Download - Simple high-order Galerkin finite element scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides

Simple high-order Galerkin finite element scheme for

the investigation of both guided and leaky modes in

anisotropic planar waveguides

H . P . U R A N U S 1; 2; * , H . J . W . M . H O E K S T R A 1 A N D E . V A N G R O E S E N 2

1Integrated Optical MicroSystems Group, MESA+ Research Institute, University of Twente, P.O. Box 217,

7500 AE Enschede, The Netherlands2Applied Analysis and Mathematical Physics Group, MESA+ Research Institute, University of Twente,

P.O. Box 217, 7500 AE Enschede, The Netherlands

(*author for correspondence: E-mail: [email protected])

Abstract. A simple high-order Galerkin finite element scheme is formulated to compute both the guided

and leaky modes of anisotropic planar waveguides with a diagonal permittivity tensor. Transparent

boundary conditions derived from the Sommerfield radiation conditions are used to model the fields at the

computational boundaries that allow the radiation into the high index cladding/substrate and decay into

the low index cladding/substrate, hence work for both guided and leaky modes. Richardson’s extrapo-

lation is employed to achieve high-order accuracy by only using simple first-order-polynomial basis

functions. Schemes up to sixth-order of accuracy in the effective index are demonstrated. The resulted non-

linear sparse matrix eigenvalue equation is solved using an iterative procedure. The ability of the scheme to

compute leaky and guided modes of various structures with isotropic and anisotropic materials, step and

graded index profiles is demonstrated; including its applications to investigate the properties of ARROW

structures.

Key words: anisotropic waveguides, finite element method, guided modes, high-order scheme, leaky

modes, transparent boundary conditions

1. Introduction

Optical waveguides that are made on a high index substrate are particularlyinteresting. This class of waveguides includes structures made on a semi-conductor wafer. Structures that are composed of silicon compounds (siliconoxynitride, silicon nitride, silica, etc.) grown on top of silicon substrate do notonly take benefits from the low cost of silicon wafer, but also share the welldeveloped technologies used by the microelectronics industries, and offerbetter prospect for integration between the optical and electronic circuits(Wong 2002). It has been shown that structures made of these materials havea wide range of available refractive indices (Worhoff et al. 1999). Besides,arbitrary refractive index profiles can be made by precise computer control ofthe fabrication parameters as has been demonstrated by the realization of arugate filter (Lim et al. 1996). This feature gives more degrees of freedom in

Optical and Quantum Electronics 36: 239–257, 2004.

� 2004 Kluwer Academic Publishers. Printed in the Netherlands. 239

refractive index profile engineering, i.e. tailoring of the refractive indexprofiles to meet certain desired properties of the waveguide, like bandwidth(Ishigure et al. 2002), mode profiles (Thompson and Weiss 1996), phasematching of modes of different wavelengths (de Michele 1983; de Sande et al.2002), etc. Hence, numerical investigations of this class of structure areimportant.

To analyze this kind of structures, computational methods that are able tosolve leaky structures are needed. Among others, the finite element method(FEM) (Grant et al. 1990; Hernandez-Figueroa et al. 1995; Tartarini 2000),the finite difference method (FDM) (Huang et al. 1996), the transverse matrixmethod (TMM) (Kubica et al. 1990; Chen et al. 2000), the transverse reso-nance method (TRM) (Huang et al. 1992), the WKB method (Ghatak 1985),and the imaginary distance (ID) BPM (Tsuji and Koshiba 2000; Obayya et al.2002) with either perfectly matched layers (PML) or transparent boundaryconditions (TBC) have been used to solve such problem. Solvers that areoriginally made for 2-D cross-section problems are applicable to generalstructures, but might be too expensive for 1-D problem modeling tasks likethose required in film deposition studies. Methods that can handle leakyplanar structures with arbitrary index profile efficiently are needed for thiskind of purpose. Since some materials exhibit birefringence (Worhoff et al.1999), the method should be able to handle anisotropic dielectric permittiv-ity. The TMM is known as an exact method for the step index planar case.However, the requirement of doing root searching in the complex planemakes this method not very easy to implement (Petracek and Singh 2002).Simplification of the characteristic equation, in order to do root searchingonly on the real axis by assuming certain phase relation at the outer interface,has been proposed for ARROW structure with small losses (Liu et al. 1999).For graded-index structures, TMM becomes too tedious, while its solutionwill not be exact any longer. For these structures, a staircase approximationto the graded index profile has to be performed, which leads to an accuracy ofonly second-order, unless precautions like the extrapolation as we proposedin this paper are taken.

