Arbitrary-order interface conditions for slab structures and their applications in waveguide...

Arbitrary-order interface conditions forslab structures and their applications in

waveguide analysis

Yih-Peng Chiou1,2* and Cheng-Han Du1

1Graduate Institute of Photonics and optoelectronics, National Taiwan University, Taipei106-17, Taiwan

2Also with Department of Electrical Engineering, National Taiwan University, Taiwan

*[email protected]

Abstract: We extend the continuity relations of field derivatives acrossan abrupt interface to arbitrary orders for transverse electric and magneticwaves in slab structures. Higher-order finite-difference formulation is thenobtained by combining the systematically-obtained interface conditionswith Taylor series expansion. Generalized Douglas scheme is also adoptedto further enhance the convergence of truncation errors by two orders. Weapply the derived finite-difference formulation, up to nine-points in thispaper, to solve the guided modes in simple a slab waveguide and multiplequantum well waveguides. The results shows the truncation error is muchhigher, up to tenth order, as expected. Using those higher-order schemes,accurate results are obtained with much fewer sampled points, and hencewith tremendously less computation time and memory.

© 2010 Optical Society of America

OCIS codes: (000.4430) Numerical approximation and analysis; (040.4200) Multiple quantumwell; (130.2790) Guided waves; (230.7400) Slab Waveguides.

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analysis and assessment,” J. Lightwave Technol. 10, 295–305 (1992).6. C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” Proc. Inst. Elect. Eng.

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waveguides with step-index profiles,” J. Lightwave Technol. 20, 1609–1618 (2002).8. Y.-P. Chiou, Y.-C. Chiang, C.-H. Lai, C.-H. Du, and H.-C.Chang, “Finite-difference modeling of dielectric

waveguides with corners and slanted facets,” J. Lightwave Technol. 27, 2077–2086 (2009).9. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Finite-difference frequency domain analysis of 2-D photonic cry-

tals with curved dielectric interfaces,” J. Lightwave Technol. 26, 971–976 (2008).10. G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation I: beam propagation,”

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#122529 - $15.00 USD Received 13 Jan 2010; revised 24 Jan 2010; accepted 30 Jan 2010; published 16 Feb 2010

(C) 2010 OSA 1 March 2010 / Vol. 18, No. 5 / OPTICS EXPRESS 4088

12. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditionsin the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. 18, 243–251 (2000).

13. H. Zhang, Q. Guo, and W. Huang, “Analysis of waveguide discontinuities by a fourth-order finite-differencereflective scheme,” J. Lightwave Technol. 25, 556–561 (2007).

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1. Introduction

Simple implementation makes finite-difference method (FDM) a popular solution for variousproblems governed by differential equations. Moreover, finite-difference-based formulation hasgreat flexibility and versatility as they can be easily modified to treat special problems, such asgraded-index structures, field and structure discontinuities, and nonlinear behaviors. Numerousresearch has greatly expanded capability of finite-difference method for general problems, suchas in-plane and out-of-plane mode solvers, beam propagation analysis, and finite-differencetime-domain method.

Semivectorial [1] and full-vectorial [2] formulations for graded-index structure were pro-posed and applied in commercial tools. Such techniques were further extended to beam prop-agation analysis [3–5]. Although graded-index approximation is applicable to structures withminimal index difference, it has difficulty in accurately modeling field behaviors near abruptinterfaces. Moreover, recent advancement of nanoprocess makes step-index structures practi-cal solutions for next-generation optical design, indicating that a modified modeling formu-lation dedicated to step-index problem is required. Vassallo [6] proposed formulation for one-dimensional (1D) and two-dimensional (2D) guiding structure in scalar and semivectorial cases,which takes boundary continuity relations of field and derivatives into consideration. Chiou etal. have extended this technique to full-vectorial waveguide mode solver and photonic crystalanalysis [7–9]. Other modeling algorithm considering abrupt interface conditions are developedas well.

