Yevick et al. Vol. 12, No. 1 /January 1995 /J. Opt. Soc. Am. A 107

Optimal absorbing boundary conditions

David Yevick and Jun Yu

Department of Electrical Engineering, Queen’s University, Kingston, Ontario K7L 3N6, Canada, andAdvanced Technology Laboratory, Bell Northern Research Ltd., P.O. Box 3511, Station C, Ottawa,

Ontario K1Y 4H7, Canada

Yosef Yayon

Department of Physics, Technion University, Haifa 32000, Israel

Received March 15, 1994; revised manuscript received August 29, 1994; accepted September 6, 1994

We examine three simply implemented boundary conditions for finite-difference electric field propagation.We find that although so-called transparent boundary conditions are appropriate for highly collimated beams,properly tailored absorbers are generally better adapted to realistic field distributions. Our analysis in-corporates a simple procedure for constructing nearly optimal absorber functions as well as a novel mixedtransparent /absorbing boundary condition.

1. INTRODUCTIONThe applicability of split-operator propagation methodsto strongly guiding discontinuous refractive-index dis-tributions such as rib waveguides has recently beenestablished.1–3 Of the methods proposed in initial stud-ies, implicit split-operator finite-difference techniques,which permit considerable flexibility in the choice ofboundary conditions and grid-point spacing, have provedespecially convenient.4 However, although radiatedpower can be removed in two-dimensional calculations,5–7

the relevant procedure is somewhat cumbersome and can-not be easily generalized to three dimensions.8 In thispaper we therefore consider three simple yet widely appli-cable boundary conditions. The first of these is obtainedby the addition of a fictitious absorbing refractive-indexcomponent in the immediate vicinity of the computationalwindow boundaries, and the second is the transparentboundary condition in which the field at each boundarypoint is assumed to be dominated by a single plane-wavecomponent with an angle obtained from the ratio of theboundary field amplitude to its neighboring value.9,10 Asa result, an additional longitudinally varying complex re-fractive index at the window edge is obtained. Finally,we propose a hybrid boundary condition that generallyleads to significant improvements.

2. THEORETICAL METHODWe now analyze each of the three procedures in turn.One can obtain a typical family of absorbers by adding

Dsn2d ­ 2in0g

k0

√√√1.0 2

(cos

"psNw 1 1 2 kd

Nw

#)b!!!

(1)

to the square of the refractive index at the Nw gridpoints k ­ 1, . . . , Nw closest to the computational windowedge, where k0, n0, and g are the vacuum wave vector,a reference refractive index, and the absorber strength,

0740-3232/95/010107-04$06.00

respectively.11 Equation (1) has generally been imple-mented with b ­ 1.

In this paper we empirically determine optimal ab-sorber strengths for two representative electric fieldsby numerically minimizing the computed reflection co-efficient. To propagate an arbitrary field distributionEsx, zd, we can select a representative, typically Gauss-ian, excitation, preferably with a spatial and angulardistribution similar to Esx, zd, and calculate the optimalstrength as a function of the propagation angle. The re-sulting function is often easily parameterized. We thenevolve Esx, zd while determining the absorber strength ata longitudinal distance z from the local propagation angle

sin ubsxb, zd ­kx

k0nb­

1ik0nbEsxb, zd

≠Esx, zd≠x

Éx­xb

(2)

at each boundary grid point xb where we have definednsz, xbd ­ nb.

We consider next the transparent boundary condition,which we obtain by assuming that the field approximatesa plane wave near the computational window edge.8–10

Accordingly, for the left window boundary,

Esx0dEsx1d

øEsx1dEsx2d

; R , (3)

so that the second-difference operator at the first gridpoint may be replaced by fs22 1 RdEsx1d 1 Esx2dgysDxd2.However, unless E(0) is set to zero whenever the value ofR indicates that kx . 0 at the left boundary, the fieldwill eventually diverge at large propagation distances.Analogous considerations are of course valid at the rightboundary.

Although the above procedure is perfectly adapted tofully collimated beams, its effectiveness is degraded, aswe demonstrate explicitly below, for fields containing alarge wave-vector spread, even if spatial averaging isemployed in the determination of R. Recently, some-what refined formulations of the transparent boundary

1995 Optical Society of America

108 J. Opt. Soc. Am. A/Vol. 12, No. 1 /January 1995 Yevick et al.

conditions have been proposed,10 but in our experiencethese do not offer significant advantages. To combinethe favorable properties of both the absorbing and thetransparent boundary conditions we therefore propose toimplement both methods simultaneously. That is, weapply both an absorber and the transparent boundary con-dition at the computational window boundaries. The op-timal absorber strength for the mixed boundary conditionis considerably smaller than that of the pure absorbingboundary condition, because a fraction of the field is re-moved at the end points.

