K D Cherednichenko
email: cherednichenkokd@cardiff.ac.uk

S Guenneau
email: guen-neau@liverpool.ac.uk

Bloch wave homogenisation for spectral asymptotic analysis of the periodic Maxwell operator

Keywords: Electromagnetic waves, periodic structures, two-scale convergence, spectrum

This paper is devoted to the asymptotic behaviour of the spectrum of the three-dimensional Maxwell operator in a bounded periodic heterogeneous dielectric medium T = [-T, T ] 3 , T > 0, as the structure period η, such that η -1 T is a positive integer, tends to 0. The domain T is extended periodically to the whole of R 3 , so that the original operator is understood as acting in a space of T-periodic functions. We use the so-called Bloch wave homogenisation technique which, unlike the classical homogenisation method, is capable of characterising a renormalised limit of the spectrum (called the Bloch spectrum). The related procedure is concerned with sequences of eigenvalues Λη of the order of the square of the medium period, which correspond to the oscillations of high-frequencies of order η -1 . The Bloch-wave description is obtained via the notion of two-scale convergence for bounded self-adjoint operators, and a proof of the "completeness" of the limiting spectrum is provided.

Introduction

The problem of determining the effective properties of a composite periodic medium in the context of electrostatics is an old one, and has been studied extensively, starting with the paper [START_REF] Lorenz | Experimentale og theoretiske undersøgelser over legemernes brydningsforhold, I[END_REF] by Lorenz. However, when the electric fields exhibit time-dependence one moves into the area of electromagnetism, which is a less-developed subject, at least in the context of the effective response of periodic composite media. In this respect it may be helpful to recall that although practitioners often make use of the traditional static formulae in the dynamic regime, this technique, generally known as the "quasi-static limit", tacitly relies on a set of assumptions about the fields in the material; namely, that their wavelengths are long in comparison with the typical size of heterogeneity. In this respect it seems to be of interest to investigate the material response when the wavelengths and the average period of variations in the conducting properties become comparable, thus inducing significant scale interactions. Whether the material then supports the imposed oscillations, such as an externally applied electric signal, becomes a non-trivial question, requiring a new averaging procedure of some sort.

There are several recent examples of similar kind, where the limiting procedure has exhibited a strong dependence on the precise way in which the underlying length-scales become small, see e.g. [START_REF] Poulton | Noncommuting limits in electromagnetic scattering: asymptotic analysis for an array of highly conducting inclusions[END_REF], [START_REF] Zhikov | On an extension of the method of two-scale convergence and its applications[END_REF]. Most of them are due to some sort of contrast in the properties at different points of the composite, which makes it possible to have comparatively short waves in selected parts of the medium. It seems equally plausible to imagine, however, that the external frequency is high enough already to warrant the existence of scale interaction effects in the medium. Several works investigating this possibility have been written by Allaire and Conca, in the context of fluid-solid structures [START_REF] Allaire | Bloch wave homogenization for a spectral problem in fluid-solid structures[END_REF], [START_REF] Allaire | Boundary layers in the homogenization of a spectral problem in fluid-solid structures[END_REF], as well as for a scalar elliptic equation with Dirichlet boundary conditions [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]. The present paper deals with an analogous problem in the context of electromagnetic waves. In order to avoid technical complications unrelated to the aim of the present study, we restrict ourselves to the case of periodic boundary conditions, thus excluding a possibility of the emergence of a "boundary layer spectrum", see e.g. [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]. Throughout the paper we use bold type to denote various spaces of vector functions, such as L 2 , or C ∞ per ; the subscript "per" will refer to the respective spaces of periodic functions. Given an operator A, we use the notation σ(A) for its spectrum.

Consider an isotropic dielectric medium occupying the domain T = [-T, T ] 3 ⊂ R 3 of fixed period 2T > 0, which is extended periodically to R 3 and composed of (η -1 2T ) 3 periodicity cells with a small period η ∈ Ξ, where the set Ξ is defined by Ξ := {η > 0 : η -1 T ∈ N}. Such a composite medium can be a model of what is known as the photonic crystal [START_REF] Yablonovitch | Photonic crystals: semiconductors of light[END_REF]. The analysis of electromagnetic modes in such a medium amounts to looking for pairs 1 (Λ η , H η ) ∈ IR + × H per (curl, T), H η ≡ 0, such that

curl ε -1 (x, x/η)curlH η = Λ -1 η H η , (1.1)
where ε(x, x/η) is the (matrix) relative permittivity of the medium at the point x. The set of all first elements Λ η of such pairs is referred to as the spectrum of (1.1), and we denote it by σ η . We assume that ε(x, y) ∈ C per T, [L ∞ (Y )] 9 , where Y := [0, 1] 3 , and that ε(x, y) ≥ I for a.e. (x, y) ∈ T × Y, which, in the case when the medium is isotropic, means that ε(x, y) is bounded below at almost every point by the permittivity of vacuum. It follows from the above two conditions that the operator in the left-hand side of (1.1) is uniformly elliptic (or "coercive"). The equation (1.1) is understood in the weak sense, i.e.

T ε -1 (x, x/η)curlH η (x) • curlϕ(x)dx = Λ -1 η T H η (x) • ϕ(x)dx
for any ϕ ∈ C ∞ per (T). Note in particular that for any solution of (1.1) one has div H η = 0. In order to study the spectrum of (1.1) for each η ∈ Ξ, consider the Green operator G η in L 2 (T) defined for every f ∈ L 2 (T) by G η f = u η , where u η is the unique solution (by the Lax-Milgram lemma, see e.g. [START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF]) in H per (curl, T) of the problem

curl ε -1 (x, x/η)curlu η + u η = f . (1.2)
Clearly, G η is self-adjoint and it can also be shown that it is compact, we can thus write σ(G η ) = {0, 1} { Λk η } k≥1 , where for each η the sequence Λk η converges to 0 as k → ∞. In the above union we include every eigenvalue as many times as is its multiplicity, so that to each Λk η is associated a normalised eigenfunction 2 u k η ∈ L 2 sol (T) such that u k η L 2 (T) = 1 and the family {u k η } k≥1 is an orthonormal basis of L 2 sol (T). The eigenvalues Λk η are related to the eigenvalues of (1.1) by the formula Λk η = (Λ k η + 1) -1 Λ k η . The aim of this paper is to study the asymptotic behaviour of the spectrum σ η when the period η goes to zero along the set Ξ. In what follows we focus on high frequencies, but it seems worth giving an outline of the main result for the "low-frequency" limit. In this case it is known (see e.g. [START_REF] Guenneau | Homogenization of three-dimensional finite photonic crystals[END_REF], [START_REF] Guenneau | Homogénéisation des quasi-cristaux et analyse des modes dans des fibres optiques de type cristal photonique[END_REF]) that when f ∈ L 2 (T), the solutions u η of (1.2) converge strongly in L 2 (T) to a limit u hom , which is a unique solution in H per (curl, T) of the homogenised problem:

curl ε -1 hom (x)curlu hom + u hom = f , (1.3)
where the effective matrix ε hom (x) is given by 3

ε hom (x)ξ • ξ = inf Φ∈H 1 per (Y ) Y ε(x, y) ξ + ▽Φ(y) • ξ + ▽Φ(y) dy (1.5)
for any ξ ∈ R 3 . Further, as η → 0, the restrictions to L 2 sol (T) of the operators G η converge in norm to the restriction to L 2 sol (T) of the limit operator G defined for every f ∈ L 2 (T) by Gf = u hom , where u hom is the solution of (1.3). This fact follows from the weak compactness of the unit ball in L 2 (T) and the above mentioned result that when a sequence of

f η ∈ L 2 sol (T) weakly converges to f ∈ L 2 sol (T) one has the strong convergence in L 2 (T) of G η f η to Gf .
[The proof of an analogous statement can be found in [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]Theorem 2.2].] Clearly, G is a compact self-adjoint operator and one has σ(G) = {0, 1} { Λk } k≥1 , where the eigenvalues Λk are listed in the decreasing order and lim k→+∞ Λk = 0. The min-max principle (see e.g. [START_REF] Conca | Fluids and periodic structures[END_REF]) implies that | Λk η -Λk | ≤ G η -G . Therefore, the above convergence in norm yields the convergence of each individual eigenvalue Λk η labelled in the decreasing order, and therefore of each eigenvalue Λ k η of (1.1). In particular, when η → 0, the set 1 The space Hper(curl, T) is defined as the closure of the set C ∞ per (T) of T-periodic infinitely smooth vector functions ϕ with respect to the norm ϕ L 2 (T) + curlϕ L 2 (T) . 2 The space L 2 sol (T) is defined to consist of those functions in L 2 (T) whose divergence vanishes, with the Hilbert structure induced from L(T). Denote by H 1 per (T) the closure of C ∞ per (T) in the norm ϕ L 2 (T) + ∇ϕ L 2 (T) . Then the divergence divu of a vector function u ∈ L 2 (T) is a functional on H 1 per (T) defined by the formula divu, ϕ = - 3 This representation is consistent with the classical homogenisation result in two dimensions for a sequence uη = u 3 η (x 1 , x 2 )e 3 (transverse electric polarisation), involving the inverse of the matrix ε(x, y) in the definition (1.5), see [START_REF] Zolla | A duality relation for the Maxwell system[END_REF]. This can be obtained by examining the dual form of (1.5) associated to ε -1 hom (x) :

R T u(x) • ∇ϕ(x)dx.
ε -1 hom (x)ξ • ξ = inf Ψ∈Hper(curl,Y ) Z Y ε -1 (x, y) " ξ + curlΨ(y) " • " ξ + curlΨ(y) " dy. (1.4)
σ η converges (in the Hausdorff sense4 ) to (1 -Λ) -1 Λ : Λ ∈ σ(G) . However, this convergence result alone does not say anything about the asymptotic behaviour of sequences of eigenvalues Λ k(η) η when η → 0, where k(η) → ∞ (which corresponds to high frequencies Λ -1 η ), and this is the issue that we address in the present work. It turns out that the behaviour of the eigenvalues Λ η as η → 0 depends on the way they are scaled with η. In particular, three different limiting operators can be constructed, depending on whether the limit of η -2 Λ η is finite, infinite, or zero. The first case, which is of greatest interest in applications, corresponds to a limiting operator with a band-gap spectrum.

We now outline the plan of the paper. In Section 2, we recall the concept of two-scale convergence for sequences in L 2 (T) and adapt it to sequences of bounded self-adjoint operators. In Sections 3 and 4, we characterise the limit spectrum of the type lim η→0 η -2 σ η . Following [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF], we will study the eigenvalues of order η 2 . Via a change of spectral parameter in (1.1), we look for pairs (µ η , H η ) ∈ IR + × H per (curl, T),

H η ≡ 0, such that η 2 curl ε -1 (x, x/η)curlH η + H η = µ -1 η H η . (1.6)
This transformation of (1.1) into (1.6) keeps unchanged the eigenfunctions and changes the eigenvalues

Λ η to µ η = (η 2 + Λ η ) -1 Λ η , ensuring that µ η ∼ 1 if Λ η ∼ η 2 .
The problem (1.6) generates an operator S η ∈ L(L 2 (T)) defined for each f ∈ L 2 (T) by S η f = v η , where v η is the unique solution in H per (curl, T) of the problem

η 2 curl ε -1 (x, x/η)curlv η + v η = f .
It can be shown that S η converge weakly to the identity operator, but this fact does not provide any information on the behaviour of the spectra of S η . The latter issue can be handled using the notion of twoscale convergence [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], whereby the limit operator S K defined for every K ∈ N acts on the space L 2 (T × KY ). The (renormalised) limit of σ η will then be determined by studying the spectra σ(S K ), K ∈ N, via the Bloch wave decomposition. For this, we introduce a family of limit problems, whereby for every (x, θ) ∈ T × Y , we look for pairs (µ(x, θ), v(y)) ∈ R + × H per (curl, Y ) satisfying the equation

curl y ε -1 (x, y)curl y v(y)e 2πiθ•y + v(y)e 2πiθ•y = µ(x, θ) -1 v(y)e 2πiθ•y .
(

For the spectrum σ(x, θ) of the problem (1.7) we have σ(x, θ) = {0, 1} {µ k (x, θ)} k≥1 , where the sequence µ k (x, θ) converges to 0. It can be shown that for each fixed k, the eigenvalue µ k (x, θ) is a continuous function of (x, θ) (cf. [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]). We then define the Bloch spectrum (band spectrum) by

σ Bloch := {0, 1} ∪ k≥1 min (x,θ)∈T×Y µ k (x, θ), max (x,θ)∈T×Y µ k (x, θ) (1.8)
Note that σ Bloch is the spectrum of a non-compact operator that maps every vector-function f (x, y) ∈ L 2 (T × Y ) to the solution w(x, y) ∈ L 2 T, H per (curl, Y ) of curl y ε -1 (x, y)curl y w(x, y) + w(x, y) = f (x, y) a.e. x ∈ T.

