Analysis of dielectric waveguides by a modified fourier

1614 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 19, NO. 10, OCTOBER 2001

Analysis of Dielectric Waveguides by a ModifiedFourier Decomposition Method With Adaptive

Mapping ParametersJ. G. Wangüemert-Pérez and I. Molina-Fernández

Abstract—The adaptive modified Fourier decompositionmethod (A-MFDM), a recently proposed technique to performthe modal analysis of linear and nonlinear two-dimensionaldielectric waveguides, is now extended to three-dimensional scalarstructures for the first time. The method includes a strategy toautomatically find quasioptimum numerical parameters of thetechnique, which, on the one hand, improves the accuracy inrelation to the known Fourier decomposition method (FDM) and,on the other, makes the method self-adaptive. To validate theperformance of the method, two linear dielectric waveguides withexact solutions have been defined and analyzed. Finally, the modalanalysis of an asymmetrical rib waveguide directional coupler iscarried out and the obtained results confirm the clear superiorityof the A-MFDM in realistic situations.

Index Terms—Dielectric waveguides, directional coupler,Fourier series, optimization method.

I. INTRODUCTION

M ODAL analysis of dielectric waveguides is a key issuein optical device design. To perform this task with ac-

curacy, different numerical methods can be used. Among them,the Fourier decomposition method (FDM) is a very well-knowntechnique [1]. Nonetheless, although the method is simple andefficient, it has serious and important limitations: its accuracyis highly dependent on the size of the enclosing computationalwindow (period of the basis functions over which the electricfield is expanded), which must be fixed before applying themethod. To avoid this truncation of the original infinite domain,the modified Fourier decomposition method (MFDM) was pro-posed in [2] for three-dimensional (3-D)/scalar cases and morerecently in [3] for 3-D/semivectorial situations. This method re-lies on a tangent-type variable transformation, which convertsthe original infinite spatial domain into a finite one, where theFourier method is then applied. Although the MFDM not onlyovercomes the aforementioned problem but also improves theaccuracy of the FDM, it still has serious limitations.

1) Due to the symmetry of the applied variable transforma-tion, the performance of the method is greatly reducedwhen analyzing asymmetrical situations. This occurs inmany linear waveguides of practical interest (as, for ex-

Manuscript received December 23, 1999; revised April 11, 2001. This workwas supported by the Spanish C.I.C.Y.T. under Projects TIC96-1072-C04-04and TIC2000-1245.

The authors are with the Departamento de Ingeniería de Comunicaciones,E.T.S.I. Ingenieros de Telecomunicación, Universidad de Málaga, Málaga29071, Spain.

Publisher Item Identifier S 0733-8724(01)07762-3.

ample, in the asymmetrical directional coupler) and isalso the case when studying nonlinear Kerr-type wave-guides.

2) The MFDM avoids the problem of determining thesize of the computational window, but in doing so, newscaling parameters are introduced in the variable trans-formations, which must also be subsequently determined,so the method is not fully automatic.

Recently, to overcome these limitations, the adaptive-MFDM(A-MFDM) has been proposed in [4] and [5], where it was suc-cessfully applied to linear and nonlinear planar [two-dimen-sional (2-D)] dielectric waveguides, under scalar approxima-tion. In this method, a modified tangent-type transformation wasproposed, which increases accuracy in asymmetrical situations.Also, an adaptive optimization algorithm was provided to au-tomatically find the quasioptimum numerical parameters of thevariable transformation.

In this paper, the natural extension of the A-MFDM to 3-Dstructures is performed. Although the presented technique hasonly been developed to analyze waveguides under scalar ap-proximation, due to its close formulation, it can be easily ex-tended to semivectorial and vectorial situations.

This paper is structured as follows. Section II develops ageneral formulation of the A-MFDM based on the concept ofa “matrix operator,” for the purpose of transforming the modalwave equation of a 3-D/scalar dielectric waveguide into a matrixeigenvalue equation. Section III focuses on the adaptive proce-dure to find the quasioptimum parameters of the variable trans-formation. First, the optimization criterion, which gives the bestresults, is presented and then the details of the necessary algo-rithm to put it into practice are explained. In Section IV resultsare shown for a great variety of situations (frequency, modes,geometry, etc.), which confirm not only the validity of the pro-posed criterion but also the clear superiority of the A-MFDM incomparison to the FDM.

II. FORMULATION OF THE A-MFDM

A. Normalized Wave Equation

Under scalar approximation, the wave equation that governsthe propagation of stationary solutions through a general-in-variant 3-D/scalar dielectric waveguide is given by

(1)

0733–8724/01$10.00 © 2001 IEEE

WANGÜEMERT-PÉREZ AND MOLINA-FERNÁNDEZ: ANALYSIS OF DIELECTRIC WAVEGUIDES 1615

where is the wavenumber in the vacuum and is therefractive index profile. Once these parameters are specified, ourinterest lies in determining the spatial distribution of the electricfield and its propagation constant, which represent,respectively, the eigenfunctions and eigenvalues of the modalwave equation (1).

It is usually preferable to normalize (1) by defining the fol-lowing normalized parameters: the normalized axesand , thenormalized frequencies and , the asymmetrical coefficient, the normalized propagation constant, and the normalized

refractive index . For a general waveguide composedof three different media—film ( ), substrate ( ), and cover( )—where , they are defined as

(2)

(3)

(4)

(5)

(6)

The normalization magnitudes and , which must be pre-viously specified, can be chosen equal to any value. Typically,the half-widths of the waveguide core (in each direction) are theselected normalization magnitudes.

By introducing the normalized parameters into (1), an equiva-lent form of the modal wave equation is obtained. This is knownas the normalized wave equation and can be finally written as

(7)

B. Variable Transformation and Discretization

Making use of the following variable transformation:

(8)

the normalized wave equation (7) is converted into

(9)

where the exact expressions of the functions , ,, and can be easily obtained by means of the chain

rule [4]. Note that in doing so, the infinite domain hasbeen mapped to a finite one in the plane by means of thetangent-type transformation. Also notice that before applyingthe method, four transformation parameters must be specified:the scaling factors ( ), which control the mappingstrength, and the offset parameters ( ), which determinethe mapping center. The latter were not present in the originalformulation of [2] and [3], and, as will be shown, they make itpossible to efficiently deal with asymmetrical situations.

Finally, an approximate solution to the resulting wave equa-tion in the transformed domain (9) is obtained by applying aprocess similar to the one performed in the FDM: first, the un-known electric field is expanded in a finite Fourier series

(10)

where now, unlike the FDM, the computational window com-prises the whole transformed domain, that is, and

, so the period of the basis functions is fixed at. Second, the Galerkin method is applied to obtain an al-

gebraic system of equations, which relates theunknown spectral coefficients . To write down this systemin a compact form, the spectral coefficients , which nat-urally define a matrix [ ], are reallocated as a column vector

, and thematrix operatorapproach, introduced in [4], is ap-plied. As a result, the following equation is obtained:

(11)

where , , , and are, respectively, thesecond derivative with respect to, first derivative withrespect to , second derivative with respect to, and firstderivative with respect to matrix operators, while ,

, , , and are theproduct by a functionmatrix operators applied to the functionsincluded in parentheses. Closed expressions to calculate theseoperators are obtained in the Appendix.

Equation (11) thus obtained can now be written in the formof a matrix eigenvalue equation

(12)

and its respective eigenvectors and eigenvalues can besolved by using any standard subroutine pack. Once this hasbeen done, modal field profiles and propagation constants canbe easily obtained: the normalized propagation constants of the


guided modes are directly the eigenvalues in the range, while their corresponding field profiles are obtained

by returning each of the calculated eigenvectors to its ma-trix form and evaluating (10) at the desired points of thetransformed domain . If such points are evenly spaced, thistask can be quickly performed by using the two-dimensional fastFourier transform (FFT). Finally, making use of the inverse pro-posed variable transformation

(13)

the equidistant field samples over the plane are convertedinto nonequidistant ones over the original domain.

III. A DAPTIVE SETTING OF MAPPINGPARAMETERS

A. Optimization Criterion

The MFDM was initially proposed to avoid the dependenceof the FDM’s accuracy on the size of the enclosing computa-tional window. However, as was pointed out by Hewlettet al.[2], if the whole space is compressed into a finite one by meansof a variable transformation, the accuracy of the spectral methodis now dependent on the transformation parameters (andfor the MFDM), so the method will only work properly if theyare adequately selected. Although Hewlettet al. [2] suggestedthat good convergence could be obtained by setting the scalingparameters equal to the half-width of the waveguide core, thisrule only applies for linear and symmetrical waveguides. There-fore, in general situations, this problem can only be partiallyovercome following ana posterioristrategy based on the visualinspection of the obtained solution and iterating until a satisfac-tory result is reached.

The four-parameter dependence ( , and ) of theproposed technique makes it almost impossible to apply such astrategy in practical situations; therefore a closed procedure todetermine these parameters must be provided. One of the keyissues of this paper is that an optimization criterion to automat-ically find the quasioptimum transformation parameters for theA-MFDM is proposed and verified in a great variety of situa-tions. The optimization criterion proposed in this paper is theextension to 3-D structures of the previously published one for2-D cases [4], and it is based on the same reasoning: it is clearthat the electric field spatial distribution in the transformed do-main and therefore the spectral coefficients willdepend on the chosen parameters, so it seems reasonable thatthe best results should be obtained with those parameters thatmost strongly concentrate the spectrum of the function into thelower spatial frequencies (i.e., into the spectral coefficients withlower indexes). Just as in the 2-D case [4], it has been stated thatthe best results are obtained with those parameters that mini-mize the spectral width of the functionand that the variance is a good measure of this spectral width.Now, taking into account the relation between the spectral coef-ficients of the proposed function and the spectral coefficients of

the electric field, the final expression to be minimized is givenby

Var (14)

where stands for the four-dimensional vector of transforma-tion parameters, i.e., . Note that in thisexpression, the dependence of on the vector of transfor-mation parameters is explicitly shown tohighlight this fact. However, for the sake of clarity, in the nextsection this dependence will be expressed through a superscript.

B. Optimization Algorithm

To put the previous criterion into practice, it is necessary todesign an iterative algorithm, which, starting from an arbitraryset of transformation parameters, will be able to automaticallyreach a quasioptimum set of values. The main steps of the im-plemented algorithm can be summarized as follows.

1) By simple physical arguments, the transformation param-eters are initialized: .

2) With these transformation values, the eigenvalue problem(12) is solved and the Fourier field profile coefficients

together with the normalized propagationconstant are obtained.

3) Now, a first guess for the optimum values that yieldthe minimum variance must be obtained. To do this, itis first necessary to provide a method that, starting fromthe previously calculated solution , makes it possibleto estimate the variance of the spectrum for any arbitraryvalue of , i.e., it is necessary to specify exactly how tocompute the function Var Var . This can be done bymeans of a direct method: applying the inverse and di-rect tangent-type transformation, with parameter values

and , respectively, to the previously calculated solu-tion , an estimation of the spectrum can be ob-tained (this aspect will be further explained later). Fromthis, its variance Var Var can be easily com-puted. Once the function to be minimizedVar hasbeen defined, any standard optimization procedure canbe used to obtain the optimum transformation parame-ters . However, as no closed-form expressions for thegradient of the function are available, direct methods thatdo not rely on the derivatives are preferred. Throughoutthis work, results have been obtained by means of a directsearch method known as the “simplex search method”or “polytope algorithm,” as implemented in MATLAB’sfunction fmins[6].

4) The optimum parametersare now used to repeat the process from the secondstep until no change is observed in two consecutiveiterations.

For complete understanding of the previous algorithm, thequestion of how can be estimated for any arbitrary value


of the parameters starting from an initial known solution(point 3 of the previous algorithm) must be conveniently clari-fied.

Suppose that are the obtained Fourier coefficients whenthe eigenvalue problem (12) is solved with some transforma-tion parameters , and are the Fourier coefficients that weintend to determine from the former for some arbitrary trans-formation parameters , in order to evaluate its bandwidth.Once the coefficients are known, the corresponding fieldprofile in the space can be evaluated in an arbi-trary grid of points by means of the inverse discreteFourier transform (10). These points can be conveniently chosenso that when performing the double transformation, fromto [by means of the inverse tangent transformation (13)]and from to [by means of the direct tangent trans-formation (8)] to obtain , the new set of sample points

are uniformly spaced. In this manner, a set of uniformlyspaced samples of are determined from and thedesired coefficients can be easily calculated by means of theFFT. To understand the basis of the aforementioned process, agraphical representation is shown in Fig. 1 for a simpler 2-D sit-uation. Its practical implementation is carried out by followingthe following steps.

1) A grid of equidistant samples over the plane is defined.

2) The respective grid of nonequidistant samples isobtained by means of the inverse and direct proposed vari-able transformation [(13) and (8), respectively], that is

3) The matrix of field amplitudes of sampledat equally spaced points defined by , that is,

, is calculated from the Fourier coefficientsof the function applying the inverse dis-

crete Fourier transform (11) over the nonequidistant gridobtained in the previous step, since the ma-

trix thus obtained is equal to the matrix.

4) Finally, the desired spectral coefficients ofare obtained, performing the FFT to the matrix of ampli-tudes .

IV. RESULTS

To validate the proposed technique and its optimizationstrategy, results will be presented relative to two different situa-tions.

Fig. 1. Relationship between the equidistant samples of� (u) where we wishto determine the field and those nonequidistant of� (u) that represent the sameamplitudes of the electric field.

1) First, 3-D structures with known analytical solutions willbe analyzed. These structures, which have refractive in-dexes of separable type, do not apply to realistic situationsbut are useful to validate a numerical method because theymake it possible to easily calculate the precision of the ob-tained solution.

2) Second, to show the performance of the proposed tech-nique in more realistic situations, the method will be ap-plied to an asymmetrical rib waveguide directional cou-pler, which is a complex structure frequently used in agreat variety of devices.

A. Separable Type Waveguides

Assessment of the method can be done by analyzing 3-Dwaveguides with known solutions and comparing the error ob-tained using the A-MFDM to that obtained using the FDM. Forthis task, two linear dielectric waveguides with analytical solu-tions have been defined. They have been obtained making use ofthe separation of variable method [7]. The first one is a symmet-rical waveguide formed by adding two step-index slabs, whilethe second one is an asymmetrical waveguide formed by addingtwo exponential graded-index slabs [8]. Both structures havebeen plotted in Fig. 2(a) and (b), respectively. The reason forsuch a choice is that in the case of Fig. 2(a), and due to the sym-metry, the A-MFDM can be first verified without the necessityof the offsets (i.e., ). Once this has been done, theoptimization strategy can then be proved in the more general sit-uation of Fig. 2(b) where, due to the asymmetry, the four trans-formation parameters are needed and must be simultaneouslyoptimized.

As a measurement of the obtained error, the mean square error(MSE) of the electric field and the relative error (RE) of the


(a)

(b)

Fig. 2. Normalized refractive index profiles of the separable type waveguides used to assess the A-MFDM and the proposed optimization strategy. (a) Symmetricalwaveguide and (b) asymmetrical waveguide.

propagation constant were used. These are defined in the fol-lowing manner:

MSE dB

(15)

RE dB (16)

where, in order to make a correct comparison betweenA-MFDM and FDM, the integration area used in (15)has been set equal to the window size for which the FDM isoptimized (that is, choosing it in a such a way that the electricfield is approximately zero over its edges).

To check the soundness of the method for the symmetricalwaveguide, the following representation has been used: the path


Fig. 3. Contour map of the MSE together with the path followed by theoptimization algorithm for the first two modes of the symmetrical waveguide:(a) TE (V = 1:5, V = 1:5, N = N = 4) and (b) TE (V = 2:5,V = 2, N = N = 6).

followed by the optimization algorithm has been superimposedonto the contour map of the MSE obtained for different valuesof the scaling parameters ( ), or onto that of the RE inthe propagation constant. This makes it possible to clearly iden-tify if the optimization algorithm has been able to converge toan optimum area and to determine the final accuracy obtainedby the A-MFDM. The obtained results for different frequencies,modes, and numbers of coefficients were always excellent. Toshow a typical situation, the aforementioned representation (forthe MSE measurement) is plotted in Fig. 3(a) and (b) for the twofirst modes of the symmetrical waveguides (TEand TE , re-spectively). In each of them, the discontinuous contour line cor-responds to a value of the MSE equal to that of the FDM withoutmapping. Therefore, in the region inside the discontinuous con-tour line, the A-MFDM is obtaining more accurate results thanthe FDM. The observed asymmetry in Fig. 3(b) is produced bythe field profile of the mode under consideration. In both cases,it can been seen that the proposed criterion is working well be-cause, even if the starting transformation parameters are located

in an area in which the A-MFDM works worse than the FDM[Fig. 3(b)], the developed algorithm automatically reaches thebest area and yields a final MSE better than30 dB. The elec-tric field spatial distributions obtained when the A-MFDM isused with such final values are shown in Fig. 4.

When the path followed by the optimization algorithm is su-perimposed onto the RE representation (Fig. 5), the contour mappresents a less smooth variation with the transformation param-eters, but even then the A-MFDM shows much better accuracyin the determination of than the FDM. As can be seen in thefigure, the final RE obtained for the two modes is better than

45 and 30 dB, respectively.With respect to the asymmetrical waveguide, the obtained

results have also been very satisfactory, showing that theproposed procedure is able to simultaneously give the fourquasioptimum parameters. Unfortunately, the four-parameterdependence makes it impossible to summarize in a singleplane the performance of the method. However, the same typeof representation can be used if two of the parameters (forexample, ( ) are kept constant while the two others arevaried around the parameter values reached by the optimizationalgorithm. As an example, in Fig. 6(a) and (b), the MSE-con-tour map and the RE-contour map are plotted versus ( ),while ( ) are kept fixed to the final reached values. Itcan be seen that in both cases, the final point is located in theoptimum area, and that an MSE of34 dB and a RE of 49dB are obtained, respectively. Notice also that, as was pointedout in the previous paragraph, the propagation constant map ismore irregular than the MSE map, but even so the A-MFDMin conjunction with the optimization procedure gives a muchbetter estimation of than the FDM.

The superiority of the A-MFDM in relation to the FDM isclearly confirmed when their convergence rates are compared.Such results are shown in Fig. 7 for different operating fre-quencies and for the asymmetrical waveguide. Note that theFDM presents worse accuracy as the frequency is decreased,due to the greater window size that must be used, while theA-MFDM has been able to overcome this drawback since itshows a less appreciable dependence on this variable. Noticealso that the greatest improvement is obtained when the numberof harmonics is small, a very interesting fact for the analysis of3-D waveguides.

Another important property observed when the proposed op-timization criterion was used together with the A-MFDM is thatregardless of the number of harmonics used in the Fourier se-ries, the optimum transformation parameters remain practicallyunchanged. This is shown in Fig. 8, where the transformationparameters finally reached by the optimization strategy, whendifferent numbers of harmonics were used, are plotted for dif-ferent operating frequencies. This desensitization of the qua-sioptimum parameters to the number of harmonics can be ad-vantageously used to reduce the computational effort requiredby the A-MFDM. This makes it possible to use a very lownumber of harmonics in all the intermediate steps of the iterativescheme to determine the optimum transformation parameters,and once these have been obtained, the number of harmonics


(a)

(b)

Fig. 4. Electric field spatial distribution of the first two modes of the symmetrical waveguide obtained with the A-MFDM: (a) TE(V = 1:5, V = 1:5,N = N = 4) and (b) TE (V = 2:5, V = 2, N = N = 6). The transformation parameters used in each case were those reached by the optimizationalgorithm, as indicated in Fig. 3.

can be conveniently increased to improve the accuracy of themethod.

B. Directional Coupler

Finally, the method has also been tested in an asymmetricalwaveguide directional coupler. This device is extensively used

in a great variety of applications, such as beam splitters, demul-tiplexors, and switches. The operation of the structure, whichis composed of two closely spaced waveguides, is based on theinteraction between the two lowest order supermodes: the sym-metric-like and the asymmetric-like one.

To prove our technique, an asymmetrical rib waveguide di-rectional coupler has been selected [9]. The geometry and the


Fig. 5. Contour map of the RE together with the path followed by theoptimization algorithm for the first two modes of the symmetrical waveguide:(a) TE (V = 1:5, V = 1:5, N = N = 4) and (b) TE (V = 2:5,V = 2, N = N = 6).

physical parameters of the structure are shown in Fig. 9. All theresults of this section correspond to field profile and propaga-tion constants obtained with the FDM or with the A-MFDM fordifferent values of the width of the rightmost waveguide (2)but maintaining all the other parameters of the structure set toa constant value. This will make it possible to test the proposedstrategy with a great variety of field profiles.

In Fig. 10, different field profiles of the symmetric-like andasymmetric-like supermodes are plotted for different values of2 . All of them have been obtained with a reduced numberof harmonics ( ), which, as will be shown later,are enough to ensure convergence of the method. In all cases,the value of has been kept fixed to a constant value (

n.u.) because this value is very close to the one obtainedby the optimization algorithm for all the analyzed situations.It must be noted that in obtaining the field profiles of Fig. 10,the A-MFDM is working in an absolutely automatic fashion,

Fig. 6. Asymmetrical waveguide: (a) MSE of the field profile and (b) RE ofthe propagation constant around the point reached by the optimization algorithm(� = � = 0:79 ando = o = �0:45). Simulation data:V = V = 3;N = N = 4.

i.e., the remaining transformation parameters ( ) havebeen calculated for each value of 2 by the optimization algo-rithm. Superimposed onto the field profile, a shaded ellipsoid isplotted, which gives a graphical representation of the optimumtransformation parameters attained by the algorithm. This el-lipsoid is obtained by transforming a centered circle of radius

, defined in the plane, to the domain by means of(13). In this manner, the ellipsoid shows the area of theplanethat is best represented in the transformed domain. It can be seenthat by varying 2 , a great variety of situations occur that leadto different field profiles centered in different positions along the

axis and that the inclusion of the center parametermakesit possible to efficiently deal with this situation. As expected, itcan be observed that this ellipsoid moves along theaxis cov-ering the area in which the field profile and the waveguidingregions are located. Therefore, the accuracy of the method is al-ways kept high with a reduced number of harmonics. It mustalso be noted that in all the cases under study, a good perfor-mance of the adaptive procedure has been observed.


Fig. 7. Asymmetrical waveguide: convergence rates of the A-MFDM andFDM versus the number of harmonics for different normalized frequencies.(a) MSE of the field profile and (b)b =b .

Because the analytical solution for this type of structure isunknown, a different strategy to check and quantify the perfor-mance of the A-MFDM in relation to the FDM was established.It exploits the property of regular convergence that must be sat-isfied by any useful numerical method. This property states thatas the number of coefficients is increased, the obtained resultswill progressively resemble the exact solution, so convergence isreached when two consecutive practically unchanged solutionsare attained. This property has been used to determine whetherthe results obtained by both methods are the same, once con-vergence has been achieved, and also to compare their rates ofconvergence. The obtained results confirm that both methodsconverge to the same solution (within a relative error of 0.015%)and that the convergence rate of the A-MFDM is much greaterthan that of the FDM, thus making it possible to decrease com-putational effort.

In Fig. 11, the effective mode index ( ) of thesymmetric-like and asymmetric-like supermodes versus thewaveguide width 2 have been plotted for different numbers

Fig. 8. Asymmetrical waveguide: variation of the final quasioptimum trans-formation parameters with the number of harmonics. Simulation data:N =

N = f4; 6; 8; 10; 12g.

of harmonics. Fig. 11(a) corresponds to the FDM and Fig. 11(b)to the A-MFDM. The obtained results are just like those ob-tained in [9]: as the width 2 increases, the modal index ofthe symmetric-like supermode approaches that of an isolatedrib waveguide of width 2 , whereas the asymmetric-likesupermode resembles that of the rib waveguide whose widthremains fixed. In the opposite direction, as 2 decreasestoward zero, the symmetric-like supermode resembles the fun-damental mode of the rib whose width remains fixed, whereasthe asymmetric-like supermode asymptotically approaches theslab solution. In the middle, when both rib waveguides havethe same width (symmetric situation), maximum couplingbetween waveguides occurs as the propagation constants of thesupermodes are very close together (almost a phase matchedcondition). To plot these curves, the numerical parameters ofboth methods have been kept fixed for all the values of thewaveguide width 2 , i.e., the computational window of theFDM has been fixed to a constant value, as has the trans-formation parameters of the A-MFDM. Specifically, for theA-MFDM, the parameters that the adaptive method determinedto be optimum for n.u. and wereselected.

The first thing that can be observed in these figures is thatthe method with variable transformation converges with a muchsmaller number of harmonics than the method of Fourier. In-deed, the convergence in the A-MFDM is obtained with only18 terms in each direction (324 unknowns) for all the values of

, while 32 terms (1024 unknowns) are necessary to reachit in the FDM. It is also observed that once convergence hasbeen reached, both methods converge, for any value of 2,to the same final values within a relative error of 0.015%. No-tice also that as the transformation parameters are optimum for

n.u., convergence of the A-MFDM is first obtainedfor the rightmost part of the graph. This is especially importantin the symmetric-like supermode, where it is observed that only


Fig. 9. Geometry and physical parameters of the rib waveguide asymmetrical directional coupler (taken from [9]).

12 terms in each direction are needed to converge for the re-gion 2 , while for the left part of the curve, i.e.,2 , 18 terms are required. This shows that theadaptive strategy used to search for the transformation parame-ters works correctly, since it is able to improve the convergencerate of the method.

Besides the adaptive algorithm, the other novelty of theA-MFDM is the introduction of the offset parameters and

to locate the origin of the transformation in the plane.It could be argued that the choice of such parameters is notvery important for the operation of the method, and that theycould be chosen by simple “intuition” (for example, locatingthem in the high-index guiding region) and kept fixed for agiven structure. This happens to be the case in many situations,and it was, for example, what was observed in Fig. 10 for theoffset parameter , which, as was already pointed out, stayedalmost constant under variations of 2, so it was fixed to aconstant value of 18 n.u. However, other occasions exist whereit is useful to determine the optimum offset parameters. Thisis, for example, the case of nonlinear waveguides, in whichthe refractive index’s dependence on the electric field makesit difficult to intuitively guess the location of the guidingregion. In other situations, as, for example, when studyingthe dependence on 2 of the proposed directional coupler,the adequate selection of the offset parameters can lead to anappreciable reduction in computational effort. This is shown inFig. 12, where the convergence rate of the effective index of thesymmetric-like supermode is plotted for a set of different offsetparameters ( ). The figure corresponds to a waveguidewidth of n.u. for which the optimum offset parametervalues obtained by the adaptive algorithm forwere n.u., n.u. From this figure,it is clear that the best convergence rate is obtained with the

parameters supplied by the algorithm, although acceptableperformance is also attained with the other values.

Finally, the assessment of the A-MFDM would be incompletewithout a discussion of computation time. It is evident fromthe previously presented results that the A-MFDM requires asmaller number of harmonics than the FDM to reach a fixedaccuracy, this being true not only for the two waveguides withanalytical solutions but also for the asymmetrical rib waveguidedirectional coupler. However, from a practical point of view,this is of little interest if the computation efficiency is not alsosuperior.

It must be remembered that the application of the A-MFDMinvolves two different steps: 1) the solution of the eigenvalueproblem for a set of mapping parameters and 2) the applicationof the optimization process to find out a new set of quasiop-timum mapping parameters. These steps are iteratively repeateduntil convergence is reached. Therefore, to evaluate the requiredtime of computation for the A-MFDM, the following formulacan be used:

(17)

wheretime required to solve the eigenvalue problem once[step 1)];mean time required to perform the optimizationprocess [step 2)];number of iterations needed to converge to the finalsolution;mean time per iteration.

In Fig. 13, the computation time expended using theA-MFDM and FDM is compared with the number of har-monics for the directional coupler with n.u. For the


(a) (b)

(c) (d)

(e) (f)

Fig. 10. Field profile contour map obtained by means of the A-MFDM for (a)–(c) symmetric-like and (d)–(f) asymmetric-like supermodes, for differentvaluesof the waveguide width 2W (N = N = 18).

FDM, only the time required to solve the eigenvalue problem isrepresented, i.e., the time required to find out the correct size ofthe computational window has not been included. In the case ofthe A-MFDM, the figure shows, on one hand, the time requiredto solve the eigenvalue problem ( ) and, on the other, themean time per iteration ( ). The difference between thesevalues gives the mean time required for optimization ().

This figure shows that for the same number of harmonics,the time needed to solve the eigenvalue problem is almost thesame for both methods and that the extra time expended bythe A-MFDM is mostly due to the optimization strategy. How-ever, it is important to notice that when making these graphs,the worst case scenario was considered, as all the iterative op-

timization process has been carried out with the final numberof harmonics, i.e., solving in each step a full size problem of

dimensions. However, as stated in the previous sec-tion (comments to Fig. 8), it is possible to drastically reducethis optimization time by using a low number of harmonics inall the intermediate steps needed to find the optimum parame-ters. Once these optimum parameters have been obtained, thefull size problem can be solved to achieve the final accuracy.

In spite of this, it is observed that comparing those situa-tions for which the same accuracy was obtained (number of har-monics needed in each case for convergence), the mean time periteration of the A-MFDM (40 s for 18 harmonics) is approxi-mately five times shorter than the FDM time of computation


Fig. 11. Asymmetrical rib waveguide directional coupler. Variationof the effective refractive index (N = �=k ) of the two lowestorder supermodes with the width of the second rib 2W , for varyingnumbers of coefficients. (a) FDM and (b) A-MFDM. Simulationdata: (a) computational window forx = 20 n.u., y = 80 n.u.and (b) transformation parameters fpr symmetric-like supermodeppp =(� ; o ; � ; o ) = (2:76; 1:76; 10:1; 18); asymmetric-like supermodeppp = (� ; o ; � ; o ) = (2:67; �0:66; 11:26; 18).

(approximately 200 s for 32 harmonics). As the mean number ofiterations needed to reach the optimum transformation parame-ters was found to be , it is clear that even in the worstcase scenario the A-MFDM gives similar computational effi-ciency as the FDM. However, for the same case, if the optimiza-tion procedure is carried out with only 12 harmonics (which isenough to find out the quasioptimum transformation parame-ters) and the eigenvalue problem is solved with 18 harmonics(to retain the same final accuracy), then the total computationaltime is only 60 s. Under this situation, the A-MFDM computa-tional efficiency is more than three times better than that of theFDM.

Fig. 12. Asymmetrical rib waveguide directional coupler. Convergence of theeffective refractive index (N = �=k ) for the symmetric-like supermodewhen the offset parameters are varied around the optimal parameters. (a) Vary-ing o and (b) varyingo .

Fig. 13. Asymmetrical rib waveguide directional coupler. Comparison of thecomputation time of FDM and A-MFDM versus the number of coefficients.(Symmetric-like supermode:2W = 4 n.u. Computer: Pentium III 500 MHz;256 MB RAM.)


V. SUMMARY

In this paper, the natural extension of the A-MFDM to3-D/scalar dielectric waveguides is performed and results arepresented for linear structures. The method is robust, as itincludes an optimization strategy that assures the automaticdetermination of quasioptimum numerical parameters. Thenew technique has been tested in several situations, includingan asymmetrical rib waveguide directional coupler, and theobtained results confirm the clear superiority of the method inrelation to the classical FDM. As the method is self-adaptive, itcould be especially useful for the analysis of situations whereprevious knowledge of the expected results is not possible.

APPENDIX

MATRIX FORMULATION OF OPERATORS

Matrix operators appearing in (11) can be obtained by fol-lowing a similar procedure as established in [4] for the simpler2-D case.

Suppose that the functions and defined in thedomain are related through

(A1)

where denotes any operation such as those that appear in(9), i.e., the “first derivative with respect toor ,” the “secondderivative with respect to or ,” or “the product by a knownfunction.” If both functions are expanded into a finite Fourierseries, with and harmonics in each of the transverse di-rections

(A2)

(A3)

and next, the spectral coefficients and , which natu-rally define matrices since they are denoted as a unique com-bination of the two spatial frequencies running in theanddirections ( and ), are rearranged ascolumn vectors to be referenced as a single index ( and

), the objective of this Appendix is to find the square-ma-trix operator [ ] in the Fourier series domain so that (A1) willbe transformed into

(A4)

with , , and.

Rearrangement of the spectral coefficient matrix [] (or [ ])into a column vector (or ) requires the conversion ofthe pair index into the single index (or, equivalentlyinto ), and vice versa. This can be done by columns or by rows.

If the first reallocation is adopted, the following mathematicalrelations must be employed:

(A5)

(A6)

(A7)

where stands for the remainder after division operationand stands for the round toward zero operation.

Once these definitions have been established, and following[4], the matrix operators needed to transform (9) into (11) canbe easily obtained.

1) First derivative with respect to matrix operator[ ]:Suppose , so that (A1) becomes

(A8)

From basic Fourier theory, their spectral coefficients arerelated by the expression

(A9)

so the matrix operator [ ] can be easily identified as adiagonal matrix

(A10)

2) First derivative with respect to matrix operator[ ]:Suppose , so that (A1) becomes

(A11)

Again from basic Fourier theory, their spectral coeffi-cients are related through

(A12)

so the matrix operator [ ] can also be identified as adiagonal matrix

and

(A13)

3) Second derivative with respect toand matrix opera-tors, : Taking into account that the second


derivative can be written as twice the first derivative op-eration, the following simple relations can be established:

(A14)

(A15)

4) Product by a function matrix operator : Sup-pose that the function defined over the interval

is intended to be multiplied bythe function , so that (A1) becomes

(A16)

As is well known from properties of the Fourier series, the spec-tral coefficients of the function are directly obtained,making the convolution between the spectral coefficients of thefunctions that are being multiplied, that is

(A17)

and

and

where are the Fourier series coefficients of , whichcan be calculated in advance. Therefore, taking into account therelation between with and with , the elements oftheproduct by a function matrix operatorare given by

(A18)

with , , and.

ACKNOWLEDGMENT

The authors wish to thank the reviewers for their commentsand J. M. Yanes Montiel and M. A. Luque Nieto for their assis-tance in the development of the computer code used to obtainthe results presented in this paper.

REFERENCES

[1] C. H. Henry and B. H. Verbeek, “Solution of the scalar wave equationfor arbitrarily shaped dielectric waveguides by two-dimensional Fourieranalysis,”J. Lightwave Technol., vol. 7, pp. 308–313, Feb. 1989.

[2] S. J. Hewlett and F. Ladouceur, “Fourier decomposition method appliedto mapped infinite domains: Scalar analysis of dielectric waveguidesdown to modal cutoff,”J. Lightwave Technol., vol. 13, pp. 375–383, Mar.1995.

[3] K. M. Lo and E. H. Li, “Solutions of the quasivector wave equationfor optical waveguides in a mapped infinite domains by the Galerkinsmethod,”J. Lightwave Technol., vol. 16, pp. 937–944, May 1998.

[4] I. Molina-Fernández and J. G. Wangüemert-Pérez, “Variable trans-formed series expansion approach for the analysis of nonlinear guidedwaves in planar dielectric waveguides,”J. Lightwave Technol., vol. 16,pp. 1354–1363, July 1998.

[5] J. G. Wangüemert-Pérez and I. Molina-Fernández, “Improved variabletransformed Fourier method to analyze nonlinear dielectric waveg-uides,” inProc. SPIE, vol. 3491, 1998, pp. 414–420.

[6] J. E. Dennis, Jr. and D. J. Woods,New Computing Environments: Mi-crocomputers in Large-Scale Computing, A. Wouk, Ed. Philadelphia,PA: SIAM, 1987, pp. 116–122.

[7] T. Tamir, Guided-Wave Optoelectronics. Berlin, Germany: Springer-Verlag, 1998, pp. 62–63.

[8] D. Mihalache and D. Mazilu, “Propagation phenomena of nonlinearguided waves in graded-index planar waveguides,”Proc. Inst. Elect.Eng. J, vol. 138, no. 6, pp. 365–372, Dec. 1991.

[9] T. M. Benson and P. C. Kendall, “Equipartition method for the analysisof two nonidentical closely separated rib waveguides,”Proc. Inst. Elect.Eng. J., vol. 140, no. 1, pp. 62–65, Feb. 1993.

J. G. Wangüemert-Pérezwas born in Las Palmasde Gran Canaria, Spain, in 1968. He received theIng.Tel. degree from the Universidad Politécnicade Madrid, Madrid, Spain, in 1992 and the doctoraldegree from the Universidad de Málaga, Málaga,Spain, in 1999.

Since 1993, he has been been with the ETS In-geniería de Telecomunicación of the Universidad deMálaga as an Assistant and then Associate Professor.His research interests are the development of numer-ical techniques to be applied in the analysis and de-

sign of optical devices.

I. Molina-Fernández received the Ing.Tel. degreefrom the Universidad Politécnica de Madrid, Madrid,Spain, in 1989 and the doctoral degree from theUniversidad de Málaga, Málaga, Spain, in 1993.

Since 1989, he has been been with the ETS In-geniería de Telecomunicación of the Universidad deMálaga as an Assistant and then Associate Professor.His current research interests deal with microwaveand optical components and systems.

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Analysis of dielectric waveguides by a modified fourier