Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors

We show that any pair of real symmetric tensors and can be realized as the effective electric permittivity and effective magnetic permeability of a metamaterial at a given fixed frequency. The construction starts with two extremely low-loss metamaterials, with arbitrarily small microstructure, whose existence is ensured by the work of Bouchitté and Bourel and Bouchitté and Schweizer: one having, at the given frequency, a permittivity tensor with exactly one negative eigenvalue, and a positive permeability tensor; and the other having a positive permittivity tensor, and a permeability tensor having exactly one negative eigenvalue. To achieve the desired effective properties, these materials are laminated together in a hierarchical multiple rank laminate structure, with widely separated length scales, and varying directions of lamination, but with the largest length scale still much shorter than the wavelengths and attenuation lengths in the macroscopic effective medium.

3 the possible real tensor pairs (ε, µ) that can exist at one given frequency. The construction uses hierarchical microstructures with structure on many widely separated length scales, and component metamaterials with arbitrarily low loss. It seems likely that the same effective properties can be achieved with more realistic designs, but it is doubtful that their properties could be calculated analytically.

Starting materials and Tartar's formulae for the effective tensors of laminates
In this paper, we assume an idealized world where the continuum and Maxwell's equations extend down to arbitrarily small length scales. We also assume the existence of two metamaterials. Material A has arbitrarily small microstructure and an effective permittivity tensor with a bounded real part (ε A ) that is diagonal, with exactly one negative and two positive diagonal elements, and an effective permeability tensor with a real part (µ A ) that is not necessarily diagonal but bounded and strictly positive definite, i.e. there exist constants β A > α A > 0 such that The imaginary parts (ε A ) and (µ A ) of these two tensors are assumed to be vanishingly small, but nonzero, i.e. there exist constants η > ν > 0 such that where the loss parameters η and ν are vanishingly small. (Strictly speaking, we should talk about a sequence of materials A, parameterized by η, and consider what happens in the limit as η → 0, when (ε A ) and (µ A ) do not depend on η, while ν and the scale and material constants of the microstructure within the metamaterial do depend on η.) Material B has arbitrarily small microstructure and an effective permittivity tensor with a real part (ε B ) that is strictly positive definite, i.e. satisfying the bounds for some choice of β B > α B > 0, and with an effective permeability with a bounded real part (µ B ) that is diagonal, with exactly one negative and two positive diagonal elements. The imaginary parts (ε B ) and (µ B ) of these two tensors are assumed to be vanishingly small, but nonzero, i.e. satisfying the bounds where, without loss of generality, η and ν are the same parameters that appear in (2). (Again, strictly speaking, we should consider a sequence of materials B, parameterized by η, and consider what happens in the limit as η → 0.) Material A could be a metamaterial comprised of a cubic lattice of well-separated cubes, where each cube has a microstructure of highly conducting rods aligned in the x 1 -direction. A rigorous mathematical proof that such a material has the desired effective properties, with a more precise description of the needed microgeometry, has been given by Bouchitté and Bourel [49]. Based on the results of Pendry et al [50] and Bouchitté and Felbacq [51], one might think that a periodic array of highly conducting thin rods aligned in the x 1 -direction would suffice for material A. However, Bouchitté and Felbacq [52] show that in a finite sample of such a material, the associated effective equations can have non-local behavior.
Material B could be a metamaterial comprised of a periodic lattice of highly conducting split-ring resonators with axes aligned in the x 1 -direction, with one split-ring per unit cell. The split rings behave like polarizable magnetic dipoles, and if one is just above resonance these can have negative permeability in the x 1 -direction. This method of creating artificial magnetism has been discussed by Schelkunoff and Friis [53], and although their formula showed negative permeability in the x 1 -direction, they did not draw attention to this fact. The microstructure was rediscovered by Pendry et al [54], who realized the significance of negative permeability. A rigorous mathematical proof that such a material has the desired effective properties has been given by Bouchitté and Schweizer [55], following earlier work [56]- [60] (see also the Introduction of [61]). Bouchitté and Schweizer also prove that one can obtain isotropic effective magnetic permeability tensors with negative permeability.
