Computing Floquet-Bloch modes in biperiodic

slabs with Boundary Elements ∗†

Johannes Tausch

Abstract

A new variational formulation for the boundary element computa-

tion of Floquet-Bloch solutions to the Helmholtz equation is derived.

The method is applicable to geometries that are layered in the vertical

and biperiodic in two horizontal directions. The discretization leads

to a nonlinear hermitian eigenvalue problem which is solved using a

homotopy approach. Numerical examples demonstrate that with the

boundary element approach a high accuracy can be achieved with a

small number of degrees of freedom in the discretization.

This paper is dedicated to the occasion of Klaus Bohmer’s 75th birthday

1 Introduction

Many wave propagation phenomena are characterized by nontrivial solutionsto the Helmholtz equation

∆u+ k2n2u = 0, (1)

in three-space. Here k is the wave number and n is a piecewise constantfunction that characterizes the medium of propagation.

The focus in this article is on a medium that is layered in the z-directionand has a perturbation of small z-extent that is biperiodic in the xy-plane.

∗This material is based upon work supported by the National Science Foundation under

grants DMS-0915222 and DMS-1115931†Preprint of an article that will appear in J. Comp. Appl. Math. 2013

1

The motivation comes from photonics and solid-state physics, where, for in-stance, photonic slab waveguides or periodic thin-film structures are of inter-est. Of course, many physically relevant problems are governed by the time-harmonic Schrodinger and Maxwell’s equations but we use the Helmholtzequation here as a simple model problem.

Wave propagation in periodic media is described by Floquet-Bloch theory,which is discussed extensively in the literature, see for instance [6, 7]. By now,there is a wealth of papers on computing Floquet-Bloch solutions numerically,we refer to [1, 2, 4] as representative examples. The common theme in thementioned papers is that by using Floquet’s theorem, the problem in infinitespace is reduced to a finite symmetry cell. If the PDE is discretized withfinite elements or differences, the spectrum can be computed by solving asequence of linear eigenvalue problems. Using other discretization methodsit is possible to arrive at nonlinear eigenvalue problems and use path followingmethods [10] to compute dispersion relations quickly.

Unlike photonic crystals, photonic slabs have an infinite symmetry celland therefore there is a continuous in addition to a discrete spectrum. More-over, the numerical solution is more involved, as the computational domainmust be truncated, and artificial boundary conditions must by imposed bya Dirichlet-to-Neumann (DtN) map. Since this map itself depends on thewave number, the resulting eigenvalue problem is always nonlinear. Thereare several options to handle these complications, although some are limitedto the single-periodic case, see [5, 8, 13] and the survey [12]. It was shownin [14] that the boundary element method is an effective tool to solve theseproblems, because the size of the eigenvalue problems that have to be solvedis drastically reduced. However, the approach there was based a collocationdiscretization, which implies that the self-adjointness of the continuous prob-lem does not carry over to the discrete one. The primary goals of this paperare to present a new integral equation formulation in variational form anda conforming discretization which leads to a hermitian, nonlinear eigenvalueproblem. We further propose a homotopy method to follow eigensolutionsbeginning with a planar structure.

In section 2 we give a more precise statement of the problem and providesome background material. A more detailed discussion can be found in theabove mentioned literature, the goal of this section is to fix notations andmake the paper self-consistent. The following section 3 derives the variationalform of the boundary integral formulation and shows its equivalence with theoriginal problem. We then discuss in section 4 suitable finite-element spaces

2

for the discretization which is followed by the description of the homotopymethod. Finally, section 6 presents the dispersion curves of some biperiodicstructures that were obtained with an implementation of the method.

2 Preliminaries

Geometry Description We assume that n > 0 is a piecewise constantbiperiodic function

n(x+ lxdx, y + lydy, z) = n(x), (lx, ly) ∈ Z2,

where x = (x, y, z) and Ω = (0, dx)×(0, dy)×R is a fundamental periodic cell.Furthermore, we assume that Ω can be decomposed into two semi-infinitedomains Ω+ and Ω− and a finite domain Ω0 = (0, dx) × (0, dy) × (z−∞, z

+∞)

such thatΩ = Ω− ∪ Ω0 ∪ Ω+.

In the semi-infinite regions Ω± the function n assumes the constant valuesn±. The domain Ω0 contains the biperiodic perturbation from the planargeometry, which can be decomposed into domains Ωl where n assumes theconstant nl such that

Ω0 = ∪Ll=1Ω

l .