Physically, the optimal g is that which minimizesthe sum of two contributions to the field after eachreflection. The first contribution arises from the firstand last grid points and for a cosine-shaped absorberapproximately equal to exps2gLay2 tan ubd times theincident field for an absorber width La. The secondcontribution is caused by diffraction from the rapidlyvarying segment of the absorber and instead grows withincreasing absorber strength. Thus the optimal absorp-tion curve depends somewhat on the absorber profile.Typically, however, the reflection coefficient is insensitiveto changes in the absorber parameters for near-optimal g.

Because the computer time and memory required byeither the transparent or the absorbing boundary condi-tion are typically far less than those demanded by thepropagation steps, little numerical efficiency is sacrificedin implementing our mixed boundary condition. We noteparenthetically that ub fluctuates rapidly as a function ofz if two or more electric field components are incident atxi with appreciably different angles. It is then advanta-geous to average over a number of longitudinal or trans-verse grid points or to select a representative value for ub.

3. RESULTSWe now examine in detail the three boundary condi-tions introduced in Section 2. Our first example is thatof a Gausssian beam with a vacuum wavelength l0 ­0.832 mm and an amplitude given by exps2x2ys2d fors ­ 10 mm propagating in vacuum sn ­ 1d at an angleu ­ ub ­ 21.8± with respect to the longitudinal z axis. Wehave selected a Wa ­ 200 -mm-wide computational win-dow, Nx ­ 1024 grid points, a propagation step lengthDz ­ 0.4 mm, and a propagation distance of Z ­ 250 mm,over which the propagating beam experiences a single re-flection from the boundary. We further set n0 ­ n cos u

in accordance with the considerations of Ref. 12. Our re-sults for the reflection coefficient, defined as the power re-maining in the computational window after the field hasfully reflected once from a window boundary, are thengraphed as a function of the ratio

J ­La

lx­

Lanb sin ub

l0(4)

of the absorber thickness La to the average transversewavelength lx of the electric field. In this manner thecalculated reflection coefficient is effectively indepen-dent of ub if the electric field width and the propa-gation length are simultaneously scaled with lx. Toillustrate, in Fig. 1 we compare the power in the compu-tational window for the 10-mm Gaussian beam (solid

curve) with the corresponding curve for a 10-mmysin u-wide beam propagating at u ­ 11.3± through a dis-tance of Z ­ stan 21.8±ytan 11.3±) 3 250 mm (dashedcurve). To make the scaling explicit, we adjust the hor-izontal axis so that the unity-normalized propagationdistance coincides with the total propagation distance forboth beams. The slight differences in the two sets ofresults can be attributed to the computational windowlength and the grid-point spacings, which have not beenadjusted for the transverse wavelength change.

Next in Fig. 2(a) we display for the single 10-mm beamof Fig. 1 the optimal value of g in inverse micrometers forboth the pure absorbing boundary condition (dotted curve)and the mixed absorberytransparent boundary condition(dotted–dashed curve) as a function of normalized ab-sorber width J. The dashed and the solid curves are thecorresponding results, again with Z ­ 250 mm, ub ­ 21.8±,and n0 ­ cos ub, for a sum of two right-propagating10-mm Gaussian beams, one displaced a distance212.5 mm from the coordinate origin and propagatingat an angle of 26.8± and the second displaced 112.5 mmfrom the coordinate origin and propagating at 16.8±. Asexpected, the transparent boundary condition is prefer-able to the absorbing boundary condition for small La, sothat the mixed and transparent methods become identi-cal in this limit.

The reflection coefficients at the optimal values of g arenext displayed as a function of normalized absorber widthin Fig. 2(b), which can be compared with coefficients of2.82 3 1027 for a single beam and 7.95 3 1024 for thetwo interfering beams computed with the standard trans-parent boundary condition. The optimal coefficients areroughly an order of magnitude lower for the one-beamthan for the two-beam configuration and are similarly anorder of magnitude lower for the mixed boundary condi-tion with respect to the pure absorber. The pure trans-parent boundary condition is superior for absorber widthssomewhat less than a transverse wavelength.

Fig. 1. Power in the computational window as a function ofdistance for a 10-mm-(solid curve) and a 10-mmysin-u (dashedcurve)-wide Gaussian beam propagating at angles of u ­ 21.8±

and u ­ 10.4±, respectively. At unity-normalized distance, bothfield distributions experience a single full reflection from theboundary.

Yevick et al. Vol. 12, No. 1 /January 1995 /J. Opt. Soc. Am. A 109

(a)

(b)Fig. 2. (a) Dependence of optimal absorber strength g ininverse micrometers, on the normalized absorber width J fora single Gaussian beam and the pure absorbing boundarycondition (dotted curve), one beam and the mixed boundarycondition (dotted–dashed curve), a pure absorber and twointerfering 10-mm-wide beams (dashed curve), and two beamsand the hybrid condition (solid curve). (b) Corresponding curvesfor minimum-power reflection coefficients.