One critical property of finite-difference method is its convergence of truncation error. Theconvergence of commonly used three-point formulation is normally no more than second order(O(h2)), since only two adjacent sampled points are referenced to approximate second-orderderivative in Helmholtz equation. Higher-order convergence is desired in certain situations toachieve higher accuracy and less computation. One effective method to increase the order is us-ing generalized Douglas (GD) scheme. Hadley [10] uses generalized Douglas scheme to reduceerror of beam propagation method on step-index problems, while modification for graded-indexproblem has also been reported [11]. Chiou et al. [12] has developed three-point formulationconsidering interface conditions with generalized Douglas scheme and successfully achievedfourth-order convergence (O(h4)) for mode solving of step-index structures. Such formula-tion is adopted in waveguide discontinuities [13] and extended three-dimensional semivecto-rial formulation, by including two-transverse directions, to solve wide-angle beam-propagation



method [14] and surface plasmon metallic waveguides [15].Convergence more than O(h4) may be desired when we want to achieve even higher ac-

curacy or even less computation. Hadley proposed a finite-difference formulation magicallyfree of truncation error [16]. It is an exact solution for eigenmode in one-dimensional problemby placing the sampled points at interfaces. It appears about 40% faster than matrix method.However, it can not be directly applied to beam propagation analysis, since the eigen valuen2 from ∂ 2/∂ z2 in Helmholtz equation is inside the trigonometric functions. The derivation offinite-difference formulation is back to the commonly used Taylor series expansion. To achieveconvergence of higher orders, we may include more sampled points to evaluate higher-orderderivatives. However, the formulation is usually derived in homogeneous region, and it doesnot consider the discontinuity across abrupt interfaces. Therefore, when there are abrupt in-terfaces, the convergence is still of low order. It might not be problematic when the numberof interfaces is small, since it is still of high order in the rest homogeneous regions. Whenthe number of interfaces are large, like multiple quantum well structures, the convergence willbe lowered, especially in structures of higher index difference. Therefore, a generalized higher-order finite-difference formulation considering the interface conditions is required in such casesto improve the numerical performance.

In this paper, we derive generalized interface conditions of arbitrary orders for TE andTM modes calculation. Such conditions can directly be incorporated into the finite-differencemethod by combining with Taylor series expansion. Higher-order finite-difference formulationfor second-order derivative is then obtained with more sampled points, which is implemented inthe mode solver based on Helmholtz equation. We apply the formulation in the analysis of slabwaveguide modes. Generalized Douglas scheme is also adopted to further increase the order.All the derivation is expressed in matrices, which is simple and easily extended to higher orders.This paper is organized as follows. In next section, we derive the interface conditions, and, bymathematical induction, show it is valid for all orders if the derivative exist. In Section 3, wegive finite-difference implementation in waveguide mode solvers. In Section 4, the truncationerrors of propagation constants in a simple slab waveguide and multiple quantum waveguidesare investigated. Finally, a conclusion is drawn in the last section.

2. Derivation of Higher-Order Interface Conditions

For monochromatic electromagnetic wave propagating of angular frequency ω in source-free,isotropic, and piecewisely homogeneous, Maxwell equations in derivatives form are expressedas

∇×E = − jωB (1)

∇×H = jωD (2)

∇ ·D = 0 (3)

∇ ·B = 0. (4)

The Helmholtz equations for electrical and magnetic fields are satisfied in a piecewiselyhomogeneous region as

∇2E+ k20n2E = 0 (5)

and∇2H+ k2

0n2H = 0, (6)

where k0 = ω/c. For an interface in a slab structure, we assume electric permitivity ε andmagnetic permeability are homogeneous in y- and z-directions, and the discontinuity is along



x-direction as in Fig. 1. We also assume the propagation is along z-direciton. A propagatingmode with propagation constant β and modal field distribution E = E(x) and H = H(x), theelectric and magnetic fields can be expressed as

E = E exp(− jβ z) (7)

andH = H exp(− jβ z), (8)

respectively. Helmholtz equations then become

∂ 2E∂x2 +ω2μεE = β 2E (9)

and∂ 2H∂x2 +ω2μεH = β 2H. (10)

The interface conditions are the continuity of tangential components of electrical and mag-netic field intensities, Ey, Ez, Hy, and Hz, and the continuity of normal components of electricaldisplacement Dx = εEx and magnetic flux density Bx = μHx as in Fig. 1. From the continuity

Fig. 1. Continuity of fields on interface and sample points.

of the field components, we will derive the continuity of their derivatives.

