Bérangère Delourme

Patrick Joly

Elizaveta Vasilevskaya

Existence of guided waves due to a lineic perturbation of a 3D periodic medium

Keywords: guided waves, periodic media, spectral theory AMS codes : 78M35, 35J05, 58C40

In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order ε > 0, is supposed to be small. We prove that, for ε small enough, shrinking the section of one line of the grating by a factor of √ µ (0 < µ < 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to ε) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as ε tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly.

Statement of the problem

Let ω 1 , ω 2 and ω 3 be three Lipschitz bounded domains of R 2 of same area (|ω 1 | = |ω 2 | = |ω 3 |) containing the origin (0, 0), let ε > 0 be a parameter (that is going to be small), and let a 1 , a 2 and a 3 be three positive real numbers. We denote by (e i ) i∈{1,2,3} , the standard basis of R 3 . For any (k, ) ∈ Z 2 , we consider the three dimensional domain D ε k, ,3 defined by

D ε k, ,3 = (x 1 , x 2 , x 3 ) ∈ R 3 such that (x 1 -a 1 k)/ε, (x 2 -a 2 )/ε ∈ ω 3 ,
which is an unbounded cylinder of constant cross section εω 3 . It is infinite along the e 3 direction (invariant with respect to x 3 ) and contains the point (a 1 k, a 2 , 0). Similarly, for any (k, ) ∈ Z 2 , we define the domains

D ε k, ,1 = (x 1 , x 2 , x 3 ) ∈ R 3 such that (x 2 -a 2 k)/ε, (x 3 -a 3 )/ε ∈ ω 1 , D ε k, ,2 = (x 1 , x 2 , x 3 ) ∈ R 3 such that (x 1 -a 1 k)/ε, (x 3 -a 3 )/ε ∈ ω 2 ,
and we consider the periodic domain Ω ε given by

Ω ε = i∈{1,2,3} (k, )∈Z 2 D ε k, ,i . (1)
The domain Ω ε is a three dimensional grating of equi-spaced parallel pipes (of constant cross section) oriented along the three orthogonal directions e 1 , e 2 and e 3 . It is a j -periodic with respect to x j , j = 1, 2, 3. Moreover, the points (ka 1 , a 2 , ma 3 ), (k, , m) ∈ Z 3 , belong to Ω ε .

In order to create guided modes, we introduce a linear defect (see [START_REF] Ammari | Guided waves in a photonic bandgap structure with a line defect[END_REF]-[5]- [START_REF] Fliss | A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides[END_REF]) in the periodic structure by modifying the section size of one pipe of the grating (it is conjectured that guided modes cannot appear in the purely periodic structure, see [START_REF] Friedlander | On the spectrum of a class of second order periodic elliptic differential operators[END_REF] for the proof in the case of a symmetric medium). More precisely, we assume that the domain D ε 0,0,3 is replaced with the domain where µ is a positive parameter. In other words, we enlarge (µ > 1) or shrink (0 < µ < 1) the section of one pipe of the domain by a factor µ (see Fig 1a). The corresponding perturbed domain is denoted by Ω µ ε . Its precise definition is given by

D ε,µ 0,0,3 = (x 1 , x 2 , x 3 ) ∈ R 3 such that x 1 /( √ µε), x 2 /( √ µε) ∈ ω 3 ,
Ω µ ε =   i∈{1,2} (k, )∈Z 2 D ε k, ,i     (k, )∈Z 2 \{(0,0)} D ε k, ,3   D ε,µ 0,0,3 . (2)
Ω µ ε is still a 3 -periodic with respect to x 3 . However, the presence of the perturbed pipe D ε,µ 0,0,0 breaks the periodicity with respect to x 1 and x 2 . We emphasize that the domain Ω µ ε (as well as Ω ε ) tends to a 3D periodic graph as ε tends to 0.

