Mathematical and numerical techniques for open periodic

waveguides∗

Johannes Tausch

Department of Mathematics

Southern Methodist University

Dallas, TX 75275

tausch@smu.edu

Abstract

The propagation of electromagnetic waves in dielec-tric slab waveguides with periodic corrugations is de-scribed by the spectrum of the Helmholtz operator onan infinite strip with quasiperiodic boundary condi-tions. This paper reviews the basic properties of thisspectrum, which typically consists of guided modes,radiation modes and leaky modes. A great deal ofattention will be devoted to planar waveguides whichshare some of the important features of the periodiccase. To compute the eigenmodes and the associ-ated propagation constants numerically, one usuallytruncates the domain that contains the grating andimposes certain radiation conditions on the artificialboundary. An alternative to this approach is to de-compose the infinite strip into a rectangle, which con-tains the grating, and two semi-infinite domains. Theguided and leaky modes can be computed by match-ing the Dirichlet-to-Neumann operator on the inter-faces of these three domains. The discretized eigen-value problem is nonlinear because of the appearanceof the propagation constant in the artificial boundarycondition. We will discuss how such problems can besolved by numerical continuation. In this approach,one starts with an approximating planar waveguideand then follows the solutions by a continuous tran-sition to the multilayer periodic structure. The chap-

∗Preprint of an article that will appear in Progress in Com-putational Physics (PiCP) 50-72,Matthias Ehrhardt, editor,Bentham (2010).

ter is concluded with a brief description of how theperfectly matched layer can be used to compute theguided modes of a waveguide.

1 Introduction

Periodic media are of considerable interest becausethere are certain directions and frequencies in whichwaves cannot propagate. This phenomenon is knownas a spectral band gap and occurs because of cancella-tion due to coherent scattering. In electromagnetics,materials in which the refractive index varies peri-odically are called photonic crystals. They can beclassified as one- two or three dimensional, depend-ing on the number of space directions in which therefractive index varies.

A classical example of a one-dimensional photoniccrystal is the Bragg mirror which is an infinite stackof alternating dielectric layers. Lord Rayleigh wasable to demonstrate that the mirror exhibits bandgaps [39]. Two dimensional photonic crystals consistof a square lattice of infinite dielectric rods. In thiscase it usually suffices to consider polarized waveswhich reduces the Maxwell problem to the scalarHelmholtz equation in the transverse plane. An ex-ample of a three dimensional crystal is the so-calledwoodpile where parallel dielectric logs are stacked inalternating directions [24, 43]. For an excellent intro-duction to photonic crystals and their use in control-ling the propagation of light we refer the reader to

1

the recent book [28].The modes of a periodic structure are the non-

trivial solutions of the Maxwell equations withoutsource terms. The appropriate mathematical tool inthis context is Floquet Theory which addresses dif-ferential equations with periodic coefficients. Thistopic has been surveyed in [31] and [32]. The typeof solutions that are possible in periodic media isdescribed by Floquet’s Theorem, which states thatthe modes are the product of periodic function and aplane wave. Thus the problem of finding the modescan be reduced to computing the eigenfunctions ofthe Maxwell operator on the periodic cell with quasiperiodic boundary conditions.

In the earlier mentioned examples of photonic crys-tals the periodic cell is finite and the resulting differ-ential operator has a discrete spectrum. The eigen-values are the frequencies that are allowed to prop-agate and depend on the wave number of the planewave. This dependence is usually referred to as thedispersion relation. In order to describe the propaga-tion characteristics of a given periodic structure, theeigenvalue problem must be discretized and solved forall wave vectors in the Brillouin zone. The frequen-cies that are not an eigenvalue for any wave vectorconstant make up the band gaps. There are numer-ous papers that describe this process and its analysis.Representative examples are [4, 16, 17, 29].

Another important class of photonic structures arelayered materials. The best known example is thedielectric slab waveguide, which consists of a highindex core sandwiched between two infinite lower in-dex materials. Such a structure can support a finitenumber of guided waves that propagate in a directionparallel to the core. It is a standard textbook exer-cise to derive the dispersion relation of such a struc-ture [34, 52]. The situation can be complicated byincreasing the number of layers and by adding a peri-odic perturbation, for instance, a grating. Open andperiodic waveguides can be found in many integrated-optics devices, such as semiconductor lasers, waveg-uide couplers, and leaky-wave antennas.

Gratings have been studied for a long time, mainlyin the context of scattering [38, 51]. In a grating therefractive index varies in two directions, but is pe-riodic only in one direction. While Floquet’s Theo-

rem is still applicable to describe the spectrum of theMaxwell operator in such a geometry, there are con-siderable differences to photonic crystals. The maindifficulty is the fact that the periodic cell is an un-bounded region, namely an infinite strip in the trans-verse direction of propagation. Thus the spectrumof the Maxwell operator typically consists of a finitenumber of discrete eigenvalues and a continuum ofeigenmodes. Leaky modes are a third type of phys-ically relevant eigensolutions. They are unboundedsolutions and result from constructive scattering of aguided mode into the semi-infinite regions.

The goal of this chapter is to review some mathe-matical and numerical methods for planar and peri-odic dielectric waveguides. The history of this prob-lem goes back to the engineering analysis of openwaveguides in the 1960s (see the references in [37]).The first mathematical existence studies are muchmore recent, and appear to be limited to guidedmodes and the continuous spectrum [7]. The firstnumerical methods appear in the first engineering pa-pers, see, again [37] and the review paper [25]. Thefollowing decades have seen vast improvements of dis-cretization methods and iterative methods for solvingthe nonlinear eigenvalue problem.

After fixing the mathematical notations in Sec-tion 2 we will first discuss planar non-periodic waveg-uides in 3. We devote a considerable amount of spacefor this topic as it is important for the understand-ing of the more complicated periodic case. Moreover,because of numerical instabilities, the computation ofthe modes is much more difficult than one might haveexpected. Section 4 reviews some analytical resultsabout the existence of the continuous and discretespectrum of periodic waveguides. The following Sec-tions 5 and 6 describe several discretization methodsfor the operator equations.

In numerical computations the infinite periodic cellmust be truncated and suitable conditions must beimposed on the artificial boundary. Since the fre-quency goes into the boundary condition the result-ing eigenvalue value problem is non-linear. A popularway to solve this type of problem is the continuationmethod. In Section 7 we describe how the modes of aperiodic structure can be computed by following thesolutions of an averaged planar structure.

2

In the recent years, the perfectly matched layer(PML) has become a very popular method to trun-cate infinite domains in scattering problems. Thismethodology has also been applied to periodic struc-tures and leads to very different computational issues.This will be mentioned briefly in the concluding Sec-tion 8.

2 Preliminaries

Throughout this article, we assume that the materi-als in which the electromagnetic fields exist are linear,non-dispersive and isotropic. In this case, the rela-tionship between the electric displacement and theelectric field is D = εE and between the magneticflux density and field is B = µH. Here, the electricpermittivity ε and the magnetic permeability µ arescalars that may be functions of the position. Al-though this assumption is only an approximation, itis sufficiently accurate in many application of inter-est. In photonics applications the materials are usu-ally non-magnetic, hence

µ = µ0, ε = ε0εr and n =√εr.

Here, ε0 and µ0 are the permittivity and permeabilityof free-space, εr is the relative permittivity and n isthe refractive index.

In most applications the electromagnetic fieldsare time-harmonic, i.e., the time dependence isexp(−iωt), where ω is the frequency. Furthermore,there are no currents and charges. Under these as-sumptions, Ampere’s and Faraday’s law are

∇×H = −iωεE. (1)

∇×E = iωµH. (2)

In photonics problems one is usually interested inthe type of solutions that can occur in a structurewith a given refractive index. Mathematically, this isan eigenvalue problem where ω is the eigenvalue andE,H are the non-trivial eigenfunctions. Upon takingthe divergence of (1) and (2) it becomes immediatelyclear that

∇ · (εE) = 0, (3)

∇ · (µH) = 0. (4)

must hold. Equations (1-4) are the source-free time-harmonic Maxwell’s equations which appear in thisform in many textbooks. At first glance, equations(3) and (4) are superfluous since they follow directlyfrom (1) and (2). Nevertheless, they must be satis-fied explicitly to avoid difficulties with the infinitelydimensional nullspace of the curl operator.

In this article we limit our attention to structuresthat are invariant in the y-direction. From basic elec-tromagnetic theory it is well known that in this casethe eigensolutions of the Maxwell equations come intwo polarizations:

• Transverse Electric (TE) waves. The elec-tric field is parallel to the y-axis E(x, y, z) =u(x, z) ey, where the scalar function u(x, z) sat-isfies the Helmholtz equation

∆u+ n2k2u = 0. (5)

where k = ω√ε0µ0 is the wave number, which is

the unknown eigenvalue. From Ampere’s law itfollows that the magnetic field is given by

H(x, y, z) =i

µω

(

uz(x, z)ex − ux(x, z)ez

)

.