2.1. TE case

In transverse electric (TE) modes, Ey, Hx, and Hz are not zero. From the continuity of thetangential electrical and magnetic fields, we have

EyR = EyL (11)

andHzR = HzL (12)

where subscript L and R represent infinitesimally close to the interface on the left and right,respectively, as shown in Fig. 1. From (1), we have

∂Ey

∂x= − jωμHz. (13)



Since Hz is continuous across the interface, we have continuity relation for the first-order deriva-tive

∂EyR

∂x=

μR

μL

∂EyL

∂x. (14)

The continuity of field and its first derivative in (11) and (14) are commonly used. To achievehigher-order convergence in the finite-difference method, the continuity conditions of higher-order derivatives should be derived. From one-dimensional Helmholtz equation

∂ 2Ey

∂x2 +ω2μεEy = β 2Ey (15)

we have

Ey =1

β 2

[∂ 2

∂x2 +ω2με]

Ey. (16)

From (11), we have the continuity relation for the second-order derivative[

∂ 2

∂x2 +ω2μLεL

]EyR =

[∂ 2

∂x2 +ω2μRεR

]EyL (17)

or

∂ 2EyR

∂x2 =[

∂ 2

∂x2 +ω2μLεL

]EyL −ω2μRεREyR =

[∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]

EyL. (18)

Substituting (18) and (16) into (11) and (14) recursively, we have generalized continuity rela-tions for even- and odd-order derivatives expressed as

∂ 2n

∂x2n EyR =[

∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]n

EyL. (19)

and∂ 2n+1

∂x2n+1 EyR =μR

μL

∂∂x

[∂ 2


EyL. (20)

respectively. We prove (19), and (20) can be proved similarly. First, the statement is true n = 0from (11). Second, we assume the statement is true for n = k, that is,

∂ 2k

∂x2k EyR =[

∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]k

EyL. (21)

Substituting (16) into both sides of (21) we have

∂ 2k

∂x2k

{1

β 2

[∂ 2

∂x2 +ω2RμRεR

]EyR

}=

1β 2

{∂ 2k+2EyR

∂x2k+2 +ω2μRεR∂ 2kEyR

∂x2k

}

=[

∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]k {

1β 2

[∂ 2

∂x2 +ω2μLεL

]EyL

}. (22)

Substituting (21) into (22), we have

∂ 2k+2EyR

∂x2k+2 =[

∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]k {[

∂ 2

∂x2 +ω2μLεL

]EyL

}

−ω2μRεR

{[∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]k

EyL

}



=[

∂ 2

∂x2 +(ω2μLεL −ω2μRεR)]k+1

EyL. (23)

The statement is also true for n = k +1. Therefore, by the principle of mathematical induction,the statement holds for all non-negative integers.

Equations (19) and (20) up to 2N-th order can be written in matrix form as

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ψR

ψ ′R

ψ ′′R

ψ(3)R

ψ(4)R...

ψ(2N−1)R

ψ(2N)R

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 · · · 00 a 0 0 0 · · · 0b 0 1 0 0 · · · 00 ab 0 a 0 · · · 0b2 0 2b 0 1 · · · 0...

......

......

. . ....

0 CN−10 abN−1 0 CN−1

1 abN−2 0 · · · 0CN

0 bN 0 CN1 bN−1 0 CN

2 bN−2 · · · CNN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ψL

ψ ′L

ψ ′′L

ψ(3)L

ψ(4)L...

ψ(2N−1)R

ψ(2N)L

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

orΨR = MR:LΨL, (25)

where a = μR/μL and bk = ω2(μLεL − μRεR). We will use this equation to drive higher-orderfinite-difference formulation.

2.2. TM case

Derivation in transverse magnetic modes is quite similar to that in TE case. Hy, Ex and Ez arenonzero in TM mode. Continuity relations of the tangential fields

HyR = HyL (26)

andEzR = EzL, (27)

while (2) leads to∂Hy

∂x= − jωεEz. (28)

From (27) and (28), we have continuity relation of the first-order derivative

∂HyR

∂x=

εR

εL

∂HyL

∂x(29)

Similar to (19) and (20), generalized continuity relations are expressed as

∂ 2n

∂x2n HyR =[

∂ 2


HyL. (30)

and∂ 2n+1

∂x2n+1 HyR =εR

εL

∂∂x

[∂ 2


HyL. (31)

Similarly, a matrix form expression for (30) and (31) is obtained as that for (24) and (25), exceptthat a = εR/εL in TM case.