We look for guided modes, i.e. solutions to the wave equation ∂ 2 t u-∆u = 0 in Ω µ ε , satisfying homogeneous Neumann boundary conditions on ∂Ω µ ε (see [START_REF] Nazarov | On the spectrum of the Laplace operator on the infinite Dirichlet ladder[END_REF] for the investigation of the Dirichlet case), that propagate along the defect pipe D ε,µ 0,0,0 (i.e. in the e 3 direction) but stay confined in the transversal directions. More precisely, denoting by B µ ε the restriction of the domain Ω µ ε to the band

|x 3 | < a 3 /2, B µ ε = {(x 1 , x 2 , x 3 ) ∈ Ω µ ε such that |x 3 | < a 3 /2} , (3)
we search solutions of the form u(

x 1 , x 2 , x 3 , t) = v(x 1 , x 2 , x 3 )e iωt-βx3
, where β is a real parameter and v(x

1 , x 2 , x 3 ) ∈ L 2 (B µ ε ) is an a 3 -periodic function in x 3 .
In fact, it is easily seen that the β-quasiperiodic fonction v(x 1 , x 2 , x 3 )e -iβx3 is an eigenfunction of the operator

A µ ε (β) : D (A µ ε (β)) ⊂ L 2 (B µ ε ) → L 2 (B µ ε ) , A µ ε (β) = -∆u in B µ ε , (4)
with

D (A µ ε (β)) = u ∈ H 1 ∆ (B µ ε ) , u| Σ + = e -iβ u| Σ -, ∂ x3 u| Σ + = e -iβ ∂ x3 u| Σ -, ∂ n u| ∂B µ ε \Σ ±= 0 ,
where

H 1 ∆ (B µ ε ) = u ∈ H 1 (B µ ε ), s.t. ∆u ∈ L 2 (B µ ε ) and Σ ± = {(x 1 , x 2 , x 3 ) ∈ ∂B µ ε , x 3 = ±a 3 /2} .
To study the spectral properties of A µ ε (β), we investigate its (formal) limit A µ (β) as ε tends to 0. The operator A µ (β) is defined on the limit graph G (see Fig. 1b) and its spectrum can be explicitly computed. In particular, its spectrum has infinitely many gaps (Lemma 2.1), i.e. open intervals (a, b) ⊂ R such that the intersection of [a, b] with the spectrum is reduced to {a, b}. Moreover, for µ < 1, there is at least one eigenvalue in each gap (Lemma 2.5). Since, in addition, for ε > 0 sufficiently small, the spectrum of A µ ε (β) is close to the spectrum of A µ (β), the existence of guided modes is guaranteed (Theorem 3.1).

2 The spectrum of the limit operator A µ (β)

2.1 Definition of the limit operator A µ (β)

The limit operator A µ (β) is defined on the infinite periodic graph G = ε>0 B µ ε obtained as the limit of B µ ε as ε tends to 0: G is made of the vertices {v k, = (ka

1 , a 2 , 0), v ± k, = (ka 1 , a 2 , ±a 3 /2), (k, ) ∈ Z 2 } connected by the edges {e k+1/2, = (v k, , v k+1, ), e k, +1/2 = (v k, , v k, +1 ), e ± k, = (v k, , v ± k, ), (k, ) ∈ Z 2 }.
It is a 1 -periodic with respect to x 1 and a 2 -periodic with respect to x 2 (see Fig. 1b).

For any function u defined on G, we denote by u k, (resp. u ± k, ) its value at the vertex v k, (resp. v ± k, ). The restriction of u to the edge e k+1/2, (resp. e k, +1/2 and e ± k, ) is denoted by u k+1/2, (x 1 ) (resp. u k, +1/2 (x 2 ) and u ± k, (x 3 )).

The definition of A µ (β) also requires the introduction of the function spaces L µ 2 (G) and H 2 (G) defined as

L µ 2 (G) = u : u L µ 2 (G) < +∞ , H 2 (G) = u ∈ C(G) : u H 2 (G) < +∞ , (5)
where,

u 2 L µ 2 (G) = (k, )∈Z 2 w µ k ± u ± k, 2 L2(e ± k, ) + u k+ 1 2 , 2 L2(e k+ 1 2
, ) + u k, + 1 2 2 L2(e k, + 1
2

) , (6)
u 2 H 2 (G) = (k, )∈Z 2 ± u ± k, 2 H 2 (e ± k, ) + u k+ 1 2 , 2 H 2 (e k+ 1 2
, ) + u k, + 1 2 2 H 2 (e k, + 1 2
) ,

and, for any (k, ) ∈ Z 2 , w µ k, is the weight coefficient equal to µ for k = = 0 and 1 otherwise.

