P. G. Dinesen and J. S. Hesthaven Vol. 17, No. 9 /September 2000 /J. Opt. Soc. Am. A 1565

Fast and accurate modeling of waveguidegrating couplers

P. G. Dinesen

Department of Optics and Fluid Dynamics, Risø National Laboratory, OFD-128, P.O. Box 49,DK-4000 Roskilde, Denmark

J. S. Hesthaven

Division of Applied Mathematics, Brown University, Box F, Providence, Rhode Island 02912

Received March 16, 2000; revised manuscript received May 30, 2000; accepted June 6, 2000

A boundary variation method for the analysis of both infinite periodic and finite aperiodic waveguide gratingcouplers in two dimensions is introduced. Based on a previously introduced boundary variation method forthe analysis of metallic and transmission gratings [J. Opt. Soc. Am. A 10, 2307, 2551 (1993)], a numericalalgorithm suitable for waveguide grating couplers is derived. Examples of the analysis of purely periodicgrating couplers are given that illustrate the convergence of the scheme. An analysis of the use of the pro-posed method for focusing waveguide grating couplers is given, and a comparison with a highly accurate spec-tral collocation method yields excellent agreement and illustrates the attractiveness of the proposed boundaryvariation method in terms of speed and achievable accuracy. © 2000 Optical Society of America[S0740-3232(00)01709-9]

OCIS codes: 050.0050, 050.1960, 000.4430.

1. INTRODUCTIONIn recent years an increasing interest in the design anduse of diffraction-based integrated optics has led to a needfor fast and accurate numerical methods for the analysisand design of diffraction-dominated structures. The for-mulation and development of such methods are severelycomplicated by the rigorous treatment of the vector-fieldbehavior in the resonant regime, where the wavelength ofthe optical field is comparable with the size of the geo-metrical features.

For inherently periodic structures of infinite extent,rigorous coupled-wave analysis has been widely used formore than a decade.1 However, the need to analyzestructures with aperiodic features and of finite extent hasled to the need for methods that are capable of dealingwith such structures, and the use of the finite-element,2

boundary element,3 finite-difference time-domain,4,5 andspectral collocation6 methods has been proposed. Thetwo last-named methods both compute a direct solution ofthe time-domain vectorial Maxwell equations and arequite general in being adaptable to a wide variety of ge-ometries and physical settings. As the need to modelproblems of realistic size and complexity becomes morepressing, however, the memory and computational timerequirements of such direct volume methods quickly be-come limiting factors not only for the design process butalso for analyses of particular structures.

In this paper we propose to take a different road,guided by the work of Bruno and Reitich.7 They have es-tablished that solutions to electromagnetic diffraction bya periodic structure depend analytically on the variationsof the interface. In other words, diffraction from a peri-

0740-3232/2000/091565-08$15.00 ©

odic grating can be determined from knowledge of reflec-tion and refraction at a plane interface. Using this re-sult, Bruno and Reitich proposed a high-orderperturbation scheme for finite-size perturbations and suc-cessfully used it in modeling diffraction by two- and three-dimensional metallic and transmission gratings.8,9

We propose a further development that leads to a for-mulation for the analysis of waveguide grating couplersin which a guided wave in a thin-film waveguide iscoupled to free-space radiation through surface relief, assketched in Fig. 1. Moreover, we extend the analysis tostructures of finite extent and demonstrate, by compari-son with a highly accurate spectral collocation method,that the method of boundary variation provides surpris-ingly accurate results for the modeling of radiation fromwaveguide grating couplers.

The remainder of this paper is organized as follows: InSection 2 we present the modified boundary variation(BV) method as applied to the waveguide grating couplerand discuss the details of the numerical implementation,including postprocessing by free-space integration of theradiated field. In Section 3 we demonstrate the use ofthe proposed code for the analysis of periodic waveguidegrating couplers, and in Section 4 we turn to the analysisof finite-length focusing grating couplers (FGC’s) withaperiodic grating functions. We present a comparison ofthe proposed method with a spectral collocation code todemonstrate the accuracy of the BV method for the mod-eling of nontrivial waveguide grating couplers. We con-clude with a discussion of the superior computational ef-ficiency of the proposed approach and a few remarks ondirections of future research.

