3 A pair of 2D corners of negative refractive index can focus a source back on to itself. This system can be mapped on to a layered system with a periodic set of sources.
We will consider the imaging axis to be the z axis. Thus we see that the system is anti-symmetric with respect to the z = 0 plane. It turns out that such a system also transfers the image of a source placed at the z = −d to the z = d plane in the same exact sense that it includes both the propagating and evanescent components [15]. Thus to an observer on the right-hand side, it would appear as if the region between z = −d and z = d did not exist. We will refer to such media with the same sense of transverse spatial variation but with opposite signs as optical complementary media, and the effect of any such pairs of complementary media on radiation is null. Using a general method of co-ordinate mapping, one can map Maxwell's equations to other geometries and obtain perfect lenses in other geometries. For instance, using a mapping from a layered structure consisting of layers of complementary media together with a periodic set of line sources, it was observed in [15,16] that two negative 2D corners sharing the same corner combine to make an optical system within which light radiating from a line source is bent around a closed trajectory and is refocused back on to the line source (see figure 2).
In this paper, we first present a proof of the generalized lens theorem based on the symmetries of Maxwell's equations. Then we explore imaging effects through negative refraction in checkerboard structures of positive-negative refraction index materials, and later map them on to 3D corner lenses which focus a source back on to itself.

Proof of the generalized lens theorem
Consider Maxwell's equations in a material medium Let us list the transformations which leave these equations invariant. The transformations fall into the following categories S1 (Generalized conformal invariance): The combination of any of these symmetries is again a symmetry of the system of equations. Then we can assert that 'if the fields in a particular region of space can be mapped on to another region of space through the symmetry transformations S1-S6 while preserving the respective boundary conditions, then the transformed fields solve the field equations whenever the original fields do'. Now consider a homogeneous slab of medium for −d < z < 0 with dielectric permittivity and magnetic permeability tensors For propagation along the z direction and origin at the interface, let us use the symmetry operations S5 and S3(α = −1), followed by S1(A) with We will call this sequence of operations a mirror operation. This choice of A preserves the continuity of E and µ −1 B across the boundary. Then the resulting complementary medium on the right for 0 < z < d is theñ which is the result obtained in [15]. The entries inε andμ could also be spatially varying along x and y.
This checkerboard is invariant along the direction, periodic of period π in the θ and directions.

6
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT First of all, it is clear that a single source generates an image inside every other cell (positive or negative) of the checkerboard (see figure 3). This can be easily shown by considering the imaging along, say, the θ direction and using the generalized lens theorem [15], whereby the condition of complementarity is satisfied for the layers along the imaging θ direction with varying refractive index n in the direction transverse to the imaging direction. Thus for a source at θ = θ 1 , = φ 1 in the first positive cell, we have a set of images along a = φ 1 line at θ = ±nπ ± θ 1 and − θ 1 , where n is a positive integer. Similarly, applying the generalized lens theorem along the direction, we can show now that the entire set of image points would be reproduced along the direction. Thus we have an image point in every cell of the checkerboard structure corresponding to the source placed in any one cell.
Since the ε and µ are periodic with period π in the θ and directions, the system has an additional invariance under a translation by a lattice vector (nπ and nπ, where n is an integer). The corresponding symmetry operations for the imaging in the checkerboard are a translation by a lattice vector followed by a mirror operation S1(A)· S5· S3(α = −1) as in the case of the slab lens. Then it follows that the fields in every pair of cells or pairs of blocks of cells are the transformed versions of the fields in the cell with the source, thus proving the imaging properties of the checkerboard system.
The eigenfunctions of the checkerboard can easily be written down. These are the periodic functions with periodicity of [π, π] in θ ∈ [−π/2, π/2], ∈ [−π/2, π/2] and where [k , κ θ , κ ] are the eigenvalues satisfying k 2 − κ 2 θ − κ 2 = ω 2 /c 2 . k is obtained by box-normalization conditions in the direction. Thus we have two conditions and three variables to be determined. Hence all modes with κ θ and κ leading to the same k are degenerate at a given frequency and the density of modes diverges. Note that propagating modes are included here as well for imaginary κ θ and κ .
The imaging properties of the checkerboard are actually quite counter-intuitive. A plain ray analysis (see figure 3) reveals only the images along the line = φ 1 and θ = θ 1 (imaging across slabs) and two (depending on the position of the source within the cell) of the four possible images in the diagonal neighbouring cells (imaging by the 2D corner lens). But it will not reveal the rest of the images. This is one more instance of the failure of the ray picture in the context of such resonant positive-negative systems as pointed out earlier by Pendry [17]. But the generalized lens theorem is an exact statement and transcends any such ray analysis.
Note here that we make no claims about the temporal order in which the images form when the source is switched on sharply. These are the single-frequency solutions to Maxwell's equations for steady sources. These checkerboard systems are extremely singular and contain a very large number of corners between positive and negative cells where the density of plasmon states actually diverges [15]. This has also been numerically verified [16]. Any source that is sharply switched on would have a frequency spread which would result in exciting resonant surface plasmon modes of the system that would ring continually in the absence of dissipation [18] and a perfect image would form only asymptotically at infinite time.
The presence of dissipation is well known to affect the imaging badly and the singularity in the density of modes at the corners only makes it worse [16]. The ratio of the wavelength to the smallest resolvable lengthscale in the image depends logarithmically on the magnitude of the absorption coefficient [8] and the extent of sub-wavelength resolution obtained for the slab lens in experimental realizations is about λ/2.5 to λ/6 [20]- [22]. However, there is also 7 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT the possibility to compensate for the dissipation in the regions with negative refractive index by using media with optical gain (or active elements) in the regions with positive refractive index [19]. Although it is yet to be implemented experimentally, this combination of amplifying media and negative material parameters can also lead to the stimulated emission of surface plasmons [23,24] (termed as Spason) and is independent of the dimensionality of the system. Indeed it is this possibility that makes the discussion of such singular checkerboard systems meaningful and interesting.