The unbounded limit operator in L µ 2 (G) has domain

D (A µ (β)) = u ∈ H 2 (G) : ∀(k, ) ∈ Z 2 , u + k, = e -iβ u - k, , (u + k, ) (a 3 /2) = e -iβ (u - k, ) (-a 3 /2) , u k+ 1 2 , (ka 1 ) -u k-1 2 , (ka 1 ) + u k, + 1 2 ( a 2 ) -u k, -1 2 ( a 2 ) + w µ k, (u + k, ) (0) -(u - k, ) (0) = 0 , ( 8
)
and is defined by

∀ u ∈ D (A µ (β)) , A µ (β)u = -u on any edge of the graph G. (9)
The functions of D (A µ (β)) are continuous on G and β quasi-periodic. Moreover, they satisfy the Kirchhoff conditions (8) that enforce the weighted sum of the outward derivatives of u to vanish at each vertex v k, ((k, ) ∈ Z 2 ). We point out that the perturbation, which results from a geometrical modification of the domain for the problem (4), is taken into account at the limit by means of the Kirchhoff condition (8) at the vertex v 0,0 (w µ 0,0 = µ). The formal derivation of the limit model can be found in [START_REF] Kuchment | Convergence of spectra of mesoscopic systems collapsing onto a graph[END_REF]. It is easily verified that the operator A µ (β) is self-adjoint (for the weighted scalar product associated with ( 6)), see also [START_REF] Kuchment | Quantum graphs: an introduction and a brief survey, Analysis on graphs and its applications[END_REF]. The objective of the following two sections is to study the spectrum of A µ (β).

Characterization and properties of the essential spectrum of A µ (β)

By a compact perturbation argument, one can prove that σ ess (A µ (β)) = σ(A(β)), where A(β) = A 1 (β) is the purely periodic operator corresponding to A µ (β) for µ = 1. The computation of its spectrum relies on the Floquet-Bloch theory (see [START_REF] Reed | Methods of modern mathematical physics v. I-IV[END_REF]). More precisely, we can prove that λ = ω 2 ∈ σ(A(β)) if and only if either ω = 0 and β = 0 or ω = 0 and there exists (k 1 , k 2 ) ∈ [0, π] 2 such that sin (ωa 2 ) sin (ωa 3 ) (cos (ωa 1 ) -cos k 1 ) + sin (ωa 3 ) sin (ωa 1 ) (cos (ωa 2 ) -cos k 2 ) + sin (ωa 1 ) sin (ωa 2 ) (cos (ωa 3 ) -cos β) = 0. [START_REF] Vasilevskaya | Open periodic waveguides. Theory and application[END_REF] Based on the previous characterization, we prove that the operator A(β) has a countable infinity of gaps that can be separated into three categories (see [START_REF] Vasilevskaya | Open periodic waveguides. Theory and application[END_REF] for the proof): Lemma 2.4 Assume that ω / ∈ {πZ/a 1 } ∪ {πZ/a 2 } ∪ {πZ/a 3 } and that λ = ω 2 / ∈ σ ess (A µ (β)). Then, λ is an eigenvalue of A µ (β) if and only if

µ = 1 -F β (ω) where F β (ω) = 1 4π 2 (0,2π) 2 φ β (ω) φ β (ω) -f (ξ, η, ω) dξdη -1 . ( 14
)
The study of the behavior of the function F β leads to the existence of at least one eigenvalue in each gap of A µ (β) as soon as µ < 1, the minimal number of eigenvalues in each gap depending on the type of gaps (cf. Lemma. 2.1-3 for the classification):

Lemma 2.5 For µ > 1, the operator A µ (β) has no eigenvalue. For 0 < µ < 1, let (ω 2 b , ω 2 t ) be a spectral gap of the operator A µ (β):

(a) If (ω 2 b , ω 2 t ) is a gap of type (i), then A µ (β) has at least two eigenvalues λ 1 = ω 2 1 and λ 2 = ω 2 2 that satisfy ω b < ω 1 < ω 0 < ω 2 < ω t (see Lemma. 2.1-3 for the definition of ω 0 ). (b) If (ω 2 b , ω 2 t ) is a gap of type (ii) or (iii), then A µ (β) has at least one eigenvalue λ 1 = ω 2 1 such that ω b < ω 1 < ω t .
The sketch of the proof of the previous lemma is the following, a complete proof being available in [START_REF] Vasilevskaya | Open periodic waveguides. Theory and application[END_REF] A similar argument works for a gap of type (iii).