(1.9)

In Section 5 we characterise the limit spectrum for other scalings. An appropriate multi-scale generalisation of the two-scale convergence method [START_REF] Allaire | Multiscale convergence and reiterated homogenization[END_REF] is used to study the spectra of the types a -2 η σ η , where a η ≫ η or a η ≪ η. Section 6 is devoted to numerical illustration of the spectrum convergence result in the "critical" case a η ∼ η. Finally, the Appendix contains some of the technical material related to the results of the paper.

Two-scale convergence for operators

The notion of two-scale convergence was introduced by Nguetseng and Allaire (see [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenisation[END_REF], [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]) for sequences of bounded vector functions in L 2 -norm. In what follows, we recall its definition, some of it basic properties and adapt it to sequences of bounded self-adjoint operators, see also [START_REF] Guenneau | Homogénéisation des quasi-cristaux et analyse des modes dans des fibres optiques de type cristal photonique[END_REF].

Definition 1. Consider a bounded set T ⊂ R N and an integer K ∈ N.

i) A vector function ϕ ∈ L 2 (T × KY ) is said to be admissible if the functions ϕ η (x) = ϕ(x, x/η), η ∈ (0, 1], are uniformly bounded in L 2 (T) and satisfy the condition

lim η→0 T ϕ η (x) 2 dx = 1 K N T KY ϕ(x, y) 2 dydx.
ii) Denote by η a sequence of positive numbers converging to zero. A sequence u η of functions in L 2 (T) is said to weakly two-scale converge to u 0 (x, y) ∈ L 2 (T × KY ), in which case we write u η ⇀ ⇀u 0 , if for any admissible ψ e.g. ψ(x, y) ∈ C T, C ∞ per (KY ) , one has

lim η→0 T u(x) • ψ(x, x/η)dx = 1 K N T KY u 0 (x, y) • ψ(x, y)dydx.
iii) A sequence u η of functions in L 2 (T) is said to strongly two-scale converge to u 0 (x, y) ∈ L 2 (T×KY ), and we write u η → →u 0 , if any of the two equivalent statements holds: a)

u η ⇀ ⇀u 0 and lim η→0 T u η (x) 2 dx = 1 K N T KY u 0 (x, y) 2 dydx; b) For any sequence v η such that v η ⇀ ⇀ v ∈ L 2 (T), T u η (x) • v η (x)dx η→0 -→ 1 K N T KY u 0 (x, y) • v(x, y)dydx.
iv) A sequence S η of operators in L 2 (T) is said to strongly two-scale converge to an operator S in

L 2 (T × KY )) if for any sequence f η ∈ L 2 (T) such that f η → → f ∈ L 2 (T × KY ) one has S η f η → → Sf .
The crucial property of the two-scale convergence that makes it useful in studying elliptic PDEs is the compactness of bounded sets in L 2 (T), see e.g. [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. The usual argument therefore is to establish a bound on a solution sequence and then infer an equation on the limit of its subsequence. The notion of two-scale operator convergence, in turn, allows one to simplify the derivation of the associated spectral convergence. In fact, given a sequence of bounded linear self-adjoint operators S η , one can state the following result on strong two-scale convergence for the families of the associated spectral projections {E λ (S η )} λ∈R (cf. [START_REF] Guenneau | Homogénéisation des quasi-cristaux et analyse des modes dans des fibres optiques de type cristal photonique[END_REF]). We recall that for a given operator A, the function E λ (A) is non-decreasing, upper semi-continuous, non-negative, and satisfies the identity E λ (A)E µ (A) = E min{λ,µ} (A) for any λ, µ ∈ R (see e.g. [START_REF] Conca | Fluids and periodic structures[END_REF], [START_REF] Reed | Methods of Modern Mathematical Physics IV: Analysis of Operators[END_REF]).

Lemma 1. Let S η be a sequence of bounded linear self-adjoint operators in L 2 (T) that strongly two-scale converges to an operator S in L 2 (T × KY ). Then S is linear and self-adjoint, and for any µ ∈ R that is not an eigenvalue of S, one has E µ (S η ) → → E µ (S) as η → 0.

Proof. Without loss of generality, we can assume that µ = 0, for otherwise, the sequence S η -µI can be considered. The linearity and self-adjointness of S is clear from that of S η and the fact that S η two-scale converge to S. Further, for any sequence

f η ∈ L 2 (T) such that f η → → f ∈ L 2 (T × KY ) we have S η f η → → Sf and S 2 η f η → → S 2 f
. It is clear that by iterating this procedure we prove that for any polynomial P the convergence P (S η ) → → P (S) holds as η → 0.

Since the operators S η , η ∈ Ξ, are bounded, and for any f ∈ L 2 (T), S η f → Sf in L 2 (T), by the Banach-Steinhaus theorem (see e.g. [START_REF] Conca | Fluids and periodic structures[END_REF], [START_REF] Riesz | Functional Analysis[END_REF]) S η are uniformly bounded. Hence, there exists a constant C > 0 such that σ η ⊂ [-C, C] for any η ∈ (0, 1], and by the spectral theorem the operators S η , η ∈ (0, 1], and S can be composed with any continuous function F on [-C, C]. Notice that for any polynomial P and a sequence

f η ∈ L 2 (T), f η → → f ∈ L 2 (T × KY ), we have F (S η )f η -F (S)f = F (S η )f η -P (S η )f η + P (S η )f η -P (S)f + P (S)f -F (S)f .
Further, there exists c > 0 such that

F (S η )f η -P (S η )f η L 2 (T) ≤ f η L 2 (T) sup s∈[-C,C] F (s) -P (s) ≤ c F -P L ∞ ([-C,C]) , and
P (S)f -F (S η )f L 2 (T) ≤ f L 2 (T×KY ) P (S) -F (S) L(L 2 (T×KY )) ≤ f L 2 (T×KY ) F -P L ∞ ([-C,C]) .
By the Weierstrass theorem there is a sequence of polynomials P n such that

F -P n L ∞ ([-C,C]) → 0. Hence we deduce that F (S η )f η → → F (S)f . (2.10) Now, consider a continuous function G on R such that G(λ) = 1 when λ ≤ 0, G(λ) = 0 when λ ≥ 1 and |G(λ)| ≤ 1 for λ ∈ [0, 1].
For any n ∈ N, the spectral theorem (see e.g. [START_REF] Conca | Fluids and periodic structures[END_REF], [START_REF] Reed | Methods of Modern Mathematical Physics IV: Analysis of Operators[END_REF]) implies5

G(nS η )f η = ∞ -∞ G(nλ)dE λ (S η )f η = 0 -∞ dE λ (S η )f η + E 1/n (S η ) -E 0 (S η ) f η .
(2.11)

We claim that the last term in (2.11) converges to zero strongly in L 2 (T) as n → ∞, and the convergence is uniform in η. Indeed, consider a continuous function

ψ on R such that ψ(λ) = 1 when 0 ≤ λ ≤ 1, supp ψ ⊂ [-1, 2]
, and ψ(λ) ≥ 0 for any λ ∈ R. Then for any δ > 0 and any subsequence η n one has

∞ -∞ ψ(δ -1 λ)dE λ (S)f , f = lim n→∞ ∞ -∞ ψ(δ -1 λ)dE λ (S ηn )f ηn , f ηn ≥ lim n→∞ 1/n 0 dE λ (S ηn )f ηn , f ηn = lim n→∞ E 1/n (S ηn ) -E 0 (S ηn ) f ηn 2 L 2 (T)
,

where the angle brackets denote the usual scalar products in L 2 (T×KY ) and L 2 (T), and the first expression tends to zero as δ → 0 in view of the assumption that µ = 0 is not an eigenvalue of S. Since the subsequence η n was arbitrary, the above claim follows. Combining it with the straightforward observation that

G(nS)f converge in L 2 (T) to E 0 (S)f as n → ∞ and invoking (2.10) with F (•) = G(n•) we conclude that the convergence E 0 (S η )f η → → E 0 (S)f holds.
The convergence of spectral families gives little information about the pointwise convergence of the spectra. As an alternative to the above lemma we have the following result.

Lemma 2. Consider a family of operators S η ∈ L(L 2 (T)) and an operator S ∈ L(L 2 (T × KY )). If S η → → S then for any µ ∈ σ(S) there exists a sequence µ η ∈ σ(S η ) such that µ η → µ as η → 0.

Proof. Suppose the opposite is true, i.e. µ ∈ σ(S) and there exists δ > 0 such that for any sequence

µ η ∈ σ(S η ) one has |µ η -µ| > δ. For any f ∈ C ∞ per (T × KY ) we can then write δ f (x, x/η) L 2 (T) ≤ S η f (x, x/η) -µf (x, x/η) L 2 (T) = sup ϕ∈C ∞ per (T×KY ) T S η f (x, x/η) -µf (x, x/η) • ϕ(x, x/η)dx.
Passing to the two-scale limit as η → 0 in the above inequality yields

δ f L 2 (T×KY ) ≤ sup ϕ∈C ∞ per (T×KY ) T KY Sf (x, y) -µf (x, y) • ϕ(x, y)dydx = Sf -µf L 2 (T×KY ) . (2.12)
By the density of C ∞ per (T × KY ) in L 2 (T × KY ) and the boundedness of S we obtain the inequality (2.12) for any f ∈ L 2 (T × KY ), which is a contradiction with the assumption that µ ∈ σ(S).

Bloch-wave homogenisation

In this section we prove the convergence of spectral families associated to sequences of eigenvalues of the order η 2 for the problem (1.1). The physical application of this model is based on the property of the possible appearance of photonic band gaps in the spectrum, i.e. a range of frequencies for which no wave can propagate within the medium. This result is based on the following statement.

Theorem 1. Consider the domain T = [-T, T ] 3 , and η ∈ Ξ. We look for u η ∈ H per (curl, T) such that

η 2 curl ε -1 (x, x/η)curlu η + u η = f η , (3.13)
with a coercive matrix ε(x, y) whose entries are in C per T, L ∞ (Y ) , and f η ∈ L 2 (T).

i) The solution of (3.13) exists and is unique in

H per (curl, T). ii) Suppose that K ∈ N and f η → → f ∈ L 2 (T × KY ).
Then there is a subsequence of problems (3.13), for which we keep the same index η, such that u η → → u 0 and ηcurlu η ⇀ ⇀ curl y u 0 , where u 0 ∈ L 2 (T, H per (curl, KY )) and for almost every x ∈ T the function u 0 (x, •) is a solution of

curl y ε -1 (x, y)curl y u 0 ) + u 0 = f (x, •). (3.14)
Proof. See the Appendix.

Remark 1. It can be shown via a standard argument that the convergences u η → → u 0 and ηcurlu η ⇀ ⇀ curl y u 0 , hold for the whole original sequence of indices η, not just for a subsequence.

Corollary 1. Let S η be a sequence of operators defined on the space L 2 (T) by S η f = u η , where u η ∈ H per (curl, T) is the solution of (3.13). Then i) For every K ∈ N the strong two-scale convergence S η → → S K holds. Here S K is defined on L 2 (T×KY ) by the formula S K f = u 0 , where u 0 ∈ L 2 (T, H per (curl, KY )) is the unique solution of (3.14).

ii) If µ ∈ R is not an eigenvalue of S K , then E µ (S η ) → → E µ (S K ), and for any K ∈ N the inclusion σ(S K ) ⊂ lim η→0 σ(S η ) holds.

Proof. The part i) follows form the fact that u η → → u and the definition of the strong two-scale convergence of operators. The part ii) is then an immediate consequence of Lemma 1 and Lemma 2.

The above corollary provides only a part of the proof of the spectral convergence, namely the statement that all Bloch eigenvalues are included in the limiting (rescaled) spectrum. Indeed, the convergence of S η to S K cannot be uniform, for S η is compact but S K is not (cf. [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]). Hence, we can only deduce the lower semi-continuity of the spectrum i.e. that for any K ∈ N one has σ(S K ) ⊂ lim η→0 σ(S η ). In order to analyse the spectra of the operators S K , a discrete Bloch wave decomposition (see [START_REF] Conca | Fluids and periodic structures[END_REF]) is used, which allows one to diagonalise S K via a representation S K = B * T K B where B is an isometry between L 2 (T×KY ) and [L 2 (T×Y )] K 3 defined by Bu = {û j }, when u(x, y) = 0≤j≤K-1 ûj (x, y) exp(2iπK -1 j•y), cf. [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]. As K → ∞, the Bloch frequencies K -1 j are dense in Y and it is natural to introduce θ ∈ Y that parameterises the resolvent operators T θ,x ∈ L L 2 (T) associated with the problems (1.7), so that

T K = diag[T j -1 K ] 0≤j≤K-1 , where for any θ ∈ Y, the operator T θ maps f ∈ L 2 (T × Y ) to the (unique) solution u(x, y) ∈ H per (curl, T) of the equation (cf. (1.7)) curl y ε -1 (x, y)curl y u(x, y)e 2πiθ•y + u(x, y)e 2πiθ•y = f (x, y)e 2πiθ•y .
Finally, noting that the eigenvalues µ(x, θ) of T θ,x are continuous (see e.g. [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF] and references therein), we conclude that lim K→∞ σ(S K ) = θ∈Y σ(T θ ) = x∈T,θ∈Y σ(T θ,x ) ≡ σ Bloch , where σ Bloch is defined by (1.8). Hence, σ Bloch ⊂ lim η→0 σ(S η ). In the next section we show that in fact lim η→0 σ(S η ) \ σ Bloch = ∅.

Completeness of the limiting spectrum

In order to substantiate the claim made at the end of the previous section, for any µ ∈ σ Bloch we will: 1) Construct "approximate eigenfunctions" for the permittivity matrix ε(x, y) "frozen" at a certain value x = x 0 , and then 2) Use the so-called "Bloch measure" technique (cf. [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF]) to show that as η → 0, the Bloch components of these approximate eigenfunctions correspond to Bloch eigenvalues concentrating near µ. Hence, we start be proving the following "macroscopic localisation" lemma. Lemma 3. Let µ η be a sequence of eigenvalues for the problem

η 2 curl ε -1 x, x/η)curlu η + u η = µ -1 η u η , (4.15)
that converges to a limit µ, and u η ∈ H per (curl, T) is the sequence of the corresponding eigenfunctions.

Then there exists a subsequence of η j , a point x 0 ∈ T, and a sequence ũj ∈ H per (curl, T), ũj L 2 (T) = 1, of quasi-eigenfunctions for the matrices ε -1 (x 0 , x/η j ), i.e. T-periodic solutions of an equation of the form

η 2 j curl ε -1 (x 0 , x/η j )curlũ j + ũj = µ -1 ηj ũj + r j ,
where r η is a remainder term, which is T-periodic and is such that6

w j L 2 (T) + curlw j L 2 (T) -1 r j , w j H -1 ,Hper(curl,T) η→0 -→ 0 (4.16)
for any sequence of non-zero w j ∈ H per (curl, T).

Proof. See the Appendix.

We are now in a position to execute the second step of the plan proposed at the beginning of the section, which we formulate next.

Theorem 2. Let u η ∈ H per (curl, T), η ∈ Ξ, be a sequence of eigenfunctions for the problem (4.15), such that u η L 2 (T) = 1 and lim η→0 µ η = µ. Then the limit µ belongs to the Bloch spectrum σ Bloch .

Proof. Using Lemma 3, select first a subsequence η j such that the matrices ε -1 (x 0 , x/η j ) for some x 0 possess quasi-eigenfunctions ũj . Consider the respective Bloch-wave decompositions of the functions ũj in the torus T as the union of periodicity cells of size η j :

ũj (x) = m u m j (x/η j ) exp(πiT -1 m • x), (4.17)
where u m j ∈ H per (curl, Y ) for every multi-index m and the summation in (4.17) is performed over all multi-indices m with components m s , s = 1, 2, 3, running from 0 to η -1 j 2T -1. Further, there is a sequence of (complex) coefficients α k j (m), k = 1, 2, ..., such that

u m j (y) = k≥1 α k j (m)v k (K -1 j m, y),
where

K j := η -1 j 2T = 2 2lj
, and the functions v k (θ, y), k = 1, 2, ... are the L 2 (T)-normalised θ-quasiperiodic eigenfunctions of the operator curl y ε -1 (x 0 , y)curl y defined on H per (curl, Y ), cf. (1.7). Consider a uniformly bounded sequence {ψ k (θ)} ∞ k=1 of real-valued continuous Y -periodic functions, which will be specified at a later stage, and define

Mũ j (x) := m k≥1 ψ k (K -1 j m)α k j (m)v k (K -1 j m, x/η j ) exp(πiT -1 m • x).
Clearly, the function Mũ j belongs to H per (curl, T) and satisfies the same a priori estimates as ũj :

Mũ j L 2 (T) + η j curlMũ j L 2 (T) ≤ C
for some C > 0. Furthermore, recalling the expression (7.40) for the term r j , we have

r j , Mũ j H -1 ,Hper(curl,T) = η 2 j T ε -1 (x 0 , x/η j )curlũ j • curlMũ j dx + (1 -µ -1 ηj ) T ũj • Mũ j dx = (2T ) 3 m k≥1 ψ k (K -1 j m) α k j (m) 2 1/µ k (x 0 , K -1 j m) -µ -1 ηj , (4.18)
hence the last expression tends to zero as j → ∞. Now define a sequence {ν k j } ∞ k=1 of Bloch measures on Y , associated with the sequence ũj via

ν k j = (2T ) 3 m α k j (m) 2 δ θ=K -1 j m ,
where δ θ=θ0 is the Dirac mass at the point θ 0 . In view of the fact that the function ũj has a unit norm in L 2 (T), we get k≥1 Y dν k j (θ) = 1 for every j.

In particular, for every k, the set {ν k j } j≥1 is bounded in the space of Radon measures on Y and hence contains a subsequence that converges weakly 7 as j → ∞ to a measure ν k . Note that

k≥1 Y dν k j (θ) = 1, (4.19)
for if we assume the opposite then we should be able to find a constant δ > 0, and sequences j l and k l such that

k≥k l Y dν k l j l (θ) ≥ δ.
On the other hand, for any j,

η 2 j curlũ j 2 L 2 (T) 3 = (2T ) 3 m k≥1 α k j (m) 2 curlv k (K -1 j m, y) + 2πiK -1 j m × v k (K -1 j m, y) 2 L 2 (Y ) ,
7 More precisely, "weakly * " if the space of Radon measures is considered as the dual of C per (Y ).

and since in view of the equation defining the functions v k (θ, y) one has

curlv k (θ, y) + 2πiθ × v k (θ, y) 2 L 2 (Y ) = 1/µ k (x 0 , θ) -1,
for any k, θ, we deduce

η 2 j curlũ j L 2 (T) ≥ δ min θ∈Y (1/µ k l (x 0 , θ) -1),
which goes to +∞ as l → ∞ since for any θ ∈ Y, we have lim k→∞ µ k (x 0 , θ) = 0, hence contradiction with the fact that η j curlũ j is bounded in L 2 (T). As we noted above (see (4.18) and the text that follows),

k≥1 Y ψ k (θ)(1/µ k (x 0 , θ) -1/µ ηj )dν k j (θ) → 0 as j → ∞,
which, having passed to the limit, can be re-written as

k≥1 Y ψ k (θ)(1/µ k (x 0 , θ) -1/µ)dν k (θ) = 0. (4.20)
Since in view of (4.19) at least one of the measures ν k is non-zero, by choosing appropriately the sequence ψ k (θ) we can ensure that (4.20) implies the existence of k and θ such that µ = µ k (x 0 , θ).

Corollary 2. For any sequence of η ∈ Ξ, one has lim η→0 η -2 σ η = (1 -Λ) -1 Λ : Λ ∈ σ Bloch .

"Non-critical" scalings

In this section we study pairs (µ η ,

H η ) ∈ R + × H per (curl, T), H η ≡ 0, such that a 2 η curl(ε -1 (x, x/η)curlH η ) + H η = µ -1 η H η , (5.21)
where a η → 0. The equation (5.21) is obtained from (1.1) by setting µ k η = (a 2 η + Λ k η ) -1 Λ k η and keeping the corresponding eigenfunctions unchanged. Analogously to what was done in Section 4, we introduce an operator S η ∈ L(L 2 (T)) defined for every f ∈ L 2 (T) by S η f = v η , where v η is the unique solution in H per (curl, T) of a 2 η curl ε -1 (x, x/η)curlv η + v η = f . The analysis of the convergence of the sequence of operators S η can be carried out in the same way as before. However, outside the scaling regime a η ∼ η there is no interaction between the homogenisation scale η and the singular perturbation scale a η . One of two things can happen instead: either η is much smaller than a η in which case the homogenisation occurs first and the singular perturbation concerns the homogenised system; or η is larger than a η and it turns out that the singular perturbation occurs first at a microscopic scale making the homogenisation irrelevant. An appropriate tool to tackle these two cases is the notion of multi-scale convergence, introduced in [START_REF] Allaire | Multiscale convergence and reiterated homogenization[END_REF]. We recall the main results of [START_REF] Allaire | Multiscale convergence and reiterated homogenization[END_REF] and adapt them, where appropriate, to the three-scale convergence of function and operator sequences.

Large scales: 1 ≫ a η ≫ η

Definition 2. Consider a bounded set T ⊂ R N and Z := [0, l] N , where l ∈ N. Let a η be a sequence of positive numbers converging to zero, η ∈ Ξ.

i) A function ϕ ∈ L 2 (T × Z × Y ) is said to be (three-scale) admissible if the sequence ϕ η (x) := ϕ(x, lx/a η , x/η) is bounded in L 2 (T) and satisfies

lim η→0 T ϕ η (x) 2 dx = 1 l N T Z Y ϕ(x, z, y) dydzdx.
ii) A sequence u η of functions in L 2 (T) is said to weakly three-scale converge to u 0 (x, z, y) ∈ L 2 (T×Z × Y ), and in this case we write u η ⇀ ⇀ ⇀ u 0 , if for any admissible ψ ∈ L 2 (T×Z ×Y ), e.g.

ψ ∈ C(T, C ∞ per (Z ×Y )), one has lim η→0 T u η (x) • ψ(x, lx/a η , x/η)dx = 1 l N T Z Y u 0 (x, z, y) • ψ(x, z, y)dydzdx.
Theorem 3. Consider T := [-T, T ] 3 , T > 0, and η ∈ Ξ. For a sequence of positive numbers a η , η ∈ Ξ, such that lim η→0 a η = 0 and lim η→0 η -1 a η = ∞, we look for u η ∈ H per (curl, T) that are weak solutions of

a 2 η curl ε -1 (x, x/η)curlu η + u η = f η , (5.22) with ε(x, y) ∈ C per T, [L ∞ (Y )] 9 coercive and f η ∈ L 2 (T).
i) The equation ( 5.22) has a unique solution u η ∈ H per (curl, T).

ii) Let l ∈ N and suppose that

f η → → f ∈ L 2 (T × Z).
There is a subsequence of problems (5.22), for which we keep the same index η, such that u η ⇀ ⇀ ⇀ u 0 + u 1 and a η curlu η ⇀ ⇀ ⇀ curl z u 1 + u 2 , where8

u 0 ∈ L 2 T × Z, L 2 pot (Y ) , u 1 ∈ L 2 T, H per (curl, Z) , u 2 ∈ L 2 T × Z, L 2 sol (Y )
, and for almost every x ∈ T the function u 1 (x, •) is the unique solution of

curl z ε -1 hom (x)curl z u 1 (x, z) + u 1 (x, z) = f (x, z).
(5.23)

where ε hom (x) is given by the formula (1.5) (or, equivalently, (1.4)).

iii) Let u η be the three-scale convergent subsequence from ii). Then u η → → u 1 , hence the operator sequence S η defined in L 2 (T) by S η f η = u η strongly two-scale converges to S l defined in L 2 (T × Z) by the formula S l f = u 1 , where u 1 ∈ L 2 T, H per (curl, Z) is the unique solution of (5.23).

