fast convergent and unconditionally stable galerkin's method with adaptive hermite-gauss...

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P. Sarrafi and K. Mehrany Vol. 26, No. 1 /January 2009 /J. Opt. Soc. Am. B 169

Fast convergent and unconditionally stableGalerkin’s method with adaptive Hermite–Gauss

expansion for guided-mode extraction intwo-dimensional photonic crystal based

waveguides

Peyman Sarrafi and Khashayar Mehrany*

Department of Electrical Engineering, Sharif University of Technology, P.O. Box, 11365-8639 Tehran, Iran*Corresponding author: [email protected]

Received September 16, 2008; accepted October 24, 2008;posted November 12, 2008 (Doc. ID 101664); published December 24, 2008

It has been recently shown that guided modes in two-dimensional photonic crystal based structures can be fastand efficiently extracted by using the Galerkin’s method with Hermite–Gauss basis functions. Although quiteefficient and reliable for photonic crystal line defect waveguides, difficulties are likely to arise for more com-plicated geometries, e.g., for coupled resonator optical waveguides. First, unwanted numerical instability maywell occur if a large number of basis functions are retained in the calculation. Second, the method could havea slow convergence rate with respect to the truncation order of the electromagnetic field expansion. Third, spu-rious solutions are not unlikely to appear. All these three important issues are here resolved by applying theunconditionally stable S-matrix propagation method, by proposing an adaptive algorithm to expedite the con-vergence rate of the expansion through duly scaled Hermite–Gauss basis functions, and by providing an effec-tive algorithm for the elimination of spurious modes. © 2008 Optical Society of America

OCIS codes: 130.0130, 130.5296, 000.4430.

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. INTRODUCTIONhere are a variety of schemes available to extract guidedodes in two-dimensional photonic crystal based struc-

ures: the finite-difference time-domain method (FDTD)1–3], the beam propagation method (BPM) [4,5], the-matrix method [6], the S-matrix method [7,8], the im-edance matching method [9], the plane wave expansionethod [10], etc. Still, the Galerkin’s method based on the

et of Hermite Gauss basis functions counts among theastest and most efficient methods to extract guidedodes in two-dimensional photonic crystals with straight

ine-defects [11]. Although the Maxwell’s equations are inhis fashion converted into a standard and easy-to-solvelgebraic eigenvalue problem, numerical difficulties areikely to arise when more complicated geometries, flat

odes with small group velocity, or H-polarized guidedodes with discontinuous profiles are to be considered.irst, unwanted numerical instability can occur particu-

arly if a large number of Hermite–Gauss basis functionsre retained in the calculation. Second, the convergencepeed of the method could be quite low particularly if theermite–Gauss basis functions are badly scaled. Third,

he projection of Bloch space harmonics in the Hilbertpace spanned by the basis functions could result in theppearance of nonphysical spurious modes [12]. All theseroblems are carefully studied and duly resolved. In Sec., the idea of an S-matrix propagation algorithm [13] isdapted to the proposed Galerkin’s method, and an un-

0740-3224/09/010169-7/$15.00 © 2

onditionally stable technique is put forth to extract theought-after guided modes. In Section 3, an adaptive ap-roach is suggested to properly scale the Hermite–Gaussasis functions. Finally, an efficient algorithm is providedo eliminate spurious modes in Section 4.

. STABILITY AND THE GENERALIZED-MATRIX PROPAGATION ALGORITHM

n this section, the origin of the unwanted numerical in-tability is described and then an unconditionally stablecheme based on the idea of the S-matrix propagation al-orithm is proposed.

. Origin of Instabilityo apply the Galerkin’s method in photonic crystal basedaveguides similar to the one schematically shown inig. 1, the Bloch spatial harmonics are to be expanded in

erms of N basis functions �n�2x /L� [11],

��

�� = �a0�z� a0�z� ¯ aN�z�

b0�z� b1�z� ¯ bN�z��0�2x/L�

�1�2x/L�

]

�N�2x/L�� , �1�

here ai�z�s and bi�z�s are the unknown coefficients ofeld expansions, L is the scaling factor to be discussed inection 3, and

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170 J. Opt. Soc. Am. B/Vol. 26, No. 1 /January 2009 P. Sarrafi and K. Mehrany