2000 Optical Society of America

1566 J. Opt. Soc. Am. A/Vol. 17, No. 9 /September 2000 P. G. Dinesen and J. S. Hesthaven

2. BOUNDARY VARIATION FOR GRATINGCOUPLERSIn the following, we shall outline the elements of the pro-posed algorithm without going into details of the algebrainvolved in the derivation of the scheme but rather dwellon the properties of a field propagating in a thin-filmwaveguide. It is an understanding of these propertiesthat allows us to develop a method of boundary variationsuitable for waveguide grating couplers based on themethod for periodic dielectric interfaces.8

We restrict ourselves to the analysis of TE-polarized,monochromatic fields for which the E field remains paral-lel to the grating, E 5 Eyy, and therefore is determinedby a single scalar quantity u 5 u(z, x) that satisfies thehomogeneous Helmholtz equation

Du 1 k2u 5 0, (1)

where k is the local wave number. We distinguish be-tween the field that is radiated into the free space abovethe grating coupler, u1, and the field radiated downwardinto the waveguide, u2, as illustrated in Fig. 2.

We furthermore assume that the incident field is givenby the fundamental TE mode for the unperturbed thin-film waveguide. For a multilayer waveguide the field ineach individual layer is given in the form

Ei 5 Ai exp~ikix 2 ibz ! 1 Bi exp~2ikix 2 ibz !, (2)

where ki is given by

ki2 1 b2 5 ni

2k02, (3)

where k0 is the free-space wave number and b is thepropagation constant determined by the layer thicknessesand the refractive indices.10 Ai and Bi in Eq. (2) are con-stants, which are given once the propagation constant iscalculated.

The TM case can be solved in a similar way by use ofHx rather than Ey . Although the boundary conditionswill be different, the extension to the TM case is straight-forward.

A. Numerical AlgorithmLet us now consider the case where the boundary betweenthe top cladding layer and free space above the waveguideis perturbed in a way described by the function f and areal number d such that

x 5 fd~z ! 5 df~z ! (4)

Fig. 1. Thin-film optical waveguide comprising a surface-reliefgrating for coupling from guided-wave to free-space radiation.

describes the upper surface of the top cladding layer. As-suming that the incoming field takes the form of Eq. (2),the radiation fields u1 and u2 must satisfy the continuityconditions

u1 2 u2 5 A0 exp@ik0df~z ! 2 ibz#

1 B0 exp@2ik0df~z !2ibz#

2 A1 exp@ik1df~z ! 2 ibz#

2 B1 exp@2ik1df~z ! 2 ibz#, (5)

]u1

]n2

]u2

]n5

]

]n$A0 exp@ik0df~z ! 2 ibz#

1 B0 exp@2ik0df~z ! 2 ibz#%

3]

]n$2A1 exp@ik1df~z ! 2 ibz#

2 B1 exp@2ik1df~z ! 2 ibz#%, (6)

at x 5 df(z). By requiring that the field vanish for x→ `, we have A0 5 0.

The central idea underlying the BV method is that theradiated fields u1 and u2 can be expanded in powers of d :

u6~z, x, d! 5 (n50

`

un6~z, x !d n. (7)

Bruno and Reitich established the validity of this power-series expansion, which is a result of u6 being analytic inits variables.11 u, because it is a solution to the Helm-holtz equation, can be expanded in a Rayleigh series:

u6~z, x, d! 5 (r52`

`

Br6~d!exp~6iar

6x 2 ibr z !, (8)

and likewise for un6 :

un6~z, x ! 5 (

r52`

`

dn,r6 exp~6iar

6x 2 ibr z !. (9)

Given

dn,r6 5

1

n!

dnBr6

dd n ~0 !, (10)

we recover the power-series expansion

Fig. 2. Power in the 21st diffraction order as a function of grat-ing depth for a grating period of 0.7036 corresponding to perpen-dicular diffraction outcoupling.