+ b (1 -F β (ω)) = lim ω→ω - t (1 -F β (ω)) =
3 Guided modes for the operator A µ ε (β): an asymptotic result Finally, thanks to the general result [START_REF] Post | Spectral convergence of quasi-one-dimensional spaces[END_REF] (Theorem 2.13 convergence of the spectrum of A µ ε (β) toward the spectrum of A µ (β)), we can prove the following result of existence of eigenvalue for the operator A µ ε (β):

Theorem 3.1 Let µ ∈ (0, 1), (λ b , λ t ) be a spectral gap of the operator A µ (β) and λ 0 ∈ (λ b , λ t ) be a (simple) eigenvalue of this operator. Then, there exists ε 0 > 0 such that if ε < ε 0 the operator A µ ε (β) has an eigenvalue λ ε inside a spectral gap (λ ε b , λ ε t ). Moreover, λ ε = λ 0 + O( √ ε).

1 (a) The perturbed domain Ω µ ε 1 (

 11 b) The graph G.

Figure 1 :

 1 Figure 1: Illustration of the perturbed periodic domain Ω µ ε and the limit graph G.

1 and that lim ω→ω0 ( 1 -( 1 -( 1 -

 ω→ω0111 F β (ω)) = 0. By continuity of F β inside the gap, (a) directly results from the intermediate value theorem and (14). If (ω 2 b , ω 2 t ) is a gap of type (ii), the intermediate value theorem also permits us to conclude since lim ω→ω b F β (ω)) ≤ 0 and lim ω→ω - t F β (ω)) = 1.

  (Theorem 5.2.1): First, one can verify that F β (ω) ≥ 0 in any gap, which, together with (14) proves that A µ (β) has no eigenvalue for µ > 1. Then, if (ω 2 b , ω 2

t ) is a gap of type (i), one can show that lim ω→ω

Lemma 2.1 The following properties hold :

1-σ 1 ∪ σ 2 ∪ σ 3 ⊂ σ(A(β)), where σ i = (πn/a i ) 2 , n ∈ Z for i ∈ {1, 2}, and σ 3 = ((±β + 2πn)/a 3 ) 2 , n ∈ Z .

2-For any β ∈ [0, π], the operator A(β) has infinitely many gaps whose ends tend to infinity.

If an interval (ω 2 b , ω 2 t ) is a spectral gap of A(β), then, one of the following possibilities holds:

and there is a unique

Computation of the discrete spectrum

Let us now determine the discrete spectrum of A µ (β). If λ = ω 2 is an eigenvalue of A µ (β), then the corresponding eigenfunction u ∈ D(A µ (β)) satisfies the linear differential equation u + ω 2 u = 0 on each edge of the graph G.

Solving explicitly this equation (on each edge), taking into account the quasi-periodicity of u and the Kirchhoff conditions (8), we can replace the eigenvalue problem A µ (β)u = λu with a set of finite differences equations for

where we have defined g β (ω) = 1 tan (ωa 1 ) + 1 tan (ωa 2 ) + cos (ωa 3 ) -cos β sin (ωa 3 ) .

As well-known, finite difference schemes may be investigated using the discrete Fourier transform

where F is an isometry between 2 (Z 2 ) and L 2 ((0, 2π) 2 ). This, together with Lemma 2.2, provides the following characterization for the discrete spectrum of A µ (β):

where φ β (ω) = cos (ωa 3 ) -cos β sin (ωa 3 ) and f (ξ, η, ω) = cos ξ -cos (ωa 1 ) sin (ωa 1 ) + cos η -cos (ωa 2 ) sin (ωa 2 ) .

Under the assumption of Lemma 2.3, (10) can be written as f (ξ, η, ω) -φ β (ω) = 0. It follows that λ = ω 2 does not belong to σ ess (A µ (β)) if and only if, for any (ξ, η) ∈ [0, 2π] 2 , φ β (ω) -f (ξ, η, ω) does not vanish. As a consequence, as soon as λ = ω 2 / ∈ σ ess (A µ (β)), the function (ξ, η) → φ β (ω)/(φ β (ω) -f (ξ, η, ω)) is continuous and bounded. Then, the inverse discrete Fourier transform can be applied to (13) to obtain u k, =

(1 -µ)u 0,0 4π 2 (0,2π) 2 φ β (ω) φ β (ω) -f (ξ, η, ω) e -i(kξ+ η) dξdη, ∀ (k, ) ∈ Z 2 .

Writing the previous relation for k = = 0 yields the following criterion of existence of an eigenvalue: