Trapped modes in thin and infinite ladder like domains. Part 1 : existence results
The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter ) whose limit (as tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that is small enough. Numerical experiments illustrate the theoretical results.
Keywords: spectral theory, periodic media, quantum graphs, trapped modes
Photonic crystals, also known as electromagnetic bandgap metamaterials,
are 2D or 3D periodic media designed to control the light
propagation. Indeed, the multiple scattering resulting from the periodicity of
the material can give rise to destructive interferences at some range
of frequencies. It follows that there
might exist intervals of frequencies (called gaps) wherein the monochromatic waves
cannot propagate. At the same time, a local perturbation of the
crystal can produce defect mid-gap modes, that is to say
solutions to the homogeneous time-harmonic wave equation, at a fixed frequency located inside one gap,
that remains strongly localized in the vicinity of the
perturbation. This localization phenomenon is of particular interest
for a variety of promising
applications in optics, for instance the design of highly efficient waveguides [26, 27].
From a mathematical point of view, the presence of gaps is theoretically explained by the band-gap structure of the spectrum of the periodic partial differential operator associated with the wave propagation in such materials (see for instance [12, 32]). In turn, the localization effect is directly linked to the possible presence of discrete spectrum appearing when perturbing the perfectly periodic operator. A thorough mathematical description of photonic crystals can be found in . Without being exhaustive, let us remind the reader about a few important results on the topic. In the one dimensional case, it is well-known  that a periodic material has infinitely many gaps unless it is constant. By contrast, in 2D and 3D, a periodic medium might or might not have gaps. Nevertheless, several configurations where at least one gap do exist can be found in [17, 18, 25, 42, 44, 3, 29, 30] and references therein. In any case, except in dimension one, the number of gaps is expected to be finite. This statement, known as the Bethe Sommerfeld conjecture is fully demonstrated in [49, 50] for the periodic Schrödinger operator but is still partially open for Maxwell equations (see ). For the localization effect, [15, 16, 1, 33, 37, 45, 8, 9] several papers exhibit situations where a compact (resp. lineic) perturbation of a periodic medium give rise to localized (resp. guided) modes. It seems that the first results concern strong material perturbations : for local perturbation [15, 16] and for lineic perturbation [33, 37]. There exist fewer results about weak material perturbations : [8, 9] deal with 2D lineic perturbations. Finally, geometrical perturbations are considered in [40, 45], where the geometrical domain under investigation is exactly the same as ours but with homogeneous Dirichlet boundary conditions on the boundary of the ladder. As in our case, changing the size of one or several rungs of the ladder can create eigenvalues inside a gap (see also remark 7).
The aim of this paper is to complement the references mentioned above by proving the existence of localized midgap modes created by a geometrical perturbation of a particular periodic medium. We shall use a standard approach of analysis (used in [17, 44]) that consists in comparing the periodic medium with a reference one, for which theoretical results are available. To be more specific, we are interested in the Laplace operator with Neumann boundary condition in a ladder-like periodic waveguide. As the thickness of the rungs (proportional to a small parameter ) tends to zeros, the domain shrinks to an (infinite) periodic graph. More precisely, the spectrum of the operator posed on the 2D domain tends to the spectrum of a self-adjoint operator posed on the limit graph ([54, 39, 55, 51, 48]). This limit operator consists of the second order derivative operator on each edge of the graph together with transmission conditions (called Kirchhoff conditions) at its vertices ([14, 10, 39]). As opposed to the initial operator, the spectrum of the limit operator can be explicitly determined using a finite difference scheme ([2, 13]). From a mode of the limit operator, we construct a so called quasi-mode and we are able to prove that, for sufficiently small, the diminution of the thickness of one rung of the ladder gives rise to a localized mode. Moreover, diminushing , it is possible to create as many eigenvalues as one wants. We point out that the analysis of quantum graphs has been a very active research area for the last three decades and we refer the reader to the surveys [34, 35, 36] as well as the books [4, 52] for an overview and an exhaustive bibliography of this field.
2 Presentation of the problem
In the present work we study the propagation of waves in a ladder-like periodic medium (see figure (a)a). The homogeneous domain (we will call it ladder) consists of the infinite band of height minus an infinite set of equispaced rectangular obstacles. The domain is -periodic in one space direction, corresponding to the variable . The distance between two consecutive obstacles is equal to the distance between the obstacles and the boundary of the band and is denoted by .
Some extensions We can change the distance between 2 consecutive obstacles from to . The study will depend on this new parameter but its conclusions remain the same.
The aim of this work is to find localized modes, that is solutions of the homogeneous scalar wave equation with Neumann boundary condition
which are confined in the -direction.