Proof. See the Appendix. Corollary 3. For any sequence a η , η ∈ Ξ, as in Theorem 3,

lim η→0 σ(S η ) = [0, 1],
(5.24)

where the limit is understood in the Hausdorff sense (see the footnote on p. 3). Equivalently, the limit of a -2 η σ η is the entire positive real axis. Proof. From Lemma 2, we deduce that for a relevant subsequence, σ(S l ) ⊂ lim η→0 σ(S η ).

(5.25)

Moreover, for any l ∈ N, the spectrum σ(S l ) is obtained from σ(S 1 ) by a simple transformation since the coefficient matrix in (5.23) does not depend on z. Labelling in increasing order the eigenvalues γ k l , k ≥ 1, of S l , they satisfy the identity

γ k l = l 2 γ k 1 + (1 -γ k 1 ) -1 l 2 γ k 1 .
Using this and the fact that γ k 1 → 0 as k → ∞, it can be shown that the set {γ k l , k, l ∈ N} ≡ l≥1 σ(S l ) is dense in [0,1]. In view of (5.25), via a simple argument, this implies (5.24).

Remark 2. Given a sequence a η , η ∈ Ξ, the statement of Corollary 3 clearly also holds for any subsequence of a η , η ∈ Ξ.

Small scales: a η ≪ η

Definition 3. Consider a bounded set T ⊂ R N and Z := [0, l] N , where l ∈ N. Let a η , η ∈ Ξ, be a sequence of positive numbers such that lim η→0 a η = 0.

i) Let u η be a sequence of functions in L 2 (T) that weakly three-scale converges to u 0 (x, z, y)

∈ L 2 (T × Z × Y ). Suppose that lim η→0 T u η (x) 2 dx = 1 l N T Z Y u 0 (x, z, y) 2 dydzdx.
Then u η is said to strongly three-scale converge to u 0 (x, z, y), and we write u η → → → u 0 .

ii) A sequence S η of operators in L 2 (T) is said to strongly three-scale converge to the operator S in

L 2 (T × Z × Y )) if for any sequence f η ∈ L 2 (T) such that f η → → → f one has S η f η → → → Sf .
Theorem 4. Suppose that the sequence a η , η ∈ Ξ is such that lim η→0 η -1 a η = 0. Consider T := [-T, T ] 3 , T > 0, and for any η ∈ Ξ look for u η ∈ H per (curl, T) satisfying (in the weak sense)

a 2 η curl ε -1 (x, x/η)curlu η + u η = f η , (5.26) with ε(x, y) ∈ C per T, [L ∞ (Y )] 9 coercive and f η ∈ L 2 (T).
i) The equation ( 5.26) has a unique solution u η ∈ H per (curl, T).

ii) Suppose that

f η → → → f ∈ L 2 (T × Z × Y ).
For any l ∈ N there is a subsequence of problems (5.26), for which we keep the same index η, such that u η ⇀ ⇀ ⇀ u 0 and a η curlu η ⇀ ⇀ ⇀ curl z u 0 , where u 0 ∈ L 2 T × Y, H per (curl, Z) and for almost every x ∈ T, y ∈ Y, the function u 0 (x, •, y) is the unique solution of

curl z ε -1 (x, y)curl z u 0 + u 0 = f (x, •, y).
(5.27)

iii) The operator sequence S η defined in L 2 (T) by S η f η = u η strongly three-scale converges to S l defined in L 2 (T × Z) by the formula S l f = u 0 , where u 0 is the solution of (5.27).

The proof of the above theorem goes along the lines of Theorem 1 and Theorem 3 with some simplifications, and we will not dwell on it here. In the same way as before, a statement on the convergence of the spectra follows.

Corollary 4. For any sequence a η , η ∈ Ξ, as in Theorem 4, lim η→0 σ(S η ) = [0, 1], or, equivalently, the limit of a -2 η σ η is the entire positive real axis. The same is true for any subsequence of a η , η ∈ Ξ.

Numerical example

In this section we illustrate our spectral convergence result in the critical case a η ∼ η (see Sections 2-4) by considering the propagation of TM-polarised electromagnetic waves through a periodic array of parallel infinitely long cylinders. The corresponding analogue of the problem (1.1) in this case is

-∆E(x) = Λ -1 η ε(x/η)E(x), (6.28)
where x ∈ [-1, 1] 2 , and E is the only non-zero component of the electric field, which is parallel to the cylinders. For simplicity, in (6.28) we drop the dependence of the permittivity matrix on the "slow" variable x, assume that the medium is isotropic at every point so that ε(y) is a scalar for any y ∈ Y, and set T = 1.

In our calculations, we set the relative permittivity ε = 12 inside and ε = 1 outside the cylinders, which corresponds to the situation of straight fibres of a semi-conductor surrounded by air or, more generally, any two materials with a permittivity contrast of 12 (the permittivity of the majority of dielectric materials found in nature ranges from 1 to 14). Here we consider fibres with square cross-section to preserve the four-fold symmetry in the unit cell, hence it is sufficient to consider only a quarter of the Brillouin zone Y * = [0, 2π] 2 ("first reduced Brillouin zone").

The Bloch spectrum of (6.28) is obtained by studying a family of spectral problems on Y associated with (6.28). For any κ ∈ Y * consider pairs (Λ(κ), E(y)) ∈ R + × H 1 per (Y) such that (cf. (1.7))

-∆ E(y)e iκ•y = Λ(κ) -1 ε(y)E(y)e iκ•y .

For every κ ∈ [0, 2π] 2 the related eigenvalues Λ(κ) form a sequence, which we denote by Λ k (κ) k≥1 . The frequencies Λ k (κ) -1/2 , k = 1, ..., 10, as functions of κ ∈ Y * are shown by the surface dispersion diagram in Figure 1. As it may be difficult to visualise these plots, it is customary to restrict oneself to "boundary" dispersion curves, which correspond to κ ∈ ∂Y * , assuming

min κ∈Y * Λ k (κ), max κ∈Y * Λ k (κ) = min κ∈∂Y * Λ k (κ), max κ∈∂Y * Λ k (κ) . (6.29)
The validity of this assumption appears to be an open problem, but since it helps getting a better idea of the stop band properties, in Figure 2 we plot such dispersion curves corresponding to Figure 1. We emphasise that we first numerically checked that all of the surfaces shown satisfy the conjecture (6.29). We feel that such a preliminary test on dispersion surfaces should be systematically performed before using the rule (6.29).

We would now like to look at the spectrum associated with the sequence of problems (6.28), as η goes to zero. Consider the corresponding differential operator with periodic boundary conditions on the cell [-1, 1] 2 and look for the convergence of its spectrum when the number of inclusions increases and their size tends to zero so that their overall surface remains constant. It is apparent from Figures 3 to 6 that the spectrum becomes denser as η goes to zero. It is worth noting that the location of the band gaps can be determined with good accuracy as early as the second step of the Bloch wave homogenisation process i.e. when η = 1/4. The mechanism leading to the appearance of the stop bands involves a relaxation of the constraint on the periodic condition: when the number of inclusions increases, the wave satisfies both a periodic condition on the sides of the cell and some phase shift between inclusions. The larger the number of inclusions, the more prominent the phase shift between the inclusions. The consequence is that it is enough to build a large super-cell with periodic boundary conditions to filter Bloch waves within a periodic array. This observation might well prove useful for numerical purposes when implementation of Floquet-Bloch conditions is not straightforward within a package.

Γ X M κ 1 κ 2 Λ k (κ 1 , κ 2 ) 1
Figure 1: Surface dispersion diagram for TM-waves propagating within a periodic array of square cylinders of a semiconductor (ε = 12) surrounded by air (ε = 1); the cross section of the cylinders is 0.6 and the structure period is 1. On the vertical axis we show the frequency 1/ Λ η (κ), while the horizontal axes correspond to the components of the Bloch vector κ, which spans the area of the reduced Brillouin zone, i.e. the right-angle triangle with vertices Γ = (0, 0), X = (π, 0) and M = (π, π).

Discussion

We have studied the high-frequency asymptotic behaviour of the spectra σ η of operators curlε -1 (x, x/η)curl as η → 0 under the standard ellipticity conditions on the periodic permittivity matrix ε -1 (x, y). We have shown that the sets a -2 η σ η converge in the Hausdorff sense either to the set [0, +∞) or to the (Bloch) Here we plot the first 40 rescaled eigenvalues Λ k 1/2 . The spectrum is denser compared to Figure 3. Notably, some of the eigenvalues are already located on the edge of the band gaps. The location of the stop bands is well determined already for η = 1/2. Nevertheless, the eigenvalues remain sparsely located between the gaps. spectrum of the operator curl y ε -1 (x, y)curl y . The proof of this fact consists of two parts: 1) We introduce the notion of two-scale operator convergence, which, having been applied to the problem in question, implies that rescaled sequences of eigenvalues converge to points in the spectrum of an operator that is "limiting" in some sense; 2) In the case when a η ∼ η, a technique based on the construction of "quasi-eigenfunctions" and the associated "Bloch measures" allows us to show that any point in the Bloch spectrum can be attained as the limit of a sequence from 1).

There are several possible directions in which our results could be developed. First, the issue of whether non-periodic boundary conditions could be treated in a similar way, which is of special importance in the electromagnetic context. A crude analysis shows that, although Dirichlet-type conditions could be studied in the same way as in [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF], the Neumann-type boundary presents a significant challenge in the description of what is called the "boundary-layer" spectrum. Secondly, the high-frequency spectrum analysis can be considered in the context of high-contrast periodic composites, which has received considerable attention recently in the finite-frequency regime, see [START_REF] Zhikov | On an extension of the method of two-scale convergence and its applications[END_REF], [START_REF] Cherednichenko | Non-local homogenised limits for composite media with highly anisotropic periodic fibres[END_REF]. Depending on the exact relationship between the contrast parameter Here we plot the first 160 rescaled eigenvalues Λ k 1/4 . The four band gaps are now clearly standing out in the dense set of eigenvalues. Comparing Figure 4 and Figure 5 we note that the eigenvalues on the edges of the gaps start degenerating (multiplicity greater than one). One can follow some of the eigenvalues in the limiting process, for instance Λ 15 1/4 = Λ 16 1/4 = 16Λ δ, the period of oscillations η and the wavelength a δ,η , the limiting problem could possess some or other type of spectrum, possibly leading to a number of new effects in the limit of vanishing parameter values. The band-gap features of the "high-contrast high-frequency" case can then be exploited in creating localised modes in the spectrum by adding a compact perturbation to the limiting operator. This will translate in a suitably rescaled perturbation of the original composite with finite δ, η, which possesses the mentioned localisation properties in a controlled way as δ and η vary. These aspects of the high-frequency spectral asymptotics for periodic media will be studied elsewhere.

Appendix: proofs of some of the main statements Proof of Theorem 1

i) The existence and uniqueness of a solution u η ∈ H per (curl, T) to the problem (3.13) follows from the standard argument invoking the Lax-Milgram lemma.

ii) The sequences u η and ηcurlu η are bounded in L 2 (T). Indeed, the weak formulation of (3.13) implies

T η 2 ε -1 (x, x/η) curlu η (x) 2 dx + T u η (x) 2 dx = T f η (x) • u η (x)dx. (7.30)
In view of the assumption that ε(x, y) is coercive and noticing that

T f (x, x/η) • u η (x)dx ≤ f η L 2 (T) u η L 2 (T) ,
it follows from (7.30) that

u η L 2 (T) ≤ f η L 2 (T)
and

ηcurlu η L 2 (T) ≤ ν -1/2 f η L 2 (T) ,
where ν > 0 is a coercivity constant for the matrix ε -1 (x, y). Hence up to a subsequence, u η ⇀ ⇀ u 0 and ηcurlu η ⇀ ⇀ v, where u, v ∈ L 2 (T × Y ) (cf. [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], [START_REF] Allaire | Bloch wave homogenization and spectral asymptotic analysis[END_REF], [START_REF] Conca | Fluids and periodic structures[END_REF]). Next we establish a relation between the two-scale limits u 0 and v. Using integration by parts, for any ϕ ∈ C ∞ per (T × KY ) we write Since ϕ(x, x/η) converges two-scale strongly to ϕ(x, y), we can pass to the limit in (7.31) as η → 0, which yields

T KY v(x, y) • ϕ(x, y)dydx = T KY u 0 (x, y) • curl y ϕ(x, y)dydx (7.32)
Setting ϕ(x, y) = Φ(x)Ψ(y), where Φ ∈ C ∞ per (T) and Ψ ∈ C ∞ per (KY ) are arbitrary, we deduce from (7.32) that for almost every x ∈ T one has curl y u 0 (x, y) = v(x, y). In particular, we get u 0 (x, y) ∈ L 2 (T, H per (curl, KY )).

Furthermore, (3.13) implies

T ε -1 (x, x/η)ηcurlu η (x) • ηcurl x ϕ(x, y) + curl y ϕ(x, y) y=x/η dx + T u η (x) • ϕ(x, x/η)dx = T f η (x) • ϕ(x, x/η)dx. ( 7
.33)

Passing to the limit in (7.33) as η → 0, we get

T KY ε -1 (x, y)curl y u 0 (x, y) • curl y ϕ(x, y)dydx + T KY u 0 (x, y) • ϕ(x, y)dydx = T KY f (x, y) • ϕ(x, y)dydx.
Setting, again, ϕ(x, y) = Φ(x)Ψ(y), where Φ ∈ C ∞ per (T) and Ψ ∈ C ∞ per (KY ), we conclude that (3.14) holds. Finally, in order to prove that u η → → u we show that lim sup

η→0 T |u η (x)| 2 dx ≤ 1 K 3 T KY u 0 (x, y)| 2 dydx. (7.34)
To this end notice that the minimising properties of u η and u 0 (x, x/η) imply

T curlu η (x) 2 dx + T u η (x) 2 dx = T f η (x) • u η (x)dx (7.35)
for any η ∈ Ξ, and

1 K 3 T KY curl y u 0 (x, y) 2 dydx + 1 K 3 T KY u 0 (x, y) 2 dydx = 1 K 3 T KY f (x, y) • u 0 (x, y)dydx, (7.36
) respectively. In view of the fact that the right-hand side of (7.35) converges to the right-hand side of (7.36) as η → 0, we get lim sup

η→0 T u η (x) 2 dx ≤ 1 K 3 T KY curl y u 0 (x, y) 2 dydx + 1 K 3 T KY u 0 (x, y) 2 dydx -lim inf η→0 T curlu η (x) 2 dx. ( 7

.37)

The inequality 1

K 3 T KY curl y u 0 (x, y) 2 dydx ≤ lim inf η→0 T curlu η (x) 2 dx
in conjunction with (7.37) immediately implies (7.34).

Proof of Lemma 3

Consider the sequence η l such that η -1 l 2T = 2 2l . Then the periodic cell T is covered by cubes P l i , i = 1, ..., 2 3l , of size √ 2T η l , whose interiors are disjoint. Denote by x l i the centre of the cube P l i , i = 1, ..., 2 3l , and by i l one of those indices i for which the norm u η L 2 (P l i ) attains its maximum over all indices i = 1, ..., 2 3l . The sequence x l i l , being bounded, has a convergent subsequence, which corresponds to some indices l j and whose limit we denote by x 0 . In what follows we use the notation x j for this subsequence, and also denote u η l j by u j , P L 2 (T) ϕ j u j , where ϕ j (x) := ϕ β -1 j (xx j ) . Further, denote D j := supp(ϕ j ). We begin by suggesting the following auxiliary statement. Lemma 4. There is a positive constant C such that for any j,

u j L 2 (Dj ) ≤ C u j L 2 (Pj ) (7.38) and η j curlu j L 2 (Dj ) ≤ C u j L 2 (Pj ) . (7.39)
Proof. The estimate (7.38) follows from the fact that D j is covered by at most 27 cubes from the set {P lj i , i = 1, ..., 2 3lj } and u j L 2 (P l j i )

attains its maximum for i = i lj . In order to verify (7.39) consider the function ψ j (x) := ϕ j (x/2) and multiply the equation (4.15) by ψ 2 j u j . Integrating by parts we get

η j T ψ 2 j ε -1 (x, x/η j )curlu j • curlu j dx = (µ -1 ηj -1) T ψ 2 j u j • u j dx -2η 2 j T ε -1 (x, x/η j )ψ j curlu j • (u j × ∇ψ j )dx.
Using the fact that the matrices ε -1 (x, x/η), η ∈ Ξ, are uniformly positive-definite, we get

η 2 j ψ j curlu j 2 L 2 (T) ≤ C η 2 j ψ j curlu j L 2 (T) u j × ∇ψ j L 2 (T) + ψ j u j L 2 (T) ≤ C η 2 j β -1 j ψ j curlu j L 2 (T) u j L 2 (Dj ) + u j 2 L 2 (Dj ) ≤ C η j ψ j curlu j L 2 (T) u j L 2 (Dj ) + u j 2 L 2 (Dj )
for some constant C > 0, from which (7.39) immediately follows, taking into account (7.38).

Proceeding with the proof of the above proposition, notice that the sequence ũj defined above is a sequence of quasi-eigenvectors for the matrix ε -1 (x, x/η j ). Indeed, for the vector-function rj defined by rj := η 2 j curl ε -1 (x, x/η j )curlũ j + (1µ -1 ηj )ũ j , and a sequence w j ∈ H per (curl, T), we have rj , w j H -1 ,Hper(curl,T) = ϕ j u j -1

L 2 (T) η 2 j T ε -1 (x, x/η j )curl(ϕ j u j ) • curlw j dx -η 2 j T ε -1 (x, x/η j )curlu j • curl(ϕ j w j ) dx = ϕ j u j -1 L 2 (T) η 2 j T ε -1 (x, x/η j )∇ϕ j • (u j × curlw j -w j × curlu j )dx.
Therefore, we have rj ,

w j H -1 ,Hper(curl,T) ≤ u j -1 L 2 (Pj ) η 2 j β -1 j u j L 2 (Dj ) curlw j L 2 (T) + w j L 2 (T) curlu j L 2 (Dj ) .
In view of the above lemma, the last estimate implies rj , w j H -1 ,Hper(curl,T) ≤ Cη j β -1 j w j L 2 (T) + η j curlw j L 2 (T) , , m hence (4.16) with r j = rj . Finally, to prove that ũj is a sequence of quasi-eigenfunctions for the matrix ε -1 (x 0 , x/η j ), note that for r j defined by r j := η 2 j curl ε -1 (x 0 , x/η j )curlũ j + (1µ -1 j )ũ j (7.40) and a sequence w j ∈ H per (T) such that the norms w j L 2 (T) + η j curlw j L 2 (T) are uniformly bounded, one has r j , w j H -1 ,Hper(curl,T) = rj , w j H -1 ,Hper(curl,T) +η 2 j T ε -1 (x 0 , x/η j )ε -1 (x, x/η j ) curlũ j • curlw j dx. (7.41) The first term in the right-hand side of (7.41) vanishes as j → ∞ in view of the fact that ũj is a sequence of quasi-eigenfunctions for the matrix ε -1 (x, x/η j ), while the second term is bounded by

η 2 j curlw j L 2 (T) curlũ j L 2 (T) sup x∈Dj ε -1 (x 0 , •) -ε -1 (x, •) [L ∞ (Y )] 9 ,
which also vanishes as j → ∞, since D j = supp(ϕ j ) concentrates near the point x 0 .

Proof of Theorem 3

i) The existence and uniqueness of a solution u η ∈ H per (curl, T) to (5.22) follows from the Lax-Milgram lemma (see e.g. [START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF]).

ii) The sequences u η and a η curlu η are bounded in L 2 (T). The proof is of this fact is analogous to that given in Theorem 1. Hence, for a fixed l ∈ N and up to a subsequence9 , u η ⇀ ⇀ ⇀ v 0 and a η curlu η ⇀ ⇀ ⇀ v 1 , where v 0 , v 1 ∈ L 2 (T × Z × Y ). We start by studying in more detail the structure of v 0 and the relationship between v 0 and v 1 .

To this end, notice first that for any ϕ ∈ C ∞ per (T × Z × Y ), Note that v 0 (x, z, y) y = u 1 (x, z), therefore u 1 ∈ L 2 T, H per (curl, Z) and curl z u 1 (x, z) = v 1 (x, z, y) y .

The identity (7.42) can now be rewritten as where the last term vanishes in view of the fact that u 0 is a potential vector in the variable y. Hence, v 1curl z u 1 ∈ L 2 T × Z, L 2 sol (Y ) . We next use the weak formulation of (5.22), and for any ϕ 0 ∈ C ∞ per (T × Z), ϕ 1 ∈ C ∞ per (T × Z × Y ) write T ε -1 (x, x/η)a η curlu η (x) • a η curl x ϕ 0 (x, z) + curl z ϕ 0 (x, z) z=x/aη) dx + T ε -1 (x, x/η)a η curlu η (x)• ηcurl x ϕ 1 (x, z, y)+a -1 η ηcurl z ϕ 1 (x, z, y)+curl y ϕ 1 (x, y, z) z=x/αη,y=x/η dx + T u η (x)• ϕ 0 (x, x/a η )+a -1 η ηϕ 1 (x, x/a η , x/η) dx = T f η (x)• ϕ 0 (x, x/a η )+a -1 η ηϕ 1 (x, x/a η , x/η) dx.

T Z Y
Passing to the limit as η → 0 yields

Figure 2 :

 2 Figure 2: Dispersion curves for TM-modes of Figure1when the Bloch vector κ runs along the edges of the reduced Brillouin zone ΓM X. The horizontal axis represents the value of κ 1 from Γ to X, the value of κ 1 + κ 2 = π + κ 2 from X to M and the value ofκ 2 1 + κ 2 2 from M to Γ. Note the presence of four stop bands (shaded regions) for normalised frequencies 1/ Λ k (κ) within the intervals [1.39, 1.67], [2.38, 2.66],[3.62, 3.83],[4.27, 4.53].

1 Figure 3 :

 13 Figure3: Inverse square root of the first 12 eigenvalues in the discrete spectrum associated with the spectral problem (6.28) on the unit cell [-1, 1] 2 with periodic boundary conditions. In this case the cell contains one square inclusion with side length 0.6. The isolated eigenvalues sit well outside the band gaps of Figure2, given by shaded regions.

Figure 4 :

 4 Figure4: The same setting as in Figure3, but the cell contains four inclusions with side length 0.3 (η = 1/2). Here we plot the first 40 rescaled eigenvalues Λ k 1/2 . The spectrum is denser compared to Figure3. Notably, some of the eigenvalues are already located on the edge of the band gaps. The location of the stop bands is well determined already for η = 1/2. Nevertheless, the eigenvalues remain sparsely located between the gaps.

4 Figure 5 :

 45 Figure 5: The same setting as in Figure 4 but the cell contains 16 inclusions with side length 0.015 (η = 1/4).Here we plot the first 160 rescaled eigenvalues Λ k 1/4 . The four band gaps are now clearly standing out in the dense set of eigenvalues. Comparing Figure4and Figure5we note that the eigenvalues on the edges of the gaps start degenerating (multiplicity greater than one). One can follow some of the eigenvalues in the limiting process, for instance Λ15 1/4 = Λ16 1/4 = 16Λ 3 1/2 ∼ 1.38.