P. G. Dinesen and J. S. Hesthaven Vol. 17, No. 9 /September 2000 /J. Opt. Soc. Am. A 1567

Br6~d! 5 (

n50

`

dn,r6 d n (11)

for Rayleigh coefficients Br6 . To obtain the coefficients in

this power-series expansion, we need the Fourier expan-sions

f l~z !

l!5 (

r52lF

lF

Cl,r exp~iKrx ! (12)

for f l/l! for l up to some N. In Eq. (12), K represents thesmallest grating vector that we wish to resolve. For a pe-riodic cosine surface relief, K is simply the grating wavenumber, K 5 2p/L, where L is the period of the grating.For that case, the Fourier expansions need extend only toF 5 1. In the case of an aperiodic grating of finitelength, K 5 2p/L, where L is the total length of the com-putational domain.

The wave vectors for the diffracted fields are given by

br 5 b 1 rK, (13)

~ar6!2 1 br

2 5 ~ki!2, (14)

where i 5 0 for ar1 and i 5 1 for ar

2 . Only a finite num-ber of Rayleigh modes Br

6 will be propagating because arwill have nonzero imaginary parts for large r.

The derivation of the recursive expressions for dn,r isthoroughly described in Ref. 8 for diffraction at an inter-face between two dielectrics. Here we need to modify theoriginal derivation by using the boundary conditions for agrating coupler, Eqs. (5) and (6).

To proceed, we perform an n-times differentiation ofthe boundary conditions, Eqs. (5) and (6), with respect to dand evaluate the result at d 5 0. For Eq. (5) we there-fore get

(k50

n f~z !n2k

~n 2 k !! F ]n2k

]xn2k S 1

k!

]ku1

]d k D ~z, 0, 0 !

2]n2k

]xn2k S 1

k!

]ku2

]d k D ~z, 0, 0 !G5

1

n!@2B0~ik0!n 1 A1~2ik1!n

1 B1~ik1!n#f~z !n exp~2ibz !. (15)

With

nd 51

~1 1 d 2f8~z !2!1/2 ~2df8~z !, 1!, (16)

n differentiations of Eq. (6) with respect to d, evaluated atd 5 0, yield

(k50

n f~z !n2k

~n 2 k !! F ]n2k11

]xn2k11 S 1

k!

]ku1

]d k D ~z, 0, 0 !

2]n2k11

]xn2k11 S 1

k!

]ku2

]d k D ~z, 0, 0 !G2 (

k50

n21 f8~z !f~z !n2k21

~n 2 k 2 1 !! F ]n2k

]xn2k21]z S 1

k!

]ku1

]d k D ~z, 0, 0 !

2]n2k

]xn2k21]z S 1

k!

]ku2

]d k D ~z, 0, 0 !G5

1

n!$2B0@ibn~ik0!n21f8~z !f~z !n21 1 ~ik0!n11f~z !n#

1 A1@ibn~2ik1!n21f8~z !f~z !n21 2 ~2ik1!n11f~z !n#

1 B1@ibn~ik1!n21f8~z !f~z !n21 1 ~ik1!n11f~z !n#%

3 exp~2ibz !. (17)From Eq. (7) we recover

uk6~z, x ! 5

1

k!

]ku6

]d k ~z, x, 0 ! (18)

such that Eqs. (15) and (17) give

un1 2 un

2

51

n!@2B0~ik0!n 1 A1~2ik1!n 1 B1~ik1!n#f~z !n

3 exp~2ibz ! 2 (k50

n21 f~z !n2k

~n 2 k !!S ]n2kuk

1

]xn2k 2]n2kuk

2

]xn2k D ,

(19)]un

1

]x2

]un2

]x

51

n!$2B0@ibn~ik0!n21f8~z !f~z !n21 1 ~ik0!n11f~z !n#

1 A1@ibn~2ik1!n21f8~z !f~z !n21 2 ~2ik1!n11f~z !n#

1 B1@ibn~ik1!n21f8~z !f~z !n21

1 ~ik1!n11f~z !n#%exp~2ibz !

