Asymptotic behaviour of the spectra of systems of Maxwell equations in periodic composite media with high contrast

We analyse the behaviour of the spectrum of the system of Maxwell equations of electromagnetism, with rapidly oscillating periodic coefficients, subject to periodic boundary conditions on a"macroscopic"domain $(0,T)^d, T>0.$ We consider the case when the contrast between the values of the coefficients in different parts of their periodicity cell increases as the period of oscillations $\eta$ goes to zero. We show that the limit of the spectrum as $\eta\to0$ contains the spectrum of a"homogenised"system of equations that is solved by the limits of sequences of eigenfunctions of the original problem. We investigate the behaviour of this system and demonstrate phenomena not present in the scalar theory for polarised waves.


Introduction
The behaviour of systems (of Maxwell equations) with periodic coefficients in the regime of "high contrast", or "large coupling" i.e. when the ratio between material properties of some of the constituents within the composite is large, is understood to be of special interest in applications. This is due to the improved band-gap properties of the spectra for such materials compared to the usual moderate-contrast composites. A series of recent studies have analysed asymptotic limits of scalar high-contrast problems, either in the strong L 2 -sense (see [10], [11]) or in the norm-resolvent L 2 -sense, see [2]. These have resulted in sharp operator convergence estimates in the homogenisation of such problems (i.e. in the limit as the period tends to zero) and have provided a link between the study of effective properties of periodic media and the behaviour of waves in such media, in particular their scattering characteristics. This suggests a potential for applications of the abstract operator theory to the study of such problems. The studies have also highlighted the need to extend the classical compactness techniques in homogenisation to cases when the symbol of the operator involved is no longer uniformly positive definite, thus leading to "degenerate" problems. The work [5] has opened a way to one such extension procedure, based on a "generalised Weyl decomposition", from the perspective of the strong L 2 -convergence.
The set of tools developed in the literature is now poised for the treatment of vector problems with degeneracies such as the linearised elasticity equations and the Maxwell equations; these examples are typically invoked in the physics and applications literature, and are prototypes for wider varieties of partial differential equations (PDE). The recent work [9] has studied the spectral behaviour of periodic operators with rapidly oscillating coefficients in the context of linearised elasticity. It shows that the related spectrum exhibits the phenomenon of "partial" wave propagation, depending on the number of eigenmodes available at each give frequency. This is close in spirit to the work of [6], where "partial wave propagation" was studied for a wider class of vector problems, with a general high-contrast anisotropy.
The high-contrast system of Maxwell equations poses an analytic challenge in view of the special structure of the "space of microscopic oscillations" (using the terminology of [5]), which consists of the functions that are curl-free on the "stiff" component, in the case of a two-component composite of a "stiff" matrix and "soft" inclusions. In the work [1] we analysed the two-scale structure of solutions to the high-contrast system of Maxwell equations in the low-frequency limit, and derived the corresponding system of homogenised equations, by developing an appropriate compactness argument on the basis of the general theory of [5]. In the present paper we consider the associated wave propagation problem for monochromatic waves of a given frequency by constructing two-scale asymptotic series for eigenfunctions. We justify these asymptotic series by demonstrating that for each element of the spectrum of the homogenised equations their exist convergent eigenvalues and eigenfunctions for the original heterogeneous problem. Our analysis is set in the context of a "supercell" spectral problem, i.e. the problem of vibrations of a square-shaped domain with periodicity conditions on the boundary (equivalently seen as a torus). The problem of the "spectral completeness" of the homogenised description in question remains open: it is not known, for the full-space problem, whether there may exist sequences of eigenvalues converging to a point outside the spectrum of the homogenised problem. We shall address this in a future publication, using the method we developed in [2].

Problem formulation and main results
In this paper we consider Maxwell equations for a three-dimensional two-component periodic dielectric composite when the dielectric properties of the constituent materials exhibit a high degree of contrast between each other. We assume that the reference cell Q := [0, 1) 3 contains an inclusion Q 0 , which is an open set with sufficiently smooth boundary. We also assume that the "matrix" Q 1 := Q\Q 0 is simply connected Lipschitz set.
We also assume moderate contrast in the magnetic permeability, and for simplicity of exposition we shall set µ ≡ 1. We consider the open cube T := (0, T ) 3 and those values of the parameter η for which T /η ∈ N. By re-scaling the spatial variable (which can also be viewed as non-dimensionalisation) we assume that T = 1 and that η −1 ∈ N. We shall study the behaviour of the magnetic component H η of the electromagnetic wave of frequency ω propagating through the domain T occupied by a dielectric material with permittivity ǫ η (x/η). More precisely, we consider pairs ω η , Notice that solutions of (2.1) are automatically solenoidal, i.e. divH η = 0. We seek solutions to the above problem in the form of an asymptotic expansion where the vector functions H j (x, y), j = 0, 1, 2, ..., are Q-periodic in the variable y. Substituting (2.2) into (2.1) and gathering the coefficients for each power of the parameter η results in a system of recurrence relations for H j , j = 0, 1, 2, ..., see Section 4. In particular, the function H 0 is an eigenfunction of a limit ("homogenised") system of PDE, as described in the following theorem.
Theorem 2.1. Consider the constant matrix where the vector-function N is a solution to the "unit-cell problem" where n is the exterior normal to ∂Q 0 . Suppose that ω ∈ R + and H 0 (x, y) = u(x) + ∇ y v(x, y) + z(x, y), where the triplet 1) There exists at least one eigenfrequency ω η for (2.1) such that |ω η −ω| < Cη, with an η-independent constant C > 0.
The matrix A hom is described by solutions to certain degenerate "cell problems", as follows. Consider the spaces and Then Existence and uniqueness of N ξ is discussed in Section 4. Notice that Indeed, for the functional where P V ⊥ is the orthogonal projection onto V ⊥ . Therefore, without loss of generality, F ξ can be minimised on V ⊥ for which (2.9) is the corresponding Euler-Lagrange equation.
The variational formulation (2.10) allows one to obtain a representation for the matrix ǫ hom stiff such that ǫ hom stiff ξ · ξ := inf Notice that for each vector v in (2.12) there exists [4, pp. 6-7], and hence It follows that for all ξ ∈ R 3 one has 3 On the spectrum of the limit problem In this section we study the set of values ω 2 such that there exists a non-trivial triple (u, v, z) solving the two-scale limit spectral problem (2.4)-(2.6).

Equivalent formulation and spectral decomposition of the limit problem
Let G be the Green function for the scalar periodic Laplacian, i.e. for all y ∈ Q one has −∆G(y) = δ 0 (y) − 1, where δ 0 is the Dirac delta-function supported at zero, on Q considered as a torus. Then, as the functions v, z solve (2.5), we have v(x, ·) = G * (div y z)(x, ·), and (2.6) takes the form (3.1) For the case ω = 0 the set of solutions z to (3.1) subject to the condition z(x, y) = 0, x ∈ T, y ∈ ∂Q 0 , is clearly given by The Euler-Lagrange equation for (2.11) is as follows: find u such that ∇u = −ξ in Q0 and The equivalent "strong" form of the same problem is to find a Q-periodic function u such that div (ǫ1 (∇u + ξ)) = 0 in Q1, Further, for ω = 0, as (3.1) is linear in u(x) and curl y ∇ y = 0, we set where B is a 3 × 3 matrix function whose column vectors where e j , j = 1, 2, 3, are the Euclidean basis vectors and a(B j ) is the "circulation" of B j , that is defined as the continuous extension, in the sense of the H 1 norm, of the map given by a( 3 , subject to the constraints (3.4)-(3.6), such that the following identity holds: Substituting the representation (3.2) into (2.4) and using the fact that leads to the operator-pencil spectral problem where Γ is a matrix-valued function that vanishes at ω = 0, and for ω = 0 has elements We denote by H 1 the space of vector fields in [H 1 # (Q)] 3 that satisfy the conditions (3.4)-(3.6). It can be shown 3 that there exist countably many pairs (α k , r k ) ∈ R × H 1 such that r k [L 2 (Q)] 3 = 1 and 3 . Therefore, the equation curl ǫ −1 0 curl u = λu, u ∈ H1, can be written as λ −1 u = Ku in the sense of the "energy" inner product generated by the norm ||| · ||| and K is a compact self-adjoint operator in (H1, ||| · |||). The claim then follows by a standard Hilbert-Schmidt argument.
Moreover, the sequence (r k ) k∈N can be chosen to form an orthonormal basis of the closure H 1 of H 1 in [L 2 (Q)] 3 and, upon a suitable rearrangement, one has Performing a decomposition 4 of the functions B j , j = 1, 2, 3, with respect to the above basis yields where r k j , j = 1, 2, 3, are the components of the vector r k , k ∈ N.
and satisfy the orthonormality conditions where ∇ 2 G is the Hessian matrix of G. Using the formula we obtain the following representation for Γ :

Analysis of the limit spectrum
Now, we consider the Fourier expansion for the function u in (3.8): where the integral is taken component-wise. As u solves (3.8), the coefficientsû(m) satisfy the equation with the matrix-valued function M is given by where e j , j = 1, 2, 3 are the Euclidean basis vectors. Here ε is the Levi-Civita symbol: We writeû(m) in terms of the basis ẽ 1 (m),ẽ 2 (m),m , as follows: Finding a non-trivial solution to the problem (3.12), We have thus proved the following statement.
Proposition 3.1. The spectrum of the problem (2.4)-(2.6) is the union of the following sets.
1. The elements of {α k : k ∈ Z} such that at least one of the corresponding r k has zero mean over Q.
These are eigenvalues of infinite multiplicity and the corresponding eigenfunctions H 0 (x, y) are of the form w(x)r k (y) for an arbitrary w ∈ L 2 (T).
An immediate consequence of the above analysis is the following result.
Proof. Since M admits the spectral decomposition a necessary condition for pairs (m, ω) such that (3.12) has a solution is as follows: This is not possible since Λ ′ (m) is positive-semidefinite and, by assumption, the matrix Γ(ω) and, consequently, the matrix C ′ (m)Γ(ω)C ′ (m) are negative-definite.

Examples of different admissible wave propagation regimes for the effective spectral problem
In this section we explore the effective wave propagation properties of high-contrast electromagnetic media. We demonstrate that the sign-indefinite nature of the matrix-valued function Γ gives rise to phenomena not present in the case of polarised waves. Suppose that the inclusion is symmetric under a rotation by π around at least two of the three coordinate axes, then the matrices A hom and Γ(ω) are diagonal (see Appendix): A hom = diag(a 1 , a 2 , a 3 ), Γ(ω) = diag β 1 (ω), β 2 (ω), β 3 (ω) . Here a i are positive constants and β i are real-valued scalar functions. Notice that, since |m| = 1, the eigenvalues λ 1,2 (m) of M(m) are the solutions to the quadratic equation We will now solve the eigenvalue problem (3.12), equivalently (3.14), for particular examples of such inclusions.
Remark 3.1. Recently, there has been several works on the analysis of problems with "partial" or "directional" wave propagation in the context of elasticity, where at some frequencies, propagation occurs for some but not for all values of the wave vector: the analysis of the vector problems for thin structures of critical thickness [8], the analysis of high-contrast [9], and partially high-contrast [6] periodic elastic composites. To our knowledge, the effect we describe here is the first example of a similar kind for Maxwell equations.
Remark 3.2. When the "size" T of the domain T increases to infinity, the spectrum of (2.4)-(2.6) converges to a union of intervals ("bands") separated by intervals of those values ω 2 for which the matrix Γ(ω) is negative-definite ("gaps", or "lacunae"). As above, we say that ω 2 belongs to a weak band gap (in the spectrum of (2.4)-(2.6)) if at least one eigenvalue of Γ(ω) is positive-semidefinite and at least one eigenvalue of Γ(ω) is negative.
Taking into account, via Lemma 4.1, that the leading-order term H 0 is of the form and substituting (4.9) into the equations (4.3)-(4.4), we find that the coefficient H 1 has the representation H 1 (x, y) = N (y) curl u(x) + H 1 (x, y), up to the addition of an element of V . Here the term H 1 (x, y) satisfies 11) and N = N (y) is a Q-periodic matrix-valued function whose columns N r = N r (y), r = 1, 2, 3, are solutions to the problems curl ǫ −1 1 (y) curl N r (y) + e r = 0, y ∈ Q 1 , ǫ −1 1 (y) curl N r (y) + e r × n(y) = 0, y ∈ ∂Q 0 , (4.12) where e r is the rth Euclidean basis vector. It is shown ( [3], [5]) that (4.12) admits a unique solution in V ⊥ , the orthogonal complement to V in the space [H 1 # (Q)] 3 . Looking for H 1 (x, ·) ∈ [H 1 # (Q)] 3 and taking into account the identity curl x ∇ y = −curl y ∇ x together with (4.10)-(4.11), we infer that for all x ∈ T the function h(x, ·) In particular, the function h belongs to the space V . Therefore, one has up to the addition of an element of V . (As we discuss in Remark 5.1 below, one can specify the divergence div y H 1 (x, y). This, along with the condition that the y-average of H 1 vanishes, defines this additional element of V in a unique way.) Further, multiplying the equation (4.5) by an arbitrary test function φ ∈ V and integrating over Q 1 yields (4.14) We integrate by parts in the left-hand side of (4.14) to determine that Now we perform integration by parts in the individual terms in the right-hand side of (4.14).
Taking into account the representations (4.9) and (4.13), we find that where we again make use of the identity curl x ∇ y = −curl y ∇ x . Finally, equations (4.14)-(4.17) imply In what follows we derive the system (2.4)-(2.6) by considering different choices of the test function φ in the identity (4.18).
Step 3. Choosing φ(y) ≡ 1 in the identity(4.18) we find, using the representation (4.9) once more, that (2.4) holds, where the matrix A hom emerges as the result of integrating the expression curl y N (y) + I with respect to y ∈ Q 1 .
In the next section we use the above formal construction of the series (2.2) to justify the two claims of Theorem 2.1.