Figure 6 :

 6 Figure 6: The same setting as in Figure 5 but there are 196 inclusions with side length 0.0075 in the cell (η = 1/16). Here the first 210 eigenvalues Λ k 1/16 are shown. We only plot eigenvalues located within the range [0, 1.8] for clarity (one should plot over 500 eigenvalues to cover the range [0, 5]).

ηT

  curlu η (x)•ϕ(x, x/η)dx = T ηu η (x)•curl x ϕ(x, y) y=x/η dx+ T u η (x)•curl y ϕ(x, y) y=x/η dx.(7.31)

  lj i l j by P j , and η lj by η j . Also, we set β j := 2 1-lj T. Consider a positive scalar cut-off function ϕ ∈ C ∞ 0 (R 3 ) such that supp(ϕ) ⊂ [-1, 1] 3 and ϕ(x) = 1 when x ∈ [-1/2, 1/2] 3 , and define ũj := ϕ j u j -1

T

  curlu η (x) • a η ϕ(x, x/a η , x/η)dx = T u η (x) • a η curl x ϕ(x, z, y) z=x/aη,y=x/η dx + T u η (x) • curl z ϕ(x, z, y) z=x/aη,y=x/η dx.Passing to the limit in this identity yieldsT Z Y v 0 (x, z, y) • curl y ϕ(x, z, y)dydzdx = 0, which implies v 0 = u 0 + u 1 , where u 0 is the y-gradient of a function from L 2 T × Z, H 1 per (Y ) , and u 1 in independent of y. Next, for any ϕ ∈ C ∞ per (T × Z × Y ), we have T a η curlu η (x) • ϕ(x, x/a η , x/η)dx = T a η u η (x) • curl x ϕ(x, z, y) z=x/aη,y=x/η dx T u η (x) • curl z ϕ(x, z, y) z=x/aη,y=x/η dx + T η -1 a η u η (x)• curl y ϕ(x, z, y) z=x/aη,y=x/η dx.Passing to the limit in the last identity yieldsT Z Y v 1 (x, z, y) • ϕ(x, z, y)dydzdx = T Z Y v 0 (x, z, y) • curl z ϕ(x, z, y)dydzdx + lim η→0 T η -1 a η u η (x) • curl y ϕ(x,z, y) z=x/aη,y=x/η dx. (7.42) Setting in (7.42) ϕ = ϕ(x, z), we get T Z v 1 (x, z, y) y • ϕ(x, z)dzdx = T Z v 0 (x, z, y) y • curl z ϕ(x, z)dzdx.

v 1

 1 (x, z, y)curl z u 1 (x, z) • ϕ(x, z, y)dydzdx = T Z Y u 0 (x, z, y) • curl z ϕ(x, z, y)dydzdx + lim η→0 T η -1 a η u η (x) • curl y ϕ(x, z, y) z=x/aη,y=x/η dx. (7.43)Finally, setting in (7.43) ϕ(x, z, y) = ∇ y ψ(x, z, y), where ψ ∈ C ∞ per (T × Z × Y ), and using the identitycurl z ∇ y ψ = -curl y ∇ z ψ, yields T Z Y v 1 (x, z, y)-curl z u 1 (x, z) •∇ψ(x, z, y)dydzdx = T Z Yu 0 (x, z, y)•curl y ∇ z ψ(x, z, y)dydzdx,

T Z Y ε - 1

 1 (x, y)v 0 (x, y, z) • curl z ϕ 0 (x, z) + curl y ϕ 1 (x, z, y) dydzdx+ T Z u 0 (x, z) • ϕ 0 (x, z)dzdx = T Z f (x, z) • ϕ 0 (x, z)dzdx. (7.44)Via a standard argument, we deduce from (7.44) that u 1 satisfies the equation (5.23) for almost every x ∈ T.Notice that u η ⇀ ⇀ ⇀ u 0 + u 1 clearly implies u η ⇀ ⇀ u 1 , where the two-scale convergence is taken with respect to the variables x and z, and similarly a η curlu η ⇀ ⇀ curl z u 1 . Arguing as in the proof of Theorem 1, we prove that lim supη→0 T u η (x) 2 dx ≤ T Z u 1 (x, z) 2 dzdx, hence u η → → u 1 .The statement concerning the strong two-scale convergence of the operators S η follows.

We say that ση converges to σ in the Hausdorff sense if max{supλ 1 ∈ση inf λ 2 ∈σ |λ 1λ 2 |, sup λ 2 ∈σ inf λ 1 ∈σ η |λ 1λ 2 |} → 0 as η → 0.

The integrals with respect to the spectral projections are understood in the sense of Stieltjes, see e.g.[START_REF] Riesz | Functional Analysis[END_REF].

We define H -1 as the space of linear bounded functionals on Hper(curl, T), and •, • H -1 ,H per (curl,T) denotes the duality between H -1 and H per (curl, T).

The space L 2 pot (Y ) is defined as the closure of the set {∇ϕ : ϕ ∈ C ∞ per (Y )} in the norm of L 2 (Y ).

Here we make use of the fact that three-scale convergence possesses a compactness property analogous to that for two-scale convergence, cf. the text after Definition 1.

Acknowledgements

KDC would like to acknowledge the EPSRC financial support, on the project EP/C52392X/1. SG was funded by Centre National de la Recherche Scientifique (CNRS), and he also received support from the EU-funded project PHOREMOST under Grant FP6/2003/IST/2-511616.


Louis Adolphe

Instrument making in Lyon and Paris around 1600

Keywords:

The recent meeting on regional schools of instrument making, on the occasion of the scientific research done on the instruments preserved in the Cathedral of Freiberg, seems to be a sort of Ŗnew waveŗ of interest for this subject. In any case it is a good occasion to focus on two French centres and to present what we know about them today.

Biographical data concerning the makers

Paris and Lyon: first research by scholars at the end of the nineteenth century I will start in 1893, a date when were published successively Constant Pierreřs Ŗclassicalŗ book Les facteurs d'instruments de musique 1 , and at the same time Henry Coutagneřs fundamental study on Gaspard Duiffoproucart et les luthiers lyonnais du XVI e siècle, 2 a booklet preceded one year before by Albert Jacquotřs monograph on Pierre Woeiriot, 3 the famous engraver, author of the exceptional portrait of this maker. Constant Pierreřs book is devoted to France, with a strong predominance of Ŗfacts, remarks, anecdotes, critics, biographical notes, technical detailsŗ 4 on Parisian makers; Coutagneřs booklet, in the other hand, deals with regional aspects, focusing on one of the most prominent makers of all times, Ŗheroŗ of Lyon, emblematic figure of the European lutherie.

Organographie [START_REF]Lyon Ŕ String manufacturers and dealers (6) BELLET (BILLET)[END_REF] and Louis Antoine Vidal in 1874 in Les instruments à archet (vol. 1) [START_REF]Lyon Ŕ Keyboards (2 or 3) Ŕ Organs (5) 64 BB = Administration communale BONTEMPS, Nicolas[END_REF] Ŕ Pierre is the first scholar to make reference to the first hand source preserved in the Bibliothèque nationale. In Chapter III, Research on ancient makers (15th-18th centuries), only two pages concern makers active during the sixteenth century. All makers are organ builders working in the French provinces. None is active in Paris. The first Parisian makers listed in his book were worked around the middle of the century (Pierre Le Duc, Jacques Dumesnil, Simon Bongars, Pierre Aubry, Nicolas Chéron). Henry Coutagne, on the other hand, a scholar of Lyon, presented his research on Gaspard Duiffoproucart (also known as Kaspar Tieffenbrucker) for the ceremony admitting him to the Academy for Sciences, Literature and Art of Lyon in March 1893. The title of his historical study says that it contains Ŗsupporting documents and a portraitŗ, the main archives documents being transcribed Ŗin extensoŗ. With the usual rigour of archivists, he demonstrates, how, from a Ŗsubtle articleŗ written by Ernst Ludwig Gerber 8 , subtle because based only on elements inferred from the portrait engraved by Woieriot, a totally fantasist biography was built up, amplified, repeated, watered down by Jean-Baptiste Roquefort-Flamericourt [START_REF] Roquefort | Gaspard Dieffoproucart[END_REF] , François Castil Blaze, [START_REF] Blaze | Chapelle-musique des rois de France[END_REF] François-Joseph Fétis, [START_REF] Fétis | Antoine Stradivari et recherches sur l'origine et les transformations des instruments à archet[END_REF] Jules Gallay, [START_REF] Gallay | Les luthiers italiens aux XVII e et XVIII e siècles[END_REF] Antoine Vidal, [START_REF] Vidal | Les instruments à archet[END_REF] Albert Jacquot [START_REF] Jacquot | La musique en Lorraine, étude rétrospective d'après les archives locales[END_REF] and Gustave Chouquet, 15 to quote only French musicographers. They all contributed to build the legend and to obscure the Ŗheroicŗ career of the famous maker. Some of them pretended that Duiffoproucart was commissioned by François Ier in 1515 in Bologna to build the set of instruments for his chapel in Paris; and because he could not stand the cold and foggy weather of the capital, he asked to retire to Lyon! Coutagne, in return, starts with an iconographical and organological analysis of the makerřs portrait, brings to light a dozen mentions and comments about his lifestyle, his situation in the city, and important moments in his career (the documents are related to taxes on imported wine; acquisition of land; citizenship etc.). Probably because he does not have a commercial interest in lutherie, Coutagne feels free to emphasise that Roquefort was not only the first charlatan biographer but also the first to list instruments attributed to Duiffoproucart. A socalled bass gamba, sold to Vuillaume (later part of the Musée Instrumental du Conservatoire de Bruxelles [START_REF] Mahillon | Catalogue descriptif et analytique du Musée Instrumental du Conservatoire royal de musique de Bruxelles[END_REF] ), copied by him in 1827, was the starting point for the increase in forgeries. Coutagne then discusses (pp. 34Ŕ41) with discernment the authenticity of three bass gambas and one violin (for more than a century present in the Paris museum), adding with perspicacity:

ŖWe react above all against the easy way used by certain authors to build the history of sixteenth century lutherie on the basis of apocryphal makers and dates. We want to stress that doubts were already expressed before us about that point by Vidal and Hart. In fact, authentic violins from the sixteenth century are so few, that they are quite unobtainableŗ. 17 Coutagne demonstrates that Duiffroproucart is related to lute makers of the Tieffenbrucker family quoted by Ernest Gottlieb Baron in his treatise, 18 a dynasty with many instruments preserved and shown in 1892 in the Musical and Theatrical Exhibition organised in Vienna 19 . With pragmatism, he points out the fact that Ŗone can figure a priori that an artist of this importance settled in a centre where instrument making was flourishing […] a hypothesis confirmed by research carried out in local archivesŗ. 20 Thanks to archival documents, he offers us also an amount of first hand biographical data on a dozen instrument makers: Nicolas Bontemps, Honoré de Loeuvre, Benoît Lejeune, André Vinatte, Jehan Helmer and Philippe Flac, these last two of German origin and stringed instrument makers; Pierre Le Camus, Simon, Mathelin de La Noue, Luc Gentil and François Furet specialised in wind instruments. With intuition and pragmatic deductive demonstration, Coutagne underlines that violin making is claimed in Lyon, thanks to archival documents, no earlier than 1548 (before that date only Ŗrebecsŗ and Ŗrebequetsŗ Ŕ rebec players Ŕ are mentioned); he deduces from this that Duiffoproucart cannot be considered as the Ŗinventorŗ of the violin in France, a legend still alive among luthiers during the twentieth century! Concerning the origin and development of the violin in Europe, he is convinced that Ŗwe need to extend the geographic field of research concerning instrument making, obstinately confined to Italyŗ. 21 Given that three makers among the professionals working in Lyon are of German origin, he means that everything suggests that, if our craftsmen come from foreign schools, we should seek their headquarters in the countries of the North, not the South. Can we conclude that Coutagne already had an intuition concerning the importance of the Lech valley and of Füssen for the dissemination of European lutherie? Six years after these Ŗrevelationsŗ, so often used and quoted, Georges Tricou continued this pioneering work and, from Coutagneřs notes, published new documents, especially about the Layolle dynasty, a family of organ makers, and about the Rafi dynasty, Ŗfleustierŗ (flute makers). The collection of documents is copious, giving all the essential biographical elements known today about this latter family. Tricou only knew of one flute by Rafi, Ŗbearing the authorřs markŗ, a specimen coming from the Correr-Contarini collection, belonging already at that time to the Musée Instrumental du Conservatoire de Bruxelles. 22 It is mainly on aspects concerning other specimens, their localisation and technical analysis, that studies about Rafi have progressed during the past decades.