1 (k50

n21 f8~z !f~z !n2k21

~n 2 k 2 1 !!S ]n2kuk

1

]xn2k21]z2

]n2kuk2

]xn2k21]zD

2 (k50

n21 f~z !n2k

~n 2 k !!S ]n2k11uk

1

]xn2k11 2]n2k11uk

2

]xn2k11 D . (20)

By substituting the Rayleigh expansions for un6 [Eq. (9)]

into Eq. (19), we recover the coefficients dn,r from a recur-rence in dk,r , k , n, and from the Fourier coefficientsCk,r , in the form

(r52`

`

~dn,r1 2 dn,r

2 !exp~2ibr z !

5 @2B0~ik0!n 1 A1~2ik1!n 1 B1~ik1!n#

3 (r52nF

nF

Cn,r exp~2ibr z !

2 (k50

n21 F (r52~n2k !F

~n2k !F

Cn2k,r exp~iKrz !G3 H (

r52`

`

@~iar1!n2kdk,r

1

2 ~2iar2!n2kdk,r

2 #exp~2ibrz !J .(21)

1568 J. Opt. Soc. Am. A/Vol. 17, No. 9 /September 2000 P. G. Dinesen and J. S. Hesthaven

In a similar fashion, substituting Eq. (9) into Eq. (20)yields

(r52`

`

~iar1dn,r

1 1 iar2dn,r

2 !exp~2ibr z !

5 (r52nF

nF

Cn,r$2B0@ib~ik0!n21~iKr ! 1 ~ik0!n11#

1 A1@ib~2ik1!n21~iKr ! 2 ~2ik1!n11#

1 B1@ib~ik1!n21~iKr ! 1 ~1ik1!n11#%exp~2ibr z !

1 (k50

n21 F (r52~n2k !F

~n2k !F

Cn2k,riKr exp~iKrz !G3 H (

r52`

`

@~iar1!n2k21~2ibr!dk,r

1

2 ~2iar2!n2k21~2ibr!dk,r

2 #exp~2ibr z !J2 (

k50

n21 F (r52~n2k !F

~n2k !F

Cn2k,r exp~iKrx !G3 H (

r52`

`

@~iar1!n2k11dk,r

1

2 ~2iar2!n2k11dk,r

2 #exp~2ibr z !J . (22)

After some manipulations, utilizing the fact that dk,q6

5 0 for uqu . kF, we recover following recursive formu-las for the coefficients dn,r

6 :

dn,r1 2 dn,r

2

5 @2B0~ik0!n 1 A1~2ik1!n 1 B1~ik1!n#Cn,r

2 (k50

n21

(q5max@2kF,r2~n2k !F#

min@kF,r1~n2k !F#

Cn2k,r2q@~iaq1!n2kdk,q

1

2 ~2iaq!n2kdk,q2 #, (23)

iar1dn,r

1 1 iar2dn,r

2

5 $2B0~ik0!n21@bKr 1 ~ik0!2#

1 A1~2ik1!n21@bKr 1 ~2ik1!2#

1 B1~ik1!n21@bKr 1 ~ik1!2#%Cn,r

1 (k50

n21

(q5max@2kF,r2~n2k !F#

min@kF,r1~n2k !F#

Cn2k,r2q$@iK~r 2 q !#

3 ~2ibq!@~iaq1!n2k21dk,q

1 2 ~2iaq2!n2k21dk,q

2 #

2 @~iaq1!n2k11dk,q

1 2 ~2iaq2!n2k11dk,q

2 #%. (24)

Once the power series expansion coefficients dn,r6 are

determined, the Rayleigh expansion coefficients Br6

can be computed from the power series, Eq. (11). The ra-dius of convergence, however, of this Taylor series israther small. To overcome this limitation we recast, assuggested in Ref. 8, the expansion as a Pade approxima-tion, which significantly enhances the radius of conver-gence. We find that, in general, using an @M/M# approx-imant, i.e., using the same order of polynomial in thenumerator as in the denominator, yields the fastest con-vergence for our problem. As for the computation of theFourier spectrum of the surface relief f(z) [Eq. (12)], weuse the fast Fourier transform for enhanced computa-tional speed.

It is important to observe that for the analysis of grat-ing couplers the BV method is approximate, as it does notaccount for the attenuation of the guided wave caused bythe loss of energy by coupling to free space. For gratingcouplers of moderate length this lack of precision is notbelieved to be a serious problem, since the overall diffrac-tion efficiency is typically less than 5%. However, ifhighly diffracting structures were to be analyzed, a feed-back mechanism could possibly be built into the model.

Another approximation lies in the fact that the methoddoes not account for any reflections at lower-lying bound-aries of the downward radiation that is also subsequentlycoupled to free space. As we shall demonstrate, however,the method provides highly accurate solutions to simpleas well as to nontrivial test problems.

B. PostprocessingIn principle, the diffracted field can be recovered any-where above the grating coupler from the Rayleigh seriesexpansion [Eq. (8)] directly. However, because of the pe-riodicity that is inherently assumed in the formulation ofthe scheme, we find it more convenient to evaluate thediffractive field along an aperture covering the gratingcoupler and to use this field to recover the near- and far-field radiation from the structure through the use of thesurface-equivalence theorem.12 To maintain high accu-racy we compute the diffracted field on a set of quadra-ture points on which high-order integration can be per-formed.

3. ANALYSIS OF PERIODIC GRATINGCOUPLERSIn the following, we shall demonstrate the convergence ofthe proposed scheme for purely periodic gratings of infi-nite extent and give further examples of the analysis fa-cilitated by the proposed method. Below, all waveguidedimensions as well as length parameters are normalizedto the free-space wavelength l of the incident field.

As the basic waveguide parameters for the numericalexamples, we use values that are representative of opticalmaterials such as glass and polymers. We consider awaveguide structure consisting of a core layer with refrac-tive index n 5 1.45 and thickness d1 5 0.8 sandwichedbetween two cladding layers of refractive index n 5 1.4.The top cladding layer has a finite thickness d2 5 1, andabove this layer is air with n 5 1. For the fundamentalTE mode this geometry yields an effective index of 1.4213.

P. G. Dinesen and J. S. Hesthaven Vol. 17, No. 9 /September 2000 /J. Opt. Soc. Am. A 1569

We consider a cosine surface relief

fd ~z ! 5 A cosS 2p

Lz D . (25)

Let us first study the convergence of the scheme. Aswe do not have an analytic solution to compare with, weare unable to compute the error. Rather, we look at thepower coupled to the 21st diffraction order as we increasethe number of terms in the Pade approximation to theTaylor series expansion [Eq. (11)]. Table 1 confirms theconvergence in the case of A 5 0.1 as the number ofterms in the power series expansion increases.

To investigate the sensitivity of the convergence fur-ther, we consider the convergence for increasing ampli-tudes of the surface relief. As we increase A, the error ofthe scheme increases in the sense that the number of sig-nificant digits for P decreases. To give an estimate of theerror, we compute the average and the standard deviationfor M ranging from 15 to 49 for increasing A, as shown inTable 2, which illustrates that, as we increase the ampli-tude of the surface relief, the standard deviation in-creases. However, even for an amplitude of 0.4, yieldinga height-to-period ratio of the surface relief of more than1, the solution remains bounded with a reasonable error.We find that, when the amplitude is increased beyond 0.4,the scheme fails to converge.

It is worthwhile making another observation fromTable 2, which clearly shows that there appears to be amaximum for the radiation into the 21st diffraction or-der. To analyze this, we show in Fig. 3 the power in the21st order as a function of grating height, which confirms

Table 1. Power in the 21st Diffraction Order forDifferent Numbers of Terms @M, M# in the PadeApproximation to the Power-Series Expansion

M P(31022)

2 1.2085893 1.2186184 1.2208345 1.2202406 1.2202477 1.2202428 1.2202439 1.220243

10 1.220243

Table 2. Convergence for Periodic GratingCoupler Analysisa

A P(31022) s

0.1 1.220243 8 3 1027

0.2 2.410162 6 3 1026

0.3 2.3120 0.0040.4 2.14 0.11

a P is the average power in the 21st diffraction order and s is the stan-dard deviation over the length of the power series ranging from 15 to 49.

that, as expected, more power is radiated when the heightof the surface-relief grating is increased. However, thepower has a maximum at a grating height of 0.23 abovewhich the grating coupler becomes less efficient. A pos-sible explanation for this is that, when the surface reliefbecomes deep, the propagation of the guided wave in thethin-film waveguide becomes heavily disturbed by thesurface relief, leading to a reduced coupling to the 21stdiffraction order. Certainly, as we shall see for the FGC,the near-field radiation from a deep surface relief is se-verely distorted.