In this work, we propose a simple high-order 1-D Galerkin FEM scheme.By using TBC derived from the Sommerfeld radiation conditions andallowing the transverse wave number to have complex value, the schemeallows light to leak into the high index substrate/cladding, and to decay intothe low index substrate/cladding, and is hence able to compute both theguided and leaky modes. The inclusion of Richardson’s extrapolation and asimple mesh-adjustment scheme leads to high-order schemes by using onlyfirst-order-polynomial basis functions. The sparse non-linear matrix eigen-value equation produced by the scheme can be solved using a simple iterationscheme. Hence, the scheme turns out to be very simple, easy to implement,but highly accurate. The method is suitable for leaky planar waveguides of

240 H.P. URANUS ET AL.

arbitrary index profile with a diagonal permittivity tensor. Using the scheme,we will present more insights into the properties of ARROW structures.

2. Description of the method

2.1. FINITE ELEMENT FORMULATION

For anisotropic planar waveguides where the principal axes of the anisotropyare parallel with the Cartesian coordinate system of the waveguide, thepermittivity tensor can be expressed in a diagonal form as

��e ¼ e0

n2xx 0 00 n2yy 0

0 0 n2zz

24 35 ð1Þ

with e0 denoting the free space permittivity, while nxx, nyy and nzz arerefractive indices associated with the x, y, and z components of the electricfield, respectively. By assuming that the waveguide is composed of non-magnetic and source-free material and that the refractive indices changingonly in the x direction while the z-axis is the propagation direction, we getuncoupled wave equations

oxx þ k20 n2yyðxÞ � n2eff� �h i

EyðxÞ ¼ 0 ð2Þ

for TE, and

ox1

n2zzðxÞox

� �þ k20 1� n2eff

n2xxðxÞ

� �� HyðxÞ ¼ 0 ð3Þ

for TM polarization. In Equations (2) and (3); k0, neff, Ey , and Hy denote thefree space wave number, mode effective index, electric and magnetic fieldsparallel to the y-axis, respectively. Following the Galerkin procedure, doingpartial integration to terms that contain second-order derivatives, and usingof the continuity of Ey , Hy , oxEy , and ð1=n2zzÞoxHy across material interfaces,the weak formulation of the corresponding wave equations can be written as

woxujoX þZX

�ðoxwÞðoxuÞ þ k20ðn2yy � n2effÞwuh i

dx ¼ 0 ð4Þ

for TE, and

SIMPLE HIGH-ORDER GALERKIN FINITE ELEMENT SCHEME 241

1

n2zzwoxu

��oX

þZX

� 1

n2zzðoxwÞðoxuÞ þ k20 1� n2eff

n2xx

� �wu

� �dx ¼ 0 ð5Þ

for TM polarization, with w, X, and oX denote the weight function, com-putational interior domain, and computational boundary, respectively. Fornotational simplicity, u has been used to denote both the Ey in Equation (4)and Hy in Equation (5). By discretizing the interior domain X into N ele-ments, Equations (4) and (5) can be written as

woxujoX þXNe¼1

ZXe

�ðoxwÞðoxuÞ þ k20 n2yy � n2eff� �

wuh i

dx ¼ 0 ð6Þ

and

1

n2zzwoxu

��oX

þXNe¼1

ZXe

� 1

n2zzðoxwÞðoxuÞ þ k20 1� n2eff

n2xx

� �wu

� �dx ¼ 0 ð7Þ

respectively. Approximating the function u in the interior domain byexpanding it in terms of first-order-polynomial basis functions, for eachelement e, we can write

ue � ~ue ¼X2l¼1

Nl;eul;e ¼ N1;e N2;e½ � u1;e u2;e� T ð8Þ

with basis functions

N1;e ¼ 1� x� x1;eDxe

ð9aÞ

N2;e ¼x� x1;eDxe

ð9bÞ

and Dxe ¼ x2;e � x1;e as the local mesh size. Subscripts 1 and 2 in Equations(8) and (9) denote the local node number within the element. By substituting(8) and (9) into (6) and (7), using the same basis functions as the weightfunctions and evaluating the integral, we can express the approximation toEquations (6) and (7) in matrix equations