• Transverse Electric (TM) waves. The mag-netic field is parallel to the y-axis H(x, y, z) =u(x, z) ey, where the scalar function u(x, z) sat-isfies the Helmholtz equation

∇ · ( 1

n2∇u) + k2u = 0. (6)

The electric field is given by

E(x, y, z) =1

iεω

(

uz(x, z)ex − ux(x, z)ez

)

,

which is a consequence of Faraday’s law.

In the literature there is an apparent disagreementabout which plane is transverse. Here, transverserefers to the plane perpendicular to the direction ofthe periodic variation, i.e., the xy-plane. This is theconvention found in, e.g. [37]. Another possibility isto denote the plane perpendicular to the grooves asthe transverse plane. In this case, the terms TE andTM must be switched. This is the choice of. e.g [7].

3

For both polarization the problem is considerablysimpler than the Maxwell problem, since only a scalarequation in two spatial variables must be solved.Moreover, the nullspaces of the Helmholtz operatorsare finite-dimensional, hence there is no need to in-corporate the conditions (3) and (4).

The equations for the TE and TM polarizationscan be unified into one, by writing

Lu := −∇ · p∇u = k2qu (7)

where

p(x, z) =

1, TE case,n−2(x, z), TM case,

and

q(x, z) =

n2(x, z), TE case,1, TM case.

A typical grating structure can consist of several,but finitely many stratified layers and a layer thatcontains a periodic grating. The refractive index is aperiodic function in z

n(x, z) = n(x, z + Λ)

where Λ is the grating period. With the exception ofthe grating layer, the index in each layer is a piece-wise constant function of x only, and constant out-side the finite interval x ∈ [0, w]. Usually, the twosemi-infinite intervals are referred to as the super-state and substrate and the region in between is calledthe stack. In many applications the refractive indicesin the two semi-infinite layers are the same, in oth-ers, they are different. We will see that the lattercase complicates matters somewhat. We orient thedirection of the x-axis such that the “negative” semi-infinite layer contains the smaller refractive index. Atypical geometry is shown in Figure 1.

For a piecewise constant index, (7) is understoodin the distributional sense. This implies that on theinterface of two layers the solution is continuous

u+ = u−. (8)

The conditions for the derivative in the normal direc-tion n of the interface depend on the polarization

∂u+

∂n=

∂u−

∂n, TE case,

εεεεε 2

substratesuperstrate

0 31 4

stack

Ligh

t

Radiation

x0 w

z

Figure 1: Typical waveguide with grating layer.

1

n2+

∂u+

∂n=

1

n2−

∂u−

∂n, TM case.

3 Planar Waveguides

In a planar waveguide there is no grating layer, thusthe refractive index is a function of x only

n(x, z) = n(x),

where n(x) is a piecewise constant function with afinite number of discontinuities inside the interval[0, w]. While planar waveguides lack some of theinteresting features of periodic waveguides, they arestill helpful for the understanding the spectral prop-erties of open structures: as in the periodic case,planar waveguides can have a set of discrete eigen-modes as well as a continuous spectrum. In addition,both types of waveguides have so-called leaky modes,which are characterized by complex propagation con-stants.

The periodic waveguide may be regarded as a per-turbation of the planar waveguide, where inside thegrating region the refractive index is replaced by itsaverage value

n2g =

1

wgΛ

∫ Λ

0

∫ x+

x−

n2(x, z) dxdz. (9)

4

Here, wg is the length of the interval [x−, x+] on thex-axis that is occupied by the grating region. Thisconcept is useful for computing the modes of a pe-riodic structure by continuation from the averagedplanar structure. In such an approach the modes ofthe structure with index

n2v(x, z) = vn2(x, z) + (1 − v)n2

g (10)

are followed in the interval v ∈ [0, 1]. This will bediscussed in more detail in Section 7.

Since the structure is invariant in z-direction, thesolutions of (7) have the form

u(x, z) = exp(iβz)Φ(x)

hence substitution into (7) leads to

(

p(x)φ′(x))′

+ k2q(x)φ(x) = β2p(x)φ(x), (11)

Since x ∈ R this is a singular Sturm-Liouville prob-lem, where the propagation constant β is the un-known eigenvalue, and k is a fixed parameter. Ofcourse, one could also revert the roles of β and k andwould obtain a similar problem.

A material consisting of J + 1 layers is completelydescribed by J + 1 refractive indices n0, . . . , nJ andthe positions x1, . . . xJ of the J interfaces, c.f. Fig-ure 2. The layer with index n0 is the cover and layernJ is the substrate. The other layers are the interiorlayers. To simplify notations we shift the origin tothe first interface, i.e., x1 = 0 and write xJ = w.

x1 = 0 x2 xJ−1xJ = w

n0 n1 nJ−1 nJ

. . .-

Figure 2: The geometry parameters

In each layer the functions p and q are constant,therefore the solution for both polarizations is a linearcombination of left and right going harmonics

φ(x) ∈ spanexp(±iαjx), xj ≤ x ≤ xj+1.

Here, the transverse wave numbers are

αj =√

k2n2j − β2, (12)

where we adopt the convention that the square rootfunction has a branch cut on the negative real axis.Thus for a real value of β the propagation constantis either positive real or positive imaginary.

It is convenient to write the solution in the j-thlayer in terms of the solution of the left endpoint ofthe layer

φ(x) = cos(αj(x − xj)φ(xj )

+sin(αj(x− xj))

αjφ′(xj). (13)

The state vector

~Φ(x) =

[

φ(x)p(x)φ′(x)

]

is continuous across layer interfaces. Using (13) itfollows that the state vectors at the interfaces xj andxj+1 transforms according to the formula

~Φ(xj+1) = Tj~Φ(xj)

where Tj is the matrix

Tj =

[

cos(αjwj) sin(αjwj)/αj

− sin(αjwj)αj cos(αjwj)

]

,

andαj = pjαj

where pj denotes the value of the function p(x) inthe jth layer. The translation matrix is always in-vertible because the determinant of Tj is unity. Thetranslation from 0 to w is simply the product

T = TJ−1 · . . . · T1

The matrix T is useful to construct solutions of (11).We will demonstrate this now.

3.1 Continuous Spectrum

The value of α0 is real if β2 ≤ n20k

2. Thus the left andright going harmonics exp(±iα0x) are two linearly in-dependent bounded solutions in the left semi-infinitelayer x ≤ 0. Using translation operators, both har-monics can be extended to a solution of (11) on the

5

whole real axis. In the right semi-infinite layer thesolution has the form of (13) for j = J .

Recall that the orientation of the x-axis is such thatn0 ≤ nJ , hence the value of αJ is real for the givenrange of β2 and thus the two solutions are boundedand are usually called two-sided radiation modes. If0 < β2 ≤ n2

0k2 the modes propagate in z direction, if

β2 < 0 the modes are evanescent, that is, they decayin z.

Now consider a value of β2 in the interval n20k

2 <β2 ≤ n2

Jk2. Here the value of α0 is imaginary

whereas the value of αJ is real. Hence there is onlyone linearly independent bounded mode in the leftsemi-infinite layer, namely exp(−iα0x). Translationthrough the layers will result in a bounded solutionon the whole real axis. These solutions radiate onlyin the right semi-infinite layer and are therefore calledsubstrate modes.

3.2 Discrete Spectrum

So far, we have found bounded solutions for everyvalue of β2, thus the interval β2 ≤ n2

Jk2 is the con-

tinuous spectrum. The situation is different whenβ2 > n2

Jk2, because then the transverse propaga-

tion constants in both semi-infinite layers are purelyimaginary. In this case there is only one boundedexponential on either side. In general, the boundedsolution in the left layer will translate to a combina-tion of the bounded and the unbounded exponentialon the right side, unless the coefficient of the un-bounded exponential vanishes. If that happens, themode is exponentially decaying in both semi-infinitelayers and is called a guided mode.

In the following we derive a condition to character-ize such a mode. Suppose that the mode is normal-ized to satisfy φ(0) = 1 then

φ(x) = exp(−iα0x) for x ≤ 0

and

φ(x) = φ(xJ ) exp(iαJ (x− xJ )) for x ≥ xJ

In each layer of the stack a guided mode is again ofthe form (13). Using translation operators it follows

that the state vectors at the points x = 0 and x = wmust satisfy the relationship

T

[

1−iα0

]

= φ(w)

[

1iαJ

]

where αj = pjαj and p0 and pJ are the values of p(x)in the semi-infinite layers. Note that the variables T ,α0 and αJ are functions of β. Simple algebra showsthat the above relationship is equivalent to findingthe roots of the characteristic function

F (β) = t12α0αJ − t21 + i(

t11αJ + t22α0

)

= 0, (14)

where the tij are the coefficients of the matrix T .Since a guided mode has exponential decay it is

a function L2(R). By multiplying (11) with φ andintegrating by parts it follows that

−∫

R

p(φ′)2 + k2

∫

R

qφ2 = β2

∫

R

pφ2

from which it can be concluded that

(

β

k

)2

≤∫

Rqφ2

∫

Rpφ2

.

For both polarizations the right hand side can be es-timated by

n2M := max

x∈R

n2(x).

Hence there are guided modes only in the intervaln2

Jk2 ≤ β2 ≤ n2

Mk2. We will see in Section 3.3 thattheir number is finite. The types of solutions to (11)are summarized in Figure 3.

3.3 Leaky Modes

In addition to real solutions in the interval n2Jk

2 <β2 < n2

Mk2 the characteristic equation (14) can alsohave complex roots. A complex value of β impliesthat the transverse wave numbers α0 and αJ are inthe complex plane and thus the branch choice of thesquare root in (12) becomes an important issue. Thecharacteristic function is four-valued depending onthe signs of the square root. Note that the sign choiceof the interior αj ’s is irrelevant because the transla-tion operators Tj are even functions of αj .

6

-0 k2n2

0 k2n2J k2n2

M β2

evanescentmodes

-

two sidedradiation modes

- substratemodes

- guidedmodes

-

continuous spectrum -

Figure 3: Classification of the spectrum of a planarwaveguide.

The solutions that have physical significance arethose that radiate energy away from the stack andare known as leaky modes, see [13]. Although leakywaves have infinite energy they are physically sig-nificant and have been verified experimentally in fi-nite regions of the waveguide [49]. The leaky modesform a discrete set of expansion functions in the stackand can represent field solutions in this region [35].Leaky-wave analysis has the advantage that in therepresentation of a field the superposition integral ofradiation modes is replaced by a discrete sum of leakymodes. In practice, only a few modes are necessaryto obtain good approximations. This type of analysishas been applied in several photonics applications, es-pecially for waveguide transitions. See, e.g. [33] andthe references cited therein.

An elegant way to deal with the four-valuedness ofthe characteristic function is to make the change ofvariables

z =1

2i

(

√

k2n20 − β2 +

√

k2n2J − β2

)

. (15)

which was suggested in the paper [41]. Simple algebrashows that

α0 = i

(

z +δ2

4z

)

(16)

αJ = i

(

z − δ2

4z

)

(17)

whereδ2 = k2(n2

J − n20). (18)

Because of α2j = k2(n2

j −n20)+α2

0 all transverse wavenumbers are analytic functions of z when z 6= 0. Fur-thermore, the coefficients of the translation matricesare analytic functions of αj and therefore they arealso analytic in z 6= 0. Thus the characteristic func-tion is an analytic function of z in C \ 0 and hasessential singularities in z = 0 and z = ∞. If n0 = nJ

then δ = 0 and the transformation is simply

α0 = αJ = iz

and the characteristic function is an entire functionwith an essential singularity at z = ∞.

Transformation (15) maps the interval β2 ∈[n2

Jk2, n2

Mk2] to a finite interval on the positive realz-axis that does not contain the origin when δ 6= 0.Since the function is analytic, there is at most a finitenumber of roots in this interval.

Another result about the z-roots that can be ob-tained from the Great Picard Theorem of complexanalysis. Because of the essential singularities it fol-lows that characteristic function either has no rootat all, or an infinite number of roots that accumu-late at infinity and, in addition, if δ 6= 0, accumu-late in the origin. If the frequency k is large enough,and nM > maxn0, nJ it well-known that there areguided modes and hence there is an infinite numberof roots. The spectrum in this situation is illustratedin Figure 4. The situation depicted here appears tobe the generic case. However, the author of this chap-ter is not aware of results that guarantee existence ofroots under more general conditions.

So far, we have considered the eigenvalue β for agiven value of the frequency k. In the analysis ofa waveguide one is usually interested how the l-theigenvalue changes with the frequency k. The func-tion βl(k) is called the dispersion relation and de-scribes the way in which the propagation speed andattenuation varies with the frequency of the mode.

As the frequency increases, the number of guidedmodes increases. The smallest frequency for which agiven mode is guided is called the cut-off frequencyfor that mode. This condition is characterized by abifurcation of two dispersion curves in the complexplane [22].

7

−10 −5 0 5 10−20

−15

−10

−5

0

5

10

15

20

Re(z)

Im(z

)

Figure 4: Roots of the characteristic equation in thecomplex z-plane for k = 9.929. Guided modes arecolored in red. The circle has radius δ/2. Arrowsindicate the motion of the roots with increasing fre-quency.

0 5 10 150

1

2

3

4

5

6

7

8

9

10

β

k

Continous Spectrum

β=nJ k

β=nM k

Mode

Figure 5: Guided modes as a function of the fre-quency for the planar waveguide in Figure 4.

8

We use the guide

n0 = 1.0,

n1 = n4 = 1.66,

n2 = 1.53,

n3 = 1.60,

n5 = 1.66,

n6 = 1.5,

wj = 0.5, 1 ≤ j ≤ 5,

to illustrate the dependence of the eigenmodes onthe frequency. This structure appears frequently inthe literature [13]. The real dispersion curves areshown in 5. The dispersion of the leaky modes arebest visualized in the z-plane. As k increases, themodes emanate from the origin in complex conjugatepairs until they bifurcate into two real solutions atz = δ/2. Only the solution z > δ/2 represents aguided solution. The other solution is non-physicalas it increases exponentially in one layer. At the sametime, there are complex conjugate pairs of solutionsthat move near the imaginary axis towards the pointz = −δ/2. Here the pairs bifurcate into another set ofnon-physical real solutions. The motion of the rootsin the complex planes is indicated by arrows in Fig-ure 4.

3.4 Variational Formulation

The previous discussion was based on deriving a char-acteristic equation of (11) using translation opera-tors. A different way to treat the problem of findingthe guided and leaky modes of a planar multilayerwaveguide is the variational approach. Unlike thecharacteristic function method, this approach gener-alizes to arbitrary index profiles as long as the indexvaries only in a finite interval. Furthermore, we willshow in the following section that numerical methodsbased on a variational formulation are more stablethan algorithms that find the roots of a characteris-tic function.

Since guided and leaky modes are defined by onlyone complex exponential in the semi-infinite layersthey solve (11) in the interval [0, w] with Robin-type

boundary conditions at the endpoints. Thus

(

pφ′)′

+ k2qφ = β2pφ, (19)

p1φ′(0) + ip0α0φ(0) = 0, (20)

pJ−1φ′(w) − ipJαJφ(w) = 0. (21)

Since the transverse wave numbers α0 and αJ dependin a nonlinear way on the eigenvalue β2 the system(19-21) does not constitute a regular Sturm-Liouvilleproblem.

The actual nature of this problem becomes moreapparent if the unknown β2 is replaced by the z-variable in (15). It was recently discovered [44] thatwith this change of variables the variational form of(19-21) transforms into a quartic eigenvalue problem.

To derive this result we multiply (19) by a testfunction ψ and integrate by parts. This leads to

(ψ′, pφ′) − k2(ψ, qφ) + β2(ψ, pφ)

−pJ−1ψ(w)φ′(w) + p1ψ(0)φ′(0) = 0.

Here, (·, ·) denotes the usual complex L2[0, w]-innerproduct. Incorporating the boundary conditions (20)and (21) leads to

(ψ′, pφ′) − k2(ψ, qφ) + β2(ψ, pφ)

−iαJ ψ(w)φ(w) − iα0ψ(0)φ(0) = 0. (22)

The problem at hand is to find β and φ 6= 0 such thatthe above equation is satisfied. Since α0, αJ dependon β (22) is a nonlinear eigenvalue problem in β.

To obtain a formulation in the z variable, note thatit follows from (12), (16) and (17) that

β2 =k2

2(n2

0 + n2J ) − 1

2(α2

0 + α2J )

=k2

2(n2

0 + n2J ) +

(

z2 +δ4

16z2

)

.

Thus (22) is transformed to

(ψ′, pφ′) − k2(ψ, qφ) +k2

2(n2

0 + n2J)(ψ, pφ)

+

(

z − δ2

4z

)

ψ(w)φ(w) +

(

z +δ2

4z

)

ψ(0)φ(0)

+

(

z2 +δ4

16z2

)

(ψ, pφ) = 0.

9

We introduce the bilinear forms

a(ψ, φ) = (ψ′, pφ′) − k2(ψ, qφ)

+k2

2(n2

0 + n2J)(ψ, pφ) ,

a±(ψ, φ) = ψ(0)φ(0) ± ψ(w)φ(w) .

After multiplying with z2 and sorting out equal pow-ers of z we obtain

δ4

16(ψ, pφ) − z

δ2

4a−(ψ, φ) + z2a(ψ, φ)

+z3a+(ψ, φ) + z4(ψ, pφ) = 0, (23)

which is a quartic eigenvalue problem. In the caseδ = 0 the above simplifies to the quadratic eigenvalueproblem

a(ψ, φ) + za+(ψ, φ) + z2(ψ, φ) = 0. (24)

3.5 Computation of modes

The most obvious approach to computing the guidedand leaky modes of a planar waveguide is to find theroots of the characteristic equation (14) with New-ton’s method. Because of the square root in (12) thisfunction has two branch cuts in the complex plane.If one of the roots is near a branch cut, then iteratestend to jump across branch cuts without converging.To avoid such difficulties one can use the change ofvariables (15).

In applications one is usually interested in howmany guided modes a given structure has or whichthe dominant leaky modes are. Therefore it is impor-tant to have a numerical method which is capable tofind all roots at least in a specified range. EmployingNewton’s method with different initial guesses is nota very satisfactory approach as it would be hard de-cide when to stop searching for new roots. A betterapproach is to use the argument principle of com-plex analysis, which relates the number of roots ina domain to a contour integral over its boundary.Delves and Lyness [15] propose a method which isbased on this principle and guarantees all roots ofan analytic function in a given region of the complexplane. Variations of this idea with application to di-electric waveguides are discussed in [2, 3, 12] and [42].

Another approach to obtain all roots in a givenrange is to use a continuation method [1]. Petracekand Singh were the first to apply this technique forplanar dielectric waveguides [30]. Their idea is totruncate the structure in the semi-infinite layers andto impose homogeneous Dirichlet boundary conditionat the endpoints. The truncated problem is a regularSturm-Liouville problem which can be formulated asa standard eigenvalue problem. Its solutions are theinitial points that are followed by a in a smooth tran-sition from the closed to the open structure. A moredetailed description of continuation methods will begiven later in Section 7 in the context of periodicwaveguides.

Methods that compute the roots of (14) are capableof finding the propagation constants of certain struc-tures, but they have their limitations because the ex-ponential scaling of the characteristic function. Thisproperty causes significant cancellation errors whenevaluating the function in floating point arithmetic.

−2 0 2 4 6 8−0.5

0

0.5

1

1.5

2

2.5

x

n(x)φ(x)

−2 0 2 4 6 8−0.5

0

0.5

1

1.5

2

2.5

x

n(x)φ(x)

Figure 6: Refractive index and dominant guidedmode for k = 9 and β/k = 1.63986.

To illustrate how this happens consider the waveg-uide whose refractive index and dominant guidedmode is shown in Figure 6. When k = 9 the modehas propagation constant β/k ≈ 1.64. Hence thetransverse wave number α4 is purely imaginary inlayer 4. The translation operator T4 has eigenval-

10

ues exp(±iα4w4), and the eigenvectors Ψ± are statevectors that are either magnified or reduced whentranslated across this layer. The state vector of theeigenmode is a linear combination of Ψ±, but sincethe mode decays in layer 4, the weight of Ψ− must bevery small. However, in the presence of floating pointerrors, this weight can be increased with the resultthat the function value of the numerically computedcharacteristic function is significantly different fromzero. This effect becomes more noticeable when thewidth of the layer or the frequency is increased. Thisis illustrated in Figure 7 which shows the numeri-cally computed characteristic function in the intervalof β/k where the guided modes occur. For k = 5 theroots are clearly visible, but when k = 7 or k = 9,the roots disappear in numerical noise.

1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66

10−8

10−6

10−4

10−2

100

β/k

|F(β

)|

k=9k=7k=5

Figure 7: Modulus of the characteristic function com-puted in double precision arithmetic of the structurewith index profile shown in Figure 6. Three frequen-cies, TE polarization.

A different approach that does not rely on findingthe roots of the characteristic function is the finiteelement method. Here, one begins with a partitionof the interval [0, w]

0 = ξ0 < ξ1 < . . . < ξN = w

where the set of node points ξj contains the layerinterfaces. The mesh width is h = maxj(ξj+1 − ξj),

and the finite element space Sh consists of func-tions that are piecewise polynomial on the partitionand globally continuous. The finite element methodseeks the solution of the variational formulation inthe space Sh. Since this space has finite dimension,the problem is reduced to a matrix problem.

If we work with the variational form in the β-variable (22), the resulting task is to find the valuesof β such that

A(β) is singular. (25)

This is a highly non-linear eigenvalue problem andwas considered in a slightly more general settingin [50]. This approach avoids the numerical instabili-ties associated with the evaluation of the characteris-tic function, but does not guarantee that a completeset of solutions is found, since the solutions dependon the quality of the initial guess.

The alternative is to work with the variational formin the z-variable(23), in which case one obtains thequartic eigenvalue problem: Find z such that

A0 + zA1 + z2A2 + z3A3 + z4A4 is singular.

When the finite elements are the piecewise linear hatfunctions, the matrices A0, A2 and A4 are tridiago-nal, and

A1 =δ2

4diag[1, 0, . . . , 0,−1],

A3 =δ2

4diag[1, 0, . . . , 0, 1].

There are several numerical methods to treat polyno-mial eigenvalue problems, they are reviewed in [48].The standard approach is to solve the equivalent gen-eralized eigenvalue problem: Find z such that

−I−I

−IA0 A1 A2 A3

+ z

II

IA4

is singular.Thus a generalized eigenvalue problem of size 4N

must be solved. For the latter the direct method,based on the QZ-factorization can be used. However,

11

since the companion matrix is sparse, it is preferableto use Arnoldi methods, which are implemented inARPACK.

The advantage of solving the quartic eigenvalueproblem is that all dominant eigenvalues can be foundby solving one eigenvalue problem. Iterative methodsfor (25) can only provide one eigenvalue at a time.Thus the numerical effort to compute several modescan be much higher and there is no guarantee thatno modes have been missed.

4 Periodic Waveguides

We now turn our attention to waveguides that con-stitute a periodic perturbation of a multilayer pla-nar structure. The discrete spectrum of the periodicstructure is the perturbation of the spectrum of theplanar structure.

However, because of coherent scattering, a guidedmode can be changed into a leaky mode as the fre-quency is increased. Such a mode radiates energyaway from the grating region into free space. Thiseffect is equivalent to the occurrence of stop-bandsin the dispersion relation and is characterized by acomplex propagation constant.

In this section we only give a minimal description ofthe spectral properties for the periodic case to makeour presentation self consistent. For more detail andmathematical rigour we refer to Stephen Shipman’schapter in the same book [40].

4.1 Floquet’s Theorem

The modes of a grating structure such as the oneshown in Figure 1 are the solutions of the eigenvalueproblem (7), which is posed in the xz-plane. Therefractive index has discrete translational symmetryin z-direction, that is, it remains unchanged undertranslations of an integer multiple of the grating pe-riod Λ. This allows us to consider the eigenvalueproblem on the fundamental domain of the symme-try, which is one period of the grating.

The mathematical result behind this reduction isknown as Floquet’s theorem, which states that theeigenfunctions are quasi-periodic in the z-direction.

This result can be seen by applying the Floquet trans-form, which, for a z-periodic medium is given by

uβ(x, z) := Uu(x, z, β) =∑

l∈Z

u(x, z − Λl) exp(iβΛl)

The transformed variable β is the quasimomentum,or propagation constant. It is obvious from the defi-nition that the inverse transform is

u(x, z) =Λ

2π

∫ π/Λ

−π/Λ

uβ(x, z)dβ.

Furthermore,

uβ(x, z + Λ) = exp(iβΛ)uβ(x, z), (26)

uβ+2π/Λ(x, z) = uβ(x, z). (27)

Moreover, the Floquet transform commutes with theoperator L defined in (7), hence it satisfies the differ-ential equation

Luβ = k2quβ in ΩΛ (28)

uβ|ΓΛ= eiβΛ uβ|Γ0

(29)

∂uβ

∂z

∣

∣

∣

∣

ΓΛ

= eiβΛ ∂uβ

∂z

∣

∣

∣

∣

Γ0

(30)

Here ΩΛ = R×[0,Λ] is an infinite strip of width Λ andΓ0 = R × 0 and ΓΛ = R × Λ are its boundaries.In the following we will drop the subscript β when itis clear that a solution of (28-30) is meant.

Because of (27) we only have to consider propaga-tion constants in the so-called Brillouin zone

−π

Λ< Reβ ≤ π

Λ. (31)

Since the fundamental domain of the symmetry isinfinite, it is possible that both the propagation con-stant or the frequency are complex. Thus the disper-sion relation k(β) of a mode is a complex function.Strictly speaking, such solutions are not part of thespectrum of the operator L as they are unboundedin the xz-plane. However, they are of great physicalinterest, as they represent waves that leak out en-ergy away from the grating region as they travel inz-direction.

There are two equivalent viewpoints to characterizethe spectrum of L.

12

1. Fix the propagation constant β ∈ R and find thepossibly complex values of k such that (28-30)has non-trivial solutions.

2. Fix the wavenumber k ∈ R and find the corre-sponding values of β, which again can be real orcomplex.

For theoretical purposes it more convenient to workwith the first alternative as this viewpoint leads toa more standard eigenvalue formulation. Since prob-lem (28-30) is posed in an infinite domain, the spec-trum for a given β usually consists of a discrete anda continuous part. The real discrete eigensolutionsare the guided modes of the structure, whereas thecontinuous spectrum consist of solutions of a scatter-ing problem with an incoming quasi-periodic planewave. The eigenvalue problem for the guided modesmay also have an infinite number of discrete complexsolutions.

The second viewpoint is often preferred in the en-gineering literature as the solutions with real k andcomplex β have the physical interpretation of a leakymode, provided that certain radiation conditions aresatisfied.

4.2 DtN Maps

For both theoretical and numerical purposes it is nec-essary to reduce the eigenvalue problem on the infi-nite strip to an equivalent problem on a finite domain.To that end consider two x-coordinates outside thegrating region x± and define the subregions

Ω− = (−∞, x−) × (0,Λ),

Ω0 = (x−, x+) × (0,Λ),

Ω+ = (x+,∞) × (0,Λ)

and the interfaces

Γ± = x± × [0,Λ].

In the exterior regions Ω± the refractive index is apiecewise constant function of x and hence it is pos-sible to write down the solution and the Dirichlet-to-Neumann (DtN) operator in closed form.

For numerical purposes it desirable to make Ω0 assmall as possible, which means that x± are the end-points of the grating region. However, to keep thediscussion simple, we begin by setting x− = x0 = 0and x+ = xJ = w, so that the exterior regions havea constant refractive index, n− = n0 and n+ = nJ ,respectively. The modifications for allowing uniformlayers in Ω± are discussed later.

Because of quasiperiodicity, the Fourier expansionof a solution in an exterior region Ω± is

u(x, z) =∑

l∈Z

u±l exp(

± iα±l x+ iβlz

)

, (32)

where

βl =2πl

Λ+ β, (33)

α±l =

(

n2±k

2 − β2l

)12 (34)

and n± is the refractive index in Ω±. The coefficientsin (32) are

u±l =1

Λ

∫ Λ

0

u(x±, z) exp(

− iβlz)

dz.

For the uniqueness in the expansion (32) the appro-priate branch choice of the square root in (34) mustbe determined. When β is real, the standard defini-tion of the square root, with

√−1 = i and the branch

cut on the negative real axis, ensures that solutionsare either outgoing or exponentially decaying in theexterior regions. The latter case happens for an infi-nite number of l’s.

To determine the branch choice for complex β onestipulates that the transverse wavenumber dependscontinuously on β [23]. This is based on the assump-tion that a guided mode of a planar waveguide un-dergoes small changes when a small periodic pertur-bation is introduced.

To clarify this concept consider the function

ζ 7→(

κ2 − ζ2)

12

for some κ > 0 in the complex plane. This func-tion has two branch cuts at (−∞,−κ) and (κ,∞).An analytic continuation of the function can be de-fined on a two-sheeted Riemann surface, with the top

13

sheet characterized by the positive square root andthe bottom sheet by the negative square root. Alongthe branch cut (κ,∞) the first quadrant of the topsheet and the forth quadrant of the bottom sheet areconnected. Furthermore, the forth quadrant of thetop and first quadrant of the bottom are connected.The connection along the other branch cut is similar.

The DtN-map is an infinite-valued function be-cause each α±

l possesses one such Riemann surface.The appropriate sheet in the presence of a periodicperturbation is the neighborhood of the real axis onthe top sheet, because this the choice for a real β inthe planar case. This is illustrated in Figure 8.

-Re ζ

6Im ζ

−κ κ− − + + + + +

+ + + + + + − −

Figure 8: The neighborhood of the top real axis onthe Riemann surface for ζ 7→ (κ2 − ζ2)

12 . Plus signs

indicate the top sheet, minus signs the bottom sheet.

Since in (33) the imaginary part of βl is the samefor all l ∈ Z, all terms in expansion (32) are either ex-ponentially increasing or decreasing in z-direction. Ifwe pick a decreasing mode ( Imβ > 0), then it can beseen from Figure 8 that the positive square root mustbe selected when Reβl < κ and the negative squareroot must be selected when Reβl > κ. Thus the so-lution u(x, z) in the exterior regions is composed ofthree types of waves, depending on the behavior inthe semi-infinite regions:

I: Reβl ∈ (−∞, 0) ⇒ decreasing, outgoing,II: Reβl ∈ (0, n±k) ⇒ increasing, outgoing,III: Reβl ∈ (n±k,∞) ⇒ decreasing, incoming.

Case I and III occurs for an infinite number of l ∈Z, whereas Case II only finitely many times or not at

all in one or both of the exterior regions.With the appropriate branch choice clarified, the

exterior DtN map is uniquely defined for any functionon the interfaces Γ± by

u(z) =∑

l∈Z

u±l exp(

iβlz)

⇒

T e±(β, k)u(z) =

∑

l∈Z

iα±l u

±l exp

(

iβlz)

. (35)

We now address the modifications when there areseveral uniform layers in the exterior regions. Welimit ourselves to Ω+ as the treatment of Ω− com-pletely analogous. The expansion of u and its deriva-tive is for (x, z) ∈ Ω+

u(x, z) =∑

l∈Z

u±l φl(x) exp(iβlz),

∂u

∂x(x, z) =

∑

l∈Z

u±l φ′l(x) exp(iβlz)

where the functions φl satisfy(

pφ′l)′ +

(

qk2 − pβ2)

φl = 0, in (x+,∞)

φl(x+) = 1.

Just as in the planar case, the boundary value prob-lem above can be solved using translation operators.The state vector at x = x+ is given by

[

φl(x+)p(x+)φ′l(x+)

]

= T−1l+

[

1iαJ,l

]

where Tl+ is the product of translation operatorsfrom x+ to xJ . Then the coefficients in DtN operatorof (35) is given by

α+l =

φ′l(x+)

φl(x+). (36)

In Section 3 we have seen that the evaluation of trans-lation operators can suffer from numerical instabili-ties. In this case it is better to solve the two-pointboundary value problem

(

pφ′l)′ +

(

qk2 − pβ2)

φl = 0, in (x+, xJ ),

φl(x+) = 1,

φ′l(xJ ) − iαJ,lφl(xJ ) = 0

using a finite-element discretization.

14

4.3 The Spectrum of the Periodic case

Just like we did for planar waveguides, we beginwith the continuous spectrum of (28-30). Thesemodes consist of scattering solutions of plane waveapproaching the grating region from one of the semi-infinite layers. To simplify notations, we only con-sider a plane wave approaching from Ω+, in whichcase the incoming field is

uinc(x, z) = exp(−iαx+ iβz). (37)

Here, k and β are real and the transverse wavenumberis given by

α =√

n2Jk

2 − β2.

Because of the Floquet theorem, we can limit our-selves to values of β in the Brillouin zone. Further-more, by the incoming wave condition α must be non-negative, thus the restrictions

|β| ≤ nJk, |β| ≤ π

Λ(38)

must hold for any incoming field.In Ω+, the total field is the superposition of the

incoming and the reflected field where both fieldsare quasi-periodic. The reflected field uref solves theHelmholtz equation with the outgoing radiation con-dition enforced by the branch choice in (34). On theinterface Γ+, the total field u satisfies

pi ∂u

∂n

i

= pe

∂u

∂n

inc

+∂u

∂n

ref

= pe

∂u

∂n

inc

+ T e+(β, k)

(

ui − uinc)

Here, the superscript i and e denote a limit ap-proached from the interior or the exterior of Ω0.

With the incoming field given by (37), the last con-dition simplifies to

pi ∂u

∂n

i

− peT e+(β, k)u(z) = 2peiα exp(iβz). (39)

In Ω−, the total field is the transmitted wave. Bycontinuity of u and p ∂u

∂n it follows that on Γ−

pi ∂u

∂n

i

− peT e−(β, k)ui = 0 (40)

holds. Thus the total field in Ω0 must satisfy theboundary value problem

Lu = k2qu in Ω0, (41)

u|ΓΛ= eiβΛ u|Γ0

, (42)

∂u

∂z

∣

∣

∣

∣

ΓΛ

= eiβΛ ∂u

∂z

∣

∣

∣

∣

Γ0

, (43)

pi ∂u

∂n

∣

∣

∣

∣

Γ±

= peT±(β, k) u|Γ±+ f±. (44)

Here f± denote the right hand sides in (39) and (40).Once a solution of (41-44) has been found, it can beextended to a solution of (28-30) by expansion (32).

The existence of solutions to the scattering prob-lem, and therefore the nature of the continuous spec-trum of (28-30) is linked to the solvability of (41-44).These questions can be settled by formulating thepartial differential equation in variational form.

Before we do that we recall the following spaces ofquasiperiodic functions.

1. C∞β (R2) is the space of C∞-functions which sat-

isfy (26) and vanish for large |z|.

2. C∞β (Ω0) consists of restrictions of functions in

C∞β (R2) to Ω0.

3. H1β(Ω0) is the closure of C∞

β (Ω0) in the H1-norm.

4. On the interfaces Γ± the spaces Hsβ(Γ±) are de-

fined by the Fourier transform

Hsβ(Γ±) =

∑

l∈Z

vleiβlz :

∑

l∈Z

(

1 + β2l

)s |vl|2 <∞

The spaceH12

β (Γ±) consists exactly of the traces of all

functions of H1β(Ω0) on Γ±. Furthermore, the DtN

operators T e± : H

12

β (Γ±) → H− 1

2

β (Γ±) are continuous.

The variational form is easily obtained by multi-plying (41-44) with v ∈ H1

β(Ω0) and applying the di-vergence theorem. Note that the contribution of theboundaries Γ0 and ΓΛ cancel because of the quasi-periodicity condition.

15

The resulting problem is to find k and u ∈ H1β(Ω0)

such thata(β, k; v, u) = (v, f) (45)

holds for all v ∈ H1β(Ω0). Here, β ∈ R is given and

a(β, k; v, u) =

∫

Ω0

p∇v · ∇u− k2

∫

Ω0

q vu

−∫

Γ±

p v T e±(β, k)u.

The following existence and uniqueness resultshave been established in [7]. For more related resultssee also [5, 20, 19].

1. Problem (45) has at least one solution u ∈H1

β(Ω0) and the solution set is a finite dimen-sional affine subspace.

2. If the solution of (45) is non-unique then k iscalled a singular frequency. This condition isequivalent to the existence of a non-trivial solu-tion of the homogeneous equation. For such asolution the transverse wave numbers α±

l in (32)are either zero or positive imaginary.

3. The singular frequencies form at most a count-able sequence with no accumulation point.

By the first result, there is a nontrivial solution of(28-30) corresponding to any plane wave approachingthe grating from Ω+. Likewise, there is a solutionfor any plane wave approaching from Ω−. Thus thecontinuous spectrum is the region

k ≥ |β|max(n0, nJ)

=|β|nJ

(46)

which is called light cone.If kn0 ≤ β ≤ knJ then the transverse wave num-

ber α+0 is real whereas all α−

l are purely imaginary.Modes in this interval are substrate radiation modes.

Below the light cone 0 ≤ β ≤ n0k there is no in-cident field, because it would violate condition (37).By the second and third result, it is possible thatthis range contains a finite number of guided modes.Furthermore, the stated existence results do not ex-clude the possibility of further real frequencies above

the light cone, although this appears to be an excep-tional case. The reason is that (45) is nonhermitianand nonlinear because of the appearance of the DtNoperators.

The derivation of variational formulation (45) isbased on the assumption that β is real. In order tofind leaky modes we must allow for complex β, inwhich case the boundary terms on Γ0 and ΓΛ do notcancel. Thus the variational formulation is

a(β, k; v, u) = 0 (47)

where

a(β, k; v, u) = a(β, k; v, u) − e−2ImβΛ

∫

Γ0

pv∂u

∂n.

A leaky mode is characterized by a complex β and areal k for which (47) has a nontrivial solution.

4.4 A simple example

We illustrate the spectral properties of the simplegrating structure with grating period Λ = 1, which isshown in Figure 9.

-

n = 1 n =√

10 n =√

2.3

x-0.2

-2π

Figure 9: Geometry of the waveguide. Λ = 1

Before we present the dispersion curves for thisstructure we consider the dispersion of the approx-imating planar structure, where the grating layer hasbeen replaced by a uniform layer with the averagedrefractive index (9). The dispersion curves of thisplanarized structure can of course be obtained withthe techniques of Section 3. On the other hand, theresults of the previous section apply as well since theplanarized structure is also invariant with respect totranslations of length Λ along the z-axis.

16

Suppose now that φ is an eigenfunction of

(pφ′)′ + (qk2q − pβ2l )φ = 0.

Then u(x, z) = exp(iβlz)φ(x) satisfies Lu = k2q andhas β-periodic boundary conditions on Γ0 and ΓΛ.The converse holds as well. We see that if k = k(β)is a dispersion of the planarized structure then k(βl),l ∈ Z, are all eigenvalues of (28- 30). Thus the k-βplot is obtained by periodically wrapping the disper-sion curves into the in the Brillouin zone.

Figure 10 shows the resulting picture. Here wechoose the interval [0, 2π/Λ] in order to clarify howmodes in the periodic structure will couple. Note thatthe planarized structure is an example where there isan infinite number of guided modes inside the lightcone.

Now we return to the periodic structure, whichmay be regarded as a perturbation of the planarstructure.

Figure 11 shows the dispersion curves for the pe-riodic structure. To keep the information given inthis plot manageable, we display only modes ±1 inblue and red and ±2 in cyan and magenta. Modes±1 start out as guided modes until they couple atβ = π, the first Bragg condition. After that, theybecome guided again, until they leave the light cone.The two modes couple again at β = 2π, the secondBragg.

The behavior of the dispersion relations near thefirst and second Bragg conditions is shown in Fig-ures 12 and 13. Note that the second Bragg is anisolated point where the mode is guided.

We now illustrate the field profiles Re (u(·, 0)) fora few selected modes. Figure 14 shows three exam-ples of profiles of the dominant mode inside the lightcone, inside the region of substrate modes and in-side the region of two-sided modes. The former isa guided mode, the other two radiate into the sub-strate or into both exterior regions. Note that thatlatter two modes are leaky, hence their magnitudeincreases exponentially away from the grating struc-ture. Since the rate of increase is small this is hardto see on the plots. Figure 15 shows a similar plotfor the second mode. The plots shown in this sectionhave been computed numerically by matching the in-

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

β

k

Mode +1Mode +2Mode −1Mode −2Lightline

Figure 10: k-β plot for the averaged planar waveg-uide. TE polarization

17

0 1 2 3 4 5 6Re(β)

−0.05 0 0.050

0.5

1

1.5

2

2.5

3

3.5

4

Im(β)

k

Figure 11: k-β plot for the waveguide shown in Fig-ure 9.

3 3.05 3.1 3.15 3.2 3.25Re(β)

−0.05 0 0.051.2

1.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

Im(β)

kFigure 12: Enlargement of Figure 11 near the firstBragg of the first mode.

6.15 6.2 6.25 6.3 6.35 6.4Re(β)

−0.0200.022.17

2.18

2.19

2.2

2.21

2.22

2.23

2.24

2.25

Im(β)

k

Figure 13: Enlargement of Figure 11 near the secondBragg of the first mode.

18

terior and exterior DtN operators. This method willbe described later in Section 6.

5 Discretization Methods

We now turn to numerical techniques for finding themodes of an open periodic waveguide. Most of thesemethods are based on some discretization of (41-44)and the truncation of the series expansion of the ex-terior DtN operator. The result is usually the thenonlinear eigenvalue problem: Find combinations of(β, k) such that

G(β, k) ∈ CN×N (48)

is singular. Here, N is the number of degrees of free-dom in the discretization. It is not our goal here togive a complete account of all the methods that haveappeared in the past, instead, we only mention a fewrepresentative examples and refer the reader to theoriginal papers.

1. Finite elements. The papers [5, 19] addressthe solution of the scattering problem (45) withknown k and β, but the methodology can alsobe applied to solve the eigenvalue problem.

2. Boundary elements. The papers [21, 8] considera boundary integral formulation of the partialdifferential equation in Ω0 that is coupled withthe exterior DtN operators. The advantage ofboundary elements is that there are unknownsonly on the interfaces of the dielectric materials.Thus the resulting eigenvalue problem is muchsmaller than with finite elements.

3. The eigenvalue method is based on expandingthe grating regions into the eigenfunctions ofthe differential operator. This suggests itself ifthe grating geometry is simple enough such thatthe eigenfunctions are known, and was first pre-sented by Peng, Tamir and Bertoni [37] and isprobably the oldest rigorous numerical method.A more recent review is [53]. In the followingsection we give a brief description of the basicideas.

−8 −6 −4 −2 0 2 4 6 8−2

−1

0

1

2

3

4

5Re(β)=3.31k=1.30

x

Re u

(x)

/ n(

x)

−8 −6 −4 −2 0 2 4 6 8−2

−1

0

1

2

3

4

5Re(β)=4.28k=1.60

x

Re u

(x)

/ n(

x)

−8 −6 −4 −2 0 2 4 6 8−2

−1

0

1

2

3

4

5Re(β)=6.24k=2.10

x

Re u

(x)

/ n(

x)

Figure 14: Plot of the refractive index and field pro-files of a guided, substrate radiation and two sidedradiation mode for the first (dominant) eigenvalue.The location of the modes in the k-β-plane can beseen in Figure 13. The grating region is shaded.

19

−8 −6 −4 −2 0 2 4 6 8

−4

−2

0

2

4 Re(β)=2.94k=1.82

x

Re u

(x)

/ n(

x)

−8 −6 −4 −2 0 2 4 6 8

−4

−2

0

2

4 Re(β)=3.65k=2.10

x

Re u

(x)

/ n(

x)

−8 −6 −4 −2 0 2 4 6 8

−4

−2

0

2

4 Re(β)=5.10k=2.50

x

Re u

(x)

/ n(

x)

Figure 15: Similar plot as Figure 14 for the secondeigenvalue.

5.1 Eigenvalue Method

Probably the oldest convergent method for analyz-ing periodic waveguide structures and solving for theFloquet-Bloch modes numerically was developed byPeng et al. [37] and later extended in [11]. It shouldbe mentioned that there are earlier investigationsof open periodic waveguides using Floquet Theoryare which are based on perturbation methods [27] orcoupled-mode theory [14, 34]. Since these techniquesare approximate in nature they can only provide aquantitative idea of the propagation characteristicsnear the first and second Bragg resonances. Thesemethods can be accurate when the grating region issmall, but fail to predict, for instance, a saturationeffect of the attenuation of wave propagation whenthe grating width is increased, see Fig. 8 of [37].

The derivation of the eigenvalue method beginswith the observation that the solution to (7) is quasi-periodic, which allows the following Fourier series ex-pansion in the z-variable

u(x, z) =∑

l∈Z

ψl(x) exp(

iβlz)

. (49)

The Fourier coefficient ψl(x) is called the lth spaceharmonic.