� = − j� �

�Ey,

� = Hx,

or E polarization, and

� = Hy,

� = − j� �

�Ex,

or H polarization.Once the above-mentioned field expressions are substi-

uted in the Maxwell’s equations, and are duly projectedn the Hilbert space spanned by the Hermite–Gauss basisunctions, the following set of coupled linear differentialquations is obtained [11]:

d

dz�a

b� = � 0 Q�z;��

H�z;�� 0 ��a

b� , �2�

here �a= �an�z�, �b= �bn�z� correspond to the unknownoefficients of field expansions in Eq. (1), and the polar-zation dependent Q�z ;�� and H�z ;�� matrices are givenn Appendix A.

The unit cell of the structure z0�z�z0+Lz can then beivided into M subunits and be analytically integrated bysing the differential transfer matrix technique. In thisashion, the generalized transfer matrix of the unit cell inhe Hilbert space can be written as [11]

�a�z0 + Lz�

b�z0 + Lz�� =

m=1

M

exp�� 0 Qm��

Hm�� 0 ��a�z0�

b�z0�� ,

�3�

here

Fig. 1. Typical photonic crystal based waveguide.

Qm�� = z0+��m−1�/MLz

z0+�m/M�Lz

Q�z;��dz,

Hm�� = z0+��m−1�/MLz

z0+�m/M�Lz

H�z;��dz.

hanks to the translational periodicity along the z direc-ion, the sought-after propagation constant of guidedodes, k, then reads as [11]

k =− j

Lzln�eig�

m=1

M

exp�� 0 Qm��

Hm�� 0 �� . �4�

nasmuch as the exponential function of large-size matri-es has huge eigenvalues, numerical overflow is liable toause instability. It should be noticed that this is veryuch similar to what usually happens in applying the

onventional transfer matrix based modal methods innalysis of diffraction gratings [13,14].

. Generalized Scattering Matrix Propagation Algorithmo avoid contingent numerical overflow, the set of first-rder coupled differential equations in Eq. (2) should beombined to form the following set of second-orderoupled differential equations:

d2

dz2 �a = Q�z� · H�z��a. �5�

y following the similar procedure, the unit cell can nowe divided into M subunits and the coefficients of field ex-ression for the mth layer �Lz /M��m−1��z� �Lz /M��m�an be written as

�a = �n=1

N

�cn+�vne−�nz + cn

−�vne+�nz�, �6�

�Qm M

Lz��b = �

n=1

N

�− cn+�n�vne−�nz + cn

−�n�vne+�nz�, �7�

here cn±, �vnN1 and �n are the yet unknown constants

o be determined in the course of applying the periodicoundary condition, the eigenvectors and the positivequare root of the eigenvalues of the matrix

�M

Lz�2

QmHm,

espectively. In this manner, the growing and decayingerms are kept apart and can be treated separately.

The above-mentioned equation can then be rewritten inhe following concise matrix form:

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�a

b� = �V exp�L+� V exp�L−�

�Qm M

Lz�−1

VL+ exp�L+� �Qm M

Lz�−1

VL− exp�L−��cn+

cn−� , �8�

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here V is the eigenvector matrix of

�M

Lz�2

QmHm,

nd L+, and L− are diagonal matrices with diagonal ele-ents −�nz, and +�nz.It is now possible to write down the following layer

ransfer matrix:

��cn+�m+1�

�cn−�m+1�

� = t�m��cn+�m�

�cn−�m�

� , �9a�

t�m� = W�m+1�−1 W�m��m�, �9b�

here

W�m� = �W11�m� W12

�m�

W21�m� W22

�m�� = � V�m� V�m�

Q�m�−1 V�m�L�m�

+ Q�m�−1 V�m�L�m�

− � ,

�9c�

��m� = ��exp�L�m�+ ��z=z 0

0 �exp�L�m�− ��z=z

� . �9d�

he overall transfer matrix can then be easily deter-ined:

�a�z + Lz�

b�z + Lz�� = W�1�TW�1�

−1�a�z�

b�z�� , �10�

here

T = m=1

M

t�m�, �11�

lthough the separation of growing and decaying eigen-unctions at each subunit is quite helpful, the growing ex-onential terms in Eq. (9d) could still be hard to dealith. Therefore, the S-matrix propagation algorithm isere followed to guarantee the unconditional stability ofhe guided mode extraction. In accordance to what is pro-osed in [13], the layer s matrix, s�m�, is here defined as

��cn+�m+1�

�cn−�m�

� = s�m�� cn+�m�

�cn−�m+1�

� , �12a�

s�m� = �s11 s12

s21 s22� , �12b�

nd the interface s matrix, s , is defined as
�m�
� �cn+�m+1�

�cn−�m��exp�L�m�

− ��z=z� = s�m��cn

+�m��exp�L�m�+ ��z=z

�cn−�m+1�

� .

�13�

he layer s matrix is then calculable in terms of the in-erface s matrix and the decaying exponential terms only,

s�m� = �I 0

0 �exp�L�m�+ ��z=z

�s�m��exp�L�m�+ ��z=z 0

0 I� .

�14�

n the other hand, the interface s matrix, s�m�, can be di-ectly calculated by using the W�m� matrices:

s�m� = �W11�m+1� − W12

�m�

W21�m+1� − W22

�m��−1�W11�m� − W12

�m+1�

W21�m� − W22

�m+1�� . �15�

onsequently, the layer s matrix, s�m�, is obtainable with-ut calculating the exponentially growing terms of thexp�L−

�m��.In this fashion, the following generalized scatteringatrix of the unit cell in the Hilbert space spanned byermite–Gauss basis functions, S, which can be recur-

ively calculated by using the layer s matrix, s�m�, is sta-ly available:

��cn+�M+1�

�cn−�1�

� = S� �cn+�1�

�cn−�M+1�

� , �16�

S�m� = �S11 S12

S21 S22� . �17�

he �cn+ and �cn

− coefficients, on the other hand, havehe translational periodicity along the z direction and con-equently fulfill the Bloch condition

��cn+�M+1�

�cn−�1�

� = �D− 0

0 D+�� cn+�1�

�cn−�M+1�

� , �18�

here D− and D+ are diagonal matrices with diagonal el-ments, ejkLz and e−jkLz, respectively.

It is then possible to combine Eqs. (16) and (18) to ex-ract the propagation constant k, where

e−jkLz�− S11 0

− S21 I��cn+�1�

�cn−�1�

� = �− I S12

0 S22��cn

+�1�

�cn−�1�

� , �19�

nd consequently

k =j

Lzln�eig��− I S12

0 S22�,�− S11 0

− S21 I�� . �20�

n this way, the possibility of numerical overflow and in-tability is unconditionally sidestepped.

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As a numerical example, a two-dimensional triangularhotonic crystal made of air holes in GaAs ��=12.96� isonsidered. The radius of holes is 0.3 of the lattice con-tant �r=0.3a� and the photonic crystal waveguide isade by removing a row of holes in the ��−X� direction

see the inset of Fig. 2). The propagation constant of the-polarized mode at the normalized frequency �a /2�c0.26 versus number of Hermite–Gauss basis functionsith L=0.5 is plotted in Fig. 2. Here, two different meth-ds are applied. First, the generalized transfer matrix al-eady proposed in [11] is employed and the obtained re-ults are shown via circles. Second, the here-proposedeneralized scattering matrix approach is followed andhe results are shown via dots. This figure clearly demon-trates that the former method fails to provide stable re-ults whenever the number of retained Hermite–Gaussasis functions is larger than 50.

. PROPER SCALING OF THEERMITE–GAUSS BASIS FUNCTIONSenerally speaking, the convergence rate of the Galer-in’s method depends strongly on how carefully the set ofasis functions is selected. Although the fittest set cannote prefigured in the most general case, it is possible to en-ance the overall convergence rate by changing the scal-

ng factor L of the chosen basis functions.Different strategies have thus far proposed to properly

cale the Hermite–Gauss basis functions [15–19]. It ishown that the fastest convergence rate is achieved when-ver the scaling factor L satisfies �k /�L=0 [16]. Evenhough the promise of the fastest convergence rate andhe optimum scaling factor is enticing, this strategy is un-t for photonic crystal based structures, where nonuni-orm z-variant geometries, i.e., waveguides with varyingross sections, are to be analyzed. First, it is a pricey al-orithm imposing a heavy numerical burden. The Q and

matrices are to be recalculated for every different scal-ng factor and truncation order, their corresponding ei-

ig. 2. Propagation constant at the normalized frequencya /2�c=0.26 versus the number of basis function kept in the cal-ulation; circles: the generalized transfer matrix based methodroposed in [11]; dots: the here-proposed generalized scatteringatrix approach.