CTEfug þ�ATE � n2effB

TE�fug ¼ f0g ð10Þ


and

CTMfug þ�ATM � n2effB

TM�fug ¼ f0g ð11Þ

respectively, where ATE, BTE, ATM, and BTM are sparse tridiagonal matricesthat result from the evaluation of the corresponding integral terms withinEquations (6) and (7), while CTE and CTM are matrices associated with theboundary terms. Here

fug ¼ u1 . . . uNþ1½ �T ð12Þ

where the subscript denotes the global node number, are column vectorsrepresenting the discretized electric field for Equation (10) and magnetic fieldfor Equation (11) at nodal points.

2.2. BOUNDARY CONDITIONS

In order to handle both guided and leaky modes, the boundary conditionsshould allow light with either evanescent decay or leaky behavior to trans-parently pass the computational boundary. To do this we use the Sommer-feld radiation conditions and allow the transverse wave number to havecomplex value.

By assuming the field to have a time dependence of expðixtÞ, we imposeSommerfeld radiation conditions at the computational boundaries for wavescoming from within the computational domain interior as follows

or þ ikrð ÞuðrÞ ¼ 0 ð13Þ

with r denotes the length of~r, a position vector pointing out from within thewaveguiding structure, and kr denotes the transverse wave number. Condi-tion (13) is fulfilled by

uðrÞ ¼ uð0Þ expð�ikrrÞ ð14Þ

with the transverse wave numbers at the computational boundaries obtained

from (2) and (3) are kr ¼ k0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2yy � n2eff

qjoX for TE and kr ¼ k0 nzz

nxx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2xx � n2eff

qjoX

for TM for structures with a homogeneous exterior domain. By usingEquation (14) as the Dirichlet conditions at the computational boundaries,we can then get the Dirichlet to Neumann (DtN) map at the computationalboundaries as


oxu ¼ r � xoru ¼ �ikrðr � xÞu ð15Þ

with the hat ( ) symbol denotes unit vector. By using DtN (15), the non-zeroentries of matrices CTE and CTM can be determined as follows

cTE1;1 ¼ �ikrjoX1 ¼ �ik0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2yy � n2eff

q ��oX1

ð16aÞ

cTENþ1;Nþ1 ¼ �ikrjoX2 ¼ �ik0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2yy � n2eff

q ��oX2

ð16bÞ

cTM1;1 ¼ �i1

n2zzkr

��oX1

¼ �ik0


qnxxnzz

��oX1

ð16cÞ

cTMNþ1;Nþ1 ¼ �i1

n2zzkr

��oX2

¼ �ik0


qnxxnzz

��oX2

ð16dÞ

with oX1 and oX2 denote the lower and upper computational boundaries,respectively.

Since the n2eff as the eigenvalue of the matrix eigenvalue Equations (10) and(11) also resides within matrix C, the matrix eigenvalue equations becomenon-linear, which can be solved by simple iteration scheme by using n2eff fromprevious iteration for evaluation of terms within matrix C. Hence, withineach iteration, the problem can be solved using linear matrix eigenvaluesolver. In this work, we have used ARPACK that implement implicitly re-started Arnoldi method as the eigenvalue solver. To be able to model bothguided and leaky modes, the sign of the square root within Equation (16)should be correctly chosen at each iteration step to fulfill the physical con-ditions. At a boundary where the wave should leak-out (at high index sub-strate/cladding for the leaky mode case), the sign of the square root is chosento give ReðkrÞ > 0, while at the boundary where the wave should decay,ImðkrÞ < 0.

2.3. HIGH-ORDER EXTENSION

By using the first-order-polynomial basis functions as described in Section2.1, the scheme will give second-order accuracy in neff. As has been shown inprevious work from our group for the BPM (Stoffer et al. 1999) and modesolver (Uranus et al. 2003), fourth-order accuracy can be obtained by simply


combining results of single- and double-sized mesh in evaluating the integralwithin FEM scheme using a Richardson’s extrapolation. Here, we generalizethe scheme to obtain even higher (fourth-, sixth-, eighth-, etc.) order accuracywhile keeping the simplicity of the scheme described previously.