We assume that the domains are Lipschitz with boundaries Γl := ∂Ωl.The interface between Ω0 and Ω± is denoted by Γz = Γ+

z ∪ Γ−z . The

other boundaries of Ω0 are denoted by Γx = Γ+x ∪ Γ−

x and Γy = Γ+y ∪ Γ−

y ,respectively. The surface Γ is the union of all interfaces in Ω0

Γ = ∪Ll=1Γ

l,

and Γ0 denotes all interior interfaces. A two-dimensional illustration is shownin Figure 1.

Floquet-Bloch Solutions We seek to compute the Floquet-Bloch solu-tions of the Helmholtz problem, which are nontrivial solutions of (1) of theform

u(x, y, z) = exp(iβxx+ iβyy)v(x, y, z), (2)

where β = β(k) = (βx, βy) ∈ R2 is the unknown propagation vector and v is

periodic in x and y with periods dx and dy.

3

z

x, y

Ω+

Ω4

Ω3

Ω2

Ω1

Ω−

Γ−x Γ+

x

z+∞

z−∞

Γ+z

Γ−z

Figure 1: Two dimensional illustration of the geometry

An equivalent characterization is to seek β and k such that (1) in Ω withquasi periodic boundary conditions

u(0, y, z) = eiβxdxu(dx, y, z),

u(x, 0, z) = eiβydyu(x, dy, z),

∂u

∂n(0, y, z) = −eiβxdx

∂u

∂n(dx, y, z),

∂u

∂n(x, 0, z) = −eiβydy

∂u

∂n(x, dy, z).

has nontrivial solutions. We will also use the more compact notation

B+β u = 0 and B−

β

∂u

∂n= 0 (3)

to indicate that u and its normal derivative satisfy quasi periodic boundaryconditions.

Note that the conditions (3) are periodic in β, hence it suffices to seeksolutions in the Brillouin zone

β ∈ B :=

[

0,2π

dx

]

×[

0,2π

dy

]

. (4)

4

DtN operator To obtain well-posed problems it is necessary to specifyradiation conditions for |z| → ∞. Our primary goal is to compute modesthat are guided by the structure, that is, the propagation vector β is realand the fields decay away from the grating region. This condition will enableus to truncate the computational domain in the z-direction and impose anartificial boundary condition by the DtN operator. To that end, considerfor a given frequency and propagation vector the Helmholtz equation withDirichlet conditions in the semi-infinite region Ω+

∆u+ k2n2+u = 0, in Ω+,

B−

β∂u∂n

= B+β u = 0,

u = f+, on Γ+z .

(5)

Since n+ is constant, the solution can be found by separation of variables. Itfollows that

u(x, y, z) =∑

l∈Z2

f+l exp

(

iβlxx+ iβlyy)

exp(

− αlz)

(6)

where l = (lx, ly),

βlx = βx +2πlxdx

, βly = βy +2πlydy

,

f+l =

1

dxdy

∫ dx

0

∫ dy

0

exp(

− iβlxx− iβlyy)

f+(x, y) dxdy

and

α+l =

(

β2lx + β2

ly − n2+k

2)

1

2

.

Here the branch cut of the square root is on the negative real axis. Thusthe number α+

l is either positive real or positive imaginary, corresponding toexponentially decaying or outgoing z-dependence. From the expansion in (6)it follows that the DtN operator is given by

T +f+(x, y) = −∑

l∈Z2

α+l f

+l exp

(

iβlxx+ iβlyy)

.

Likewise, the DtN for the semi-infinite Ω− is given by

T −f−(x, y) = −∑

l∈Z2

α−

l f−

l exp(

iβlxx+ iβlyy)

.

5

where α−

l and f−

l are defined analogously.For the computation of guided modes only the values of β and k are of

interest where all coefficients α±

l are positive real. Thus we seek solutionsonly in the region

L :=

(β, k) :(

β2lx + β2

ly

)1

2 ≤ kmin n+, n−, β ∈ B, l ∈ Z2

.

In (β, k)-space this is the region below four cones centered at the corners ofthe Brillouin zone. The region above L is usually denoted as the light cone.

Planar Structures It is often instructive to consider a slab structure as aperturbation of a planar structure. For instance, in the example of figure 1,one obtains a planar structure if one sets n2 = n3. The modes of a planarstructure can be obtained with the ansatz

u(x, y, z) = exp(iβxx+ iβyy)ϕ(z). (7)

Substitution into (1) leads to the singular Sturm-Liouville problem

ϕ”(z)− |β|2ϕ(z) = k2n2(z)ϕ(z), z ∈ R. (8)

The spectrum of this problem is well understood, see, e.g. [12]. For any |β|there is at most a finite number of k-values with k2 < min(n2

±) with guidedmode solutions. This is the discrete spectrum. Furthermore, every value ofk with k2 ≥ min(n2

±) has a radiation mode, which constitutes the continuousspectrum.

Of course, the planar structure can also be considered periodic. Thefunction u in (7) is a Floquet-Bloch mode in (2) if the propagation vectorssatisfy

βx = βx +2πlxdx

, βy = βy +2πlydy

,

for integer (lx, ly). Hence the k-values found for any β are also part of theFloquet-Bloch spectrum of β as long as the above relationship between β

and β is satisfied. The connection between the spectrum and the Floquetspectrum is illustrated in figure 2. Note that both spectra contain the samesolutions to the Helmholtz equation. They are only different ways of labelingthem.

6

k k

2π/00 dβ β~

continuousspectrum

Figure 2: One-dimensional illustration of the spectrum of a planar structure(left) and its Floquet spectrum (right)

3 Boundary Integral Formulation

To derive the boundary integral formulation, we begin with Green’s repre-sentation formula for a subdomain Ωl with boundary Γl := ∂Ωl

u(x) =

∫

Γl

Gl(x,y)∂u

∂n(y) ds(y)−

∫

Γl

∂

∂ny

Gl(x,y)u(y) ds(y), (9)

for x ∈ Ωl. For the Helmholtz equation of an an interior domain, no radiationconditions have to be satisfied, therefore one can choose the Green’s functionto be the real part of the free-space Green’s function. Thus we have

Gl(x,y) =cos(κl|x− y|)4π|x− y| ,

where κl = knl is the wave number in domain Ωl.Taking the trace and the normal trace on the boundary leads to two

integral equations on Γl

Vl∂u

∂n−(

1

2+Kl

)

u = 0, (10)

(

−1

2+K′

l

)

∂u

∂n+Dlu = 0, (11)

where Vl,Kl,K′l and Dl are the usual layer potentials

Vlg(x) =

∫

Γl

Gl(x,y)g(y) ds(y),

7

Klg(x) =

∫

Γl

∂

∂ny

Gl(x,y)g(y) ds(y),

K′

lg(x) =

∫

Γl

∂

∂nx

Gl(x,y)g(y) ds(y),

Dlg(x) = − ∂

∂nx

∫

Γl

∂

∂ny

Gl(x,y)g(y) ds(y).

Note that the normals here point into the exterior of Ωl. With our choice ofGl the operator K′

l is the adjoint of Kl.A standard result in the theory of boundary integral operators is that

the single layer and hyper-singular operator satisfy a Garding inequality [3].

That is, there are compact operators Cvl : H−1

2 (Γl) → H−1

2 (Γl) and Cdl :

H1

2 (Γl)→ H1

2 (Γl) and positive constants cv and cd such that

〈ψl, (Vl + Cvl )ψl〉 ≥ cv‖ψl‖2H−

1

2 (Γl), (12)

⟨

ϕl, (Dl + Cdl )ϕl

⟩

≥ cd‖ϕl‖2H

1

2 (Γl), (13)

for all ψl ∈ H−1

2 (Γl) and ϕl ∈ H1

2 (Γl). Here, 〈·, ·〉 denotes the dual-

ity pairing between H1

2 (Γl) and H−1

2 (Γl). Moreover, the operators Vl :

H−1

2 (Γl) → H−1

2 (Γl), Kl : H1

2 (Γl) → H−1

2 (Γl), K′l : H−

1

2 (Γl) → H1

2 (Γl)

and Dl : H1

2 (Γl)→ H−1

2 (Γl) are bounded.We define the spaces

Uβ =

(ϕ1, . . . , ϕL) : ϕl ∈ H1

2 (Γl), ϕl = ϕl′ on Γl ∩ Γl′ and B+βϕ = 0

,

Vβ =

(ψ1, . . . , ψL) : ψl ∈ H−1

2 (Γl), ψl = −ψl′ on Γl ∩ Γl′ and B−

βψ = 0

,

which are closed subspaces of H1

2 (Γ1)⊗ . . .⊗H1

2 (ΓL) and H−

1

2 (Γ1)⊗ . . .⊗H−

1

2 (ΓL), equipped with the norms

‖ϕ‖H

1

2:=

(

L∑

l=1

‖ϕl‖2H

1

2 (Γl)

)

1

2

and ‖ψ‖H−

1

2:=

(

L∑

l=1

‖ψl‖2H−

1

2 (Γl)

)

1

2

,

respectively.In addition to satisfying (10) and (11) a guided mode solution is also

continuous across the Γz-interfaces

∂u

∂n= T ±u, on Γ±

z . (14)

8

To derive a variational form the following observation will be useful. Itfollows easily because contributions of interior and periodic surfaces cancel.

Lemma 1. For u, ϕ ∈ Uβ and v, ψ ∈ Vβ it follows that

L∑

l=1

〈ψl, ul〉 = 〈ψ, u〉Γzand

L∑

l=1

〈ϕl, vl〉 = 〈ϕ, v〉Γz.

To obtain a variational formulation multiply (10), (11) and (14) by testfunctions ψl and ϕl, defined on the boundary Γl and add all integrals. To-gether with lemma 1 this leads to

a(ψ, ϕ; v, u) = 0, (15)

where

a(ψ, ϕ; v, u) := 〈ϕ, T u〉Γz− 1

2

(

〈ψ, u〉Γz+ 〈ϕ, v〉Γz

)

+L∑

l=1

(

〈ψl,Vlv〉 − 〈ψl,Klul〉 − 〈ϕl,K′

lvl〉 − 〈ϕl,Dlul〉)

Evidently, the bilinear form is symmetric

a(ψ, ϕ; v, u) = a(v, u;ψ, ϕ).

If u is a guided mode solution of (1), then (u1, . . . , uL) and (v1, . . . , vL)with ul = u|Γl

and vl := ∂u/∂nl is a solution to the variational problem:Find (u, v) ∈ Uβ ⊗ Vβ such that for all (ϕ, ψ) ∈ Vβ ⊗ Uβ (15) holds. Theconverse is stated in the following result.

Theorem 2. A nontrivial solution of (15) with (β, k) ∈ L extends to aguided-mode solution of (1) by

u(x) =

∫

Γl

Gl(x,y)∂u∂n(y) ds(y)−

∫

Γl

∂∂ny

Gl(x,y)u(y) ds(y), x ∈ Ωl,

∑

l∈Z2

u±l exp(

iβlxx+ iβly ∓ α±

l z)

, x ∈ Ω±.

(16)

9

Proof. The function defined in (16) satisfies the Helmholtz equation in ev-ery subdomain. It remains to verify that the traces and normal traces arecontinuous across the interfaces in the weak sense.

Let x ∈ Γ0 a point on the interface between Ωl and Ωl′ . The jumprelations of the double layer potential implies that

u(x−) = Vlvl(x)−Klul(x) +1

2ul(x), (17)

u(x+) = Vl′vl′(x)−Kl′ul′(x) +1

2ul′(x), (18)

where x± indicates taking the limit from the Ωl and Ωl′ side. From choosing in(15) ϕ = 0 and ψ to be a function that is supported in a small neighborhoodof x one can conclude that

Vlvl(x)−Klul(x)−1

2ul(x)−

(

Vl′vl′(x)−Kl′ul′(x)−1

2ul′(x)

)

= 0.

Mind that the minus sign comes from the fact that ψl(x) = −ψl′(x). Fromthis and equations (17), (18) we can conclude that u(x−) = u(x+). Nowapply the jump relations of the normal derivative. It follows that

∂u

∂nl

(x−) = K′

lvl(x) +Dlul(x) +1

2vl(x), (19)

∂u

∂nl′(x−) = K′

l′vl′(x) +Dl′ul′(x) +1

2vl′(x). (20)

From choosing in (15) ψ = 0 and ϕ to be a function that is supported in asmall neighborhood of x one can conclude that

Klvl(x) +Dlul(x)−1

2vl(x) +Kl′vl′(x) +Dl′ul′(x)−

1

2vl′(x) = 0,

thus it follows from (19), (20) that ∂u∂nl

(x−) = − ∂u∂nl′

(x+) = ∂u∂nl

(x+).

Now let x ∈ Γ+z be a point on the interface of Ω+ and some interior

domain, say Ωl. The jump relations (17) and (19) apply again, where x−

denotes a limit as the evaluation point approaches Γ+z . On the other hand,

from (15) we can conclude that

Vlvl(x)−Klul(x)−1

2ul(x) = 0,

K′

lvl(x) +Dlul(x)−1

2vl(x) + T u(x) = 0.

10

This implies that u(x−) = ul(x) and that vl(x) = T u(x). Thus the traceand the normal trace are continuous across Γ+. The continuity for Γ− iscompletely analogous.

It remains to verify the quasi continuity of u. Consider a point x ∈ Γ−x

which lies on the boundary of some subdomain, say Ωl. The periodic imagex′ := x+dxex lies on the Γ+

x which lies on the interface of a possibly differentdomain Ωl′ . By construction of u, the jump relations (17-20) hold, where x−

indicates limiting value as the evaluation point approaches x from Ωl andx+ a limiting value of approaching x′ from Ωl′ . From the variational formwe obtain

Vlvl(x)−Klul(x)− e−iβxdx (Vl′vl′(x′)−Kl′ul′(x′)) = 0, (21)

K′

lvl(x) +Dlul(x) + e−iβxdx (K′

l′vl′(x′) +Dl′ul′(x

′)) = 0. (22)

By combining (17),(18) and (21) and using ul′(x) = exp(iβxdx)ul(x) it fol-lows that

u(x+)− e−iβxdx u(x−) = 0.

Likewise, from (19),(20), (22) and vl′(x) = − exp(iβxdx)vl(x) it follows that

∂u

∂n′(x+) + e−iβxdx

∂u

∂nu(x−) = 0.

This implies quasiperiodicity for Γx. The same argument for Γy completesthe proof.

The DtN operator induces the bilinear form

〈ϕ, T u〉 = −∑

l∈Z2

(

α+l ϕ

+l u

+l + α−

l ϕ−

l u−

l

)

(23)

which is bounded in Uβ and negative

〈ϕ, T ϕ〉 ≤ 0. (24)

The solvability of the variational form is stated in the following result.

Theorem 3. The nullspace of (15) is either trivial or finite dimensional.

Proof. To understand the solvability of (15) consider the variational formu-lation

a(ψ, ϕ; v, u) := a(ψ,−ϕ; v, u) = 0. (25)

11

Clearly, (15) and (25) have the same solutions. Moreover, a is bounded inVβ ⊗ Uβ and

a(ψ, ϕ;ψ, ϕ) =L∑

l=1

(

〈ψ,Vlv〉+ 〈ϕ,Dlul〉)

+1

2

(

〈ψ, ϕ〉 − 〈ϕ, ψ〉)

− 〈ϕ, T ϕ〉

From (12) and (13) and (24) it can be concluded that there are compactoperators Cv : Vβ → Vβ and Cd : Uβ → Uβ and a positive constant c suchthat

Re a(ψ, ϕ;ψ, ϕ) + 〈ϕ, Cvϕ〉+ 〈ψ, Cdψ〉 ≥ c(

‖ϕ‖2H

1

2

+ ‖ψ‖2H−

1

2

)1

2

Thus the bilinear form satisfies another Garding inequality in the space Vβ⊗Uβ. This implies the assertion.

4 Galerkin Method for the Interior Problem

To derive the Galerkin solution of (15) consider an admissible triangulationof Γ. That is, two triangles intersect either at a common edge, a commonvertex or not at all. Note that it is possible that an edge is shared by morethan two triangles. We also assume that the triangulation is periodic in thex- and y- directions, i.e., τ is a triangle on Γ+

x if and only if there is a triangleτ ′ on Γ−

x such that τ = τ ′ + dxex and τ is a triangle on Γ+y if and only

if there is a triangle τ ′ on Γ−y such that τ = τ ′ + dyey. The triangulation

of Γ also implies that every subdomain Γl is partitioned into an admissibletriangulation.

We seek the Galerkin solution in piecewise polynomial subspaces of Uβ

and Vβ. To that end, consider the spaces S(0)l of piecewise constant func-

tions subject to the triangulation of Γl and S(1)l the space of continuous and

piecewise linear functions on Γl. Then the subspaces for the discretizationare given by

Uhβ =

(ϕ1, . . . , ϕL) : ϕl ∈ S(1)l , ϕl = ϕl′ on Γl ∩ Γl′ and B+

βϕ = 0

,

V hβ =

(ψ1, . . . , ψL) : ψl ∈ S(0)l , ψl = −ψl′ on Γl ∩ Γl′ and B−

βψ = 0

.