3. Implementation in Mode Solvers

3.1. Normal Finite-Difference Scheme

Considering a single abrupt interface and equispaced sampled points shown in Fig. 1, we firstrelate ψi+1 and its derivatives with ψi and its derivatives up to 2N-th order. Sampled points inhomogeneous medium can be described by Taylor series expansion up to 2N-th order, and wehave the relation between ψL and ψi as

⎡⎢⎢⎢⎢⎢⎣

ψL

ψ ′L

ψ ′′L...

ψ(2N)L

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣

1 p p2

2! · · · p2N/(2N)!0 1 p · · · p2N−1/(2N −1)!0 0 1 · · · p2N−2/(2N −2)!...

......

. . ....

0 0 0 · · · 1

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

ψi

ψ ′i

ψ ′′i...

ψ(2N)i

⎤⎥⎥⎥⎥⎥⎦

+HOT (32)

orΨL = ML:iΨi +HOT, (33)

where HOT stands for truncated higher-order terms. Similarly, the relation between ψi+1 andψR as

⎡⎢⎢⎢⎢⎢⎣

ψi+1

ψ ′i+1

ψ ′′i+1...

ψ(2N)i+1

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣

1 q q2

2! · · · q2N/(2N)!0 1 q · · · q2N−1/(2N −1)!0 0 1 · · · q2N−2/(2N −2)!...

......

. . ....

0 0 0 · · · 1

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

ψR

ψ ′R

ψ ′′R...

ψ(2N)R

⎤⎥⎥⎥⎥⎥⎦

+HOT (34)

orΨi+1 = Mi+1:RΨR +HOT. (35)

With (33), (33), and interface continuity conditions (25), the relation between Ψi+1 and Ψi isthus established as

Ψi+1 =Mi+1:RΨR +HOT = Mi+1:RMR:LΨL +HOT = Mi+1:RMR:LML:iΨi +HOT

=Mi+1:iΨi +HOT. (36)

Note that, there can be multiple interfaces between the sampled points ψi+1 and ψi. For exam-ple, if there are T interfaces with left and right fields represented as ψLt and ψRt at tth interface,respectively, we can have the relation between Ψi+1 and Ψi as

Ψi+1 =Mi+1:RT MRT :LT MLT :R(T−1) · · ·ML(t+1):RtMRt:Lt · · ·MR1:L1ML1:iΨi +HOT

=Mi+1:iΨi +HOT (37)

where MRt:Lt’s are from matching the interface conditions, and Mi+1:RT , ML1:i, and MR(t+1):Lt’sare from Taylor series expansion. Applying the same procedure recursively, we can obtain therelation between Ψi+ j and Ψi,

Ψi+ j =Mi+ j:i+ j−1Ψi+ j−1 +HOT = · · · = Mi+ j:i+ j−1 · · ·Mi+1:iΨi +HOT

=Mi+ j:iΨi +HOT, (38)

where j = 1, 2, , · · · , N. Similarly, we can have relation for negative js.



We can express ψi+ j in terms of ψi and its derivatives from the first row in Ψi+ j = Mi+ j:iΨi

ψi+ j = [u j,0 u j,1 u j,2 · · · u j,2N ]Ψi +O(h2N+1). (39)

In the derivation, there is no approximation in matching the boundary conditions. All theerrors are from the truncation of Taylor series expansion. Therefore, the truncation error isO(h2N+1). After applying similar procedure on all the (2N + 1) sampled points, we have therelation between the field at sampled points and ψi and its derivatives up to 2Nth order as

⎡⎢⎢⎢⎢⎢⎢⎣

ψi−N...

ψi...

ψi+N

⎤⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎣

u−N,0 u−N,1 · · · u−N,2N...

.... . .

...uN,0 uN,1 · · · uN,2N

⎤⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎣

ψi...

ψ( j)i...

ψ(2N)i

⎤⎥⎥⎥⎥⎥⎥⎥⎦

+O(h2N+1) (40)

or

Fi = UiΨi +O(h2N+1). (41)

If the higher-order terms are neglected, and the finite-difference approximation for the deriva-tives is expressed as Dj

xψi, we have

Fi = UiDi, (42)

where Di = [ψi, Dxψi, D2xψi, · · · , Dj

xψi, · · · , D2Nx ψi]T . Finally, we obtain the finite-difference

approximation of derivatives up 2N-th order considering the interface conditions at of ψi with(2N +1) sampled points

Di = BiFi, (43)

where Bi = U−1i . The third row in (43) is the approximation of second derivative expressed in

terms of the sampled points, which is O(h2N+1)/h2 = O(h2N−1) accurate. We use centered-difference scheme here, but it is not compulsory. We can also use forward-difference orbackward-difference scheme. The advantage of using centered-difference scheme is that thetruncation error can be enhanced by one order, from O(h2N−1) to O(h2N), when the medium ishomogeneous and the grids are uniformly positioned [19].

To solve the eigen problem in mode solvers, we replace the second derivative ∂ 2ψ/∂x2 in(9) and (10) with D2

xψ . Then we have the discretized Helmholtz equation

D2xψ +ω2μεψ = β 2ψ. (44)

As mentioned, the truncation error in D2xψ is O(h2N) if the medium is homogeneous between

the (2N + 1) points and the grids are uniformly positioned [19]. If the grids are nonuniformlypositioned, the truncation error is usually reduced to O(h2N−1). If there is an abrupt interfaceor interfaces, the truncation error is usually O(h2N−1). In waveguide problems, most grids areusually in homogeneous regions. Therefore, the eigen problem in (44) is O(h2N) accurate inglobal sense.

3.2. Generalized Douglas Finite-Difference Scheme

Generalized Douglas scheme can also be implemented to magically enhance the accuracy bytwo orders. With two additional derivatives included in (41), we have

Fi = UiΨi +G1ψ(2N+1)i +G2ψ(2N+2)

i +O(h2N+3), (45)



where G1 and G2 are (2N +1)×1 vectors. Multiplying the inverse matrix Bi = U−1i as in (43),

we have improved formulation

Di = BiFi = Ψi +Bi(G1ψ(2N+1)i +G2ψ(2N+2)

i )+O(h2N+1), (46)

where the third row is expressed as

D2xψi =

j=N

∑j=−N

b3, jψi+ j ≈ ψ ′′i + v1ψ(2N+1)

i + v2ψ(2N+2)i

=(

1+ v1∂ 2N−1

∂x2N−1 + v2∂ 2N

∂x2N

)ψ ′′

i , (47)

where v j is the third element of the vector V = BiG j in (46). We further approximate ∂ 2N−1

∂x2N−1

and ∂ 2N

∂x2N with D2N−1x and D2N

x in (43), and we have the generalized Douglas scheme to improvethe accuracy of second-order derivative

D2xψi ≈

(1+ v1D2N−1

x + v2D2Nx

)ψ ′′

i , (48)

which is sometimes written as

ψ ′′i =

D2x

1+ v1D2N−1x + v2D2N

x

ψi. (49)

The accuracy is increased by two orders by including two additional terms.Multiply the Helmholtz equation with

(1+ v1D2N−1

x + v2D2Nx

), we finally have improved

discretized Helmholtz equation

D2xψ +ωμε

(1+ v1D2N−1

x + v2D2Nx

)ψ = β 2 (

1+ v1D2N−1x + v2D2N

x

)ψ (50)

based on generalized Douglas scheme, which considers the interface conditions. Note that ma-trix is still banded and the bandwidth is the same, (2N+1). The computation time and memoryin the eigenvalue problem is the same, while the accuracy is enhanced by two orders. That is,the truncation error in is O(h2N+2) if the medium is homogeneous between the (2N +1) pointsand the grids are uniformly positioned. If the grids are nonuniformly positioned, the truncationerror is usually reduced to O(h2N+1). If there is an abrupt interface or interfaces, the trunca-tion error is O(h2N+1). In waveguide problems, most grids are usually in homogeneous region.Therefore, the eigen problem in (50) is usually O(h2N+2) accurate in global sense.

4. Applications in Slab Waveguides

The convergence of propagation constants in slab waveguides is investigated in this section.First, a simple waveguide with a homogeneous core is investigated. Then multiple quantumwaveguides (MQW) with more than than one-hundred interfaces in the guiding layer are inves-tigated. The eigenvalues and vectors of the guided modes are solved. By slight modification, itcan be directly applied to beam propagation analysis.

4.1. Slab Waveguide

Error analysis of slab waveguide simulation is demonstrated in this section. Refractive indicesof core and cladding are ncore =

√11.088 and nclad =

√11.044, respectively. Waveguide width

is 2 μm and wavelength is 1.55 μm. The computation domain is 40 μm in width and transparent



boundary condition (TBC) is adopted on the outer boundary. The relative error of propagationconstant β is defined as

Relative error εr =∣∣∣∣βcalculated −βexact

k0(ncore −nclad)

∣∣∣∣ , (51)

where βcalculated is calculated from finite-difference method, and βexact is from the analyticalsolution of dispersion relation.

Figures 2 and 3 illustrate the relative errors in TE and TM modes, respectively, for differ-ent grid sizes Δx. The thinnest black and blue lines are from three-point formulation based ongraded-index (GI) formulation as in [1] with or without index averaging. Intermediate thick andthickest lines are from calculation considering the interface conditions without and with gener-alized Douglas (GD) scheme as in Section 3, respectively. The approximation of a step indexinterface with graded index is only O(h0). The higher index difference the larger deviationof the approximation is. In this case, however, only two interfaces with small index differ-ence exist, and most sampled points are in homogeneous region. Therefore, using conventionalgraded-index (GI) approach can still yield O(h2) accuracy. The red lines are calculated fromthree-point formulation without and with GD scheme as proposed in [12]. Similarly, the three-point formulation in (44) and (50) are usually O(h1) and O(h3) accurate, respectively. However,most sampled pointed are in homogeneous region, the accuracy are still O(h2) and O(h4) accu-rate, respectively. Furthermore, if the interfaces are at the center between two sampled pointsor at one sampled point, the three-point formulation in (44) and (50) can be O(h2) and O(h4)accurate, respectively.

The green, pink, and cyan lines are calculated from five-, seven-, and nine-point formulation,respectively. For (2N+1)-point calculation with boundary condition matching, the convergenceis approximately O(h2N) as the intermediate thick lines, and they can be further increased up toO(h2N+2) by applying GD scheme as in the thickest lines. Overall, error analysis quite matchestheoretical expectation. Including more points in evaluating the second derivative in Helmholtzequation, we can reach desired accuracy with much fewer grid points. For example, if we wantto reach relative error εr = 10−8, the grid number in nine-point formulation with GD schemeis about 1/1000 of that in conventional three-point formulation. The required memory in theformer is only about 3/1000 of that in the latter. The computation time in finding the eigenvaluein the former is almost negligible as compared with that in the latter.

Furthermore, the limitation of accuracy can also be put into lower value. Total numerical erroris from truncation error and round-off error. Truncation error is due to the dropping of higherorder terms (HOT) in the derivation. As we can see, the P-th order truncation error, O(hP),decreases polynomially with grid size h, Round-off error is due to finite digits of the floatingpoint numbers in computers. It accumulates with the number of operations and increases withgrid number polynomially. Therefore, the lower bound of of the accuracy is limited by theround-off error. As we can see in the figure, lower-order formulations require more operationsand hence result in larger round-off error. For example, the lower bound of the relative errorin three-point formulation is about 10−9 and that in nine-point formulation with GD scheme isabout 10−14.

4.2. Multiple-Quantum-Well Waveguides

We extend to multiple quantum well (MQW) waveguides the finite-difference analysis basedon higher-order approximation and GD scheme in this subsection. The guiding region consistsof 56 barriers and 55 wells. The parameters are the same as those in [12]. Widths of each barrierand well are 0.012 μm and 0.007 μm, respectively. Refractive indices of upper cover, wells,barriers, and lower cladding are ncovr = 3.2224, nwell = 3.3704 nwell = 3.3704, nbarr = 3.2874,



10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0.001 0.01 0.1 1

Rel

ativ

e er

ror

Δx (μm)

GI w/o IAGI w/ IA

3-point w/o GD3-point w/ GD




Fig. 2. Relative error of TE mode propagation constant with respect to grid size in a simpleslab waveguide.

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0.001 0.01 0.1 1

Rel

ativ

e er

ror

Δx (μm)

GI w/o IAGI w/ IA





Fig. 3. Relative error of TM mode propagation constant with respect to grid size in a simpleslab waveguide.

and nclad = 3.2224, respectively. The relative error of propagation constant is defined as

Relative error =∣∣∣∣βcalculated −βexact

k0(nwell −nclad)

∣∣∣∣ . (52)



There are 111 thin layers in the guiding region. Ohke et al. proved that the MQW region couldbe approximately replaced with a homogeneous layer with an averaged or effective index [17,18]. The averaged index is

n2avg =

∑i

win2i

∑i

wi(53)

for TE polarization and

1n2

avg=

∑i

win−2i

∑i

wi(54)

for TM polarization, where wi is the thickness of i-th layer with refractive index ni.Figures 4 and 5 show the relative errors of propagation constants of TE and TM fundamen-

tal modes for different grid sizes, respectively. The blue thinnest lines are from conventionalmethod [1], and no index averaging is used in the guiding region. The refractive index at asampled point in the layered region is either nwell or nbarr depending on the point position. Theconvergence is relatively slow and roughly O(Δx) with relatively large fluctuations. As men-tioned, the graded index approximation is only O(h0) accuracy. The results do not convergewell since there are more than one-hundred layers.

The black thinnest lines are from conventional formulation [1], and index averaging in[17, 18] is used in guiding region. The convergence is about O(Δx2) in TE polarization. Itis worse in TM polarization and close to O(Δx). The electric field and its derivative in TEpolariation are continuous across the interface, therefore, the index averaging gives a betterapproximation. However, the derivative of magnetic field in TM polarization is discontinuousacross the interface, the index averaging does not working so well as that in TE polarization.

Intermediate thick and thickest lines are from calculation considering the interface conditionswithout and with generalized Douglas (GD) scheme as in Section 3, respectively. When the gridsize is larger than the layer thickness, there are more than one interface between two sampledpoints. In such situation, the technique in (37) is adopted.

The red lines are calculated from three-point formulation without and with GD scheme asproposed in [12]. Note that, the well width is 0.007 μm and the barrier width is 0.012 μm.Similarly, the three-point formulation in (44) and (50) are usually O(h1) and O(h3) accurate,respectively. As mentioned in [12], if the interfaces are at the center between two sampledpoints or at one sampled point, the three-point formulation in (44) and (50) can be O(h2) andO(h4) accurate, respectively. Besides, the truncation error may be positive or negative depend-ing on the interface position. Since there are many layers, the truncation error from differentlayers may be compensate each other to reach higher accuracy. Therefore, in large grid size,the accuracy fluctuates between O(h) and O(h2) for the formulation without GD scheme, and itfluctuates between O(h3) and O(h4) for that with GD scheme. When the grid is smaller, moregrids are in homogeneous region. The fluctuation becomes smaller with finer grid size and theaccuracy is approximately O(h2) and O(h4) for the formulation without and with GD scheme,respectively.

Again, the green, pink, and cyan lines are calculated from five-, seven-, and nine-point for-mulation, respectively. Similar to three-point formulation, when the grid size is large, the trun-cation error for the formulation without GD scheme is between O(h2N−1) and O(h2N) as theintermediate thick lines, and that with GD scheme is between O(h2N+1) and O(h2N+2) as in thethickest lines. In seven-point formulation with GD scheme and nine-point formulation, whenthe round-off error dominates, the grid size is still larger than layer thickness of wells or barri-ers. The lines still fluctuate and no straight section exists.



10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0.001 0.01 0.1

Rel

ativ

e er

ror

Δx (μm)

GI w/o IAGI w/ IA





Fig. 4. Relative error of TE mode propagation constant with respect to grid size in a MQWwaveguide.