For mesh located between global nodal point j (at xj) and jþ 1 (at xjþ1),using Taylor’s expansion around xjþ1

2, it can be shown that approximating the

function using an expansion in terms of first-order-polynomial basis func-tions will produce errors of

Fj�jþ1 ¼ eF linj�jþ1 þ Dx3jerr3 xjþ1

2

� �þ Dx5j err5 xjþ1

2

� �þ Dx7j err7 xjþ1

2

� �þ Dx9jerr9 xjþ1

2

� �þ � � � ð17Þ

where Fj�jþ1 and eF linj�jþ1 denote the exact and the linear-approximate value of

the integral terms of Equations (6) and (7) within the mesh, respectively,while errkðxjþ1

2Þ is just a short form notation for the function related to the

kth-order error terms. After summation, for uniform meshes, the global errorof approximating Equations (6) and (7) will be just second-order as follows.

F1�Nþ1 ¼ eF lin1�Nþ1;Dx þOðDx2Þ ð18Þ

with F1�Nþ1 and eF lin1�Nþ1;Dx denoting the exact and approximate (using Dx-

sized meshes) value for the integral over the whole interior domain. Hence, inorder to get fourth-order of accuracy, we should get rid of the third-ordererror term within each integration interval. Similarly, to get sixth-orderaccuracy, we should eliminate the third- and fifth-order error terms. In thiswork, we use Richardson’s extrapolation by combining integration resultsfrom meshes with different sizes to get high-order of accuracy while still usingthe simple first-order basis functions.

By using integration results from Dx-sized and 2Dx-sized meshes, fourth-order accuracy can be achieved by Richardson’s extrapolation as follows

F1�Nþ1 ¼4

3eF lin1�Nþ1;Dx �

1

3eF lin1�Nþ1;2Dx

þ Dx5X

l¼2;4;...N

�8err5ðxlÞ þ1

3oxxerr3ðxlÞ

� �þ h:o:t: ð19aÞ

F1�Nþ1 ¼4


1

3eF lin1�Nþ1;2Dx þOðDx4Þ ð19bÞ

In Equation (19a), l denotes the middle grid point of each group of 2 meshesof Dx size while h.o.t. denotes the higher order terms. Since there will be


X=ð2DxÞ number of these groups of meshes, the order of accuracy will befourth-order. Similarly, by using Dx-sized, 2Dx-sized, and 4Dx-sized meshes,sixth-order accuracy can be achieved by

F1�Nþ1 ¼64


20

45eF lin1�Nþ1;2Dx þ

1

45eF lin1�Nþ1;4Dx

þ Dx7X

l¼3;7;...;N�1

256err7ðxlÞ �32

3oxxerr5ðxlÞ þ

14

45oxxxxerr3ðxlÞ

� �þ h:o:t: ð20aÞ

F1�Nþ1 ¼64


20

45eF lin1�Nþ1;2Dx þ

1

45eF lin1�Nþ1;4Dx þOðDx6Þ ð20bÞ

with l denoting the middle grid point of each group of 4 meshes of Dx size.Using Dx-sized, 2Dx-sized, 4Dx-sized, and 8Dx-sized meshes, eighth-orderaccuracy can be obtained by

F1�Nþ1 ¼4096


1344

2835eF lin1�Nþ1;2Dx

þ 84

2835eF lin1�Nþ1;4Dx �

1

2835eF lin1�Nþ1;8Dx þOðDx8Þ ð21Þ

Since the integral terms will contribute only to matrices A and B, the finalway of implementing the Richardson’s extrapolation is just a matter ofmultiplication of scalars with sparse matrices and addition/subtractionoperation of sparse matrices before eigenvalue computation. Since the righthand side of Equations (10) and (11) is just a null vector, the matrices can bescaled by multiplying them with the denominator occurring in Equations(19)–(21) to get integer coefficients for combining the matrices. Moreover,since the integrations only differ in their interval sizes, the same expressionsfor evaluation of matrix entries can be reused by just plugging in the propermesh sizes. It should be noted here that since iteration is used to solve thenon-linear matrix eigenvalue problem, the convergence depth will also limitthe number of digits of accuracy. In the implementation, we have usedconvergence depth of 10�10 in n2eff; which is more than enough for mostpractical applications. Besides, the expected order of accuracy will only beeffective if the mesh size is small enough to effectively exclude the effect of thehigher order terms within Equation (17) for all mesh sizes being used in theextrapolation. Also, since the coefficients within the summation of Equation(20a) are larger than that of Equation (19a), the sixth-order scheme will bemore accurate than the fourth-order scheme if the mesh size is small enough


to compensate the effect of these coefficients. The same thing holds for anyhigher order scheme compare to the lower order one. We are aware that thisfact might limit the benefits of further higher order extension.

To let the Richardson’s extrapolation works properly, none of the inte-gration intervals should cross material interfaces. Generalizing the mesh-evenization scheme that we proposed in previous work (Uranus et al. 2003),the number of Dx-sized meshes within each layer of the structure should beadjusted to be integer multiple of 2Ord=2�1 with Ord denoting the expectedorder of accuracy, and interfaces should coincide with grids. Fig. 1 illustratesthis simple and easy to implement mesh-adjustment scheme and their cor-responding matrix structure. It should be noted that due to different meshsizes between terms in Equations (19)–(21), the combined matrices will not betridiagonal anymore, but of the form as shown in Fig. 1b.

Fig. 1. Richardson’s extrapolation with mesh-adjustment scheme: (a) meshes for sixth-order scheme; (b)

matrix structure for fourth-, sixth-, and eighth-order scheme (black dots denote the non-zero matrix

entries).


3. Computational results

The ability of the scheme to compute guided modes will be demonstratedusing a sample with exponential permittivity profile, for which exact solu-tions are available. ARROW structures, either isotropic or anisotropic onewill be used as samples to demonstrate the leaky mode computation ability.Finally, computational results for anisotropic graded index leaky waveguidewill be presented.

3.1. GUIDED MODES

As the first sample, we take a waveguide with the following relative permit-tivity profile:

n2ðxÞ ¼ n2s þ 2nsD expð�jxj=dÞ for x � 0 ð22aÞ

n2ðxÞ ¼ n2c for x > 0 ð22bÞ

with d denotes the effective depth of the exponential profile. The calculatednormalized propagation constants using present scheme as well as the exactvalues for ns ¼ 2:177, nc ¼ 1 (air), D ¼ 0:043, and V � k0d

ffiffiffiffiffiffiffiffiffiffi2nsD

p¼ 4, are

presented in Table 1. The computation is carried out for k ¼ 1 lm and acomputational window of W ¼ 9lm with the upper computational boundarylocated exactly at the waveguide–air interface. The exact values of theeffective indices for TE polarization were calculated using the exact disper-sion relation given by Conwell (1973). As shown from the table, the resultsapproach the exact values either by using finer mesh or higher order schemefor coarse mesh.

3.2. ISOTROPIC ARROW STRUCTURE

As an isotropic ARROW structure, we take the same structure as the onecalculated by Kubica et al. (1990) and Liu et al. (1999). The structure is as

Table 1. Computational results of structure with exponential permittivity profile

b ¼ ðn2eff � n2s Þ=ð2nsDÞ

Present scheme

(W = 9 lm, 900 meshes)

Present scheme

(W = 9 lm, 40 meshes)

Exact

Second-order Fourth-order Second-order Fourth-order Sixth-order

TE0 0.321176 0.321179 0.319757 0.321065 0.321134 0.321179

TE1 0.053969 0.053972 0.052618 0.053823 0.053942 0.053972

TM0 0.300843 0.300846 0.299536 0.300716 0.300794 –

TM1 0.046947 0.046950 0.045761 0.046807 0.046917 –


shown in Fig. 2a with ns ¼ 3:5, n1 ¼ 1:45, n2 ¼ 3:5, n3 ¼ 1:45, nc ¼ 1,d1 ¼ 2:0985 lm, d2 ¼ 0:1019lm, and d3 ¼ 4 lm. The computation was car-ried out for light vacuum wavelength of 1.3 lm. The calculated results for thefirst-two leaky TE and TM modes are given in Table 2 that shows agreementbetween results of present scheme with the published data. The effective meshsize being used in the table is defined as

Dxeff �1

N

XNj¼1

DxOrdj

!1=Ord

ð23Þ

This definition is based on the expectation that the error within each elementwill be OðDxOrdÞ and the fact that the mesh sizes vary slightly due to themesh-adjustment scheme.