envalue problems are to be solved, and then the gener-lized scattering matrix should be found before extractinghe propagation constant k. Second, the numerical esti-ation of the derivative is always noisy and the equation

k /�L=0 cannot be easily solved. It is therefore desirableo follow a more efficient approach that if cannot result inhe fastest convergence rate, can at least guarantee a fastnough convergence.

Here, an adaptive approach similar to that of Ortega-onux et al. [18] is applied and a semioptimized scaling

actor is obtained. First, the initial scaling factor, L0, andhe truncation order, N, are heuristically chosen. Second,he generalized scattering matrix is found in the corre-ponding Hilbert space of the truncated Hermite–Gaussxpansion, wherein the electromagnetic field profile�x ,z� is calculated. It should be, however, noticed that

he obtained field profile ��x ,z� is an approximate solu-ion calculated by keeping N basis function, suffers fromruncation error, and most probably has not yet convergedo the correct field profile. Third, the updated scaling fac-or, L1, is now found by minimizing the following costunction:

�L� =�M�N�n�2��mn��L��2

�M�N��mn��L��2, �21�

here �L� is the spectral width of projecting the fieldrofile ��x ,z� in the Hilbert space spanned by keeping N�ermite–Gauss basis functions, and �mn��L� is

�mn��L� = −�

+�

��x,�m −1

2�Lz

M��n��2x

Lz�dx. �22�

he above-mentioned spectral width indicates the nor-alized energy spectrum of the field profile, depends on

he scaling factor, and can be minimized by using the sim-lex search method implemented in MATLAB’s fmin-earch function [20].

Once the updated scaling factor is set, the whole pro-ess restarts until the scaling factor converges to theemioptimum value. The numerical experiments showhat the semioptimum scaling factor can be found at 5–10teps, is almost independent of N�, and guarantees a fastonvergence rate with respect to the truncation order. It isherefore possible to work with N��N and the resultsill be still satisfactory.As an example, the previous two-dimensional GaAs tri-

ngular photonic crystal is considered and a coupled reso-ator optical waveguide (CROW) is formed by removing aow of holes in the ��–M� direction. This is schematicallyepicted in Fig. 3(a), where the elementary cell of thetructure is plotted. The extracted propagation constantf H-polarized eigenmode at the normalized frequencya /2�c=0.22 versus the scaling factor, L, and the trunca-

ion order, N, is then plotted in Fig. 3(b). The bold line inhe figure designates the whereabouts of the optimumcaling factor Lopt=0.4.

This figure shows that the optimum scaling factor ful-lls the �k /�L=0 equation, and provides the fastest con-ergence rate. As already mentioned, however, the pro-ess necessitates solving an eigenvalue problem for everyifferent scaling factor L, and truncation order N. It is

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herefore time-consuming and inefficient. The secondtrategy, on the other hand, is numerically cost-effective,nd results in a proper scaling factor with a very fast con-ergence rate. Here, the initial scaling factor, L0, and theruncation order of the calculation, N, are set to 1.5, and5, respectively. The proposed algorithm is then followednd the scaling factor is updated by minimizing the spec-ral width of projecting the electromagnetic field profile inhe Hilbert space of N� Hermite–Gauss basis functions. Inig. 4 the updated scaling factors at different iterationteps are plotted for N�=15, 25, and 30. This figure showshat the updated scaling factors with different yet largenough N� converge to the same semioptimized scalingactor Lso�0.54, and that the converged scaling factor islmost independent of N�. Thanks to the possibility oforking with N��N, the semioptimized scaling factor

an be very quickly found.To demonstrate the efficacy of the semioptimized scal-

ng factor, the convergence of the solution is demonstratedn Fig. 5, where extracted propagation constants are plot-ed versus the truncation order N. Crosses, dots, andquares in this figure correspond to L=0.4, 0.54, and 1.5,espectively. It is obvious from this figure that the semi-ptimized scaling factor is as efficient as the optimumcaling factor, and can considerably increase the overallonvergence rate of calculation. This figure also demon-trates the occurrence of unwanted spurious modes,hich are encircled to be discriminated from the rest. Theext section is devoted to eliminate the presence of suchnwanted modes.

ig. 3. (a) Schematic of the elementary cell in the analyzedROW. (b) The propagation constant versus the truncation order, and the scaling factor L. The bold line designates the optimum

caling factor.

