design of electrooptic modulators using a multiobjective optimization approach

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 16, AUGUST 15, 2008 2969

Design of Electrooptic Modulators Using aMultiobjective Optimization Approach

Ademar Muraro Jr., Angelo Passaro, Nancy Mieko Abe, Airam Jonatas Preto, and Stephan Stephany

Abstract—This paper presents preliminary results of a mul-tiobjective approach for optimization of design parameters ofMach–Zehnder-based lithium niobate modulators. This processuses a genetic algorithm to iteratively refine candidate sets of pa-rameters, and the characterization of the modulators is performedby the finite-element method. Test cases include optimizationof a conventional Mach–Zehnder modulator and a variation ofthis kind of device using additional floating electrodes. The setof characteristics includes parameters such as the characteristicimpedance, the effective index of the microwave, and the half-wavevoltage.

Index Terms—Computer-aided engineering, electrooptic mod-ulation, finite-element methods, genetic algorithms, optimizationmethod, traveling-wave devices.

I. INTRODUCTION

O NE of the most urgent demands of the modern techno-logical society is for telecommunication services and in-

creasing bandwidth. This requirement has promoted the devel-opment of new technologies, and integrated optic (IO) devicesand optic components are among the main devices employed toincrease transmission bandwidth.

Amongst these IO devices, electrooptic modulators provideone of the main requirements in fiber-optic communication sys-tems: external modulation. The external modulation is an alter-native to eliminate the frequency chirping due to the direct mod-ulation of semiconductor lasers, which limits the transmissionbandwidth [1]. Mach–Zehnder-type modulators made of tita-nium diffused lithium niobate (Ti:LN) substrate with traveling-wave electrodes allow zero or adjustable frequency chirp, lowbias drift, and large electrooptic coefficients [2]. Technologicaladvances allied to significant investments have resulted in high-quality LN modulator components for use in many commercialand military applications. Nowadays, 2.5 and 10 Gb/s modula-tors are standard commercial products, and 40 Gb/s modulatorsare also emerging in the market [3]. The continuous demandto increase data rate further will push their operating frequencyinto the millimeter-wave range.

Furthermore, new manufacturing processes and modulatorconfigurations are continuously presented in the literature. Re-

Manuscript received August 01, 2007; revised December 28, 2007. Currentversion published October 24, 2008.

A. Muraro Jr., A. Passaro, and N. M. Abe are with the Virtual EngineeringLaboratory, Applied Physics Division, Institute for Advanced Studies, São Josédos Campos 12228-001 SP, Brazil (e-mail: [email protected]).

A. J. Preto and S. Stephany are with the Laboratory for Computing and Ap-plied Mathematics, National Institute for Space Research, São José dos Campos12227-010 SP, Brazil.

Digital Object Identifier 10.1109/JLT.2008.919479

cently, a new manufacturing process using Zn-diffusion for fab-rication of waveguides was presented [4]. Ridge-type modula-tors with Zn-diffused waveguides have been evaluated [5], andthin-film deposition technology has also been explored [6]–[8].The electrooptic effect in organic materials (polymers) has beenstudied in [9] and [10], and additional research has involved theinclusion of nanoparticles in polymers [11]. Techniques to fab-ricate more compact devices are also being developed [12], [13].

Materials, electric or magnetic fields, temperature, and geo-metric parameters affect the electromagnetic characteristics ofelectrooptic modulators. In general, the effect of these parame-ters on the device characteristics cannot be evaluated with ana-lytical approaches. The use of computational models and soft-ware tools based on numerical methods allows analyzing theperformance of the device with different materials and geo-metric design, reducing significantly both cost and time devel-opment.

The development of high-performance devices demands theanalysis of characteristics that depend on a large number of pa-rameters. Optimization techniques based on stochastic methodsare often employed to improve the process of analysis of the ef-fects of these parameters [14], [15]. The stochastic methods areof easy implementation and present good tradeoff between thequality of the solution and the processing time.

Specific optimization techniques have been developed to treatmore complex problems in which more than one characteristichas to be optimized at once [16], [17]. These are named multi-objective problems.

To the best of our knowledge, only optimization of singlecharacteristics of electrooptic modulators has been presentedin the literature, mostly optimizing either the characteristicimpedance or the half-wave voltage, independently. In [18], asingle-objective approach is presented for the optimization ofresonantly enhanced modulators in which the objective functionis a composition of two characteristics: the half-wave voltageand the characteristic impedance. In that paper, the simulatedannealing technique is applied for the optimization process.

In this paper, a multiobjective optimization of electroopticmodulators is carried out by using a stochastic technique inorder to explore the effect of some geometric parameters of thecoplanar waveguide on the performance of the modulator. A ge-netic algorithm (GA) is used for the design optimization of theelectrooptic modulator, and the finite-element method (FEM)is applied on each candidate solution generated by the GA tostudy the propagation of the quasi-static transverse electromag-netic (TEM) modes of the coplanar waveguides (CPWs). TheFEM is one of the numerical methods most used in the simu-lation of engineering problems and has been applied to analyze

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2970 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 16, AUGUST 15, 2008

low- and high-frequency electromagnetic field problems. It hasalso been used for the analysis of microwave structures and op-tical guided wave devices.

A parallel GA implementation executed in a distributedmemory parallel environment using the message-passing para-digm is used in this paper.

Section II of this paper presents the electrical parametersoften used as figure of merit of an electrooptic modulator, theFEM simulation technique used to solve the electric field, anda brief description of the employed genetic algorithms. The re-sults of the optimization are discussed in Section III. Section IVpresents the main conclusions and future work associated withthe analysis and optimization of this sort of device.

II. THEORY

The modulation of the LN modulators is produced by avoltage-induced change in the refractive index. The achiev-able index change is small; thus, either large voltages or longelectrode lengths are needed to obtain sufficient modulation.A useful figure of merit for modulation is the product of thehalf-wave or switching voltage and the electrode length

. By using traveling-wave electrodes, in which the electricalsignal propagates along the same direction as the optical wave,much higher bandwidths can be obtained, but they are limitedby the mismatch between the electrical and optical propagationconstants.

Other electrical parameters of the modulator, which aredetermined from the microwave propagation (electromagneticwave) characteristics of its electrodes, are the effective index ofthe microwave and the characteristic impedance . Thesecharacteristics can be obtained by a simple quasi-static analysisusing scalar potential function. The FEM is capable of handlingtransmission lines with rather arbitrary configurations, such asthick metal electrodes and ridge-type structures, besides theanisotropic permittivity of the LN.

Several parameters have to be analyzed in order to obtaina modulator model that satisfies the required performance. Inorder to reduce time development, design optimization pro-cesses that allow the analysis of different models and are ableto choose the more convenient models are very useful.

The design optimization processes are composed by twomechanisms: the analysis engine and the optimization/searchalgorithm. The design optimization is an iterative process, andcandidate solutions are successively generated by the optimiza-tion algorithm and evaluated by the analysis engine.

A. Mach–Zehnder-Type Modulator on Lithium Niobate

The cross-section of a conventional Mach–Zehnder modu-lator with CPW is presented in Fig. 1. In this figure, is thethickness of the buffer layer, is the electrode thickness, isthe gap between the hot electrode and the ground electrode, and

is the width of the hot electrode. The buffer layer is com-posed by a suitable dielectric medium and is used to distributethe radio-frequency (RF) wave energy in order to achieve a ve-locity match over a wide band of frequencies. On the other hand,such a layer causes a reduction of the microwave field intensityapplied to the modulation process.

Fig. 1. Cross section of the Mach–Zehnder type modulator.

A variation of the conventional Mach–Zehnder modulatorwas presented in [19]. Three extra floating electrodes, also pre-sented in Fig. 1, are included under the main electrodes and de-posited on the LN without electrical contacts with any other partof the device. The gap between pairs of floating electrodes isidentified by in Fig. 1. The purpose of these extra electrodes,according to [19], is to apply the available RF voltage directlyacross the indiffused optical waveguides, increasing the cou-pling between electric field and optical field. The expected resultis to improve modulation, reducing the required RF input. Wereproduced part of the results of [19] in a previous work [20].

An electrooptic modulator can be characterized by some elec-trical characteristics such as the characteristic impedance ,the half-wave voltage , the effective index of the microwave

, the bandwidth , and the driving power for the mod-ulator . The definition of these parameters is presented inthe following:

(1)

(2)

(3)

(4)

(5)

In (1)–(5), and are the capacitance per unit length ofthe CPW calculated with the dielectric materials and with thematerials replaced by vacuum, respectively, is the free-spacevelocity of light, is the effective index of the optical wave,

is the applied voltage, is the length of CPW electrodes,is the free-space optical wavelength, is the extraordinary

refractive index of the substrate, is the pertinent electroopticcoefficient, is the impedance of the microwave source, andis the overlap factor between the electric field of the lightwave(optical wave) and the electric field induced by the electrodes.For x-cut Mach–Zehnder modulators, the optical waveguidesare in the middle of the gap and the overlap factor is equalin both waveguides and defined as

(6)

The optical waveguides used in this sort of devices are builtby diffusing ions (usually Ti, or protons) in a dielectric substrate.

MURARO JR. et al.: DESIGN OF ELECTROOPTIC MODULATORS 2971

One of the most used substrates in integrated optical circuits isthe LN because it presents appropriate optical and mechanicalproperties [21]. In the fabrication process, a diffused channel, inwhich the refractive index varies continuously, is produced.

B. Numerical Analysis of the Microwave Properties UsingFEM

This paper uses a two-dimensional FEM in the quasi-staticapproximation (TEM modes) to numerically compute themicrowave properties generated by each different geometry ofthe modulator [22], [23]. Alternatively, the weighted residualmethod [24] can be used in order to obtain integrodifferentialequations from the differential ones, instead of the variationalprinciples.

In the quasi-static approximation, the electric field iscomputed from the electric potential obtained as the solutionof the Laplace equation [20]

(7)

where the diagonal relative permittivity tensor is given by

(8)

The FEM applied to (7) yields the matrix equation

(9)

where

and (10)

represents a row matrix, stands for a transposed matrix,and represents a complete set of base functions for the usedfinite elements.

In order to evaluate the accuracy of the FEM implementationused in this paper, some results presented in the literature werereproduced [19], [20], [25].

C. Design Optimization Approach

An optimization problem consists of finding one or more so-lutions (usually the best ones) from all solutions in a feasible re-gion that correspond to extreme values, minimum or maximum,of one or more objectives. Some basic factors in this type ofproblem are an objective function, which it is intended to mini-mize or maximize; a set of variables, or parameters, which affectthe value of the objective function; and a set of constraints that

allow some values for the variables and exclude others. For aminimization problem, it can be stated as

minimize

subject to (11)

where is the objective function, is a vector of deci-sion variables , and is a set of con-straints.

Classical optimization methods use a point-by-point ap-proach, in which one solution in each iteration is modified toa different solution, generally better than the previous one.Stochastic-based optimization methods, on the other hand,adopt a different approach, based on “populations,” and canfind multiple solutions in one iteration.

A stochastic method, the genetic algorithm, is used in thispaper for the design optimization process. GAs have beenwidely applied for optimization and for inverse problems inseveral areas. GAs mimic the survival of the fittest individualamong competing organisms [26].

In a standard GA algorithm, as the one adopted in this paper,an initial population of individuals with binary coded chro-mosomes is randomly generated. Each individual represents aparticular set of values for the geometric parameters of the mod-ulator. These values are coded in the bits of the individual chro-mosomes. Each individual is then associated to a feasible geom-etry of the modulator, and its fitness can be evaluated by com-puting the characteristic electric parameters of the modulator.Successive populations are generated, in an iterative process.At each generation, a selection process is performed in order togenerate the next population. The probability that a particularindividual will be selected is equal to the ratio of its fitness tothe best fitness of the current population. In addition, the indi-viduals with the highest fitness values are selected to guaranteethat their genetic information will be passed on to the next pop-ulation. This scheme is known as elitism. After selection, indi-viduals are paired off as parents and crossed over, according toa crossover probability. Each time a crossover occurs, the chro-mosomes are split at a random bit location and their partial bitstrings (all bits before the random bit) swapped between the par-ents. Following crossover, a mutation process is performed onthe individuals driven by a mutation probability. Each bit of theindividual chromosome is changed. The fitness of the individ-uals of the new population is evaluated, and the process is re-peated for a specific number of generations or until a specificconvergence criterion is reached.

Multiobjective optimization aims for the optimization ofmore than one characteristic of the problem. In this case, thereis one objective function for each characteristic to be optimized.It is desirable to identify a set of solutions known as Pareto-op-timal solutions in which each optimal solution corresponds toa particular tradeoff of the different objectives [16], [27]. Themultiobjective optimization tries to find as many optimal solu-tions as possible. Many of the solutions obtained by the GA aregoing to be on the Pareto front, the line formed by the optimalsolutions. The criterion for building the front is based on the


concept of dominance. In a minimization problem, a solutionis said to dominate other solution if there is at least one of

the objective functions of lower than the corresponding oneof and the remaining objective functions of are not worsethan the corresponding ones of . A Pareto-optimal solutiondominates other nonoptimal solutions. In other words, a point

, where represents the space of solutions, is Paretooptimal if for

(12)

An approach to finding the Pareto-optimal solutions thatcompose the Pareto-optimal front is the weighted sum method.This method composes a multiobjective function byperforming the weighted sum of the objective functionsof each characteristic to be optimized. The multiobjectiveoptimization is converted to a single objective optimization asfollows:

Min (13)

where is the parameter vector defined previously,is the weight of the th objective function, and

(14)

If the Pareto-optimal front has a convex distribution, thismethod usually finds solutions on the entire Pareto-optimalset. However, this method cannot find some Pareto-optimalsolutions if the objective space has a nonconvex front. Althoughthis method is simple to use, it must be applied with caution ifone does not know whether the objective space has nonconvexparts or not.

The execution of the GA is very fast. Almost all processingtime is spent by the finite-element analysis engine’s computingthe electromagnetic fields for each solutions. In order to min-imize the total computation time, the optimization processutilizes a parallel GA implementation for distributed memoryparallel environment that follows a master–slave scheme. Themaster processor executes the GA and sends the tentativesolutions to the slave processors for parallel evaluations. Theparallel machine is composed by a cluster of microcomputersinterconnected by a switched fast Ethernet network. The paral-lelization of the software employed the MPICH implementationof the message passing interface communication library [28].

III. RESULTS

In a previous work, the GA algorithm was used to optimizeone characteristic at a time of a conventional Mach–Zehnder-type modulator [29]. The parameters were the gap between elec-trodes and the thickness of the electrodes.

For engineering purposes, it may be convenient to use near-Pareto-optimal solutions in order to include not only solutionsthat are optimal in terms of the Pareto concept, but also solu-tions that are acceptable in terms of the tradeoff between dif-

ferent characteristics. In this paper, the weighted sum methodwas chosen because it is an easy and simple way to composethe multiobjective function and solve many multiobjective prob-lems. The GA searches for possible solutions in the space of so-lutions of the problem. It was implemented using a reproductionoperator and the roulette wheel with one-point crossover. Theoccurrence probability for the crossover was set to 0.8 and themutation to 0.05, and elitism of one individual was applied. Thestopping criteria of the GA algorithm were defined either by themaximum number of generations (30 for the conventional and40 for the floating) or when 51% of the population reaches thesame objective function value. A typical population was com-posed by 20 individuals (tentative solutions).

This section presents the results of two multiobjective experi-ments. The first one corresponds to the optimization of a conven-tional Mach–Zehnder modulator. Two design parameters, thegap and the thickness of the electrodes (Fig. 1), are encoded inthe individual chromosome. The range of these parameters waschosen from the limits of the current manufacturing technology.

Another modulator structure optimized in this paper isthe so-called floating electrodes modulator presented inSection II-A. In this case, three design parameters (gap andthickness of the electrodes and the gap of the floating elec-trodes) are encoded in the individual chromosome.

The parameters range adopted for both configurations are

m m

m m

m m

m

and the gap between the floating electrodes (g) for the floatingelectrodes structures

m m

The effective index of the optical wave in (4) wascomputed by using FEM for a particular diffused waveguide, aspresented in [30].

Figs. 2–8 show the results obtained in various experiments ofmultiobjective optimization with different pairs of characteris-tics. The Pareto frontier was built by varying the weights con-sidering the condition stated in (14).

The tables included in Figs. 2 and 3 and Figs. 5–8 show thecharacteristics defined in Section II-A in three points A, B, andC of the near-Pareto frontier. The wide range of values obtainedfor each characteristic can be noticed. In terms of the Paretoconcept, not all of the solutions on the near-Pareto frontier arepredominant over the other solutions, but the choice of any ofthem depends on an additional factor such as the manufactureof the geometric modeling, power consumption, losses due tocoupling, and applicability, among others. Obviously, such fac-tors can also be included as characteristics or constraints of theproblem if it is desired.

In a first step, two characteristics of a conventionalMach–Zehnder modulator were optimized at a time:and . The optimal value for the first characteristic is 0 V,while the optimal value for the latter one is the impedance of


Fig. 2. Pareto-optimal solutions for the characteristic impedance � and thehalf-wave voltage � � for conventional modulators.

Fig. 3. Near-Pareto-optimal solutions for the characteristic impedance � andthe microwave effective index � for conventional modulators.

Fig. 4. Space of solutions, showing a nonconvex region in the range ��

� � ��.

commercial microwave sources, namely, 50 . The two charac-teristics provide a clear Pareto-optimal front for this problem,as shown in Fig. 2. The set of optimal solutions converges tothe Pareto-optimal front and a sparsely spaced distribution wasobtained, which ensures a good set of tradeoff solutions thatprovides a compromise between the different objectives.

In a second study with conventional modulators, both theand the characteristics were optimized at a time. The op-timal value for is (2.142). The near-Pareto frontier ob-tained is presented in (Fig. 3). In this case, the Pareto frontieris defined by only one point, the one represented by a crosspoint. All solutions in the front are feasible but a lack of so-lutions can be observed in the range . The

Fig. 5. Near-Pareto-optimal solutions for the driving power � � and the mi-crowave effective index � for conventional modulators.

Fig. 6. Near-Pareto-optimal solutions for � and � � for floating electrodesmodulators.

behavior in this region can be evidenced with the introduction ofan additional set of dominated solutions, pertaining to the spaceof solutions of the problem, as shown in Fig. 4. Observing thetable inset in Fig. 3, it can be noticed that points A and B rep-resent devices with very good impedance. The microwave ef-fective index is not good in configuration A. On the other hand,configuration B presents the worst driving power, despite thevery good impedance matching. Finally, configuration C corre-sponds to a solution that presents the better effective index andlower driving power, despite the mismatch of the characteristicimpedance with respect to the commercial microwave sources.

The third study optimized the conventional modulator con-sidering the characteristics and . Again, all solu-tions present in the near-Pareto frontier are feasible, and thereis an indication of a nonconvex region in the range

, as shown in Fig. 5. Notice the flat Paretofrontier from point B to C. In this region, varies 9.1% and

6.2%. The value of varies 14% and varies only0.15% from 6.45 V-cm. Once again it can be noticed that a mod-ulator with low driving power and good effective index was ob-tained (configuration B), despite the mismatch of the character-istic impedance.

The minimization of both the and characteristics ofthe floating electrodes modulator structure yielded the Pareto-optimal solutions shown in Fig. 6. Again a good set of tradeoffsolutions can be noticed. As expected, these results are betterthan those obtained for the conventional structure because the


Fig. 7. Near-Pareto-optimal solutions for � and � for floating electrodesmodulators.

Fig. 8. Near-Pareto-optimal solutions for � � and � for floating elec-trodes modulator.

Pareto frontier in this case is closer to zero for both objectivesthat compose the multiobjective function.

Fig. 7 shows the near-Pareto-optimal solutions for the mini-mization of and in a floating electrodes structure modu-lator. The front is similar to that presented in Fig. 3, suggestinga nonconvex region in the range .

The near-Pareto-optimal frontier for the optimization ofand of modulators with floating electrodes presents

a nonconvex region in the range(Fig. 8). As expected, the floating electrodes improve theperformance of the modulator. For instance, comparing thevalues of the characteristics obtained in point B in Figs. 5 and8, it can be observed that almost the same value of , i.e., thesame modulator bandwidth, was obtained with smaller powerconsumption.

Tables I and II show the geometric parameters obtained foreach modulator labeled A, B, and C in the presented studies.

An interesting point that emerged from the optimization ex-periments is that modulators with good matching of the effectiveindex of the microwave and the optical wave and small drivingpower are obtained, despite the mismatch of the characteristicand source impedances.

The results presented in this section were obtained by em-ploying a cluster of 11 Athlon XP Barton 2500 monopro-cessed nodes. The master node executes the GA, and the eval-uation, i.e., the computation of the characteristics of each indi-vidual, is performed by the nodes. In terms of computational

TABLE IPARAMETERS OF THE CONVENTIONAL MODULATORS NAMED A, B, AND C

IN FIGS. 2, 3, AND 5

TABLE IIPARAMETERS OF THE FLOATING ELECTRODES MODULATORS NAMED A, B,

AND C IN FIGS. 6 –8

performance, the GA required a maximum of 720 evaluationsin the case of conventional modulators and 1200 for floatingelectrodes. Each evaluation takes about 100 s, and the determi-nation of the Pareto-optimal front typically takes between 10to 15 h. An exhaustive algorithm would require approximately11 000 evaluations to explore the complete search space for theconventional modulator and 165 000 in the case of the floatingelectrodes modulator.

IV. CONCLUSION

In this paper, a multiobjective optimization method was ana-lyzed for the design of Mach–Zehnder-type modulators. A Ge-netic Algorithm performed the optimization. Pareto and near-Pareto-optimal solutions were obtained using a multiobjectivefunction given by the weighted sum method. The finite-elementmethod was employed to model the quasi-TEM propagation ofthe microwave in the electrooptic modulator and to analyze itsperformance for each candidate solution.

In order to minimize the processing time, the optimizationemployed a parallel GA using a master–slave scheme, executedin a cluster of PCs.

Numerical results has shown that the approach was able to ob-tain the near-Pareto-optimal front for the characteristics and


, but nonconvex regions in this front were evidenced whenother pairs of characteristics were considered. This fact will re-quire an additional effort in order to analyze the adequacy ofother methods to build the Pareto-optimal front for each set ofcharacteristics to be optimized.

Notice that it is possible to include as many characteristicsand parameters as necessary or desired in the multiobjectiveoptimization. To the best of our knowledge, this work is thefirst that addresses a multiobjective analysis in this scope, andthe analysis started by composing the multiobjective function inorder to optimize only two characteristics each time. The opti-mization of traveling-wave modulator requires tradeoffs amongvarious design characteristics. The multiobjective optimizationapproach is suitable for this kind of problem since the monoob-jective approach optimizes one characteristic to the detriment ofothers. The Pareto frontier shows a set of optimal solutions. Par-ticularly, we can obtain the extreme cases (e.g., modulator con-figurations that present either optimal value of the impedance oroptimal value of effective index) and a set of solutions that are atradeoff between characteristics. The weighted sum method al-lows to weight the importance of each characteristic.

Once defined how to compose the multiobjective function, theinclusion of additional characteristics to be optimized and addi-tional design parameters, such as buffer layer thickness, mate-rials, or parameters related to the fabrication process of the op-tical waveguides, is straightforward.

The proposed method provided modulators with goodmatching of the effective index of the microwave and the op-tical wave and with small driving power, despite the mismatchof the characteristic and source impedances. However, forpractical engineering purposes, these results have to be furtherenhanced in order to optimize additional design parameters andto include formulations that allow treating conductor losses andmodulator encapsulation.

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Ademar Muraro, Jr., received the B.Sc. degree in physics and the M.Sc. degreein electrical engineering from the Universidade de Campinas, São Paolo, Brazil,in 1989 and 1998, respectively.

In 1994, he joined the Institute for Advanced Studies of Aerospace TechnicalCenter (IEAv), Brazil. Since 2002 he has been with the Virtual EngineeringLaboratory, IEAv, where he is currently a Researcher. His research interests in-clude computational optimization techniques, numerical optimization of elec-trooptic devices, high-performance parallel programming, and electromagneticfield computation.

Angelo Passaro received the B.Sc. degree in physics and the M.Sc. degree in nu-clear physics from the Instituto de Física da Universidade de São Paulo, Brazil,in 1981 and 1988, respectively. He received the doctoral degree in electrical en-gineering from the Escola Politécnica da Universidade de São Paulo, Brazil, in1998.

In 1984, he joined the Institute for Advanced Studies of Aerospace TechnicalCenter (IEAv), Brazil. Since 1999, he has been Head of the Virtual EngineeringLaboratory, IEAv. His current research interests are in high-performance par-allel programming, numerical methods, computational optimization techniques,software development, nanostructured semiconductor devices (quantum well,wires, and dots), electromagnetic field computation, and plasma simulation.

Nancy Mieko Abe received the B.Sc., M.Sc., and doctoral degrees from theEscola Politécnica da Universidade de São Paulo, Brazil, in 1989, 1992, and1997, respectively, all in electrical engineering.

In 1998, she joined the Institute for Advanced Studies of Aerospace Tech-nical Center (IEAv), Brazil, as a Research Fellow. Since 2002, she has been with

the Virtual Engineering Laboratory, IEAv, where she is currently a Researcher.Her research interests include electromagnetic field computation, plasma simu-lation, nanostructured semiconductor devices (quantum well, wires, and dots),computational optimizations techniques, numerical methods, and high-perfor-mance parallel programming.

Airam Jonatas Preto received the Ph.D. degree in computer science from theUniversity of Manchester, U.K., in 1997.

He is a Researcher with the Laboratory for Computing and Applied Mathe-matics, National Institute for Space Research, São José dos Campos, Brazil. Hehas experience in the field of high performance and parallel computing, workingon software for scientific problems in many different fields related to space andatmospheric sciences. He has been participating in three research projects to de-velop and employ parallel computing infrastructure and has published work inscientific events and specialized journals.

Stephan Stephany received the Ph.D. degree in applied computing from theBrazilian Institute for Space Research in 1997.

He is a Researcher with the Brazilian Institute for Space Research and has ex-perience in inverse problems applied to space science and technology. He alsohas been working with space and atmospheric science software aiming at per-formance optimization and parallelization and takes part in research projects re-lated to the parallel computing infrastructure of its institution. He has publishedwork in scientific events and specialized journals in fields of inverse problemsand parallel computing.

design of electrooptic modulators using a multiobjective optimization approach

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