We may further demonstrate the high angular sen-sitivity of the pure transparent boundary condition bymodeling the reflection of a 10-mm-wide Gaussian beampropagating at various angles with respect to the opticalaxis but with n0 held fixed at cos 21.8±. In Fig. 3 we dis-play the reflection coefficient as a function of angle forbeams directed at 10±–30± with respect to the optical axisas computed with the transparent boundary condition(dashed curve), an optimized absorber (solid curve), andthe hybrid boundary condition (dotted–dashed curve).The absorber width is set to La ­ 2lx. Since the angleub is not dynamically adjusted to the local direction of theelectric field at the boundary, the reflection coefficients forboth the pure transparent and the mixed boundary condi-tions are considerably larger than in Fig. 2. Although

in Fig. 3 the pure absorber is by far the least angle-sensitive boundary condition and the mixed condition isonly marginally superior to the pure transparent condi-tion, the relative accuracy of the pure transparent condi-tion improves rapidly as the normalized absorber widthis decreased.

To illustrate the practical advantages of the hybridand the absorbing boundary conditions, we examine asimple model of the field distribution in the off state of aninterferometric modulator. In particular, we consider atwo-dimensional step-index waveguide with a core thick-ness of 2.0 mm and cladding and core refractive indicesof 3.166 and 3.18, respectively, excited by an antisym-metric superposition of two Gaussian beams with 1yewidths of 0.2 mm and separated by 2.0 mm. Our com-putational parameters are l0 ­ 1.56 mm, Wa ­ 8 mm,Nx ­ 512, and Dz ­ 0.1 mm. In Fig. 4(a) we display thefield over a 30-mm propagation distance as computed withthe pure transparent boundary condition. Clearly, theinterference between the two rapidly expanding beamsgenerates considerable reflection at the boundaries. Thecorresponding results for the mixed boundary conditionwith two 150-point absorbers, Fig. 4(b), display negligi-ble error. In this calculation we adjusted our absorberstrength dynamically by first calculating ub and sub-sequently lx after each propagation step according toEq. (2). We then approximated the curves of Fig. 2by setting g ­ 0.0 for La , 0.5lx and g ­ c0slx 2 0.5dwith c0 ­ 2.0 otherwise. Increasing c0 somewhat above2.0 to simulate a pure absorber leads to little degra-dation in the electric field. For larger Wa the elec-tric field divergence at the window edges and thus thespurious reflections in Fig. 4(a) are greatly reduced.However the smaller computational window dimensionsafforded by the mixed boundary condition are highly de-sirable in three-dimensional calculations, for which thecomputational cost depends markedly on the number ofgrid points.

Fig. 3. Reflection coefficient for angles of 10±–30± with respectto the optical axis of a 10-mm perturbing Gaussian beam asdetermined with the standard transparent boundary condition(dashed curve), a pure optimized absorber (solid curve), andthe hybrid boundary condition (dotted–dashed curve). All threeboundary conditions are optimized for a 20± plane wave and anabsorber width La ­ 2lx .

110 J. Opt. Soc. Am. A/Vol. 12, No. 1 /January 1995 Yevick et al.

(a)

(b)Fig. 4. Evolution of an unguided asymmetric field in astep-index waveguide as computed with (a) the standardtransparent boundary condition and (b) the hybrid boundarycondition.

4. CONCLUSIONSIn summary, we have determined that when local ab-sorbers are tailored to the angular spectrum of thepropagating electric field, losses in waveguides can becalculated more effectively than with standard trans-parent boundary conditions. Such a procedure requiresthat the absorber width remain somewhat wider than thetransverse wavelengths of the dominant electric field com-ponents. Consequently, for low-angle fields the requirednumber of transverse grid points becomes prohibitivelylarge, and the standard transparent boundary condi-tion is preferable. The transparent condition is, on theother hand, not fully justified in three-dimensional split-

operator calculations and may be unreliable for fieldswith a number of interfering components at the bound-aries. We have therefore introduced a hybrid methodthat interpolates continuously between the transparentboundary condition for thin absorber widths and a stronglocal absorber for larger widths. Finally, we have ex-plicitly demonstrated the additional accuracy that canbe achieved through the hybrid technique in practicalcontexts.

ACKNOWLEDGMENTSSupport for this research was provided by the OntarioCenter for Materials Research, Bell Northern Research,Corning Glass, the National Sciences and Research Coun-cil of Canada, and the Solid State Optoelectronics Corpo-ration of Canada.

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