Fig. 4. Adaptive updating of the scal

. ELIMINATION OF SPURIOUS MODESs shown in the last example of Section 3, unwanted spu-ious solutions are likely to be observed in applying thealerkin’s method [12]. The origin of the appearance of

hese unwanted solutions is already discussed for longitu-inally invariant structures, i.e., for waveguides with con-tant cross sections [21]. Here, the longitudinal variationf the structure requires successive matrix multiplica-ions, and confounds the problem still further. The spuri-us modes, however, do not converge and randomlyhange as truncation order increases [12]. It is thereforeossible to increment the truncation order, to extract theew eigenvalues, and to eliminate those with large varia-ions. To demonstrate the applicability and efficiency ofhis simple strategy, the band structure of the previousxample is extracted by keeping N=80 and 81 Hermite–auss basis functions and the obtained results are plotted

n Fig. 6. This figure clearly demonstrates that the spuri-us modes, in contrast to the nonspurious ones, do notonverge as truncation order increases. It is therefore pos-ible to eliminate unwanted spurious solutions by incre-enting the truncation order N by one and removing the

igensolutions with more than 10% variation in theirropagation constants. For properly scaled Hermite–

tor versus number of iteration steps.

ig. 5. Convergence of the solution in terms of the truncationrder and for different values of scaling factors: L=0.4 (crosses),=0.54 (dots), and L=1.5 (squares). The encircled crosses, dots,nd squares are unwanted spurious modes.

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auss basis functions having a good convergence rate, theroposed strategy is not very much sensitive to thehreshold level of acceptable variation. This specific strat-gy is here applied to the preceding example, the bandtructure is recalculated, and the results are shown inig. 7. The plane wave expansion method (PWE) [10] islso employed to verify the accuracy of the proposedethod.

. CONCLUSIONlthough the Galerkin’s method with Hermite–Gauss ba-is functions has been already established as a reliableethod to extract guided modes in optical waveguidesith constant cross section, analysis of photonic wave-uide structures with longitudinal periodic variation im-oses new difficulties. These difficulties are here resolved.irst, a generalized scattering matrix propagation algo-

ig. 6. Band structure extracted by retaining N=80 (crosses),nd N=81 (circles) basis functions. The unwanted spuriousodes are encircled to be discriminated from the correct

olutions.

ig. 7. Band structure calculated by using the proposed strat-gy of removing the unwanted spurious modes (crosses), and bypplying the PWE method (circles).

ithm is proposed to secure the unconditional stability ofhe method. Second, the Hermite–Gauss basis functionsre properly scaled to speed up the convergence rate.hird, an easy-to-implement strategy is proposed to elimi-ate the unwanted spurious solutions.

PPENDIX Ahe derivation of polarization dependent Q�z ;�� and�z ;�� matrices in Eq. (2) is already discussed in [11]. To

e self-contained, however, the final result is given below.Both Q and H matrices are of the size �N+1� �N+1�,

nd can be expressed as follows:

Q = k0I,

H = −1

k0G2 − k0D,

or E polarization, and

Q = k0F−1,

H = −1

k0GD−1G − k0I,

or H polarization.The G, D, and F matrices in these expressions are

G = �gmn; gmn =2

L −�

�

�n��x��m�x�dx,

D = �dmn�z�; dmn�z� = −�

�

�m�x��n�x��r�Lx/2,z�dx,

F = �fmn�z�; fmn�z� = −�

� �m�x��n�x�

�r�Lx/2,z�dx,

nd k0 denotes the vacuum wavenumber.

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fast convergent and unconditionally stable galerkin's method with adaptive hermite-gauss...

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