As another example we keep a fixed amplitude A5 0.1 and consider the output power in the diffraction or-ders as a function of the grating period, L. Figure 3shows the power output in three diffraction orders, m5 21, 22, and 23, as a function of the grating period,and a number of things are worth noticing. First, we seethat the 21st diffraction order has a global minimum fora grating period of L 5 1/1.4213, corresponding to a per-pendicular output. This is not surprising from a physicalpoint of view because of the energy conservation. A sec-ond observation to be made is that the appearance of asecond diffraction order leads to an abrupt change in theslope of the curve of the first diffraction order. A similarchange of slope is seen for the 22nd diffraction order atthe cutoff for the 23rd diffraction order.

Fig. 3. Power output in (top) 21st, (center) 22nd, and (bottom)23rd diffraction orders as a function of grating period L of thesurface relief.

1570 J. Opt. Soc. Am. A/Vol. 17, No. 9 /September 2000 P. G. Dinesen and J. S. Hesthaven

4. FOCUSING GRATING COUPLERSLet us now demonstrate the use of the proposed BVmethod to study aperiodic surface-relief gratings couplersof finite length. Clearly, as we use a discrete Fourier se-ries expansion for the surface relief, the grating is implic-itly forced to be periodic. However, as we shall demon-strate, when a sufficiently large total length of thecomputational domain relative to the length of the finitesurface relief is chosen, the results become consistentwith those obtained by a method that deals with truly fi-nite gratings.

For the FGC surface relief we use the generic profile

fd~z ! 5 A expF2S z 2 z0

w D 2G3 cos$2p@a0 1 a1~z 2 z0!#~z 2 z0!%, (26)

where A is the amplitude, w is the width of the exponen-tially truncated relief, z0 is the center of the relief, a05 1/L for the unchirped relief, and a1 is the chirp param-eter.

To establish the necessary length to simulate a FGC offinite extent we investigate the far-field radiation patternwhen varying the total length L of the computation do-main. The number of modes in the Fourier transform isscaled with L to maintain a constant resolution. Figure4 shows the far-field radiation pattern for four values of L

Fig. 4. Far-field radiation pattern for four lengths, L, of thecomputational domain.

Fig. 5. Far-field radiation pattern for various resolutions usedin the Fourier series [Eq. (12)] to represent the surface relief. F,number of Fourier modes (24–48 here).

with the following parameters for the surface relief: A5 0.25, a0 5 1.4213, a1 5 0.005, and w 5 3. From thefigure it is clear that, once the length of the computationdomain exceeds 16, or ;5w, the sidelobes caused by theperiodicity are efficiently suppressed.

Another issue related to the Fourier transform is theresolution required for accurately resolving the structureof the surface relief, i.e., the number of terms in Eq. (12).To address this issue we performed a number of simula-tions with several resolutions for the same geometry asdiscussed above. The results are illustrated in Fig. 5,where we find that once the number of Fourier modes ex-ceeds F 5 26, corresponding to a little more than two

Fig. 6. Far-field radiation patterns for a FGC computed by theBV method (dashed curve) and by the SC method (solid curve).

Fig. 7. Near-field radiation patterns for surface-relief ampli-tudes (a) 0.1, (b) 0.2, and (c) 0.3 by the BV method (dashedcurves) and the SC (solid curves) methods.

Table 3. Key Figures for SC and Proposed BVComputationsa

AM (BV)

BV

f Time

BV SC BV (s) SC (h)

0.1 7 47.1 47.0 46 300.2 13 46.3 46.2 390 640.3 17 45.2 45.1 1081 126

a A is the amplitude of the surface relief. An @M, M# Pade approxi-mant is used for the BV method. f is the focal length normalized withthe free-space wavelength, and Time means the total computation time.

P. G. Dinesen and J. S. Hesthaven Vol. 17, No. 9 /September 2000 /J. Opt. Soc. Am. A 1571

modes per grating period for this case, the sidelobes areefficiently suppressed and the center lobe well resolved.It should be noted that the Fourier spectrum of the sur-face relief is affected by both the width of the Gaussiantruncation and the chirp of the grating period, so a higherresolution may be necessary for other values of these pa-rameters.

Having established resolution and computational-domain requirements for our problem, let us perform a di-rect comparison with a highly accurate spectral colloca-tion (SC) method.6 This method computes a rigoroussolution of the vectorial time-domain Maxwell equations,and our group has previously demonstrated its superioraccuracy.13

For the comparison, we study a longer FGC. The totallength of the computation domain is now 80l, w 5 12l,a0 5 1.4213, a1 5 0.01, and A 5 0.3, yielding a height-to-period ratio close to 1. As we can see from Fig. 6, thereis excellent agreement between the far-field patterns ofthe two methods, even for this relatively deep surface re-lief grating.

To further validate the proposed approach, we computea number of solutions, using both the BV and the SCmethods, and compare them with the near-field solutions.Figure 7 shows a line scan of the intensity in the focalplane for both methods for three values of the surface re-lief amplitude, and Table 3 lists key figures for the simu-lations. The computations were performed on a single-processor Sun Ultra-1 workstation.

We find that the excellent correspondence in the farfield is maintained in the near field even though the in-tensity is slightly lower for the BV method for all ampli-tudes, which may be due to not accounting for multiple re-flections. We also confirm that the use of deep surfacereliefs leads to a deteriorated outcoupling from the wave-guide grating coupler: While it is clear that for A 5 0.1the intensity is symmetric and Gaussian in the focalplane, we find that as the amplitude increases to A5 0.2 and A 5 0.3, sidelobes evolve and an asymmetrybecomes noticeable. We also see a shift in the focal plane(Table 3) toward the surface relief for both methods,which agree well on the focal length.

To illustrate the reduction in computation time that isachievable by the BV method, in Table 3 we also list com-putation times obtained with the two methods. For theBV method the computation time increases with thedepth of the surface-relief grating, as it is necessary to goto higher order in the power series expansion of the Ray-leigh coefficients. The time listed for the SC method isthe computation time necessary for reaching a steady-state solution for this time-domain method, after whichthe steady-state near- and far-field radiation can be com-puted. Again, as the surface relief gets deeper, it takeslonger for the problem to reach steady state, hence the in-creasing computation time. Comparing computationtimes in Table 3, we can see that the use of the approxi-mate BV method certainly pays off: While we find excel-lent agreement with the rigorous SC method, we find thata reduction in the computation time by more than a factorof 2000 is achievable, a fact that calls for use of themethod in the future as the forward solver in an optimi-zation scheme.

5. CONCLUSIONSWe have presented the development of a boundary varia-tion (BV) method for the analysis of both periodic andaperiodic waveguide grating couplers and have given ex-amples of the analysis of continuous surface-relief grat-ings. For a periodic grating we have found that thepower radiated into the fundamental 21st diffraction or-der does not increase monotonically with the gratingheight. Rather, an optimum exists, and we attribute thisto the distortion of the guided-wave propagation that re-sults from very deep surface reliefs.

For a focusing grating coupler we have found excellentagreement with the highly accurate spectral collocationmethod. A reduction in computation time of up to morethan 2000 times compared with a rigorous approach isachieved.

These highly encouraging results suggest that furtherdevelopment along the lines discussed here are worth-while, and we are currently considering the formulationof the BV-based methods for the three-dimensionalforward-scattering problem.

ACKNOWLEDGMENTSWe gratefully acknowledge fruitful discussions on the BVmethod with Oscar Bruno and Fernando Reitich. J. S.Hesthaven acknowledges partial support for this studyfrom Defense Advanced Research Projects Agency/U.S.Air Force Office of Scientific Research grant F49620-1-0426.

P. G. Dinesen’s e-mail address is palle.dinesen@risoe.dk.

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