Paris: a new generation of archivists (1950Ŕ1975)

After these pioneering works on Lyon, it was not until the post-world war era that new generations of palaeographers focused on Paris instrument makers. François Lesure, a renowned musicologist, played the leading role. With a multiple training at the Sorbonne, the Conservatoire of Paris and the École des Chartes, this music archivist was also interested in art sociology (he studied under Pierre Francastel). His Mémoire de l'Ecole pratique des Hautes Etudes (at the Sorbonne) completed in 1948, is an Introduction to the study of instrumental music in France during the sixteenth century, with a study on Instrument makers and instrument making in Paris from 1544 to 1610. 24 From that work, published in 1954 in the very new Galpin Society Journal, came his study on La facture instrumentale à Paris au XVI e siècle. 25 Almost nothing has been done since that time from first hand sources (in the guild registers and notarised minutes) about this early period of Parisian instrument making. After the book of references published by Jules Ecorcheville in 1907 on civil certificates concerning musicians, 26 Lesure works with a similar methology to that of Coutagne sixty years earlier, publishing full documents, giving an interpretation of them, as we will show later, and providing many references (in his Note on violin making during the sixteenth century, he indicates that he could gather about 50 new names of makers). 27 Between 1960 and 1973, a group of female scholars, inspired by Norbert Dufourcq (we all know his classical publications on French organ and musical life under the Bourbon kings), regularly published new collected sources about Parisian musicians in the journal Recherche sur la musique française.

Among these musicians there is an interesting number of instrument makers. Probably not easily accessible to foreign colleagues, I would like to mention the useful work done by Yolande de Brossard (1965) 28 , Colombe Samoyault-Verlet (1966) 29 and Bernadette Gérard (1973).30 Madeleine Jurgens, an archivist palaeographer who worked in the Archives nationales, offered during the same period, in 1967 and 1974, two monumental volumes, each one with more than 1 000 pages, on Documents du Minutier central [des notaires] concernant l'histoire de la musique. 1600-1650. 31 Impressive compendia of documents preserved in 20 Parisian Notarial offices (among 72), and produced between 1600 and 1650, they constitute a gigantic store of information, too few exploited, even if published. They are presented as references, together with an abstracted analysis. A selection of documents is partly transcribed. The third part of each book is devoted to instrument makers. We find here a chronological list of professional acts (for example apprenticeship contracts, 18 concern the period interesting us here). They are completed by private acts, alphabetically presented (248 acts for 72 makers for the period around 1600). The same work is done about organ makers and clock founders, completed by certain contracts entirely published. String makers (4 for our period) and instrument dealers are also quoted. Among these musicians there is an interesting number of Wind s carried out for organ makers and clock founders, supplemented by certain contracts published in full. String makers (four for our period) and instrument dealers are also quoted. In the last part of her volume, Madeleine Jurgens introduces us to the musical taste of Parisian amateurs, with a table giving an overall picture of instruments in the possession of private people, adding a chronological list of instrument collections and musical libraries. Between 1600 and 1630, she lists about 100 houses possessing musical instruments. One can imagine the amount of knowledge available through these two volumes. A third one, although written, was unfortunately never published but is available at the Minutier Central in Paris (Archives nationales).

Lyon: academic secondary reference works (1978Ŕ1992) Making extensive use of the work published by Coutagne and Tricou, several students prepared doctoral theses on the theme of musical life in Lyon. Bernard Faurie presented his own work at the Sorbonne. 32 He also examined the published inventories of the municipal archives of Lyon 33 and the inventory of the records of the Lyonřs charitable institutions. 34 These directories do not provide many more names of instrument makers, but do provide many more deeds for them. We should note that this thesis was never published or exploited. It is also ignored by Frank Dobbins in his reference book on Music in Renaissance Lyons. 35 In the part of his book entitled Musical Activity in Lyons, Dobbins includes a chapter on Instrument makers [mentioned in the Archives of Lyons]. A close examination of that chapter shows that it is a compilation of references, not the publication of the documents themselves. He also worked on second-hand references, adding to Coutagne and Tricou sources listed in the Inventaires sommaires des archives quoted above and exploited by Bernard Faurie. Dobbins lists more makers because he also had access to seventeen Ŗunnumbered manuscript notes in the possession of François Lesureŗ from the archives of Georges Tricou. Apart from these quotations, presented in a clear systematic list, and introduced by a short summarised analysis, Dobbins proposes no organological approach on preserved instruments. Concerning Gaspard Duiffoproucart, he quotes the bass gamba Ŗsigned by himŗ in the Donaldson collection 36 (currently in London, in the museum of musical instruments at the Royal College of Music), and the specimen in Brussels 37 both very controversial instruments ascribed to him. As for Claude Rafi, he refers to Mahillonřs catalogue, 38

Profession and production

What follows is an assessment of the historical interest of documents gathered by these scholars.

Free trade and guilds

In the case of Paris we have to accept with François Lesure that Ŗup to the very end of the sixteenth century, makers in the capital worked under the regime of a free trade.ŗ 42 Paris, a city of jurymen (elected representatives of a guild), had still many unorganised trades, but the royal administration, mentioning instrument makers in the list of Ŗmediocreŗ (poor) trades, urged them to propose a guild. Lesure considers that the statutes accepted by the king in 1599, Ŗare officially establishing practices already and for a long time in use.ŗ 43 Unfortunately we have no idea how this guild worked because no archives have been preserved before the end of the seventeenth century. In the case of Lyon, we have to consider that this highly commercial city, at a crossroads in Europe, with its three or four fairs a year, was a major route for exchange between Italy, Lorraine, Flanders, Bourgogne and the Ile-de-France. Claude de Rubys, in his work on the Ŗprivilèges des habitans de Lyonŗ confirms that the free statute of the city allowed anybody and even foreigners, to become a citizen and to trade, without controlled access:

Circulation of models

What kind of models were offered by makers? As pointed by Edouard Martin in his study on Lyon, the Ŗeleven makers active in Lyon come from countries having commercial and financial activities with the city, and an early culture for instrument making. Five German makers come from Füssen, and six Italian makers are from Northern Italy.ŗ 46 Lesure gives us many details about models and their circulation: In workshop inventories published by him in 1954, the following instruments are listed: 63 lutes from Padua, 17 from Venice, 24 violins from Brescia, 15 lots of strings of Firenze, 21 of Siena.[This is not a quotation by a synthesis made by myself. ]

In Claude Denisřs workshop in 1587, a Parisian maker who ordered 200 Italian lutes through Simon de Luz in Lyons, we find Ŗ14 lutes from Lyons, 35 lutes from Padua, 9 lutes from Venice, and 4 guitars from Lyonsŗ. In Robert Denisřs workshop, his son, there is two years later Ŗan old lute by Gaspart [Tieffenbrucker]ŗ and after the death of Pierre Aubry in 1596, Ŗ5 old lutes by Gaspart and 2 from Lyons.ŗ [START_REF] Boulanger | Ŗrebecquetŗ taxed in 1545[END_REF] The different Ŗfashionsŗ are also quoted: Ŗfrom Cambraiŗ for rebecs (1551) and violins (1587), from Flanders and Germany for lutes (1589), from Spain for guitars (1587), from Padua for lutes (1580), from Paris (for a majority of instruments), from Cremona and Brescia for violins.

Lesure underlined that it is not very easy to Ŗevaluateŗ the real nature of these fashions. Firstly, because no stringed instrument have been preserved from that period and these areas (the first Parisian guitar preserved is dated 1641 [START_REF] Jeune | Faiseur de luths[END_REF] and the first Parisian harpsichord 1648 49 ) 44 Claude de Rubys, Mémoire de l'histoire de Lyon par Guillaume Paradin [ŖPrivilèges, franchises et immunisez octroyées par les rois très-chestriens aux consuls, échevins, manans et habitans de la ville de Lyon et à la postéritéŗ], Lyon 1573, p. 48. 45 Nicolas de Nicolay, Description générale de la ville de Lyon et des anciennes provinces du Lyonnais & du Beaujolais, Lyon 1573, reprint Lyon 1881, ch. XX, p. 161. Quoted by Faurie, La musique à Lyon au XVI e siècle, p. 199 (see footnote 28). 46 Martin, Gaspard Tieffenbrucker et la lutherie lyonnaise (see footnote 36), p. 31. [START_REF] Boulanger | Ŗrebecquetŗ taxed in 1545[END_REF] Lesure, La facture instrumentale à Paris (see footnote 25), pp. 37, 39-41, 42-43. [START_REF] Jeune | Faiseur de luths[END_REF] It is signed by René Voboam. See Florence Gétreau, Recent research about the Voboam family and their guitars, in: Journal of the American Musical Instrument Society 31 (2005), pp. 23Ŕ26. thus depriving us of morphologic regional characteristics. Secondly, musical iconography and treatises are not sufficiently precise to provide complementary elements. But quotations in inventories sometimes do provide some more information on the technology or musical use of instruments: Ŗtroys fleustes dřallumens dont deux coupées [à deux corps?]; quatre autres piffres lřun grand et les troys aultres petitz; troys musettes et sept petitsŗ (1544); Ŗtroys flutes à tabourin dont deux petites et une grandeŗ (1551); Ŗun haultboys de moyen ton avec ung autre du clerc tonŗ (1553); Ŗdeux cistres, lřune grande vernye et lřaultre petite; deux tailles, deux dessus et une basse taille de cornetzŗ (1557); Ŗcinq poches de cane et deux aultres marquettées carréeŗ (1589); Ŗun luth en fasson de Turebe [tuorbe] à deux ransŗ (1606).

[Ŗthree German flutes, among them two Řcut downř (in two pieces?); four other fifes, one long, three others short; three musettes and seven smallŗ (1544); Ŗthree flutes for tabor, among them two small and one largeŗ (1551); Ŗa shalme of medium tone and another one of clear toneŗ (1553); Ŗtwo citterns, one large and varnished, the other one small; two tenor, two treble, and one bass tenor cornettoŗ (1557); Ŗfive kits in cane wood and two others with squared marquetryŗ (1589); Ŗa lute fashioned like a theorbo with two rangesŗ (1606).] [START_REF] Vanelle | 1529 ŖGraveurŗ; 1530 ŖFaiseur de trompettesŗ[END_REF] A forged lute, Ŗin the manner of Gaspartŗ [Ŗun luth contrefaict fasson de Gaspartŗ] (1617) 51 recall for us the lawsuit between 1557 and 1559 brought by Gaspard Duiffoproucard and his contemporary and rival Le Jeune, who used or imitated Duiffoproucardřs makerřs mark (these documents were already published by Tricou in 1903 52 ). It means that questions of paternity, authenticity and forgery were already part of commercial life during that period.

Some more examples, published by Madeleine Jurgens, selected during the three first decades of the seventeenth century, complete this panorama of models and fashions: Ŗune grande espinette organiséeŗ (1607); Ŗune grande violle de la façon dřAngleterre; dix violles, tant basses taille, basses, haulte conte que dessus, garnies de leurs archetz ; un luth painct à ramaige; un clavesin couvert de maroquin rougeŗ (1622); Ŗun clavesin, fason de Venise, garny de son tréteau; un jeu de violle à huict partye, facon de Paris; une cornemuse couverte de velour vert, garny de son souflet (type musette de cour); une pendore dřAngleterreŗ (1626); Ŗneuf luth tant de Padoue, de Gaspard [Duieffoprucard] et autres de diverses grandeurs et deux grands tuorbesŗ (1629).