Without giving a strict mathematical formulation (this will be done in the following section) a localized mode can be understood as a solution of the wave equation (1), which is harmonic in time
Problem (3) is an eigenvalue problem posed in the unbounded domain . In order to create eigenvalues, we introduce a local perturbation in this perfectly periodic domain (the delicate question of existence of eigenvalues (or flat bands) for the unperturbed problem is not addressed in this paper, see for the absence of flat bands in waveguide problems (or the absolute continuity of the spectrum) for instance [56, 22, 57] and for the existence  ). The perturbed domain is obtained by changing the distance between two consecutive obstacles from to , (see Figure (b)b in the case where ). It corresponds to modify the width of one vertical rung of the ladder from to .
As we will see such a perturbation does not change the continuous spectrum of the underlying operator but it can introduce a non-empty discrete spectrum. Our aim is to find eigenvalues by playing with the values of and , being treated as a small parameter.
Imitating the approach developped in this article, it is also possible to study sufficient conditions which ensures the existence of guided modes in a ladder-like open periodic waveguides. More precisely, the domain is minus an infinite set of equispaced perfect conductor rectangular obstacles with Neumann boundary conditions. And this domain is perturbed by a lineic defect, by changing the distance between two consecutive columns of obstacles. There exists a guided mode with a wave number if and only if there exists a localized mode in a perturbed periodic ladder where -boundary conditions are imposed. The results of the present paper can be extended and the sufficient condition which ensures the existence of guided modes remains basically the same.
3 Mathematical formulation of the problem
This section describes a mathematical framework for the analysis of the spectral problem formulated above. We introduce the operator , acting in the space , associated with the eigenvalue problem (3) in the perturbed domain:
Here . The operator is self-adjoint and positive. Our goal is to characterize its spectrum and, more precisely, to find sufficient conditions which ensures the existence of eigenvalues.
3.1 The essential spectrum of
To determine the essential spectrum of the operator , we consider the case , where the domain is perfectly periodic (see Figure (a)a). We will denote the corresponding operator . The Floquet-Bloch theory shows that the spectrum of periodic elliptic operators is reduced to its essential spectrum and has a band-gap structure [12, 53, 32]:
where, in the previous formula, the union disappears if . For , the intervals are called spectral gaps. Their number is conjectured to be finite (Bethe-Sommerfeld, 1933, [49, 50, 58]). The band-gap structure of the spectrum is a consequence of the following result given by the Floquet-Bloch theory:
Here is the Laplace operator defined on the periodicity cell (see Figure 4) with -quasiperiodic boundary conditions on the lateral boundaries and homogeneous Neumann boundary conditions on the remaining part of the boundary: for ,
For each the operator is self-adjoint, positive and its resolvent is compact. Its spectrum is then a sequence of non-negative eigenvalues of finite multiplicity tending to infinity:
In (6) the eigenvalues are repeated with their multiplicity. The representative curves of the functions are called dispersion curves and are known to be continuous and non-constant (cf. Theorem XIII.86, volume IV in ). The fact that the dispersion curves are non-constant implies that the operator has no eigenvalues of infinite multiplicity. Finally, (5) can be rewritten as
which gives (4). The conjecture of
Bethe-Sommerfeld means that for large enough the intervals
overlap or only touch.
and the operators have real coefficients, the function
are even. Thus, it is sufficient to consider
in (5). This will be
used systematically in the rest of the paper.
As expected (this is related to Weyl’s Theorem, see in [53, Chapter 13, Volume 4], [6, Chapter 9] and [15, Theorem 1]), the essential spectrum is stable under a perturbation of the thickness of one rung of the ladder. In the present case, the domains of definition of the resolvents of and are not the same. As a result, we cannot directly apply the standard results (in [6, Chapter 9]).
Let be a function such that
such that .
If is a singular sequence for the operator corresponding to the value , then there exists a subsequence of which is also a singular sequence for the operator corresponding to the value .
By definition of a singular sequence, the sequence has the following properties:
Let us show that there exists a subsequence of which has the same properties. The property 1 is verified by the whole sequence thanks to property (a) . To prove property 2, it suffices to show that there exists a subsequence, still denoted , such that
Then properties 3 and 4 imply that is bounded in . By compactness, one can thus extract a subsequence that converges weakly and strongly in to a limit which is necessarily thanks to property 3, which proves (7). For the sequel, we work with the above subsequence.
The property 3 being obvious the only thing to show is the property 4 for the sequence . We have:
The first and the last terms in the right-hand side tend to zero thanks to property 4 and (7)). Let us estimate the second term. Using first property (b), then properties (a) and (1) together with an integration by parts, we obtain
which tends to due to (7) since is bounded in as well as (by properties 3 and 4). ∎
Proof of Proposition 1.
It is sufficient to take a function in the previous lemma which does not depend on , vanishes in a neighbourhood of the perturbed edge and such that vanishes in a neighbourhood of all vertical edges. Then, it follows from Lemma 1 that any singular sequence associated to of the operator provides the construction of a singular sequence of the operator for the same and vice versa. ∎
The essential spectrum of the operator having a band-gap structure, we will be interested in finding eigenvalues inside gaps (once the existence of gaps is established).
3.2 Towards the existence of eigenvalues: the method of study.
Our analysis consists of three main steps.
First, we find a formal limit of the eigenvalue problem (3) when (Section 4.1). To do so, we use the fact that, as goes to zero, the domain shrinks to a graph . As a consequence, the formal limit problem will involve a self-adjoint operator associated with a second order differential operator along the graph. Its definition is strongly related to the fact that homogeneous Neumann boundary conditions are considered in the original problem. More precisely, at the limit , looking for an eigenvalue of leads to search an eigenvalue of . This operator, that is well known (see the works of [14, 10, 39]), will be described more rigorously in the next section.
The second step is an explicit calculation of the spectrum of the limit operator. The essential spectrum is determined using the Floquet-Bloch theory (by solving a set of cell problems) (Section 4.2) while the discrete spectrum of the perturbed operator is found using a reduction to a finite difference equation (Section 4.3). In particular, we shall show that the limit operator has infinitely many eigenvalues of finite multplicity as soon as (and no one when ), which form a discrete subset of .
Finally, when , we deduce the existence of an eigenvalue of close to the eigenvalue of as soon as is small enough (Section 5). The proof will be based on the construction of a quasi-mode (a kind approximation of the eigenfunction) and a criterion for the existence of eigenvalues of self-adjoint operators (see for instance Lemma 4 in ). It can be seen as a generalization of the well-known min-max principle for eigenvalues located below the lower bound of the essential spectrum.
An essential preliminary step is the decomposition of the operator as the sum of two operators, namely its symmetric and antisymmetric parts. To do so, we introduce the following decomposition of :
where and are subspaces consisting of functions respectively symmetric and antisymmetric with respect to the axis :
The operator is then decomposed into the orthogonal sum
Accordingly, the limit operator is decomposed as:
The key point is that, as we shall see, contrary to the full operator
whose spectrum is , both operators and have spectral gaps (an infinity of them), each of them containing eigenvalues : these are isolated eigenvalues for or , but embedded eigenvalues for
. One deduces that the operators
and have at
least finitely many spectral gaps, the number of gaps tending to when goes to 0: this is an important fact for applying the quasi-mode approach.
At this stage, it is worthwhile mentionning that the convergence of the spectrum of differential operators in thin domains degenerating into a graph is not a new subject, particularly in the case of elliptic operators. In particular, for the Laplace operator with Neumann boundary conditions and in the case of compact domains, the convergence results (which are reduced to the convergence of eigenvalues) have been known since the works of Rubinstein-Schatzman  and Kuchment-Zheng . Thanks to the Floquet Bloch theory, such results are transformed into analogous results for thin periodic domains (in [39, Theorem 5.1]), since in this case, only continuous spectrum is involved. For general unbounded domains, a general (and somewhat abstract) theory has been developed by Post in  for the convergence of all spectral components. This theory can be applied to our problem, however, for the sake of simplicity, we have chosen to use here a more direct approach (based on the construction quasi-modes).
4 Spectral problem on the graph
4.1 The operator .
As , the domain tends to the periodic graph represented on Figure 5. Let us number the vertical edges of the graph from left to right so that the set of the vertical edges is . The upper end of the edge is denoted by and the lower one by . The set of all the vertices of the graphs is then
The horizontal edge joining the vertices and is denoted by . The set of all the edges of the graph is
and we denote by the set of all the edges of the graph containing the vertex .
If is a function defined on we will use the following notation:
Let be a weight function which is equal to on the ”perturbed edge” , i.e. the limit of the perturbed rung , and to on the other edges:
Let us now introduce the following functional spaces
where denotes the space of continuous functions on :
We define the limit operator in as follows. Denoting the restriction of to ,
where stands for the derivative of the function at the point in the outgoing direction. The vertex relations in (13) are called Kirchhoff’s conditions. Note that they all have an identical expression except at the vertices . The following assertion as well as its proof can be found in [34, Section 3.3].
Proposition 2 (Kuchment).
The operator in the space is self-adjoint. The corresponding closed sesquilinear form has the following form:
As for the ladder, we introduce the following decomposition of the space into the spaces of symmetric and antisymmetric functions:
Again, the operator can be decomposed into the orthogonal sum
Thus, it is sufficient to study the spectrum of the operators and separately. The analysis of
these two operators being analogous, we will present a detailed study
of (Section 4.2 and
Section 4.3) and state the results for (Section 4.4).
4.2 The essential spectrum of the operator
We shall study the spectrum of the operator by a perturbation technique with respect to the case which corresponds to the purely periodic case. The corresponding operator will be denoted by . Indeed, based on compact perturbation arguments ( in [6, Theorem 4, Chapter 9]), we can prove the following proposition:
The essential spectra of and coincide:
This reduces the study of the essential spectrum of to the study of the spectrum of the purely periodic operator , which can be done through the Floquet-Bloch theory.
4.2.1 Description of the spectrum of through Floquet-Bloch theory
As previously explained, the spectrum of the operator can be studied using the Floquet-Bloch theory. One has then to study a set of problems set on the periodicity cell of . Since we consider the subspace of symmetric functions with respect to the axis , this enables to reduce the problem to the lower half part of the periodicity cell (see Figure 6).
We have then
where is the following unbounded operator in
In the definition of , the condition corresponds to the symmetry with respect to , (16)-(a) is the Kirchhoff’s condition with and (16)-(b) are the -quasiperiodicity conditions. For each , the operator is self-adjoint and positive and its resolvent is compact due to the compactness of the embedding . Consequently, its spectrum is a sequence of non-negative eigenvalues of finite multiplicity tending to infinity:
In the present case, the eigenvalues can be computed explicitly.
For , if and only if is a solution of the equation
If is an eigenvalue of the operator then the corresponding eigenfunction is of the form
Taking into account that , we arrive at the following linear system
The relations (22) express the continuity of the eigenfunction at the vertex . The equation (23) comes from the condition . The relation (24) corresponds to (16)-(a) while (25) and (26) correspond to (16)-(b). Adding and substracting (25) and (26) lead to which we can substitute into (22)–(24) to obtain the following system in
It is then easy to conclude since one obtains, after some computations omitted here
One then easily sees that there exists a non-trivial solution if and only if and that the corresponding eigenfunction is constant. Noticing that, for , is solution of (18) allows us to conclude. ∎
The reader will notice that when , the spectrum of has a particular structure: it is the image by the function of a periodic countable subset of . To see that, it suffices to remark that both functions at the left and right hand sides of (18) are periodic with a common period. As a consequence of (15), the spectrum of is the image by the function of a periodic subset of .
4.2.2 Characterization of the spectrum of
The following properties hold
, where and .
The operator has infinitely many gaps whose ends tend to infinity.
For or , the equation (18) is satisfied for so that belongs to .
Let (such that ), let us distinguish two cases:
(a) : the left hand side of equation (18) vanishes for all and, as , the right hand side does not. Then does not belong to the spectrum of . Since and belong to (in view of the point 1) there exists a gap which contains , strictly included in .
(b) : (this case can occur only for special values of , see remark 3), we know by point 1, that and we are going to show that it exists such that and are in the resolvant set of . This will show the existence of two disjoint gaps of the form and . Setting in relation (18) leads to
which cannot vanish for for small enough, since . This implies that for all .
The conclusion follows from the fact that the intervals are disjoint, go to infinity with , and contain one or two gaps.
The case 2.(b) of the above proof can occur only for special values of . Indeed, the reader will easily verify that the existence of such that is equivalent to the fact that
In fact, the condition (29) also influences the nature of the spectrum of . Indeed it can be shown that when does not belong to , the point spectrum of is empty (i. e. the spectrum of is purely continuous). When belongs to , it coincides with an infinity of eigenvalues of infinite multiplicity, associated with compactly supported eigenfunctions. It is worth noting that the presence of such eigenvalues is a specific feature of periodic graphs (see [35, Section 5]).
In the proof, in the case 2.(a), gaps are located in the vicinity of the points satisfying . These points are nothing else but the eigenvalues of the 1d Laplace operator defined on the vertical half edges with Dirichlet boundary condition at and Neumann boundary condition at . The presence of gaps is therefore consistent with [35, Theorem 5] dealing with gaps created by so-called graph decorations. Indeed, the vertical half edges can be seen as decorations of the infinite periodic graph made of the set of the horizontal edges .
where the functions and are defined by
In the following we reason in the -plane with an additional auxiliary variable. We introduce the domain
The domain is the domain of the -plane, -periodic with respect to , given by
The -periodicity of the domain with respect to follows from the identity . To conclude, it suffices to remark that, for a given , if varies in the interval is continuous and strictly decreasing from to while, if varies in the interval , is continuous and strictly decreasing from to . ∎
In other words, is the union of ,
, and the image by the application of the projection on the
line of the intersection of the domain
with the curve . Thanks to this geometrical
characterization, we shall be able to describe the structure of the
gaps of the operator .
Let us introduce the -periodic functions , such that, for any ,