To simplify notations we limit the discussion toTE-modes (5), the extension to the TM case isstraight forward. The technique limits the form ofthe grating to a rectangular tooth profile, hence thewaveguide consists of stratified layers, where the re-fractive index is a function of x only, and the gratingregion, where the refractive index is a function of zonly. To allow arbitrary shaped profiles, the gratinglayer must be partitioned into multiple layers withrectangular geometry, see Figure 16, but we will con-sider only the case of one grating layer.

In the grating layer the refractive index is a peri-odic function of z and may be expanded in a Fourierseries

n2(z) =∑

l∈Z

κl exp

(

i2π

Λlz

)

(50)

and

κl =

dcn2t + (1 − dc)n

2f , l = 0,

(n2t − n2

f ) sin(lπdc)πl , l 6= 0.

20

- B

BBBBBB

z?

wg

6

Λ -

-z

?

wg

6

Λ -

dcΛ -

Figure 16: Top:Rectangular tooth shape in the grat-ing region. The tooth height is wg the tooth width isdcΛ. Bottom: Approximation of an arbitrary profileby multiple rectangular regions.

Here, nt, nf are the indices inside and outside thetooth and dc is the fraction of the grating period oc-cupied by the tooth, which is commonly referred toas the duty cycle.

Substitution of (49) into (5) leads to a system ofconstant coefficient second-order ordinary differentialequations for the space harmonics

ψl”(x) − β2l ψl(x) + k2

∑

m∈Z

κl−mψm(x) = 0,

for l ∈ Z. In matrix notation, the system appears inthe form

d2

dx2~ψ − P ~ψ = ~0 (51)

where ~ψ = ~ψ(x) is an infinite vector with coefficientsψn and P is an infinite matrix with constant coeffi-cients

Plm = −β2l δlm + k2κl−m .

The boundary conditions at the endpoints of the grat-ing layer will be incorporated later. At this point weconsider a general solution, which may be assumedto be of the form

~ψ(x) = ~c exp(iρx),

where ~c is a constant vector. Substituting this forminto (51) shows that ρ and ~c are an eigenvalue andeigenvector of

P~c = ρ~c.

Suppose that ~cm is a complete set of eigenvectors

with eigenvalues ρn, then ~ψ has the representation

~ψ =∑

m

~cm[

exp(iρmx)v+m + exp(−iρmx)v

−m

]

where the v±m are coefficients that will be determinednext. From (49) it follows that the solution in thegrating region is

u(x, z) =∑

l,m

exp(iβlz)clm

×[

exp(iρmx)v+m + exp(−iρmx)v

−m

]

where clm are the coefficients of the eigenvector ~cm.Outside the grating the solution has the expansion

21

(32), thus there is another set of coefficients, u±l , thatdetermine the solution in Ω±. These coefficients pro-vide the degrees of freedom needed to match the func-tion and the derivative on the interfaces Γ± in orderto obtain a solution in the whole strip ΩΛ. This leadsto the linear system

S++~v+ + S+−

~v− = u+

S−+~v+ + S−−

~v− = u−

RS++~v+ −R−1S+−

~v− = D+u+

RS−+~v+ −R−1S+−S−−

~v− = D−u−

where the matrices are defined as

[S±,±]lm = clm exp(i± ρmx±)

Rlm = δlmρm

[D±]lm = δlmα±l

In the above system the ~u± coefficients can be elimi-

nated easily. The result is that [ ~v+, ~v−]T must be inthe nullspace of the matrix

G =

[

(R −D+)S++ (R−D+)S+−

(R−D+)S−+ (R −D+)S−−

]

.

Since the coefficients of G depend on k and β, a modeis characterized by a combination of these parameterssuch that G has a nontrivial nullspace. This is anonlinear eigenvalue problem, which can be solvedwith the methods described in Section 7.

6 Matching the Interior and

Exterior DtN Operators

In the numerical methods discussed above the sizeof the matrix is primarily determined by the numberof degrees of freedom used for the discretization ofthe interior domain. There are additional unknownsfrom the discretization of the interfaces Γ± which issmall. This section discusses a technique which re-duces the problem to an eigenvalue problem on theinterfaces [46]. Thus the number of unknowns is sig-nificantly reduced which will allow the use of densematrix techniques for the solution of the nonlineareigenvalue problem.

To that end, consider the interior problem withDirichlet boundary conditions

Lu− k2qu = 0 in Ω0 (52)

u|ΓΛ= eiβΛ u|ΓΛ

(53)

∂u

∂z

∣

∣

∣

∣

ΓΛ

= eiβΛ ∂u

∂z

∣

∣

∣

∣

Γ0

(54)

u|Γ±= ϕ± (55)

If β and k are given then the problem is uniquelysolvable with the exception of a number of discretefrequencies, called interior resonances. If k is not aresonance, then the mapping from the Dirichlet con-ditions on the interfaces to the normal derivative ofthe solution on the interface is well defined. This isthe interior DtN operator

T i(β, k)

[

ϕ+

ϕ−

]

=

[

∂u+

∂n∂u−

∂n

]

Now suppose that (β, k) is such that (41-44) has anontrivial solution. Denote its restriction to the in-terfaces by ϕ, then

[

piT i(β, k) − peT e(β, k)]

ϕ = 0.

To simplify the following notations will set

T (β, k) = piT i(β, k) − peT e(β, k). (56)

On the other hand, if (56) holds for some ϕ 6= 0, thenϕ can be extended to Ω0 by solving (52-55) and toΩ± by equation (32). Thus the eigenvalue problemis reduced to finding (β, k) such that (56) holds forsome ϕ 6= 0.

The DtN operator is only defined if (β, k) is not aninterior resonance. This is not really a problem, be-cause the singular frequencies are different from theinterior resonances, and therefore the interior DtNmaps are well defined in a neighborhood of the so-lutions of (56). However, it is possible to modifythe approach by matching the Robin-to-Robin (RtR)maps instead of DtN maps. The interior problemwith Robin boundary conditions is well posed hencethe RtR operator is well defined. This idea has beenused in the context of computing artificial boundaryconditions in, for instance [18].

22

6.1 Discretization of the DtN opera-

tor

Equation (56) is discretized using a variational ap-proach. To that end, we introduce the functions

e±l (z) =

exp(

iβlz)

, on Γ±,0 on Γ∓,

(57)

for l ∈ Z, and the 4p+ 2-dimensional space

Sp := span|l|≤pe±l .

The discretized version of (56) is to find (β, k) and0 6= ϕ ∈ Sp such that

(ψ, T (β, k)ϕ) = 0

holds for all φ ∈ Sp. In the basis (57) the variationalform reduces to a nonlinear eigenvalue problem: Find(β, k) such that Tp(β, k) is singular, where Tp is a2 × 2-block matrix, whose coefficients are

[T±,±(β, k)]l,l′ =(

e±l , T (β, k)e±l′)

.

Since this is a small matrix, direct methods are suit-able for the numerical solution. Before we discussthat, we describe how the matrix coefficients can becomputed.

From its definition in (35) it follows that the ex-terior DtN operator contributes the diagonal factoriαl± (36). In general, the interior part does not havea closed form and must be computed numerically. Toobtain the matrix coefficients of the interior DtN op-erator the interior problem (52-55) must be solved2p + 1 times with Dirichlet data φ = e±l , |l′| ≤ p.One such solution gives one column of T±,± whichconsists of the l-th Fourier coefficient of the normalderivative. There are several alternatives to solve theinterior problem. The finite element approach wasconsidered in [46]. An alternative is the boundaryelement approach, which was considered [47].

7 Solution of the Eigenvalue

Problem

We have seen that after discretization the problem offinding the modes of (7) reduces to finding the values

of β = β(k) such that the matrix G(β, k) is singular.Of course, one could also compute km(β), but if weare interested in leaky modes the first formulation ismore natural. In the case of the planar waveguidethe nonlinear eigenvalue problem can be formulatedas a quartic problem, but for the periodic structurethis is no longer possible.

Suppose now we want to compute a value for βsuch that G(β, k) is singular for a given k. As kis fixed in the following discussion, we will omit thek-dependence and simply write G(β). Since work-ing with the determinant is problematic for computa-tions, see [45], a popular way to solve this problem isby converting the matrix equation into an equivalentsystem of nonlinear equations. This can be accom-plished, for instance, by the system

F(x, β) :=

[

G(β)xb∗x − 1

]

= 0, (58)

which has N + 1 equations and unknowns. The ad-ditional unknown x ∈ CN is the eigenvector. In (58)the vector b ∈ CN must be chosen such that it isnot in the orthogonal complement of the eigenspaceof G. Since this space is unknown it is impossible togive a vector that will always work, but in practicea randomly selected b will be sufficient. Note thatit is not a good idea to replace b by x because thefunction x → x

∗x is not analytic.

The system can be solved by Newton’s method,which consists of solving a linear system with theJacobian

F ′(x, β) =

[

G(β)x G′(β)xb∗ 0

]

in each step. A few remarks are in order

1. The Jacobian involves computing G′(β), thederivative of G(β) with respect to β. For thediscretizations of the interior DtN operator it isusually easy to find this derivative analytically.For the exterior problem one can derive trans-lation operators for the for the derivative of thestate vector with respect to β, in a similar fash-ion as for the statevector itself. This is describedin [46].

