1 Statement of the problem
###### Abstract

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 , is supposed to be small. We prove that, for small enough, shrinking the section of one line of the grating by a factor of () creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to ) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as tends to , the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly.
Keywords : guided waves, periodic media, spectral theory.
AMS codes : 78M35, 35J05, 58C40

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

[0.5cm]

Bérangère Delourme, Patrick Joly, Elizaveta Vasilevskaya

[0.5cm]

: Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539, 93430 Villetaneuse, France

: POEMS, UMR 7231 CNRS/ENSTA/Inria, ENSTA ParisTech, 828 boulevard des Maréchaux, 91762 Palaiseau Cedex, France

## 1 Statement of the problem

Let , and be three Lipschitz bounded domains of of same area () containing the origin , let be a parameter (that is going to be small), and let , and be three positive real numbers. We denote by , the standard basis of . For any , we consider the three dimensional domain defined by

 Dεk,ℓ,3={(x1,x2,x3)∈R3 % such that ((x1−a1k)/ε,(x2−a2ℓ)/ε)∈ω3},

which is an unbounded cylinder of constant cross section . It is infinite along the direction (invariant with respect to ) and contains the point . Similarly, for any , we define the domains

 Dεk,ℓ,1={(x1,x2,x3)∈R3 such that ((x2−a2k)/ε,(x3−a3ℓ)/ε)∈ω1}, Dεk,ℓ,2={(x1,x2,x3)∈R3 such that ((x1−a1k)/ε,(x3−a3ℓ)/ε)∈ω2},

and we consider the periodic domain given by

 Ωε=⋃i∈{1,2,3}⋃(k,ℓ)∈Z2Dεk,ℓ,i. (1)

The domain is a three dimensional grating of equi-spaced parallel pipes (of constant cross section) oriented along the three orthogonal directions , and . It is -periodic with respect to . Moreover, the points , , belong to .

In order to create guided modes, we introduce a linear defect (see [1]-[5]-[2]) 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 [4] for the proof in the case of a symmetric medium). More precisely, we assume that the domain is replaced with the domain

 Dε,μ0,0,3={(x1,x2,x3)∈R3 % such that (x1/(√με),x2/(√με))∈ω3},

where is a positive parameter. In other words, we enlarge () or shrink () the section of one pipe of the domain by a factor (see Fig 0(a)). The corresponding perturbed domain is denoted by . Its precise definition is given by

 Ωμε=⎛⎜⎝⋃i∈{1,2}⋃(k,ℓ)∈Z2Dεk,ℓ,i⎞⎟⎠⋃⎛⎜⎝⋃(k,ℓ)∈Z2∖{(0,0)}Dεk,ℓ,3⎞⎟⎠⋃Dε,μ0,0,3. (2)

is still -periodic with respect to . However, the presence of the perturbed pipe breaks the periodicity with respect to and . We emphasize that the domain (as well as ) tends to a 3D periodic graph as tends to .

We look for guided modes, i.e. solutions to the wave equation in , satisfying homogeneous Neumann boundary conditions on (see [7] for the investigation of the Dirichlet case), that propagate along the defect pipe (i.e. in the direction) but stay confined in the transversal directions. More precisely, denoting by the restriction of the domain to the band ,

 Bμε={(x1,x2,x3)∈Ωμεsuch that|x3|

we search solutions of the form , where is a real parameter and is an -periodic function in . In fact, it is easily seen that the -quasiperiodic fonction is an eigenfunction of the operator

 Aμε(β):D(Aμε(β))⊂L2(Bμε)→L2(Bμε),Aμε(β)=−ΔuinBμε, (4)

with where

 H1Δ(Bμε)={u∈H1(Bμε),s.t.Δu∈L2(Bμε)} and Σ±={(x1,x2,x3)∈∂Bμε,x3=±a3/2}.

To study the spectral properties of , we investigate its (formal) limit as tends to . The operator is defined on the limit graph (see Fig. 0(b)) and its spectrum can be explicitly computed. In particular, its spectrum has infinitely many gaps (Lemma 2.1), i.e. open intervals such that the intersection of with the spectrum is reduced to . Moreover, for , there is at least one eigenvalue in each gap (Lemma 2.5). Since, in addition, for sufficiently small, the spectrum of is close to the spectrum of , the existence of guided modes is guaranteed (Theorem 3.1).

## 2 The spectrum of the limit operator Aμ(β)

### 2.1 Definition of the limit operator Aμ(β)

The limit operator is defined on the infinite periodic graph obtained as the limit of as tends to : is made of the vertices connected by the edges

 {ek+1/2,ℓ=(vk,ℓ,vk+1,ℓ),ek,ℓ+1/2=(vk,ℓ,vk,ℓ+1),e±k,ℓ=(vk,ℓ,v±k,ℓ),(k,ℓ)∈Z2}.

It is -periodic with respect to and -periodic with respect to (see Fig. 0(b)).

For any function defined on , we denote by (resp. ) its value at the vertex (resp. ). The restriction of to the edge (resp. and ) is denoted by (resp. and ).

The definition of also requires the introduction of the function spaces and defined as

 Lμ2(G)={u:∥u∥Lμ2(G)<+∞},H2(G)={u∈C(G):∥u∥H2(G)<+∞}, (5)

where,

 ∥u∥2Lμ2(G)=∑(k,ℓ)∈Z2(wμkℓ∑±∥u±k,ℓ∥2L2(e±k,ℓ)+∥uk+12,ℓ∥2L2(ek+12,ℓ)+∥uk,ℓ+12∥2L2(ek,ℓ+12)), (6)
 ∥u∥2H2(G)=∑(k,ℓ)∈Z2(∑±∥u±k,ℓ∥2H2(e±k,ℓ)+∥uk+12,ℓ∥2H2(ek+12,ℓ)+∥uk,ℓ+12∥2H2(ek,ℓ+12)), (7)

and, for any , is the weight coefficient equal to for and otherwise.

The unbounded limit operator in has domain

 D(Aμ(β))={u∈H2(G):∀(k,ℓ)∈Z2,u′j+12(0)−u′j−12(1)+μ(u+j)′(0)−μ(u−j)′(0)=0,∀j∈Z}u+k,ℓ=e−iβu−k,ℓ,(u+k,ℓ)′(a3/2)=e−iβ(u−k,ℓ)′(−a3/2),u′k+12,ℓ(ka1)−u′k−12,ℓ(ka1)+u′k,ℓ+12(ℓa2)−u′k,ℓ−12(ℓa2)+wμk,ℓ((u+k,ℓ)′(0)−(u−k,ℓ)′(0))=0}, (8)

and is defined by

 ∀u∈D(Aμ(β)),Aμ(β)u=−u′′on any edge of the graphG. (9)

The functions of are continuous on and quasi-periodic. Moreover, they satisfy the Kirchhoff conditions (8) that enforce the weighted sum of the outward derivatives of to vanish at each vertex (). 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 (). The formal derivation of the limit model can be found in [6]. It is easily verified that the operator is self-adjoint (for the weighted scalar product associated with (6)), see also [3]. The objective of the following two sections is to study the spectrum of .

### 2.2 Characterization and properties of the essential spectrum of Aμ(β)

By a compact perturbation argument, one can prove that , where is the purely periodic operator corresponding to for . The computation of its spectrum relies on the Floquet-Bloch theory (see [9]). More precisely, we can prove that if and only if either and or and there exists such that

 sin(ωa2)sin(ωa3)(cos(ωa1)−cosk1)+sin(ωa3)sin(ωa1)(cos(ωa2)−cosk2)+sin(ωa1)sin(ωa2)(cos(ωa3)−cosβ)=0. (10)

Based on the previous characterization, we prove that the operator has a countable infinity of gaps that can be separated into three categories (see [10] for the proof):

###### Lemma 2.1

The following properties hold :

1. , where

 σi={(πn/ai)2,n∈Z}fori∈{1,2},andσ3={((±β+2πn)/a3)2,n∈Z}.
2. For any , the operator has infinitely many gaps whose ends tend to infinity.

3. Let and .
If an interval is a spectral gap of , then, one of the following possibilities holds:

1. , , and there is a unique .

2. , and .

3. , and .

### 2.3 Computation of the discrete spectrum

Let us now determine the discrete spectrum of . If is an eigenvalue of , then the corresponding eigenfunction satisfies the linear differential equation on each edge of the graph . Solving explicitly this equation (on each edge), taking into account the quasi-periodicity of and the Kirchhoff conditions (8), we can replace the eigenvalue problem with a set of finite differences equations for :

###### Lemma 2.2

Assume that . is an eigenfunction of if and only if belongs to and satisfies

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩uk+1,ℓsin(ωa1)+uk−1,ℓsin(ωa1)+uk,ℓ+1sin(ωa2)+uk,ℓ−1sin(ωa2)−2gβ(ω)uk,l=0,∀(k,ℓ)∈Z2∖{(0,0)},u1,0sin(ωa1)+u−1,0sin(ωa1)+u0,1sin(ωa2)+u0,−1sin(ωa2)−2gβ(ω)u0,0=2(μ−1)cos(ωa3)−cosβsin(ωa3)u0,0, (11)

where we have defined

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

 F:v=(vk,ℓ)(k,ℓ)∈Z2↦F(v)=ˆv,ˆv(ξ,η)=∑k,ℓ∈Zei(kξ+ℓη)vk,ℓ,(ξ,η)∈[0,2π]2, (12)

where is an isometry between and . This, together with Lemma 2.2, provides the following characterization for the discrete spectrum of :

###### Lemma 2.3

Assume that . is an eigenfunction of if and only if the discrete Fourier transform of belongs to and satisfies

 (f(ξ,η,ω)−ϕβ(ω))ˆu(ξ,η)=(μ−1)ϕβ(ω)u0,0. (13)

where

Under the assumption of Lemma 2.3, (10) can be written as . It follows that does not belong to if and only if, for any , does not vanish. As a consequence, as soon as , the function is continuous and bounded. Then, the inverse discrete Fourier transform can be applied to (13) to obtain

 uk,ℓ=(1−μ)u0,04π2∫(0,2π)2ϕβ(ω)ϕβ(ω)−f(ξ,η,ω)e−i(kξ+ℓη)dξdη,∀(k,ℓ)∈Z2.

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

###### Lemma 2.4

Assume that and that . Then, is an eigenvalue of if and only if

 μ=1−Fβ(ω)whereFβ(ω)=(14π2∫(0,2π)2ϕβ(ω)ϕβ(ω)−f(ξ,η,ω)dξdη)−1. (14)

The study of the behavior of the function leads to the existence of at least one eigenvalue in each gap of as soon as , 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 , the operator has no eigenvalue. For , let be a spectral gap of the operator :

1. If is a gap of type (i), then has at least two eigenvalues and that satisfy (see Lemma. 2.1-3 for the definition of ).

2. If is a gap of type (ii) or (iii), then has at least one eigenvalue such that .

The sketch of the proof of the previous lemma is the following, a complete proof being available in [10] (Theorem 5.2.1): First, one can verify that in any gap, which, together with (14) proves that has no eigenvalue for . Then, if is a gap of type (i), one can show that

 limω→ω+b(1−Fβ(ω))=limω→ω−t(1−Fβ(ω))=1 and that limω→ω0(1−Fβ(ω))=0.

By continuity of inside the gap, (a) directly results from the intermediate value theorem and (14). If is a gap of type (ii), the intermediate value theorem also permits us to conclude since

 limω→ωb(1−Fβ(ω))≤0 and limω→ω−t(1−Fβ(ω))=1.

A similar argument works for a gap of type (iii).

## 3 Guided modes for the operator Aμε(β): an asymptotic result

Finally, thanks to the general result [8] (Theorem 2.13 convergence of the spectrum of toward the spectrum of ), we can prove the following result of existence of eigenvalue for the operator :

###### Theorem 3.1

Let , be a spectral gap of the operator and be a (simple) eigenvalue of this operator. Then, there exists such that if the operator has an eigenvalue inside a spectral gap . Moreover,

## References

• [1] Ammari, H. and Santosa, F., Guided waves in a photonic bandgap structure with a line defect, SIAM Journal on Applied Mathematics, Vol. 64, Num. 6, p 2018-2033, 2004
• [2] Fliss, Sonia, A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides, SIAM Journal on Scientific Computing, Vol. 35, Num. 2, p B438–B461, 2013
• [3] Kuchment P., Quantum graphs: an introduction and a brief survey, Analysis on graphs and its applications,. Ser. Proc. Sympos. Pure Math., Vol. 77, p. 291–312, Amer. Math. Soc., Providence, RI, 2008
• [4] Friedlander, L., On the spectrum of a class of second order periodic elliptic differential operators, Communications in Mathematical Physics, Vol. 229, p. 49–55, 2002
• [5] Kuchment, P. and Ong, B. S. , On guided waves in photonic crystal waveguides, Waves in periodic and random media, Vol. 339, p. 105–115, 2003
• [6] Kuchment, P. and Zeng, H., Convergence of spectra of mesoscopic systems collapsing onto a graph, Journal of Mathematical Analysis and Applications, Vol. 258, Num. 2, p. 671-700, 2001
• [7] Nazarov, S.A., On the spectrum of the Laplace operator on the infinite Dirichlet ladder, St. Petersburg Mathematical Journal, Vol. 23, p. 1023–1045, 2012
• [8] Post O. , Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré, Vol. 7, Num. 5, p. 933-977, 2006
• [9] Reed, M. and Simon, B., Methods of modern mathematical physics v. I-IV, Academic Press,1972-1978
• [10] Vasilevskaya, E., Open periodic waveguides. Theory and application, PhD thesis, 2016
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters