Ortega-Monux et al. Vol. 20, No. 3 /March 2003/J. Opt. Soc. Am. A 557

Adaptive Hermite–Gauss decompositionmethod to analyze

optical dielectric waveguides

Alejandro Ortega-Monux, J. Gonzalo Wanguemert-Perez, and Inigo Molina-Fernandez

Departamento de Ingenierıa de Comunicaciones, E.T.S.I. Telecomunicacion, Universidad de Malaga,Campus Universitario de Teatinos, 29071 Malaga, Spain

Received August 2, 2002; revised manuscript received October 7, 2002; accepted October 9, 2002

A novel spectral method with variable transformation, the adaptive Hermite–Gauss decomposition method (A-HGDM), has been developed and applied to the analysis of three-dimensional (3D) dielectric structures. Theproposed method includes an optimization strategy to automatically find the quasi-optimum numerical param-eters of the variable transformation with low computational effort. The technique has been tested by analyz-ing two typical 3D dielectric structures: the rectangular step-index waveguide and the rib-waveguide direc-tional coupler. In both cases, the A-HGDM increases the accuracy of the Hermite–Gauss decompositionmethod (HGDM), especially when the mode is near cutoff, and improves the computational efficiency of pre-viously published optimization strategies (optimized HGDM). © 2003 Optical Society of America

OCIS codes: 000.4430, 130.2790, 230.7370.

1. INTRODUCTIONIn recent years, many different numerical methods havebeen developed to analyze dielectric waveguides. A pre-liminary classification of these techniques groups theminto two families: local methods, such as finite differenceand finite elements, and global methods, based on the ex-pansion of unknown electric or magnetic fields into a se-ries of orthogonal basis functions (i.e., Fourier,1–4

Chebyshev,4 or Hermite–Gauss).5–7 One of the advan-tages of using Hermite–Gauss basis functions is that theyform a complete orthogonal set that satisfies the bound-ary condition at infinity and therefore do not require en-closing the waveguide in a computational window, unlikewhat occurs, for example, with Fourier basis functions.In addition, Hermite–Gauss functions are the exact solu-tion of a waveguide with a parabolic index profile, and, formany practical structures, they lead to accurate resultswith a reduced number of terms. In the Hermite–Gaussdecomposition method (HGDM), proposed in Refs. 5 and6, the basis functions are defined by introducing two scal-ing factors in order to adapt them to the geometry understudy. Such scaling factors are fixed to the square root ofthe normalized frequencies, but this choice has been as-sessed only for simple structures and yields accurate re-sults only when the electric field distribution is well con-fined (modes far from cutoff). To improve results whenthe mode is near cutoff, a technique based on a varia-tional approach has been proposed in Ref. 7. Thismethod, named the optimized HGDM (O-HGDM), auto-matically finds the optimum values of the scaling factorsby maximizing the propagation constant. The main limi-tation of this technique is its computational efficiency, asseveral eigenvalue problems must be solved at each stepof the optimization procedure.

1084-7529/2003/030557-12$15.00 ©

In this paper, we propose an alternative procedure tofind a quasi-optimum combination of mapping param-eters. The basic idea behind this new strategy lies in re-alizing that solving a fixed eigenvalue equation withparameter-dependent Hermite–Gauss basis functions(with unknown scaling parameters) can alternatively beseen as solving a new linearly transformed coordinate ei-genvalue equation (with unknown values of the transfor-mation parameters) with fixed Hermite–Gauss basisfunctions. In this way, the adaptive optimization tech-nique recently developed in Refs. 3 and 4 for the modifiedFourier decomposition method (named the A-MFDM) canbe easily extended to the HGDM. This novel strategy,which will hereafter be called the adaptive HGDM (A-HGDM), finds the quasi-optimum transformation param-eters in a more efficient way than that proposed by theO-HGDM and also improves the accuracy obtained by theHGDM, especially in cases near cutoff.

This paper is structured as follows. Section 2 startswith the development of a general formulation for globalmethods with variable transformation. This formula-tion, which can be applied to any basis function and vari-able transformation, applies the concept of matrixoperators3,4 to convert the scalar wave equation into amatrix eigenvalue equation. The A-HGDM is deducedfrom this general framework by particularizing the gen-eral formulation for Hermite–Gauss basis functions un-der linear variable transformation. Section 3 focuses onthe adaptive optimization technique applied to automati-cally find the quasi-optimum values of the mapping pa-rameters: First, the optimization criterion is presented,and then an algorithm is proposed to efficiently search forthe solution. Finally, in Section 4, the performance of themethod is validated by analyzing two typical three-

2003 Optical Society of America

558 J. Opt. Soc. Am. A/Vol. 20, No. 3 /March 2003 Ortega-Monux et al.

dimensional (3D) dielectric structures: the rectangularstep-index waveguide and the rib-waveguide directionalcoupler. The results obtained confirm the good perfor-mance of the A-HGDM under a broad variety of situa-tions.

2. GENERALIZED FORMULATION OFSPECTRAL METHODS WITH VARIABLETRANSFORMATIONThe scalar wave equation that governs the propagation ofstationary solutions through a z-invariant 3D dielectricwaveguide is given by

]2f~x8, y8!

]x82 1]2f~x8, y8!

]y82 1 k02n82~x8, y8!f~x8, y8!

5 b2f~x8, y8!, (1)

where k0 is the wave number of free space, n82(x8, y8) isthe refractive index, and b and f(x8, y8) are, respectively,the propagation constant and the electric field spatial dis-tribution of the different modes supported by the wave-guide.

Equation (1) can be normalized by introducing two ar-bitrary normalization parameters dx and dy , which re-late the original transversal coordinates (x8, y8) and thenormalized ones (x, y) through x 5 x8/dx and y 5 y8/dy . Doing so3 yields

1

Vx2

]2f~x, y !

]x2 11

Vy2

]2f~x, y !

]y2 1 n2~x, y !f~x, y !

5 bf~x, y !, (2)

where Vx and Vy are the normalized frequencies, a is theasymmetrical coefficient, b is the normalized propagationconstant, and n2(x, y) is the normalized refractive index.The normalization magnitudes dx and dy can be arbi-trarily selected, although they are usually chosen to beequal to the half-width of the waveguide core in each di-rection.

A. Variable Transformation and DiscretizationWhat distinguishes spectral methods with variable trans-formation from classical spectral methods is the applica-tion of a previous mapping over the spatial coordinates ofthe modal wave equation:

u 5 f~x !, v 5 g~ y !. (3)

As a result, the normalized wave equation (2) is convertedinto a new one, which is defined over the transformed do-main uv and given by

1

Vx2 F f1~u !

]2f

]u2 1 f2~u !]f

]u G 11

Vy2 Fg1~u !

]2f

]v2 1 g2~v !]f

]v G1 n2~u, v !f 5 bf, (4)

where the functions f1(u), f2(u), g1(v), and g2(v) are theresult of applying the chain rule over the second deriva-tives with respect to x and y coordinates of the original do-main. Their expressions, which clearly depend on theused transformation variables (3), are given by

f1~u ! 5 S ]u

]x D 2

, f2~u ! 5]2u

]x2 ,

g1~v ! 5 S ]v

]y D 2

, g2~v ! 5]2v

]y2 . (5)

The next step is to solve the resulting transformedwave equation (4). This is done by applying the standarddiscretization process of classical spectral methods. Thatis, the unknown electric field is first expanded into a finiteseries of orthogonal basis functions Fk(u, v), given by

f~u, v ! 5 (k50

N

fkFk~u, v !, (6)

and second, Galerkin’s method is applied to obtain a ma-trix eigenvalue equation

@M#$F% 5 b$F%, (7)

whose eigenvectors $F% and eigenvalues b are, respec-tively, the coefficients fk of the electric field series expan-sion and the normalized propagation constants of the dif-ferent modes of the structure. Note that only theeigenvalues in the range 0 , b , 1 correspond to guidedmodes.

The eigenvalue system matrix [M] can be easily ob-tained from Eq. (4) by making use of the matrix operatorapproach.3,4 In doing so, we arrive at the following ex-pression:

@M# 5 F 1

Vx2 $@P(f1~u !)#@DDu# 1 @P(f2~u !)#@Du#%

11

Vy2 $@P(g1~v !)#@DDv# 1 @P(g2~v !)#@Dv#%

1 @P(n2~u, v !)#G, (8)

where @DDu#, @Du#, @DDv#, and @Dv# are the matrix op-erators used to perform the following functions over theunknown electric field, respectively: the second deriva-tive with respect to u, the first derivative with respect tou, the second derivative with respect to v, and the firstderivative with respect to v; and @P( f1(u))#, @P( f2(u))#,@P( g1(v))#, @P( g2(v))#, and @P(n2(u, v))# are the matrixoperators used to perform the product of the unknownelectric field by the function included in parentheses.Once the set of orthogonal basis functions is defined,closed expressions for the aforementioned operators canbe calculated from the following generic expressions:

DDuui,k 5 EV0

Fi* ~u, v !]2Fk~u, v !

]u2 dudv,

DDvui,k 5 EV0

Fi* ~u, v !]2Fk~u, v !

]v2 dudv, (9)

Duui,k 5 EV0

Fi* ~u, v !]Fk~u, v !

]ududv,

Ortega-Monux et al. Vol. 20, No. 3 /March 2003/J. Opt. Soc. Am. A 559

Dvui,k 5 EV0

Fi* ~u, v !]Fk~u, v !

]vdudv, (10)

P~h~u, v !!ui,k 5 EV0

h~u,v !Fi* ~u,v !Fk~u,v !dudv,

(11)

where V0 is the domain of definition of the basis functionsand i and k are the pair of indices that reference the ele-ments of the matrices.

Once the eigenvectors $F% of Eq. (7) have been obtained,the field profile in the structure, f(x, y), can be recoveredthrough a two-step procedure: First, the field profile inthe transformed domain, f(u, v), is obtained from $F% bymeans of Eq. (6), and second, the field profile in the origi-nal domain, f(x, y), is obtained from f(u, v) by applyingthe inverse variable transformation

x 5 f 21~u !, y 5 g21~v !. (12)

The usefulness of this generalized formulation lies inthat, depending on the applied variable transformation(3) and the basis functions used in series expansion (6),different spectral methods with variable transformationcan be defined. For example, the expressions used forthe A-MFDM3,4 and for the A-HGDM are summarized inTable 1.

Note that, because of the separable nature of the basisfunctions, the index k of series expansion (6) can also bereferenced as a combination of two indices m and n run-ning in the u and v directions, respectively. In this way,the vector of the spectral coefficients, $F% 5 $fk%, canalso be written in the matrix form @F# 5 @ fm,n#. Thefirst notation, typically referred to as the unique index, issuitable for formulating the matrix eigenvalue problem,while the second is useful in formulating the direct andinverse transformations between the field distributionsf(u, v) and their coefficients [F] in compact matrix nota-

tion.

B. Adaptive Hermite–Gauss Decomposition MethodAs we stated in Section 1, Hermite–Gauss basis functionshave specific characteristics that make them useful forthe modal analysis of dielectric waveguides. However,even though they have been widely used in theliterature5–7 for this purpose, the approach followed hereis completely different from others that were publishedpreviously. In these works, instead of using a mappingover the original axis to adapt the waveguide under studyto Hermite–Gauss basis functions (as the variable-transformation approach suggests), scaling factors are in-troduced into the basis functions to adapt them to the ge-ometry of the waveguide under study. This is done bydefining the following parameter-dependent Hermite–Gauss basis functions:

Bm~x, sx! 5Hm~x/sx!exp@2~x/sx!2/2#

~p1/22mm! !1/2 ,

Bn~ y, sy! 5Hn~ y/sy!exp@2~ y/sy!2/2#

~p1/22nn! !1/2 .

(13)

When the scaling factors sx and sy are chosen in advanceto be equal to the inverse of the square root of the normal-ized frequencies ( sx 5 Vx

21/2 , sy 5 Vy21/2), the method is

commonly called the Hermite–Gauss decompositionmethod (HGDM),5,6 while if such scaling factors are opti-mized through a variational approach, it is called the op-timized HGDM (O-HGDM).7 This method attains themost accurate solution for a given number of terms by de-termining the scaling factors that maximize the normal-ized propagation constant. The disadvantages of thevariational optimization strategy employed by theO-HGDM and its comparison with the optimization crite-rion proposed in this work for the A-HGDM will be dis-cussed in Section 3.

Besides the fact that the A-HGDM is based on avariable-transformation approach, other differences exist

Table 1. Summary of Spectral Methods with Variable Transformation

MethodVariable

TransformationBasis Functions

Fk(u, v) 5 Bm(u)Bn(v)

A-MFDMu 5

2

ptan21Sx 2 ox

axD Bm~u! 5 expFjmS2p

u0DuG

v 52

ptan21Sy 2 oy

ayD Bn~v! 5 expFjnS2p

v0DvG

m P (2Nx/2,..., Nx/2) and n P (2Ny/2,..., Ny/2)k 5 (n 1 Ny/2)(Nx 1 1) 1 (m 1 Nx/2)

A-HGDM u 5 ax(x 2 ox)Bm~u! 5

Hm~u!exp~2u2/2!

~p1/22mm! !1/2

v 5 ay( y 2 oy)Bn~v! 5

Hn~v!exp~2v2/2!

~p1/22nn! !1/2

Hm(u) and Hn(v) are Hermite polynomials of orders mand n, respectively

m P (0,..., Nx 2 1) and n P (0,..., Ny 2 1)k 5 nNx 1 m

560 J. Opt. Soc. Am. A/Vol. 20, No. 3 /March 2003 Ortega-Monux et al.

with previously reported Hermite–Gauss-based methods.The first involves the proposed variable transformation

u 5 ax~x 2 ox!, v 5 ay~ y 2 oy!. (14)

This transformation not only includes two scaling factorsto control the mapping strength in each direction (ax anday) but also contains two offset parameters to allow for avariable mapping center in asymmetrical situations (oxand oy). The second difference with the A-HGDM is thatit includes a novel optimization criterion to find the trans-formation parameters that yield nearly the best results ineach particular situation. Note that the generalized for-mulation in which the A-HGDM has been introduced canalso be used for the HGDM and the O-HGDM, simply byusing the variable transformations

u 5 AVx x, v 5 AVy y (15)

for the HGDM and

u 51

sxx, v 5

1

syy (16)

for the O-HGDM.Once the variable transformation (14) is defined, the ei-

genvalue matrix [M] of the A-HGDM can be calculatedfrom Eq. (8), becoming

@M# 5 ax2

1

Vx2 @DDu# 1 ay

21

Vy2 @DDv# 1 @P(n2~u, v !)#.

(17)

Note that the obtained equation (17) is much simplerthan the general equation (8) (valid for any variabletransformation) because of the linear nature of the pro-posed variable transformation (14). In fact, it can be eas-ily seen that, in contrast to the A-MFDM, the same com-putational effort is needed to formulate the A-HGDMeigenvalue equation (17) as that for the eigenvalue equa-tion of the HGDM. Closed expressions to calculate theoperators @DDu#, @DDv#, and @P(n2(u, v))# when usingHermite–Gauss basis functions are provided in AppendixB.

Once the eigenvalue problem (7) is solved for a certainset of transformation parameters (ax , ay , ox , oy), theelectric field profiles in the original domain xy must be ob-tained from the Hermite–Gauss spectral coefficients $F%in the transformed uv domain. This can be carried outaccording to the following two-step procedure: (1) Thefield profile in a certain grid of points @UV# of the trans-formed domain uv is obtained from the previously calcu-lated spectral coefficients $F% through the inverseHermite–Gauss transform [as detailed in expressions(A3)–(A9) of Appendix A]. (2) The field samples in thetransformed domain are translated to the original domainby applying the inverse variable transformation (12).

To conclude Section 2, it must be highlighted that theaccuracy of the obtained results is highly dependent onthe mapping parameters (ax , ay , ox , oy) that are cho-sen, and, as they must all be specified a priori, it is nec-essary to have an automatic procedure to determine theoptimum transformation parameters. This procedure,called the adaptive optimization technique, has recently

been developed in Refs. 3 and 4 for the A-MFDM and willbe extended to the A-HGDM in Section 3.

3. ADAPTIVE OPTIMIZATION TECHNIQUEThe spectral methods with variable transformation ex-plained above work properly only if the transformationparameters are adequately selected. Unfortunately, thefour-parameter dependence p 5 (ax , ay , ox , oy) of theproposed technique makes it impossible to choose them ina correct way by physical arguments, and therefore it isnecessary to have a closed procedure to automatically de-termine the optimum mapping parameters for the prob-lem under study starting from an arbitrary set of trans-formation parameters. The development of such aprocedure is a key issue of this work and will be describedin detail below.

A. Optimization CriterionThe electric field spatial distribution in the original do-main, f(x, y), will produce different profiles f(u, v) inthe transformed domain, depending on the chosen trans-formation parameters. Therefore their respectiveHermite–Gauss spectral coefficients @F# 5 @ fm,n# willalso be parameter dependent. With this taken into ac-count, the basic idea behind the optimization criterioncan be summarized as follows: The best transformationparameters are those that minimize the spectral width ofthe electric field in the transformed domain, because theyattain a good representation of the field profile with a lownumber of coefficients. To clarify the basis of the pro-posed criterion, we show in Fig. 1 a graphical representa-tion for a simpler two-dimensional situation. In this fig-ure, an electric field profile f(x) in the original domain ismapped onto the u space by using two different sets oftransformation parameters: pi 5 (ax

i , oxi ) and p j

5 (a xj , o x

j ). As one can clearly observe, the resulting

Fig. 1. Basic idea behind the optimization criterion: (a)electric field distribution in the original domain; (b), (c) spectralcoefficients in the transformed domain obtained with twodifferent sets of transformation parameters pi 5 (ax

i , oxi ) and

p j 5 (a xj , o x

j ), respectively.

Ortega-Monux et al. Vol. 20, No. 3 /March 2003/J. Opt. Soc. Am. A 561

Fig. 2. Optimization algorithm flow chart.

profiles over the transformed domain, f i(u) and f j(u),will require different numbers of coefficients to be cor-rectly approximated by a Hermite–Gauss-series expan-sion approach. For example, the f i(u) profile wouldroughly need 20 terms, while only ten terms would ap-proximately be needed for the f j(u) profile. Thus it isobvious that the set of transformation parameters p j

5 (a xj , o x

j ) is better than pi 5 (axi , ox

i ), because it al-lows us to obtain a good representation of the originalelectric field with a lower number of spectral coefficients.

Just as in the A-MFDM,3,4 the variance function hasbeen used as a measure of spectral width. However, in-stead of minimizing the variance of the spectral coeffi-cients of the function ]2]2f(u, v)/]u2]v2, it was estab-lished that the best results are obtained directly with theparameters that minimize the spectral width of the trans-formed electric field f(u, v). Therefore the function tobe minimized is given by

Var~p! 5

(m

(n

m2n2ufm,n~p!u2

(m

(n

ufm,n~p!u2

, (18)

where p 5 (ax , ay , ox , oy)is the four-dimensional vectorof transformation parameters. In this expression,ufm,n(p)u2 gives the energy of the field profile that is con-tained in the m, n coefficient; thus the functionufm,n(p)u2/((ufm,n(p)u2 can be regarded as the normal-ized energy spectrum of the field profile, and therefore Eq.(18) is a measure of the width of the normalized energyspectrum.

It must be remarked that, unlike the variational opti-mization procedure used in Ref. 7, the proposed optimiza-

tion criterion does not guarantee that the obtainedvariable-transformation parameters give the best possibleresult. However, as Subsection 3.B shows, it gives an ac-curate enough solution with reduced computational ef-fort.

B. Optimization AlgorithmAn optimization algorithm must now be designed that,starting from an arbitrary set of transformation param-eters and making use of the proposed optimization crite-rion, is able to automatically find a set of quasi-optimumtransformation parameters. The main steps of the pro-posed algorithm, which closely resemble those of theA-MFDM3,4 although adapted to the specific characteris-tics of Hermite–Gauss basis functions, can be seen in Fig.2. They can be summarized as follows:

1. A set of initial transformation parameters p0 isheuristically chosen.

2. An approximate solution of the field profile spectralcoefficients @F0# 5 @F(p0)# is obtained by solving the ei-genvalue equation resulting from the transformation pa-rameters.

3. From this approximate solution, a set of quasi-optimum transformation parameters pOPT can be esti-mated by using any standard minimization procedure(the ‘‘simplex search method’’ as implemented in MAT-LAB’s function fmins is used in this work). This step iscomputationally efficient, since it is not necessary to for-mulate and solve the eigenvalue equation again. This isdue to the fact that the function to be minimized [i.e., thevariance of spectral coefficients, Var(@F i#)] can be easilyestimated for any arbitrary value of the transformation

562 J. Opt. Soc. Am. A/Vol. 20, No. 3 /March 2003 Ortega-Monux et al.

parameters pi from the previously calculated solution@F0#. Further details will be given later in this subsec-tion.

4. The obtained quasi-optimum parameters pOPT arenow used to repeat the process from the second step untilno change is observed in two consecutive iterations.

The principal advantage of this scheme is that only oneeigenvalue equation is formulated and solved per itera-tion. Tests involving a wide variety of situations and ini-tial guesses have demonstrated the method’s computa-tional efficiency: The proposed optimization algorithmtakes an average of just 5–7 iterations to reach conver-gence, while the O-HGDM algorithm proposed in Ref. 7,which is based on the maximization of the propagationconstant, requires solving a much greater number of ei-genvalue equations (50 solutions are cited by the au-thors).

Now the question of calculating the variance of thespectral coefficients, Var(@F i#), for any arbitrary value ofthe transformation parameters pi starting from the ei-genvalue solution @F0# (obtained with a different set oftransformation parameters p0) must be addressed. Thebasic idea behind the process is the following: From aknown set of series coefficients @F0#, the field profilef i(u, v) can be evaluated at any suitable set of points@UVi# through the inverse Hermite–Gauss transforma-tion [Eq. (A8) of Appendix A]. From this sampled fieldprofile, the new series coefficients @F i#, corresponding tothe transformation parameters pi, can be easily obtainedby means of the direct Hermite–Gauss transformation[Eq. (A11) of Appendix A]. Once these spectral coeffi-cients are known, their variance is easily calculated withEq. (18). The detailed procedure of the algorithm closelyresembles that applied in Ref. 3 for the Fourier-series ex-pansion; thus it will not be repeated here. The only dif-ferences arise in that, in this case, the set of samplepoints in the transformed domain @UVi#, which will beused to calculate the spectral coefficients @F i#, must beselected as the quadrature points of the Hermite–Gaussbasis functions,8 instead of using the equidistant points ofthe Fourier approach.

One should note that the computational effort neededto calculate the spectral coefficients @F i# through Eq.(A11), which requires two NxNy matrix products, is neg-ligible in comparison with the effort needed to formulateand solve the eigenvalue problem (7), where the systemmatrix is (NxNy) 3 (NxNy). This circumstance justifiesthe approach, which will be used in Section 4, of approxi-mating the total computational effort of the A-HGDM bythe time employed in formulating and solving the eigen-value equation (7) the required number of times (which,for the A-HGDM, equals the number of iterations of theoptimization algorithm).

4. RESULTSTo assess the proposed technique, we have analyzed twodifferent 3D dielectric structures: the rectangular step-index waveguide and the rib-waveguide directional cou-pler. The rectangular step-index waveguide is a simplestructure that has been used in previously published

works5–7 to verify the accuracy of the HGDM and theO-HGDM. Because of the symmetry of this structure,the offset parameters are equal to zero (ox 5 oy 5 0),and only the scaling factors (ax and ay) must be opti-mized. On the other hand, the rib-waveguide directionalcoupler is a more complex structure that is present in agreat variety of applications and is typically used as abenchmark3 to test the performance of numerical meth-ods.

A. Rectangular Step-Index WaveguideThe geometry of the rectangular step-index dielectricwaveguide is shown in Fig. 3, while Fig. 4 shows the nor-

Fig. 3. Geometry of the rectangular step-index waveguide: (a)refractive index in the original domain x8y8, (b) normalized re-fractive index in the normalized original domain xy.

Fig. 4. Rectangular step-index waveguide: normalized propa-gation constant versus normalized frequency for the fundamen-tal mode (Nx 5 Ny 5 7).

Ortega-Monux et al. Vol. 20, No. 3 /March 2003/J. Opt. Soc. Am. A 563

malized propagation constant of the fundamental modeversus the operating frequency for a certain number of co-efficients. The three curves plotted in this figure corre-spond to the results obtained when using the HGDM(with fixed mapping parameters), the O-HGDM (withtransformation parameters obtained after applying an op-timization algorithm based on the maximization of thepropagation constant), and the A-HGDM (with transfor-mation parameters obtained after applying an optimiza-tion algorithm based on the minimization of the spectralcoefficients’ variance). Although the results obtainedwhen the mode is far from cutoff (high frequencies) arepractically the same for the three methods, the HGDMdoes not work properly at low frequencies (no propagatingmode found below Vx 5 Vy 5 1) because of the incorrectselection of the transformation parameters. In contrast,the curve of the A-HGDM closely resembles the best re-sults that can be achieved with the number of coefficientsused (the curve of the O-HGDM), even at very low fre-quencies, thanks to the optimization strategy that is ap-plied for each frequency.

Figure 5 shows the convergence rates as a function ofthe number of coefficients used in the electric field seriesexpansion for a very low normalized frequency (Vx 5 Vy5 1). One can clearly observe the slow convergence rateof the HGDM compared with that of methods based on anoptimization strategy, such as the A-HGDM and theO-HGDM. For example, the HGDM needs to use at least25 coefficients in each direction to obtain similar resultsto those obtained with the A-HGDM and O-HGDM by us-ing only nine terms.

To check the usefulness of the optimization criterionproposed in the A-HGDM, we must verify two things:first, that the A-HGDM optimization algorithm is actuallyable to find a set of quasi-optimum transformation pa-rameters that approximate the optimum set of param-eters obtained with the O-HGDM, and second, that thiscan be done with less computational effort. Figure 6shows, for the fundamental mode of the rectangular step-index waveguide and Nx 5 Ny 5 7, the contour map ofthe quotient (in percentage) between the normalizedpropagation constant b obtained for different values of the

Fig. 5. Rectangular step-index waveguide (Vx 5 Vy 5 1): con-vergence rates of the normalized propagation constant versus thenumber of coefficients for the fundamental mode.

mapping parameters (ax and ay) and the maximum nor-malized propagation constant bmax (the best solution) thatcan be obtained with this number of expansion terms.The dashed contour line corresponds to the values of axand ay that yield a value of b equal to the one obtainedwith the HGDM. Therefore any combination of scalingfactors inside this dashed line gives a more accurate solu-tion than the one calculated with the HGDM. The pathfollowed by the optimization algorithm of the A-HGDMhas been superimposed onto the contour map, startingfrom an arbitrary set of transformation parameters (inthis example, those of the HGDM, i.e., ax 5 Vx

1/2 5 1 andax 5 Vy

1/2). Two aspects must be highlighted here:First, the point reached by the A-HGDM optimization al-gorithm is located at the maximum beta area (95%), andsecond, only five eigenvalue problems have been solved toreach this point. This is, in essence, the principal advan-tage of the optimization algorithm proposed in this workover the O-HGDM, which usually involves solving ap-proximately 50 eigenvalue problems to reach optimumscaling factors.7

Fig. 6. Rectangular step-index waveguide (Vx 5 Vy 5 1, Nx5 Ny 5 7): contour map of the quotient b/bmax (%). The pathfollowed by the optimization algorithm is superimposed onto themap.

Fig. 7. Geometry of a rib-waveguide asymmetric directionalcoupler (l 5 1.55 mm).

564 J. Opt. Soc. Am. A/Vol. 20, No. 3 /March 2003 Ortega-Monux et al.

Fig. 8. Symmetric rib-waveguide directional coupler (2W2 5 3 mm): (a) electric field contour maps, (b) A-HGDM with Nx 5 Ny5 10; (c), (d) HGDM with Nx 5 Ny 5 10; (e), (f ) HGDM with Nx 5 Ny 5 25.

B. Rib-Waveguide Directional CouplerThis device, composed of two closely spaced ribwaveguides, has a complex refractive-index profile thatmakes it difficult to define the normalization magnitudesdx and dy , which in turn determine the normalized fre-quencies Vx and Vy . This can have serious consequencesin a method such as the HGDM, where the scaling factorsused depend exclusively on such normalized frequencies.

Thus, for this device, it becomes even more necessary tohave an optimization strategy to automatically find quasi-optimum mapping parameters (not only the scaling fac-tors but also the offset parameters) that lead to an accu-rate solution.

The geometry of the rib-waveguide directional couplerthat is analyzed in this section is shown in Fig. 7. Thenormalization magnitudes dx and dy have been chosen by

Ortega-Monux et al. Vol. 20, No. 3 /March 2003/J. Opt. Soc. Am. A 565

using physical arguments, taking into account the areawhere the electric field profile is expected to be concen-trated.

In Fig. 8, the electric field profiles of the symmetriclikeand asymmetriclike supermodes for the symmetric rib-waveguide directional coupler (2W2 5 3 mm) have beenplotted for different values of mapping parameters andnumbers of terms. The fields of Figs. 8(a) and 8(b) wereobtained by applying the A-HGDM optimization strategywith only Nx 5 Ny 5 10 coefficients, whereas the rest ofthe field profiles in this figure have been calculated by us-ing the mapping parameters suggested by the HGDM(ax 5 Vx

1/2 , ay 5 Vy1/2 ; ox 5 0, oy 5 1), with Nx 5 Ny

5 10 coefficients in Figs. 8(c) and 8(d) and Nx 5 Ny5 25 coefficients in Figs. 8(e) and 8( f). As expected, theresults obtained with the proposed A-HGDM are accurateeven with a very low number of terms. For example, theelectric field profiles calculated with only ten coefficients[Figs. 8(a) and 8(b)] are nearly identical to those obtainedin Ref. 3 and hardly change when increasing the numberof terms, which confirms that convergence has beenreached. It must be highlighted that the A-HGDM re-sults are obviously independent of the chosen normaliza-tion magnitudes (dx and dy) as a result of the automaticadaptive feature of the method. In contrast, when we areworking with the HGDM, which uses mapping param-eters based on physical considerations, the field profilescalculated by using only ten coefficients in each directionare not accurate enough [Figs. 8(c) and 8(d)]; the numberof terms must be increased in this case to 25 [Figs. 8(e)and 8(f )] to get results that are similar to those obtainedby the A-HGDM.

The convergence of the effective mode index (Neff5 b/k0) versus the number of coefficients has been plottedin Fig. 9 for the first two modes of the structure and fordifferent widths of the second rib (2W2). In all cases, theA-HGDM presents a very good and fast convergence rate,and the solutions obtained with only ten coefficients ineach direction are quite accurate (the effective mode in-dex calculated with 10 and 20 basis functions differs byless than 0.05%). When we compare these results withthose obtained when the optimization criterion proposedin Ref. 7 is applied, a difference of less than 0.02% is ob-tained in the Neff , while the electric field profiles practi-cally coincide. Furthermore, in this case, the four-parameter dependence (p 5 ax , ay , ox , oy) of theO-HGDM optimization process makes it an impracticalstrategy from a computational point of view.

Once the proposed technique has been validated in dif-ferent situations, it is necessary to address the issue ofcomputational time. In Fig. 10, the time required to for-mulate and solve the eigenvalue problem once is plottedversus the number of coefficients. This computationaltime grows as N4, clearly showing the importance ofworking with a reduced number of terms. In this figure,three points have been marked, corresponding to the totalcomputational time that it takes to obtain the same levelof accuracy by using the HGDM, the O-HGDM, and theA-HGDM. The selected points correspond to the situa-tion presented in Fig. 5, i.e., a rectangular step-indexwaveguide under very low operating frequency. Thecomputational time for the O-HGDM has been calculated

assuming that it needs to solve 50 eigenvalue problems tofind the optimum mapping parameters (as stated in Ref.7). For the A-HGDM, six iterations have been supposed,as this is the mean number of iterations that this methodneeds to reach convergence. It can be easily seen howthe A-HGDM can significantly reduce computational ef-fort.

5. CONCLUSIONSIn this paper, a novel technique to efficiently perform themodal analysis of 3D/scalar dielectric waveguides is pre-sented. The method, which has been named the (adap-tive Hermite–Gauss decomposition method (A-HGDM), isin fact the natural extension of the recently publishedadaptive modified Fourier decomposition method (A-MFDM) to Hermite–Gauss basis functions. Instead ofusing parameter-dependent basis functions, as occurswith other Hermite–Gauss-based decomposition methodssuch as the HGDM and the optimized HGDM, theA-HGDM follows a variable-transformation approach.The proposed method also includes an efficient strategy toautomatically find a set of quasi-optimum transformation

Fig. 9. Directional coupler: convergence of the effective refrac-tive index (Neff 5 b/k0) as a function of the number of coeffi-cients.

Fig. 10. Rectangular step-index waveguide: computation timerequired to formulate and solve the eigenvalue problem once ver-sus the number of coefficients. The computer used was a Pen-tium IV 1.7 GHz.

566 J. Opt. Soc. Am. A/Vol. 20, No. 3 /March 2003 Ortega-Monux et al.

parameters with reduced computational effort. Thisstrategy is based on an optimization criterion, previouslyapplied to the A-MFDM, which involves minimizing thevariance of the field profile’s spectral coefficients. The re-sults obtained with two different 3D/scalar waveguides—the rectangular step-index waveguide and the rib-waveguide directional coupler—confirm that the proposedtechnique overcomes the drawbacks of previously pub-lished works and provides superior and powerful perfor-mance. Specifically, it has been shown that, while theO-HGDM is able to reach the optimum transformationparameters, the A-HGDM attains virtually the same re-sults with remarkably lower computational effort. It hasalso been shown that, although the HGDM has beenwidely used because of its simplicity, many practical situ-ations exist (complex waveguide structures, near-cutoffoperations, etc.) where the application of an efficient self-adaptive strategy becomes essential.

APPENDIX A: THE HERMITE–GAUSSTRANSFORMLet f(u, v) be a generic function defined over the entireuv plane V0 , which is expanded as a series of orthogonalbasis functions Fk(u, v):

f~u, v ! ' (k50

N

fkFk~u, v !, (A1)

where the spectral coefficients fk are obtained as the in-ner product:

fk 5^f~u, v !, Fk* ~u, v !&

^Fk~u, v !, Fk* ~u, v !&

5

EV0

f~u, v !Fk* ~u, v !dudv

EV0

Fk~u, v !Fk* ~u, v !dudv

. (A2)

These expressions are commonly called the inverse trans-form (A1) and the direct transform (A2) of the selected ba-sis functions.

The objective of this appendix is to find a compact ma-trix formulation to calculate the inverse Hermite–GaussTransform and the direct Hermite–Gauss transform suit-able for the numerical resolution of differential equations.

1. Matrix Formulation of the Inverse Hermite–GaussTransformIf the selected orthogonal functions Fk(u, v) are the bidi-mensional Hermite–Gauss basis functions

Fk~u, v ! 5 Bm~u !Bn~v !

5Hm~u !exp~2u2/2!

~p1/22mm! !1/2

Hn~v !exp~2v2/2!

~p1/22nn! !1/2 ,

m P ~0,..., Nx 2 1 !, n P ~0,..., Ny 2 1 !,(A3)

where Hm(u) and Hn(v) are the Hermite polynomials oforders m and n, respectively, the series expansion (A1)adopts the following form:

f~u, v ! ' (m50

Nx21

(n50

Ny21

fm,nBm~u !Bn~v !. (A4)

To convert the unique index notation (fk) used in rela-tion (A1) into the pair index notation (fm,n) used in rela-tion (A4), and vice versa, one must apply the following re-lations, respectively:

m 5 k mod Nx , n 5 k div Nx , (A5)

k 5 nNx 1 m. (A6)

To write the inverse Hermite–Gauss transform (A4)like a matrix expression, it is first necessary to specify thegrid of points @UV# over which the function f(u, v) is go-ing to be evaluated, given by

@UV# 5 $U%$V%t,

$U% 5 $u0 , u1 ,..., umax%t,

$V% 5 $v0 , v1 ,..., vmax%t, (A7)

which, introduced into relation (A4), allows us to estab-lish the following matrix form of the inverse Hermite–Gauss transform:

@ f~@UV# !# 5 @T inv~$V%!#@F#@T inv~$U%!#t, (A8)

where the matrices @T inv($U%)# and @T inv($V%)# are de-fined as

@T inv~$U%!#

5 F B0~u0! B1~u0! ¯ BNx21~u0!

B0~u1! B1~u1! ¯ BNx21~u1!

] ] � ]

B0~umax! B1~umax! ¯ BNx21~umax!

G ,

@T inv~$V%!#

5 F B0~v0! B1~v0! ¯ BNy21~v0!

B0~v1! B1~v1! ¯ BNy21~v1!

] ] � ]

B0~vmax! B1~vmax! ¯ BNy21~vmax!

G .

(A9)

2. Matrix Form of the Direct Hermite–GaussTransformThe most efficient way to calculate the spectral coeffi-cients @F# 5 @ fm,n# of a known function f(u, v) is toevaluate the integrals present in Eq. (A2) by using theGaussian quadrature scheme8:

fm,n 5

(p

(q

wpwqf~up , vq!Bm~up!Bn~vq!

(p

(q

wpwq@Bm~up!#2@Bn~vq!#2

,

(A10)

Ortega-Monux et al. Vol. 20, No. 3 /March 2003/J. Opt. Soc. Am. A 567

where wp and wq are the Gaussian quadrature weights,and up and vq are the Gaussian quadrature points in eachdirection, which, for the Hermite–Gauss-series expansionapproach, coincide with the zeros of the Hermite polyno-mials of orders Nx 1 1 and Ny 1 1, respectively.

In the same way as was done above, a matrix expres-sion for Eq. (A10) can be established:

@F# 5 @T~$V%!#@ f~@UV# !#@T~$U%!#t, (A11)

which is known as the matrix form of the direct Hermite–Gauss transform. The matrices @T($U%)# and @T($V%)#can be obtained by inverting Eqs. (A9):

T~$U%! 5 inverse~T inv~$U%!!,

T~$V%! 5 inverse~T inv~$V%!!. (A12)

APPENDIX B: THE HERMITE–GAUSSMATRIX OPERATORSThe purpose of this appendix is to provide the expressionsfor calculating the Hermite–Gauss matrix operators thatappear in the A-HGDM eigenvalue system matrix (17),i.e., the second derivative and product by a function op-erators.

1. Second Derivative with Respect to the u MatrixOperator @DDu#If the function c (u, v) is defined as

c ~u, v ! 5]2f~u, v !

]u2 , (B1)

then its spectral coefficients can be easily calculated by

$C% 5 @DDu#$F%, (B2)

where the elements of the matrix operator @DDu# are de-fined as

DDuui,k 5 EV0

Fi* ~u, v !]2Fk~u, v !

]u2 dudv

5 EV0

Br~u !Bs~v !]2Bm~u !Bn~v !

]u2 dudv.

(B3)

These integrals can be solved in a closed form9 by makinguse of the following identities:

]2Bm~u !

]u2 5 u2Bm~u ! 2 ~2m 1 1 !Bm~u !,

(B4)

E2`

`

u2Bm~u !Br~u !du 51

2@~m 1 1 !~m 1 2 !#1/2dm,r12

11

2~2m 1 1 !dm,r

11

2@m~m 2 1 !#1/2dm,r22 ,

(B5)

which finally leads to

DDuui,k 51

2@~m 1 1 !~m 1 2 !#1/2d i,k12

21

2~2m 1 1 !d i,k 1

1

2@m~m 2 1 !#1/2d i,k22 .

(B6)

2. Second Derivative with Respect to the v MatrixOperator @DDv#If the function c (u, v) is defined as

c ~u, v ! 5]2f~u, v !

]v2 , (B7)

then its spectral coefficients can be again calculated by

$C% 5 @DDv#$F%, (B8)

where the elements of the matrix operator @DDv# are

DDvui,k 512 @~n 1 1 !~n 1 2 !#1/2d i,k12Nx

212 ~2n 1 1 !d i,k 1

12 @n~n 2 1 !#1/2d i,k22Nx

.

(B9)

Note that both Eqs. (B6) and (B9) define tridiagonal ma-trices.

3. Product by a Function Matrix Operator @P( f(u, v))#Suppose that the function c (u, v) is defined as the prod-uct

c ~u, v ! 5 f~u, v !f~u, v !; (B10)

then its spectral coefficients can be obtained by

$C% 5 @P( f~u, v !)#$F%, (B11)

where the matrix operator P( f(u, v)) is defined as

P( f~u, v !)ui,k 5 EV0

f~u, v !Br~u !Bs~v !Bm~u !Bn~v !dudv.

(B12)

The numerical evaluation of this expression is in gen-eral very time expensive. However, in certain power-lawindex profile structures (including the waveguides ana-lyzed in this work), it is possible to find analytic expres-sions for the elements5 of the product matrix operator.

ACKNOWLEDGMENTThis work was supported by the Spanish Comision Inter-ministerial de Ciencia y Tecnologıa under projectTIC2000-1245.

Alejandro Ortega-Monux, the corresponding author,can be reached by e-mail at aom@ic.uma.es. E-mail ad-dresses of the other authors are gonzalo@ic.uma.es andimf @ic.uma.es.

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