The discretization of (15) is: Find (vh, uh) ∈ V hβ ⊗ Uh

β such that for all

(ψh, ϕh) ∈ V hβ ⊗ Uh

β

a(ψh, ϕh; vh, uh) = 0 (26)

12

holds. It is not hard to see that Uhβ , V

hβ are subspaces of Uβ, Vβ thus the

discretization is conforming.To obtain a basis of Uh

β and V hβ consider a box functions χτ for a tri-

angle τ and a hat function ζv for a vertex v in the triangulation of Γ. Thequasiperiodic extensions of these functions are defined as

χβτ =

χτ , if τ ⊂ Γ0 ∪ Γz,χτ − eiβxdxχτ ′ , if τ ⊂ Γ−

x ,χτ − eiβydyχτ ′ , if τ ⊂ Γ−

y ,

and

ζβv =

ζv, if v ∈ Γ0 ∪ Γz,ζv + eiβxdxζv′ , if v ∈ Γ−

x \ Γ−y ,

ζv + eiβydyζv′ , if v ∈ Γ−y \ Γ−

x ,ζv + eiβxdxζv′ + eiβydyζv′′ + ei(βxdx+βydy)ζv′′′ , if v ∈ Γ−

x ∩ Γ−y .

Here primes denote the corresponding periodic copy of the panel or vertex.Since functions in V h

β alternate signs on interior interfaces define

σlτ =

−1 if τ ∈ Ωl ∩ Ωl′ and nl < nl′ ,1 else,

then the basis functions of V hβ and Uh

β are

χβτ = (σ1

τχβτ |Γ1

, . . . , σLτ χβ,τ |ΓL

),

ζβv = (ζβv |Γ1

, . . . , ζβv |ΓL).

With this basis the discrete variational formulation (26) can be expressedin matrix form as follows: Find (β, k) ∈ L such that the matrix Ah(β, k) hasa nontrivial nullspace. The matrix is given by

Ah(β, k) =

[

a(χβτ ,χ

β

τ ′) a(χβτ , ζ

β

v′)

a(ζβv ,χ

β

τ ′) a(ζβv , ζ

β

v′)

]

where τ, τ ′ and v, v′ run through all triangles and vertices that appear in theabove definitions of χβ,τ and ζβ,v. Since the bilinear form is hermitian, thematrix Ah is hermitian.

13

Computation of the DtN operator Some of the matrix coefficients de-pend on the DtN operator of (23), whose evaluation involve an infinite series.To compute this operator, we truncate the series by adding only terms in thediamond-shaped domain

l ∈ Z2p :=

(lx, ly) ∈ Z2 : |lx|+ |ly| ≤ p

. (27)

Note that if some of the subdomains are planar it is possible to discretizeonly the irregularly shaped domains and incorporate the planar subdomainsin the semi-infinite domains. For instance, in figure 1 one would discretize Ω2

and Ω3 and move Γ−x to the Ω1-Ω3-interface and Γ+

x to the Ω3-Ω4-interface. Inthis case the DtN operator still appears in the general form of (23), but thecoefficients α±

l must be computed using planar layer transfer matrices. Sincethis process is described in the literature it is not described here. Details canbe found in, e.g., [13].

5 Solution of the Eigenvalue Problem

After discretization problem (26) reduces to a nonlinear eigenvalue problem.That is, we have a matrix Ah whose coefficients depend on β and k and wishto find those values in L for which the matrix is singular. The conventionalapproach in photonics is to fix the propagation vector β in the Brillouinzone and to compute all values of the frequency k that lie below the lightcone. However, in view of the boundary element method, it turns out thatit is more efficient to fix the frequency and solve for the values of β. Thereason is that when computing Ah one first calculates the influence coeffi-cients of the classical boundary integral operators (which depend on k) andthen forms weighted combinations (which depend on β) to form the matrixAh. The computation of the influence coefficients is much more costly, asfour-dimensional, possibly singular integrations have to be computed. If thefrequency is fixed, the expensive part of the computation has to be done onlyonce when β is varied.

Nonlinear Inverse Iteration Since β is a vector with two componentsthe solutions for a fixed k form curves in the Brillouin zone B. We firstdescribe how to obtain a point on such a curve beginning with an initialguess β and a search direction ν. Specifically, we search for the value t ∈ R

such that the matrix T (t) := Ah(β+ tν, k) is singular. The latter problem is

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a nonlinear eigenvalue problem for which several solution methods exist [9].The following nonlinear version of the inverse iteration is suitable for thispurpose.

choose t0 = 0 and x0 ∈ CN

for k = 0, 1, 2 . . .

δtk =(xk)HT (tk)xk

(xk)HT ′(tk)xk

solve T (tk)yk+1 = T ′(tk)xk

tk+1 = tk − δtkxk+1 = δtkxk

end

Note that in each iteration the matrix and its derivative must be re-assembledfor a different propagation vector and a linear system must be solved. Sincewith the boundary element method the sizes of the matrices are typicallyvery manageable, the linear solver is simply based on the LU -decompositionalthough iterative methods are feasible as well. Since T (tk) and T ′(tk) arehermitian, the iterates tk are always real, whereas the vectors xk are generallycomplex.

To obtain the initial vector x0 the conventional inverse iteration

choose x0 randomfor k = 0, 1, 2 . . .

solve T (t0)yk+1 = T ′(t0)xk

xk+1 =yk+1

‖yk+1‖endx0 ← xk

is effective. Here the matrix is fixed and hence the cost of finding the initialguess is small compared to the nonlinear inverse iteration.

Homotopy method So far we have discussed how to find a single pointon the solution curve, which, of course, depends on a good initial guess. Toobtain the full curve, a path following method will not be satisfactory sincesolutions generally consist of several disconnected components. Therefore weemploy a homotopy method based on deforming the function n from a planar

15

waveguide, where solutions can be found easily, to the biperiodic waveguideunder consideration. Specifically, we set

nλ(x, y, z) = (1− λ)nP (z) + λn(x, y, z), λ ∈ [0, 1]. (28)

where nP (z) is the material constant of the ’planarized’ geometry that resultswhen the value of n in the irregularly shaped subdomains is replaced by theirmaximum.

As discussed in section 2 the dispersion curves for the planarized waveg-uide are four circles that are centered in the corners of the Brillouin zone.The radii are the propagation constants of the planar structure that can befound numerically, for instance, with the method of [11]. We select pointson these circles which will be the initial guesses to compute the curves forthe structure with parameter 0 < λ1 < 1, which in turn will serve as initialguesses for λ2 > λ1, and so on, until the final curve has been reached. Thesearch direction ν is set to be orthogonal to the current solution curve. Todetermine this direction, assume that the dispersion curve is parameterizedby θ. If x(β) is the eigenvector, then

A(β(θ))x(β(θ)) = 0.

Differentiating with respect to θ leads to

(

∂A

∂βxx+ A

∂x

∂βx

)

∂βx∂θ

+

(

∂A

∂βyx+ A

∂x

∂βy

)

∂βy∂θ

= 0.

The latter two equations imply that

xH∂A

∂βxx∂βx∂θ

+ xH∂A

∂βyx∂βy∂θ

= 0,

hence we see that the vector

ν =

[

xH ∂A∂βx

x

xH ∂A∂βy

x

]

∈ R2

is orthogonal to the curve.

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6 Numerical Results

We have tested the method on several geometries that result from perturbinga four layer planar geometry given by

nP (z) =

√2.3, z < −0.5,

n1, −0.5 < z < 0,n2, 0 < z < 0.5,1, 0.5 < z.

In the grating region z ∈ [0, 0.5] the function is replaced either by a rectan-gular

n(x, y, z) =

1, if (x, y) ∈ [0, 0.5]2,n2, else,

or by a triangular perturbation

n(x, y, z) =

1, if 0 ≤ x, y, x+ y ≤ 1,n2, else.

The grating period is dx = dy = 1. For the values n1 = n2 = 2 the planarstructure has one guided mode below the light cone, and for n1 = n2 = 5there are two guided modes.

p n = 36 n = 154 n = 642 n = 26261 1.734263 1.742757 1.747680 1.7488212 1.725492 1.742439 1.747457 1.7485983 1.725478 1.742412 1.747453 1.7485934 1.724365 1.741744 1.747443 1.7485895 1.724362 1.741736 1.747442 1.7485886 1.724035 1.741606 1.747429 1.7485877 1.724034 1.741603 1.747427 1.7485878 1.723897 1.741414 1.747366 1.7485879 1.723896 1.741413 1.747365 1.748587

Table 1: Convergence of βx of the cubical perturbation.