[Ŗa large organised spinetŗ (1607); Ŗa large viol made in the English fashion; ten viols, tenor, bass, alt and treble with their bows, a painted lute with floral design; a harpsichord covered with red leatherŗ (1622); ŖA harpsichord, fashioned as in Venice, with its stand; a set of viols with eight parts, fashioned as in Paris; a bagpipe covered with green velvet with its bellows; an English pandoraŗ (1626); Ŗnine lutes from Padua, by Gaspard [Duiffoproucard], and others of different sizes and two large theorbosŗ (1629)]. 53 Duiffoproucartřs portrait; Raffiřs flutes and recorders Earlier than Nancy, and even more early than Mirecourt, with an incomparable development, Paris (with an intensive musical life), Lyon (with a large commercial and financial influence) were the two main centres of French instrument making. But apart from documents, we have no tangible evidence of this activity. In the absence of authentic instruments, one can understand why Gaspard Duiffoproucardřs fame passed down to posterity thought the famous, magnificent and so symbolic portrait, engraved by Woieriot in 1562 (figure). We have traced, in another context, [START_REF] Gétreau | L'image du faiseur d'instruments de musique à la Renaissance [Gaspard Duiffoprucart] (Actes des journées d'étude Les images de la musique à la Renaissance[END_REF] its rich iconography and its singularity among the rare images of instrument makers. In the same way, we remain fascinated by the reputation of the second prominent maker of Renaissance Lyon, Claude Rafi. Praised by three contemporary poet friends, one can understand that, as for Duiffoproucard, the eulogy is addressed to the elite of the makers. To the sole flute quoted by the Ŗpioneersŗ at the end of the nineteenth century, six more have been progressively listed by different scholars and studied recently by Philippe Allain-Dupré, 55 while three recorders were extensively documented in their historical, sociological and technological context by Frank Bär in 1995. 56

Conclusion

Since Madeleine Jurgens and, later, François Lesure passed away, no scholar, to my knowledge, has carried on this exceptional work of documentation to the extent they did. Even recently, in an important book on Music at the Court of François Ier, Christelle Cazaux confessed that even though François Lesure encouraged her to undertake a systematic work in the Minutier Central des notaires in Paris as Madeleine Jurgens did, she could not have committed herself to this huge task in the context of her study. [START_REF] Cazaux | La musique à la cour de François I er[END_REF] As a homage to them both, Lesureřs words may appropriately conclude this paper: ŖDocuments […] provide much more knowledge concerning the specialisation of makers and the instrumental practice of Parisian musicians. But we hope to convince scholars that publishing such inventories in Northern Italy, in Flanders, in Germany and in Great Britain may help to approach the historical role of each regional school during this dynamic period of instrument makingŗ.

40

  Edouard Martin, Gaspard Tieffenbrucker et la lutherie lyonnaise du XVI e siècle dans les courants d'échanges européens, Paris, 2003 (Mémoire de maîtrise dřhistoire économique et sociale, Université de Paris I) 41 Ibid., p. 6. 42 Lesure, La facture instrumentale à Paris (see footnote 25), p. 11. 43 Ibid., p. 11.ŖCar les mestiers de ladicte ville, hors les orfèvres, les barbiers, & les serruriers, ne sont jurez comme en plusieurs autres villes de ce royaume: & ce à lřoccasion des privilèges des foires & de la liberté du commerce, par vertu de laquelle est loisible à gens de toute sorte & de toutes nations, de venir librement habiter en ceste ville, pour y exercer leurs trains, trafics ou mestiers, sans estre astrains à aucune maistrise, acte dřexperience, ou chef dřoeuvre qui ne leur servirait qaue de despence, & longueur, & leur serait plus inutile & sans fruict, quand ils sřen voudroyent retourner en leur pays.ŗ 44 Nicolas de Nicolay, in his Description générale de la ville de Lyon, confirms that musical instrument production is part of the Ŗmerchandise usually manufactured in the city […]ŗ: ŖPlus sřy font violes, violons, cistres, guiterres, leutz, fleuste dřallemand, fleustes à neuf troux, hautbois et cornemuses et autres sortes dřinstrumentz.ŗ [ŖViols, violins, citterns, guitars, lutes, flutes, recorders with nine holes, oboes and bagpipes and all sort of instruments are also produced.ŗ] 45

Fig. Pierre

 Pierre Fig. Pierre Woeiriot de Bouzey, Portrait de Gaspar Duiffoprucar, Lyon 1562

  Catalogue descriptif et analytique (see footnote 16), vol. 2, p. 309, no. 1066. 23 See in our selected bibliography, Lesure, Byrne, Puglisi, Bär, Allain-Dupré.

	23
	17 Coutagne, Gaspard Duiffoproucart (see footnote 2), p. 55 : ŖNous réagirons surtout contre la facilité avec
	laquelle certains auteurs ont édifié lřhistoire de la lutherie du XVI e siècle sur des instruments de fabricants et de
	dates aprocryphes. On ne saurait trop insister sur les doutes émis déjà avant nous sur ce point par Vidal et Hart.
	En réalité les violins authentiques du XVI e siècle peuvent compter parmi les pièces rarissimes, disons même
	presque introuvablesŗ.
	18 Ernst Gottlieb Baron, Historische theoretisch und practische Untersuchung des Instruments der Lauten,
	Nuremberg 1727, ch. VII, pp. 88Ŕ98.
	19 Fach-Katalog der musik-historischen Abteilung von Deutschland, Wien, 1892.
	20 Coutagne, Gaspard Duiffoproucart (see footnote 2), p. 46.
	21 Ibid., p. 61.
	22 Mahillon,

  considering only one of the preserved flutes and quoting Byrneřs article Instruments by Claude Rafi in the Collection of Manfredo Settala. 39 32 Bernard Faurie, La musique à Lyon au XVI e siècle. 1528-1574, 3 vol., Paris 1978 (Thèse de 3 ème cycle, Université de Paris IV-Sorbonne). 33 Fortuné Rolle, Inventaire sommaire des archives communales antérieures à 1790, I, Paris 1865; II, Paris 1875; Fortuné Rolle, Marie-Claude Guigue, Georges Guigue and Joseph Vaesen, Inventaire sommaire des archives communales antérieures à 1790, III, Paris 1887. 34 André Steyer and Fortuné Rolle, Inventaire sommaire des archives hospitalières antérieures à 1790 (Ville de Lyon, La Charité ou Aumône générale), 5 vol. (Alphabetical Index in vol. 5), Lyon 1874Ŕ1880. 35 Frank Dobbins, Music in Renaissance Lyons, Oxford 1992. 36 Catalogue of the Musical Instruments and objects forming the Donaldson Museum, London 1894. 37 Mahillon, Catalogue descriptif et analytique (see footnote 16). 38 Ibid., vol. 2, p. 309, no. 1066. 39 Maurice Byrne, Instruments by Claude Rafi in the collection of Manfredo Settala, in: The Galpin Society Journal 18 (1965), pp. 126Ŕ127. Following in his tracks, Edouard Martin, a student at the Sorbonne, achieved a Masterřs degree in 2005 in Economic and Social History on Gaspard Tieffenbrücker et la lutherie lyonnaise du XVI e siècle. 40 He gathered all previous publications, but he also discovered that Gaspard Duiffoprucart was already established in Lyon five years earlier than indicated by Coutagne and his followers. In 1548 he rented a house from Maître Pierre Pelu, stonemason, citizen of Lyon. Gaspard is called ŖHonorable homme Gaspard Duifobrocard faiseur de luthŗ. He is indeed quoted in all documents as lute maker. 41 The updating of dictionaries The table presented here as an appendix with my selected bibliography on instrument making in Lyon and Paris shows to what degree dictionaries have or have not integrated publications produced by archivists. 1. Vannes. For Lyon, he confused the works by Coutagne and Tricou. For Paris he took into account in his second edition the publications by Lesure. 2. Lütgendorff/Drescher. For Paris, the new edition takes into account all the bibliographical references published in Recherches sur la musique française and by Madeleine Jurgens that I recommended to Thomas Drescher at the end of the 1980s. For Lyons, it lacks Dobbinsřs book published two years after this volume. 3. Langwill/Waterhouse. Dobbinsřs work is fully integrated, even though issued only one year before. 4. Boalch/Mould. Dobbinsřs work is ignored, although published three years before.

58 Appendix 1 Instrument makers quoted in the main references of our selected bibliography 1500-1630

	Paris
	Author	Wind instruments Stringed	Keyboards	Unspecified
			instruments		instruments
	1893 Pierre	0	0	0		0
	1954 Lesure	4	32	10		1
	1966 Samoyault			36
	1970 Boalch			40
	1951Ŕ59 Vannes		34
	1990 Drescher		79
	1993 Waterhouse 2
	1995 Mould			38
	Lyon
	Author	Wind	Stringed	Keyboards	Stringmakers	Unspecified
		instruments	instruments	and organs	and	instruments
					dealers
	1893 Coutagne	2	11	2		5
	1899 Tricou	3	5	2		6
	1970 Boalch			3
	1951Ŕ59 Vannes		22
	1978 Faurie	5	6	4
	1992 Dobbins	10	19 or 20	7 or 8	6	13
	1993 Waterhouse 8
	1995 Mould			3
	2004 Martin	11	19	2		8

Appendix 2 Instrument makers in Paris and Lyon, 1500-1630 from our selected bibliography 1

  . Paris Ŕ Wind instruments (4) FRELLE,Pierre (fl. 1542). Documented as a [wind] maker in Paris, sending his son Jacques to serve as apprentice to LA NOUE in Lyon.Lesure 1954, p. 21. Langwill/Waterhouse, p.123. LA CANESSIERE, Philippe de (fl. 1542Ŕ1551). 1551 Inventory after his death. Produced Ŗflûtes, hautbois, sacqueboutes, tabourinsŗ.Lesure 1954, p. 22. LA NOUE, Mathurin de (fl. 1518Ŕ1544). 30 September 1542, contract to turn a Ŗjeu de flûtesŗ; 1544, inventory after his death.Lesure 1954, pp. 20Ŕ21; Langwill/Waterhouse, p. 224. THOMASSIN, Jean (1542). Apprenticed to Mathurin de La Noue, wind instrument maker in Lyon.Lesure 1954, 19.

Louis Adolphe Le Doulcet, comte de Pontécoulant, Organographie. Essai sur la facture instrumentale.Art, industrie et commerce, Paris 1861, reprint Amsterdam 1972.

Louis Antoine Vidal, Les instruments à archet […] suivi d'un catalogue général de la musique de chambre, Paris 1876Ŕ1878.

François Lesure, Introduction à l'étude de la musique instrumentale en France au XVI e siècle. Les facteurs d'instruments de musique et la facture instrumentale à Paris de 1544 à 1610, Paris 1948.

François Lesure, La facture instrumentale à Paris, in : The Galpin Society Journal 7 (1954), pp. 11Ŕ34, reprint in: Musique et musiciens français du XVI e siècle, Genève 1976, pp. 63Ŕ103.

Jules Ecorcheville, Actes d'état civil de musiciens insinués au Châtelet de Paris (1539-1650), Paris 1907.

François Lesure, Notes sur la facture du violon au XVI e siècle, in: Revue musicale 38 (1956), pp. 30Ŕ34.

Yolande de Brossard, Musiciens de Paris. 1532-1792. Actes d'Etat civil d'après le fichier Laborde de la Bibliothèque Nationale, Paris 1965.

Colombe Samoyault-Verlet, Les Facteurs de clavecins parisiens: notices biographiques et documents(1550( - 1793( ), Paris 1966. .

 30 Bernadette Gérard, Inventaire alphabétique des actes relatifs aux musiciens parisiens conservés aux archives de Paris, in: Recherches sur la Musique française classique13 (1973), p.

181Ŕ213. 31 Madeleine Jurgens, Documents du Minutier Central concernant l'histoire de la musique[START_REF] Prin | Lütgendorff-Drescher[END_REF]( -1650( ), vol. 1, Paris 1967;; vol. 2, Paris 1974.

It is signed by Jean Denis. See Alain and Marie-Christine Anselm, Petit prélude à l'étude des clavecins français du XVII e siècle, in: Musique-Images-Instruments 2 (1997), p. 228.

great detail the Lettres de création du métier