Proof of Theorem 2.1
For each η > 0, denote by A η the operator in the space 5 L 2 #sol (T) defined in a standard way by the bilinear form (cf. (2.1)) For fixed ω in the spectrum of (2.4)-(2.5), let H 0 be a corresponding eigenfunction. Consider the (unique) solution H η ∈ H to the problem

and (cf. (2.2))
where H j , j = 1, 2, are solution of the system of recurrence relations described in Section 4. The existence of solutions H 1 , H 2 is guaranteed by a result we established in [1,Lemma 3.4]. As these solutions are unique up to the addition of an element from V , we shall choose them as in Remark 5.1.
Proposition 5.1. There exists a constant C > 0 such that the estimate Proof. Using the definition of the function H η and the recurrence relations (4.1)-(4.8) yields Here, F 1 , F 2 are elements of L 2 (T) defined for a.e. x ∈ T by Notice that the functions H 0 = H 0 (x, y), . Indeed, this is seen to be true for H 0 by Proposition 3.1; in the case of ω = α k we choose w ∈ C ∞ # (T). The assertions for H 1 and H 2 now follow from formula (4.13) for the corrector H 1 (x, y), and the boundaryvalue problem (4.5)-(4.6) for the function H 2 (x, y). It then follows from (5.5) that ||F 1 (·, η)|| L 2 (T) ≤ Cη, ||F 2 (·, η)|| L 2 (T) ≤ Cη, and by applying the Hölder inequality to (5.4) we deduce that , as required.
The above proposition implies the following statement.
Remark 5.1. Note that H (2) is not solenoidal in general, but can be defined in such a way that it is "close" to a solenoidal field, thanks to the equation (4.19) (equivalently, (2.5)) and the special choice of the function H 1 so that div x H 0 (x, y) + div y H 1 (x, y) = 0 a.e. (x, y) ∈ T × Q.
The function H (2) thus defined is η-close to the eigenspace X η in the norm of [H 1 # (Q)] 3 .

Appendix: Symmetry of A hom and Γ(ω) under rotations
where N q is the unique solution to the problem (cf. [5], [1,Lemma 3.4] and (4.12) above for a = χ 1 , the characteristic function of Q 1 ) curl A[curl N q + e q ] = 0, Here the superscript "⊥" denotes the orthogonal complement in [H 1 # (Q)] 3 . Notice that if, for fixed ζ ∈ R 3 , we multiply each of the above equations by ζ q then where the vector N ζ , whose components are N q p ζ q , is the unique solution to the problem It is clear that the matrix representation of the bounded linear mapping ζ → Q A curlN ζ + ζ is equal to A hom . The following property holds.
Corollary 5.1. If (5.9) holds for σ = σ k , where σ k is the rotation by π around the x k -axis, then A hom kl = 0, for all l = k.
Proof. Indeed, say for k = 1 (5.10) takes the form  Similarly, direct calculation proves the following statement.
By analogy with Corollary 5.1, Corollary 5.2 we obtain the following statement.

Achknowledgements
KDC and SC are grateful for the financial support of the Leverhulme Trust (Grant RPG-167 "Dissipative and non-self-adjoint problems") and the Engineering and Physical Sciences Research Council (Grant EP/L018802/2 "Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory", and Grant EP/M017281/1 "Operator asymptotics, a new approach to length-scale interactions in metamaterials.") We would also like to thank Valery Smyshlyaev for helpful discussions on the subject of this manuscript.