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0.001 0.01 0.1

Rel

ativ

e er

ror

Δx (μm)

GI w/o IAGI w/ IA





Fig. 5. Relative error of TM mode propagation constant with respect to grid size in a MQWwaveguide.

We furthermore investigate a MQW waveguide with large index difference in guiding lay-ers. Refractive indices are nwell = 3.4, nbarr = 1.6, and ncover = nclad = 1.5. Layer number andthickness of wells and barriers are the same. Relative error for TE and TM modes are illus-trated in Figs. 6 and 7, respectively. The truncation error with (2N + 1)-point formulation is



still between O(h2N−1) and O(h2N), and it is between O(h2N+1) and O(h2N+2) with general-ized Douglas scheme. When the grid size is smaller, most points are in homogeneous region, thetruncation error is almost O(h2N) and O(h2N+2) for formulation without and with GD scheme.The transverse variation is much larger due to larger index difference. The propagation constantdiverges for grid size larger than 0.1 μm.

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0.001 0.01 0.1

Rel

ativ

e er

ror

Δx (μm)

GI w/o IAGI w/ IA





Fig. 6. Relative error of TE mode propagation constant with respect to grid size in a MQWwaveguide with larger index difference.

4.3. Some Remarks

Although the derivation in this paper is for mode solving, the formulation can be directly ap-plied to beam propagation method as commonly adopted three-point formulation. No extralimitation is imposed. By replacing β 2 with ∂ 2/∂ z2 and using ψ(x,z) = Ψexp(− jβ z), we have

D2xΨ+ω2μεΨ− β 2Ψ+

∂ 2Ψ∂ z2 = j2β

∂Ψ∂ z

. (55)

In paraxial approximation, ∂ 2Ψ/∂ z2 is dropped and Crank-Nicolson scheme is adopted. The re-sultant matrices are (2N +1) banded instead of tridiagonal in three-point formulation. Note thatwide angle techniques based on Pade approximants and multistep method [20] can be imple-mented. In addition, absorption boundary conditions, reflection and transmission in waveguidediscontinuities, and bidirectional beam propagation can all be implemented as usual three-pointfinite-difference beam propagation method. (Ref. [21] gives details of propagation beam anal-ysis.) Furthermore, it can be applied to semivectorial analysis of three-dimensional guidingstructures as in [14] and [15], for both mode solver and beam propagation. With the help ofhigher-order formulation, the grid number is much less and computation time and memory aregreatly reduced again in those numerical algorithms.



10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0.001 0.01 0.1

Rel

ativ

e er

ror

Δx (μm)

GI w/o IAGI w/ IA





Fig. 7. Relative error of TM mode propagation constant with respect to grid size in a MQWwaveguide with larger index difference.

5. Conclusion

We have proposed a novel technique to systematically derive the continuity of field derivativesat a slab interface up to arbitrary orders. The derived interface conditions can combine directlywith Taylor series expansion to build finite-difference formulation for Helmholtz equation. Weapply the proposed finite-difference formulation to the analysis of simple slab waveguides andmultiple quantum waveguides. In simple slab waveguides, the truncation error with (2N + 1)-point formulation are O(h2N), and it is enhanced to O(h2N+2) with generalized Douglas scheme.In multiple quantum well waveguides, the truncation error with (2N + 1)-point formulationwithout and with generalized Douglas scheme are, at worst, O(h2N−1) and O(h2N+1), respec-tively. Adopting higher-order formulation, we can reach desired accuracy with much fewer gridpoints. Therefore, it can significantly reduce the computation time and memory. Furthermore,the ultimate accuracy is also lower due to smaller round-off error. Such higher-order formula-tion may find its applications in various optical and electromagnetic slab structures, especiallywhen high accuracy is essential in their analysis. The formulation can be directly applied tobeam propagation analysis. Three-dimensional semivectorial analysis is also possible by ex-tending one-dimensional transverse direction to two-dimensional transverse directions.

Acknowledgement

This work was supported in part by the National Science Council of Taiwan under grant NSC98-2221-E-002-170-MY3 and in part by the Excellent Research Projects of National Taiwan Uni-versity under grant 98R0062-07.



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