Figs. 3–6 present the real part of neff and the attenuation constant (relatedto the imaginary part of neff) for the first-ten leaky modes as function of d1and d3. The solid line curves of Figs. 5a and 6a have also been presented byKubica et al. (1990) and agree well with these. The strongly dispersive part ofthe curves within Figs. 3 and 5 are related to modes concentrated mostly

Fig. 2. Refractive index profile of the isotropic ARROW sample: (a) full structure, (b) structure with

ns ¼ n1 and (c) structure with nc ¼ n3.

Table 2. Computational results of the isotropic ARROW sample

Mode Present scheme (Sixth-order)

(W = 9.2 lm Dxeff = 0.01 lm)

TMM

(Kubica et al. 1990)

Simplified TMM

(Liu et al. 1999)

Re(neff) Im(neff)

(·10)3)Attenuation

(dB/cm)

Re(neff) Attenuation

(dB/cm)

Re(neff) Attenuation

(dB/cm)

TE0,L 1.44170845 )0.00060491 0.253944 1.4417085 0.25 1.4417085 0.25

TE1,L 1.41759871 )0.97220073 408.1377 1.4176 407 1.41798 270

TM0,L 1.44130390 )0.12983415 54.50543 – – – –

TM1,L 1.42164054 )5.30958274 2229.006 – – – –


within the SiO2 layer associated with the variable of the horizontal axis of thecurves, while the non-dispersive part related to modes confined within theother SiO2 layer (Huang et al. 1992). This phenomenon can be understood bycomparing the dispersion curve plot of the full structure with those of

(a)

(b)

Fig. 3. Dispersion curves by varying d1 of the isotropic ARROW structure. Other parameters are as in the

text: (a) TE and (b) TM.


structures with ns ¼ n1 (Fig. 2b) and nc ¼ n3 (Fig. 2c) as shown in Figs. 3aand 5a for TE polarization. For clarity, only modes of the full structure werelabeled in these figures. The transitions between the dispersive and non-dispersive part of the curves take place around the crossings between

(a)

(b)

Fig. 4. Attenuation as function of d1 of the isotropic ARROW sample. Other parameters are as in the text:

(a) TE and (b) TM.


dispersion curves of structures of Figs. 2b and c. For TM polarization, thisphenomenon is not so pronounced; hence, we only present the plot for thefull structure. Figs. 4a and 6a show that the top of the attenuation curvescorresponds to the attenuation curves of structure with nc ¼ n3, which indi-

(b)

(a)

Fig. 5. Dispersion curves by varying d3 of the isotropic ARROW structure. Other parameters are as in the

text: (a) TE and (b) TM.


cates that the high-losses part of the curve is related to power leakage mostlyinto the silicon substrate, i.e. for modes confined mainly within the SiO2 layernearest to the substrate.

(a)

(b)

Fig. 6. Attenuation as function of d3 of the isotropic ARROW sample. Other parameters are as in the text:

(a) TE and (b) TM.


3.3. ANISOTROPIC ARROW STRUCTURE

For the anisotropic ARROW sample, we take the same structure as the onecalculated by Chen et al. (2000) using TMM. The structure is a 5-layerARROW with isotropic substrate and cover, but with anisotropic innerlayers. The refractive indices for layers counted sequentially from substrate tocover are ns ¼ 3:85, nzz1 ¼ 1:46, nzz2 ¼ 2:3, nzz3 ¼ 1:46, nc ¼ 1. For theanisotropic inner layers nxxi ¼ nyyi ¼ 1:03nzzi for i ¼ 1, 2, 3. The vacuumwavelength k is 0.6328lm. The thickness of the anisotropic layers ared1 ¼ 3:15k, d2 ¼ 0:142k, d3 ¼ 6:3k. The calculated results for the first-five TEand TM leaky modes are given in Table 3 for fourth- and sixth-order schemeusing effective mesh size of 0.005lm, with upper computational boundarypositioned at the surface of the structure (interface between layer 3 and aircover) while the lower one at 0.5lm into the substrate. This mesh size hasbeen chosen in order to represent the thin layer with a sufficient number ofmeshes. Beside modes confined within the second cladding and the core of theARROW structure, i.e. layers 1 and 3; our scheme also captures those leakymodes quasi-guided within the first cladding, i.e. the thin anisotropic layerwith highest refractive indices.