We explore the effective permittivity and permeability tensors that can be generated from composites of these two metamaterials A and B and all rotations of them within classical homogenization in the limit as η tends to zero. The first step will be to recover the result of Bouchitté and Bourel [49] that a material with any real symmetric permittivity tensor ε * can be approximately achieved. Then the same argument applies to show that a material with any real symmetric permeability tensor µ * can be approximately achieved. Finally, by combining these two results we show that any pair of real symmetric tensors (ε * , µ * ) can be approximately achieved.
The composites we consider are multiple rank laminates. This class of composites was first introduced by Maxwell [62] and has been extensively studied in the homogenization literature (see, for example, [63] and references therein). They are obtained by laminating the starting (rank 0) materials together on an extremely small length scale to obtain rank 1 laminates and then laminating these rank 1 laminates with other rank 1 or rank 0 laminates on a much larger, but still extremely small, length scale, using a possibly different direction of lamination, to obtain rank 2 laminates and so forth. The advantage of considering this class of composites is that their effective tensors can be explicitly calculated, in the limit in which the ratio of successive length scales approaches infinity. The bounds (2) and (4) ensure that one can use reiterated homogenization theory: to calculate the effective tensor in the limit of widely separated scales of a multiple rank laminate, which is a simple laminate of two (possibly multiple) rank laminates C and D, one can replace C and D by homogeneous materials with material tensors the same as the effective tensors of C and D.
The use of classical homogenization, except inside metamaterials A and B, should be valid, provided, when we replace materials A and B by homogeneous materials with material tensors the same as the effective tensors of A and B, the wavelengths and attenuation lengths within each substructure are much larger than the microstructure at that level. For any fixed η > 0, this should be ensured by (2) and (4), in the limit in which the overall microstructure tends to zero and the ratio between scales tends to infinity.
If a set of materials are laminated in direction x 1 , then the formulae obtained by Tartar [64] (using an idea that goes back to Backus [65]) for the effective permittivity and permeability arẽ where ε(x 1 ) and µ(x 1 ) are the local permittivity and permeability tensors (that could themselves be effective tensors) that only depend on x 1 . Angular brackets denote volume averages and, for any symmetric matrix C with elements c i j , the matrixC is symmetric and has elements for all k = 1, = 1. Conversely if a symmetric matrixC is given, then C is symmetric with elements for all k = 1, = 1.
In the case where ε is diagonal, (5) and (6) reduce to the familiar harmonic and arithmetic averages with similar formulae applying when µ is diagonal.
In the limit as the imaginary part of ε approaches zero, (5) implies that the imaginary part ofε * will also tend to zero unless one is at resonance where the real part of 1/ε 11 or the real part of 1/µ 11 vanishes. We will always prevent this from happening. Then the limiting tensors ε * and µ * can be obtained by replacing ε and µ in (5) with their real parts, i.e. the vanishingly small imaginary parts do not affect the effective tensors except at resonance. From now on we will drop the primes and treat ε and µ as real, while recalling that they do have vanishingly small imaginary parts, to ensure that we can apply the rules of classical reiterated homogenization.

Realizing a material with a desired tensor ε * using lamination
Now let us explore which effective permittivity tensors ε * can be obtained by hierarchically laminating material A with itself (and its rotations by 90 • about the axes) in directions parallel to the axes, calculating the effective tensors at each stage using harmonic and arithmetic averages (8) Given prescribed values of e > 0 and g = 0, we may choose nonzero so that (9) is satisfied provided we choose δ 0 with 0 < δ 0 < |ge|/(bc) (11) to ensure that the roots of (10) are real. Since (11)  with e > 0, provided δ 0 is chosen sufficiently small to satisfy both (11) and the constraint δ 0 < a|h| (which ensures that h lies between a/δ 0 and −a/δ 0 ). So any diagonal tensor with nonzero diagonal elements, at least one of which is positive, is realizable. The only case left to treat is when h, e and g are all negative. Let t be bigger than both −e and −g. Then the tensors Diag[h, e + t, g − t] and Diag[h, e − t, g + t] are both realizable, and by laminating them together in direction x 1 , we see that the tensor Diag[h, e, g] is also realizable as an effective permittivity tensor for any nonzero finite choice of h, e and g.

Realizing any desired pair of tensors (ε * , µ * )
By rotating the material just obtained, we see that any symmetric matrix ε A * , except possibly those that are singular, is realizable as an effective permittivity tensor. The associated effective permeability tensor µ A * (in the limit as the loss goes to zero) must satisfy the classical homogenization Weiner bounds [66] µ where µ(x) is the local permeability tensor, which is locally a rotation of µ A . Since µ A satisfies (1), we conclude that µ A * also satisfies the inequality Let us call U the family of materials thus obtained. Similarly (because we can interchange the roles of ε and µ), using material B and its rotations, any symmetric matrix µ B * , except possibly those that are singular, is realizable as an effective permeability tensor, and the associated effective permittivity ε B * satisfies the bounds We let V denote this family of materials. Now we are free to take any set of materials in the set W = U ∪ V and laminate them together in direction x 1 to form a larger set of materials L W containing W . By an abuse of notation, we let W (and U and V ) also denote the set of tensor pairs (ε * ,μ * ) derived from effective tensors (ε * , µ * ) of materials in W (and U and V ), and we let L W also denote the set of tensor pairs (ε * ,μ * ) derived from effective tensors (ε * , µ * ) of materials in L W . A schematic illustration of the sets U and V is given in figure 1. Now (5) implies that the tensor pairs in L W (being arithmetic averages) correspond to all convex combinations of the tensor pairs in W , except for possibly those tensor pairs for which eitherε * 11 = 0 orμ * 11 = 0, where one is at resonance. So aside from a set of measure zero, the set of tensor pairs L W is a convex set 7 V * µ ε * U Figure 1. Sketch of the sets U and V in the (ε * ,μ * ) plane. The figure is schematic in that each axis is not really one-dimensional, but rather sixdimensional representing symmetric 3 × 3 matrices. The projection of U onto theε * 'axis' consists of all symmetric 3 × 3 matrices, whereas its projection onto theμ * 'axis' is bounded. Similarly, the projection of V onto theμ * 'axis' consists of all symmetric 3 × 3 matrices, whereas its projection onto theε * 'axis' is bounded. whose convex hull, like the convex hull of any set of points (see figure 2), is characterized by the Legendre transform, as a function of symmetric matrices P and Q. If we can show that f (P, Q) is minus infinity for every choice of P and Q, both not zero, then we can conclude that the pair of real symmetric matrices (ε * ,μ * ) is approximately realizable.
First observe that Since almost every matrix ε A * is realizable as a permittivity tensor, it follows that almost every matrixε A * is realizable: givenε A * , we realize, or almost realize, the tensor ε A * whose elements are given according to (7). In particular, we can realize a tensorε A * ≈ −λP for any value of λ, no matter how large. The associated permeability tensor µ A * satisfies (13), which implies where the latter condition follows from the positivity of the determinant of the associated 2 × 2 submatrix of µ A * . It follows that the elements ofμ A * , given according to (6), satisfy the bounds irrespective of the choice of λ. Consequently, for any fixed choice of P = 0 and Q, approaches −∞ as λ → ∞, while remains bounded. It follows that f A (P, Q) is minus infinity unless P = 0. Similarly, f B (P, Q) is minus infinity unless Q = 0, and thus f (P, Q) is minus infinity unless both P and Q are zero. We conclude that any given pair of real symmetric matrices (ε * , µ * ) is realizable to an arbitrarily high degree of approximation as the (effective permittivity, effective permeability) of a metamaterial.