Tables 1 and 2 display the computed values of βx = βy for the one-modestructure when k = 1.5 for different uniform mesh refinements and different

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p n = 36 n = 154 n = 642 n = 26261 1.681019 1.692809 1.697039 1.6979772 1.675070 1.692606 1.696843 1.6977753 1.675049 1.692569 1.696838 1.6977714 1.674311 1.692034 1.696828 1.6977655 1.674307 1.692026 1.696826 1.6977646 1.674090 1.691936 1.696816 1.6977637 1.674088 1.691933 1.696813 1.6977638 1.673997 1.691794 1.696763 1.6977629 1.673997 1.691793 1.696762 1.697761

Table 2: Convergence of βx for the triangular perturbation.

expansion orders in (27). The data makes clear that the convergence is rapidand that even with a very coarse mesh a good approximation of the actualsolution can be obtained.

Figures 3 and 4 display the dispersion curves for the planar structure inlighter shade and the curves for the cubically perturbed structure in black.The regions that are not in L are marked in gray. The computation was donewith the second refinement with 642 degrees of freedom. The homotopymethod used six steps and followed 1600 points. The overall cpu time togenerate one of the figures is about two to three hours. For the coarser meshwith 154 dof the cpu time is in the range of minutes and the figures wouldbe indistinguishable at the shown resolution. Finally, figure 5 shows theevolution of the dispersion curves under the homotopy method. For clarity,we only display only every second step.

References

[1] W. Axmann and P. Kuchment. An efficient finite element method forcomputing spectra of photonic and acoustic band-gap materials. J. Com-put. Phys., 150:468–481, 1999.

[2] G. Bao. Finite element approximation of time harmonic waves in peri-odic structures. SIAM J. Numer. Anal., 32(4):1155–1169, 1995.

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0 π 2π

π

2πk=1.6

0 π 2π

π

2πk=1.9

0 π 2π

π

2πk=2.2

0 π 2π

π

2πk=2.5

Figure 3: Dispersion curves for the one-mode structure

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0 π 2π

π

2πk=0.8

0 π 2π

π

2πk=0.9

0 π 2π

π

2πk=1.0

0 π 2π

π

2πk=1.2

Figure 4: Dispersion curves for the two-mode structure

0 1 22

3

4

Figure 5: Evolution of the dispersion curve under the homotopy method.One-mode structure, k = 1.9.

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[3] M. Costabel. Boundary integral operators on Lipschitz domains: Ele-mentary results. SIAM J. Math. Anal., 19:613–626, 1988.

[4] D. C. Dobson, J. Gopalakrishnan, and J.E. Pasciak. An efficient methodfor band structure calculations in 3D photonic crystals. J. Comput.Phys., 161(2):668–679, 2000.

[5] J. Jacobsen. Analytical, numerical, and experimental investigation ofguided waves on a periodically strip-loaded dielectric slab. IEEE Trans.Antennas and Propagation, 18(3):379–388, 1970.

[6] J.D. Joannopoulos, S.G. Johnson, J. Winn, and R. Meade. PhotonicCrystals, Molding the Flow of Light. Princeton, 2nd edition, 2008.

[7] Peter Kuchment. Floquet Theory for Partial Differential Equations.Birkhauser Verlag, Basel, 1993.

[8] S.T. Peng, T. Tamir, and H.L. Bertoni. Theory of periodic dielectricwaveguides. IEEE Trans. Microwave Theory Tech., 23(1), Jan. 1975.

[9] A. Ruhe. Algorithms for the nonlinear eigenvalue problem. SIAM J.Numer. Anal., 10(4):674–689, 1973.

[10] A. Spence and C. Poulton. Photonic band structure calculations usingnonlinear eigenvalue techniques. J. Comput. Phys., 204(1):65–81, 2005.

[11] D. Stowell and J. Tausch. Variational formulation for guided andleaky modes in multilayer dielectric waveguides. Comm. Comp. Phys.,7(3):564–579, 2010.

[12] J. Tausch. Mathematical and numerical techniques for open periodicwaveguides. In M. Ehrhardt, editor, Wave Propagation in PeriodicMedia - Analysis,Numerical Techniques and practical Applications, vol-ume 1 of Progress in Computational Physics, pages 50–72. Bentham,2010.

[13] J. Tausch and J. Butler. Floquet multipliers of periodic waveguides viaDirichlet-to-Neumann maps. J. Comput. Phys., 159:90–102, 2000.

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