Arbitrary truncation order three-point finite differencemethod for optical waveguides with stepwise

refractive index discontinuitiesSlawomir Sujecki

The School of Electrical and Electronic Engineering, The University of Nottingham,NG7 2RD, Nottingham, UK (Slawomir.Sujecki@Nottingham.ac.uk)

Received October 1, 2010; revised October 28, 2010; accepted November 4, 2010;posted November 15, 2010 (Doc. ID 136022); published December 9, 2010

An exact finite difference (FD) representation of the second-order derivative on three nodes is presented and used toobtain an FD algorithm that allows achieving an arbitrary truncation order. The FD weights are calculated analy-tically using the series that expresses the field value at a given FD node in terms of the field value and its derivativesat a neighboring node, when a stepwise discontinuity in the refractive index distribution is present between thenodes. The results obtained confirm that the proposed algorithm is accurate, efficient, and achieves the predictedimproved performance. © 2010 Optical Society of AmericaOCIS codes: 130.2790, 130.0130.

Structures with discontinuities in the refractive indexprofile are of primary importance for the realization ofmost optical devices in integrated, fiber, and bulk optics.One of the standard techniques used for the design of op-tical devices is the finite difference (FD) method. How-ever, as the standard FD method is valid only if themedium is homogeneous, there is a need for the develop-ment of FD approximations for inhomogeneous media.The first three-point FD scheme valid for a waveguidewith the discontinuous refractive index distributionwas developed by Stern [1]. The accuracy of the FD sten-cil for the discontinuous media was then improved gra-dually by other authors [2–9]. However, only recently hasthe first algorithm been developed that allows achievingarbitrary truncation orders [10]. The algorithm presentedin [10] relies on the numerical calculation of the FD sten-cil weights at each node, which reduces significantly theefficiency of calculating the FD matrix coefficients.Further, in [10] the FD stencil weights are calculated onlyapproximately, which affects the accuracy of the results.Therefore, in this Letter, we develop an algorithm thatallows for achieving an arbitrary truncation order witharbitrary positions of FD nodes and stepwise refractiveindex discontinuities. The algorithm is based on a simplethree-point FD stencil, which makes the process of meshsetup very efficient and simple to program. The FDweights are calculated efficiently and accurately usingprederived analytical expressions. The basis for the de-rivation of the presented algorithm forms the series thatallows representing the field value at a given FD node interms of the field value and its derivatives at the neigh-boring FD node, in the presence of a stepwise disconti-nuity in the distribution of the refractive index betweenthe nodes. Once the series coefficients are known, wederive the equation that exactly represents the secondderivative using three FD nodes and use it to developan algorithm that allows achieving arbitrary truncationorder on the three-point FD mesh.We consider an eigenmode problem for a slab wave-

guide using the magnetic field formulation:

s∂

∂x

�1s∂ϕ∂x

�þ�n2k20 − β2

�ϕ ¼ 0; ð1Þ

where β is the propagation constant, n is the refractiveindex, k0 stands for the wavenumber, and s equals 1 andn2 for the TE and TM modes, respectively. Following[5,10], for ϕ we can construct the following matrix equa-tion that relates the value of ϕ at two discrete points i andiþ 1 (Fig. 1) that is valid if one stepwise refractive indexdiscontinuity is present between the points:

ϕiþ1 ¼ b0ϕi þ b1ϕ0i þ b2ϕ00

i þ HOT

¼h1 q 1

2 q2 …

i×

266641 0 0 …

0 θ 0 …

η 0 1 …

..

. ... ..

. . ..

37775 ×

26664epDϕi

epDϕ0i

epDϕ00i

..

.

37775:

ð2Þ

The middle coefficient matrix in (2) contains the conti-nuity conditions for ϕ and all its derivatives at the inter-face between two regions with different values of therefractive index [5,11], q and p are defined in Fig. 1,η ¼ k20ðn2

i − n2iþ1Þ, and θ equals 1 and n2

iþ1=n2i for the

TE and TM cases, respectively. Further, D ¼ d=dx, while“HOT” stands for higher-order terms. In [5,10], only ap-proximate values for the bj coefficients were obtained;however, owing to the particular form of the coefficientmatrix, the exact values of the bj coefficients can beobtained by carrying out the multiplications in (2) and

Fig. 1. Schematic diagram of the FD grid showing the relativepositions of the refractive index profile discontinuities.

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0146-9592/10/244115-03$15.00/0 © 2010 Optical Society of America

collecting separately all the terms containing ϕi, ϕ0i, ϕ00

i ,and higher derivatives of ϕ. This yields an infinite sum foreach bj coefficient, the limit of which can be expressedusing analytical functions. So, calculating the respectiveinfinite sums gives the expressions for the bj coefficients:

bM ¼XMk¼0

1k!pkBM−kϕi: ð3Þ

The coefficients B are given by the following formulas:

B2l ¼1ηl12

I

�l − 1=2; q

ffiffiffiηp �Γðlþ 1=2Þ2lþ1=2

�q

ffiffiffiηp �lþ1=2

ð2lÞ! ;

for even values of M − k ¼ 2l and

B2lþ1

¼ 1

ηð2lþ1Þ=212

I

�lþ 1=2; q

ffiffiffiηp �Γðlþ 3=2Þ2lþ1=2

�q

ffiffiffiηp �lþ1=2

ð2lÞ! ;

for odd vales of M − k ¼ 21, where I denotes the circularBessel function and Γ is the gamma function [11]. Tocomplete the derivation of the three-point FD approxima-tion, we, similarly, obtain the series expansion for ϕi−1:

ϕi−1 ¼ a0ϕi þ a1ϕ0i þ a2ϕ00

i þ HOT; ð4Þ

where the expansion coefficients aj can be obtained fromthe respective coefficients bj by substituting −c, −d, andni−1 for p, q, and niþ1, respectively. Finally, by eliminatingthe first term containing the derivative of ϕi from (2) and(4), we obtain a three-point FD approximation of thesecond derivative at the central node:

ϕ00i ¼ Δ2ϕi −

X∞j¼3

DjϕðjÞi ;

Δ2ϕi ¼1b1ϕiþ1 −

�b0b1−

a0a1

�ϕi−1 −

1a1ϕi−1�

b2b1−

a2a1

� ;

Dj ¼�bjb1−

aja1

��b2b1−

a2a1

� : ð5Þ

Similarly, by eliminating the second derivative from (2)and (4), one can obtain an approximation of the first de-rivative on the three-point FD stencil:

ϕ0 ¼ Δϕi −X∞j¼3

CjϕðjÞi : ð6Þ

Using the identity dϕ2=dx2 ¼ ðβ2 − n2k20Þϕ to reduce thederivative order and approximating the resulting first-and second-order derivatives by (5) and (6) leads tothe following equation that gives the exact approxima-tion of the second-order derivative on a three-pointFD stencil:

ϕ00i ¼ Δ2ϕi þ

X∞k¼1

ðβ2 − n2k20ÞGkϕi;

Gkϕi ¼ ð−Dk−13 Δϕi − Dk−1

4 Δ2ϕiÞ;ð7Þ

where the coefficientsD can be obtained from the follow-ing recurrent relation:

Dkj ¼ ð−Dk−1

3 Cj − Dk−14 Cj − Dk−1

jþ2Þ;

which is initiated from D0j ¼ Dj . Substituting (7) into (1)

leads to an eigenvalue problemwith increasing powers ofβ2. For k ¼ 1, we obtain the generalized Douglas scheme[5]. However, unlike in [5], the FD coefficients in (7) arecalculated exactly. Including more terms in (7) leads to aquadratic, cubic, eigenvalue problem, which can be re-duced to a generalized eigenvalue problem that can besolved using the inverse power method. Here we usedthe transformation given in [12] that transforms an N ×N quadratic eigenvalue problem into a 2N × 2N general-ized eigenvalue problem and an N × N cubic eigenvalueproblem into a 3N × 3N generalized eigenvalue prob-lem, etc.

For demonstrating the advantages of the proposed al-gorithm, we selected the same structure as in [5], namely,a GaAs-based symmetrical slab waveguide with the re-fractive index equal to 3.3704 and 3.2874 in the coreand cladding, respectively. The width of the waveguidecore is 2 μm, while the wavelength λ ¼ 1:55 μm.Although the refractive index contrast is rather small,it was found sufficient for showing the differences be-tween the proposed algorithm and the previously usedtechniques. For the purpose of the comparison, we de-fine the relative error in the propagation constant

εβ ¼βcalculated − βexact

βexact: ð8Þ

In Figs. 2 and 3 we compared the FD schemes for k ¼ 2,3, and 4 in (7) with the formally equivalent seven-, nine-,and eleven-point FD schemes obtained using the algo-rithm given in [10]. We use the Dirichlet boundary

Fig. 2. (Color online) Dependence of the relative error in thepropagation constant on the grid size for the seven-, nine-, andeleven-point FD stencils [11] and the algorithm presented herefor k ¼ 2, 3, and 4 in (7) in the case of the TE mode.

4116 OPTICS LETTERS / Vol. 35, No. 24 / December 15, 2010

condition at the computational window edge, which isapproximately 17 μm away from the waveguide center.The FD grid nodes were distributed symmetrically, withrespect to the waveguide center. The values of the rela-tive error (8) were calculated at each minor grid line ofthe abscissa axis, which is shown in Fig. 2. It is observedthat for both mode polarizations, both the algorithm gi-ven in [10] and the one presented here achieve increasingtruncation order. However, the relative error of the re-sults obtained using the algorithm given by Eq. (7) is evenby 2 orders lower when compared with [10]. Further, theproposed algorithm does not suffer from the large errorincrease at a small grid size. This is most likely the con-sequence of the buildup of the round-off errors resultingfrom the large number of operations needed to obtain theFD weights by the algorithm given in [10] (many matrixmultiplications are needed per one FD node), whichmakes this algorithm also inefficient in comparison withthe presented one.

Finally, it is noted that the advantages offered by thepresented FD method over the previously used techni-ques are not only relevant for the 1D case. The proposedalgorithm allows for a very efficient and accurate calcu-lation of the entire modal spectrum using a large value ofthe mesh size (especially for multilayered waveguides,which are of large particular practical importance, e.g.,in laser diodes and quantum cascade lasers). Hence, itoffers a significant improvement of the performancefor the method of lines, which is a very powerful techni-que that has been very successfully used in the multidi-mensional analysis of photonic devices; see [13].

In conclusion, we presented an efficient and simple touse algorithm that allows achieving an arbitrary trunca-tion order on the three-point FDmesh with arbitrary posi-tions of mesh nodes and refractive index discontinuities.The presented results confirm that the proposed algo-rithm achieves the predicted performance.

References

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Fig. 3. (Color online) Dependence of the relative error in thepropagation constant on the grid size for the seven-, nine-, andeleven-point FD stencils [11] and the algorithm presented herefor k ¼ 2, 3, and 4 in (7) in the case of the TM mode.

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