3.4. ANISOTROPIC GRADED-INDEX LEAKY WAVEGUIDES

To demonstrate the ability of the scheme to model anisotropic leaky planarwaveguide with arbitrary refractive index profile, we take a full Gaussianleaky waveguide made by silicon oxynitride. The refractive index profile ofthe structure is given in Fig. 7. The refractive index of the silicon substrate is

Table 3. Computational results of the anisotropic ARROW structure

Present scheme TMM (Chen et al. 2000)

Fourth-order Sixth-order Re(neff) Im(neff) (·10)3)

Re(neff) Im(neff) (·10)3) Re(neff) Im(neff) (·10)3)

TE0,L 1.867833649 )0.000000000 1.867833876 )0.000000000 – –

TE1,L 1.501798936 )0.000050179 1.501798936 )0.000050179 1.501798936 )0.000050179TE2,L 1.495945498 )0.053815080 1.495945499 )0.053815130 1.495945499 )0.053815143TE3.L 1.495255343 )0.184243985 1.495255344 )0.184243812 1.495255344 )0.184243873TE4,L 1.485698163 )0.004051177 1.485698165 )0.004051177 1.485698165 )0.004051178TM0,L 1.632729635 )0.000000004 1.632729919 )0.000000004 – –

TM1,L 1.501625054 )0.002544521 1.501625054 )0.002544520 1.501625054 )0.002544521TM2,L 1.495287894 )0.576100220 1.495287895 )0.576101327 1.495287895 )0.576101022TM3,L 1.494855075 )1.189341232 1.494855077 )1.189338782 1.494855078 )1.189339701TM4,L 1.484121305 )0.197863266 1.484121307 )0.197863148 1.484121307 )0.197863211


ns ¼ 3:476, while the refractive index profile of the anisotropic graded indexSiOxNy follows

ng ¼ nlo þ D exp �ðx� x0Þ2=d2h i

ð24Þ

with nlo;xx ¼ nlo;yy ¼ nlo;zz ¼ 1:45, Dxx ¼ 0:1, Dyy ¼ Dzz ¼ 0:098, and d ¼0:5 lm. The cladding of the structure is air with nc ¼ 1. The position of thepeak of the SiOxNy index profile from the waveguide–air interface isd2 ¼ 2 lm, while d1 being varied from 2 to 8 lm. The computation results bytaking vacuum wavelength of k ¼ 1:55lm using computational window sizeof just d1 þ d2 with the computational boundaries put exactly at the inter-faces between the SiOxNy and Si substrate or air cladding and effective meshsize of 0.05lm using fourth-order scheme are given in Table 4. In this table,d1 ¼ 1 denotes the simplified structure by neglecting the silicon substrate but

Fig. 7. Refractive index profile of a Gaussian SiOxNy leaky structure.

Table 4. Computational results of the Gaussian SiOxNy leaky structure

d1 (lm) Polarization Re(neff) Im(neff) (·10)3) a (dB/cm)

2 TE 1.4876498 )0.0984765 34.673298

TM 1.4865629 )0.4558552 160.505289

4 TE 1.4880960 )0.0004157 0.146365

TM 1.4867917 )0.0021471 0.755969

8 TE 1.4880980 )0.00000001 0.000003

TM 1.4867929 )0.00000005 0.000018

¥ TE 1.4880980 0 0

TM 1.4867929 0 0


calculated using finite computational domain width with lower boundary putat 8lm from the peak of the Gaussian profile. It should be noted, that sinceour boundary conditions assume homogeneous exterior domain while thissimplified structure has inhomogeneous exterior domain, the error in repre-senting the field at the boundary will also limit the accuracy. It is shown fromthe results that the losses decrease while the real part of the effective indicesapproaching the results of the simplified lossless structure as the thickness ofthe buffer d1 is increased. Hence, the simplified lossless structure model canbe used to approximate the leaky structure for thick enough buffer.

4. Conclusion

A simple high-order Galerkin finite element scheme is proposed for thecomputation of both guided and leaky modes of anisotropic planar wave-guides with a diagonal permittivity tensor. TBC based on the Sommerfeldradiation conditions are used to model the field at the computationalboundaries, both at the one where the wave should leak and at the one wherethe wave should decay. Richardson’s extrapolation and mesh adjustmentscheme are used to extend the order of the scheme while still keep its sim-plicity. The ability of the scheme to study the properties of ARROW struc-tures was demonstrated.

Acknowledgement

This work is supported by STW Technology Foundation through projectTWI.4813.

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