Inhomogeneous and anisotropic checkerboards
Firstly, we note that the variation of ε and µ along the direction is irrelevant to the imaging. We could also have the ε and µ varying with θ and in the checkerboard cells. Provided the positive and negative cells are optically complementary, we would retain the image transfer properties of the homogeneous checkerboard system. As an example, consider a checkerboard with ε i (θ, ) = +ε i sin θ or +ε i / sin θ ∈ positive and ε i (θ, ) = −ε i sin θ or −ε i / sin θ ∈ negative, and similarly for µ i (θ, ). It is clear that such a system satisfies the condition of complementarity and the mirror anti-symmetry, and therefore it should also image in a similar manner. We will call such a system a sine-cosecant checkerboard for our later use. Similarly, we can generate a checkerboard structure with as many complementary phases as we wish, provided that they respect mirror symmetries along the main axes of the cells.

Mapping of homogeneous cubic corners on to sine-cosecant checkerboards
Now we will proceed to prove that eight 3D cubic corners with alternating refractive indices of n = ±1 (see figure 4) and sharing of the common vertex will behave analogously to the 2D corner lens proposed in [15]. Consider the mapping of coordinates: x 1 = r 0 e / 0 sin θ cos , x 2 = r 0 e / 0 sin θ sin , x 3 = r 0 e / 0 cos θ.
In the spherical (orthogonal) frame ( , θ, ), we denote byε i (respectivelyμ i ), i = l, θ, , the three non-zero components of the 'diagonal' tensorsε (respectivelyμ) associated with the 2D spatially varying anisotropic checkerboards. These are related to the piecewise constant function ε (respectively µ) for the 3D system of (x 1 , x 2 , x 3 ) as in [25] with From (12), (14) can be recast as Variation along the transverse r direction is irrelevant as before. We want for the set of eight homogeneous 3D corners This simply amounts to taking ε =μ =ε θ =μ θ = ±e r 0 sin θ, andε =μ = ± e r 0 sin θ , (18) which is the sine-cosecant checkerboard, but with a doubly periodic set of sources with period 2π along and period π along θ as shown in figure 4. Hence, the cubic corner with the homogeneous materials with alternating signs for the refractive index will also form an imaging device with an image point inside every cubic corner. Thus, we have now generalized the result of [15] for a 2D corner to a 3D corner. Analogously, there should be an infinite degeneracy of the associated surface states in this case also. We can also map a multiphase checkerboard that satisfies the mirror anti-symmetry condition to generate a multiphase cubic corner. Figure 5 schematically shows such a mapping from a four-phase checkerboard to a four-phase cubic-corner lens.

9
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT X1 X2 X3 Figure 5. Unit cell of a four-phase checkerboard (left) having the mirror antisymmetry can be mapped on to a four-phase cubic corner. The colours green and yellow are taken to represent the regions with the complementary negative parameters of those regions represented by grey and red respectively. The 3D corner generated is composed of the regions formed by the intersections of the cones of θ = ±π/4 and the planes θ = π/2 and = π/2, π. The regions extend out to infinity.
The electric and magnetic fields within this 3D corner reflector can be expressed as: where Q i is given by (15).

Mapping on to spatially varying anisotropic cubic corners
In the previous section, we mapped a cubic system into a checkerboard system which satisfied the conditions of optical complementarity. But, in general we can generate a large variety of inhomogeneous, anisotropic cubic corners that behave as imaging systems, from some corresponding checkerboard system containing a doubly periodic set of point sources. Mapping our system of generalized coordinates [ , θ, ] on to Cartesian coordinates [x 1 , x 2 , x 3 ] as follows: = arctan where r 0 is a scale factor and can be taken to be the radial position of the source. Here, denotes the radial (logarithmic) coordinate and θ and denote the longitudinal and azimuthal coordinates; we can now generate the corresponding cubic corner systems in [x 1 , x 2 , x 3 ].
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

Homogeneous isotropic 3D checkercubes
It can be easily shown by using the generalized lens theorem that a periodic homogeneous isotropic checkercube medium with adjacent cubical regions having n = +1 and n = −1 respectively will also make a similar imaging system as it satisfies the prerequisite optical complementarity conditions along all three imaging directions. A point source located within a cell will be imaged into every other cell. In this system of checkercubes of complementary media, all surface plasma modes are degenerate at frequency ω p / √ 2. At this frequency, the density of states must become infinite.

Conclusion
In conclusion, we have demonstrated that by mapping complementary checkerboard media, it is possible to design imaging configurations that are 3D. We have generalized the result of a 2D corner lens [15] to 3D corner structures which can either consist of homogeneous isotropic regions of space or be anisotropic and spatially varying. The only condition is that the eight cubical corners must satisfy the condition of mirror anti-symmetry across the interface between the positive and negative regions. These are very interesting singular configurations for which all the surface states are infinitely degenerate. In fact, this can also be taken as a technique to develop degenerate plasmonic 3D surfaces. We believe that such metamaterials may be usefully engineered for microwave frequencies if active gain elements are incorporated into the regions of positive material parameters.