23

2. The Newton iteration must be performed on theRiemann surface discussed in Section 4.2. Thusevery sheet of α±

l in (34) must be changed, when-ever a branch cut is crossed, see [47].

3. The Jacobian at the solution is invertible only ifthe dimension of the eigenspace is one. If twodispersion relations cross this assumption is nolonger satisfied and the convergence of Newton’smethod applied to (58) will be slower.

If the interior problem is discretized with a finite-element type approach then the matrices G and G′

are large but sparse which can be effectively exploitedwhen solving system (58). On the other hand, if Gis the matrix that arises from matching the interiorand exterior DtN maps, as described in Section 6then G is dense, but very small. In this setting thecost of computing the matrices G and G′ dominatesover the cost of a matrix inversion or diagonalization.It is preferable to solve the system with the Matrix-Newton method [46] instead of converting it into anonlinear system. The idea of this iteration to findthe correction σ such that the linearization at thecurrent iterate β

G(β + σ) ≈ G0 + σG1 (59)

is singular. Here, G0 = G(β) and G1 = G′(β). Thenext iterate is the smallest generalized eigenvalue (inmodulus). This can be realized numerically by com-puting the largest eigenvalue λM of G−1

0 G1 and set-ting σ = −1/λM . Then the next iterate is given byβ → β + σ. The method has several advantages oversolving (58). It only requires an initial guess for βbut not for x, furthermore, the convergence does notdeteriorate near crossing dispersion relations.

Regardless of which solver is used, it is essential tohave a good initial guess to start the iteration. Sincethere are many solutions of the nonlinear eigenprob-lem in the complex plane, the iteration often con-verges to an undesired root. Especially near pointswhere two modes couple, the region of attraction toone of the solutions is usually very small.

A good choice for the initial guess of β is the so-lution of a ’nearby’ planar structure, which was al-ready mentioned at the beginning of Section 3. If

the grating is weak and the root is sufficiently farfrom coupling modes this initial guess suffices to ob-tain convergence. However, for strong gratings andin the neighborhood of Bragg points the convergenceregions are small and the choice of an initial guess ismore delicate.

In this case it is preferable to use a continuationmethod. This is a numerical technique where onefollows the solution path u(v) of an underdeterminednonlinear system F (u, v) = 0, with F : Cn×R → Cn.The method is often used to solve nonlinear systemsF (u) = 0 by introducing a homotopy F (u, v) whereF (u, 1) = F (u) and F (u, 0) = 0 has known solutions.An introduction to continuation methods is [1], ex-amples of papers that apply this technique to nonlin-ear eigenvalue problems are [10, 36] and the referencescited therein.

In the case of a periodic grating, we already men-tioned in Section 3 that a homotopy can be obtainedby considering the eigenproblem G(β, v) that arisesfor the refractive index (10). For v = 0 the structureis planar and the solutions β(0) can be obtained withthe methods described in Section 3.5. For v = 1 thesolutions β(1) are the desired eigenvalues the givenperiodic structure.

Numerical continuation methods are generally ofthe predictor-corrector type. For an accepted solu-tion (β0, v0) a new solution is predicted by followinga certain step width ∆v in the direction of the tan-gent vector. The predicted value (βp, vp) is in generalnot on the solution curve (β(v), v). Hence a correctedpoint on the curve is found with Newton’s iteration.A popular corrector is the intersection of the curvewith the plane through (βp, vp) that is normal to thetangent.

In view of the matrix-Newton method one can alsoconsider the following predictor-corrector step. If(β0, v0) is an accepted solution, then the predictoris determined from the linearization of G(β(v), v) at(β0, v0). That is, we find the smallest value of σ suchthat

∂G

∂βσ +

∂G

∂v

is singular. This value is an approximation for dβdv .

24

Then the predicted value is

βp = β0 + σ∆v.

The value of βp is the initial guess for the systemG(β, v0 + ∆v) = 0.

8 The Perfectly Matched Layer

The perfectly matched layer (PML) was originallydeveloped by Berenger for scattering problems [6],but the idea can also be used to compute the modesof open dielectric waveguides. There are numer-ous papers that address various applications, for in-stance, planar structures are discussed [26] and peri-odic structures are discussed in [9].

The idea of the PML is to surround the core of thewaveguide by a region with lossy materials in such away that all plane waves are absorbed without anyreflection back into the core. Hence the name per-fectly matched. For the computation, the absorber istruncated after a finite distance where homogeneousDirichlet conditions are introduced. On the artificialboundary waves can reflect, but since they have totravel through a lossy medium they have no practi-cal influence when the absorber is large enough.

Consider, for instance, a structure like the oneshown in Figure 1. This structure requires two ab-sorbers in the regions x < a and x > b, where a < 0and b > w. We briefly describe the construction of anabsorber in the region x > b. To simplify notations,we limit ourselves to the TE case, which is governedby the Helmholtz equation (5). The derivation of theTM case is the same after switching E ↔ H andµ↔ ε.

The key idea is introduce a complex electric per-mittivity ε and two complex magnetic permittivitiesµx, µz in the absorber. This corresponds to a ma-terial with a finite conductivity and an anisotropic(non-physical) magnetic conductivity. We will seesoon that is important to have two different values ofthe x and z direction in order to achieve a reflectionless layer for all angles of incidence.

If the permeability is a diagonal tensor diag[µx, µz]and the permittivity is a scalar, then the equation for

the y-component for the electric field is

∂

∂x

1

µz

∂u

∂x+

∂

∂z

1

µx

∂u

∂z+ εk2u = 0. (60)

Here, µx, µz and ε are relative permeabilities and per-mittivities that depend on the position. In the neigh-borhood of the absorber we have

µx(x, z) =

1, x < b,µx, x ≥ b,

µz(x, z) =

1, x < b,µz, x ≥ b,

and

ε(x) =

εJ x < b,ε x ≥ b.

Now consider a plane wave that approaches the PMLfrom the (ε, µ)-region in the direction (kx, kz). Ingeneral, we expect to obtain a reflected and a trans-mitted wave, hence the total fields are given by

u(x, z) = exp(

ikx(x− b) + ikzz)

+Γ exp(

− ikx(x − b) + ikzz)

, x < b,

u(x, z) = τ exp(

ikx(x− b) + ikzz)

, x ≥ b.

Here, Γ is the reflection- and τ is the transmission co-efficient and (kx, kz) is the wave vector of the trans-mitted wave in the PML. These coefficients can bedetermined from the condition that u and 1

µz

∂u∂x are

continuous across the interface x = b. Simple algebraleads to the following conclusions

kz = kz (61)

1 + Γ = τ (62)

kx

(

1 − Γ)

=kx

µzτ (63)

Suppose that the material constants in the PML aregiven, then the reflection coefficient can be found bysolving (62) and (63), which gives

Γ =kx − kx

µz

kx − kx

µz

. (64)

We now design the material constants in the absorbersuch that the reflection coefficient vanishes for planewaves of all directions and frequencies.

25

From (5), (60) and (61) it follows that

k2x = εJk

2 − k2z

k2x

µ2z

=ε

µzk2 − k2

z

µxµz

In view of (64), the left hand sides of the last twoequations must agree to avoid reflection back into thewaveguide region. This can be achieved when

ε

µz= εJ ,

µxµz = 1.

For a solution we pick some absorption factor g > 0and set

ε = (1 + ig)ε

µz = (1 + ig)µ

µx = (1 + ig)−1µ

Thenkx = µzµkx = (1 + ig)kx

thus u(x, z) decays exponentially as x → ∞.In a numerical scheme the absorber has finite width

c, thus the eigenvalue problem with the PML in placeis posed in the domain [a− c, b+ c]× [0,Λ] as follows

∂

∂x

1

µz

∂u

∂x+

∂

∂z

1

µx

∂u

∂z= −εk2u

uβ|ΓΛ= eiβΛ uβ |Γ0

∂uβ

∂n

∣

∣

∣

∣

ΓΛ

= eiβΛ ∂uβ

∂n

∣

∣

∣

∣

Γ0

u|x=a−c = u|x=b+c = 0.

Suppose that β is given and that we seek the singu-lar frequencies, then the discretization with finite ele-ments will result in a non-hermitian, but linear eigen-value problem in k. Since there are standard packagesavailable to solve such problems iteratively, the prob-lem is considerably simpler than the formulations in-volving DtN operators. However, it will depend ona case to case basis which approach is more efficient,since the PML formulation involves the discretizationof the whole waveguide, whereas in the DtN formu-lations only the grating region is discretized.

In conclusion, the PML can also be combined withseveral other techniques that have been developed tosolve the eigenvalue problem associated with openwaveguides. In this case it is also possible to com-pute leaky modes to high accuracy with a carefullyselected value of the absorption factor g. This is theapproach taken in [9]. However, in this setting oneends up solving nonlinear eigenvalue problems andthus it is not clear what computational advantagesthe PML offers over methods that are based on DtNmaps.

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