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1514 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 Finite-Element Analysis of Photonic Crystal Cavities: Time and Frequency Domains Vitaly Félix Rodríguez-Esquerre, Masanori Koshiba, Fellow, IEEE, Fellow, OSA, and Hugo E. Hernández-Figueroa, Senior Member, IEEE Abstract—Finite-element analysis in time and frequency domains using perfectly matched layers and isoparametric curvilinear elements for finite-size photonic-crystal (PC) cavities is presented in this paper. The time-domain approach includes current sources, the full band scheme, and the slowly varying envelope approximation; consequently, bigger time steps can be used independent of the size of the elements. The resonant frequency, quality factor, effective modal area, and field distribution for each mode can be obtained in a single simulation. A strategy to compute the higher resonant modes by using only a quarter of the cavity and adequate boundary conditions is also presented. Index Terms—Cavity resonator, finite-element method (FEM), isoparametric element, photonic crystal (PC), factor, time-domain analysis. I. INTRODUCTION R ECENTLY photonic crystals (PCs) have attracted the attention of researchers because of their fascinating ability to suppress, enhance, or otherwise control the emission of light in a selected frequency range by judicious choice of the constant lattice, filling factor, and refractive indexes obtaining a complete photonic bandgap (PBG), where light cannot propagate through the crystal in any direction [1]–[7]. Photonic crystal structures have a number of potential applications: resonant cavities, lasers, waveguides, low-loss waveguide bends, junctions, couplers, and many more. Resonant cavities are formed by introducing point defects in the periodic lattice. These structures exhibit localized modes in the bandgap region with a very narrow factor in a small area, becoming a spectra and high-quality good candidate for laser fabrication [1]–[6]. Numerical simulation of defects in PCs is essential for the study of the localized modes, and it is divided in frequency- and time-domain methods. In this paper, the finite-element method (FEM) is applied, in both the time and frequency domains, to further investigate the properties of PC resonant cavities. The plane-wave method (PWM) [1], one of the most popular frequency-domain methods, is able to find the resonant freManuscript received January 15, 2004; revised November 22, 2004. This work was supported by the 21st Century Center of Excellence (COE) Program in Japan. V. F. Rodríguez-Esquerre and M. Koshiba are with the Division of Media and Network Technologies, Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan (e-mail: vitaly@ dpo7.ice.eng.hokudai.ac.jp; [email protected]). H. E. Hernández-Figueroa is with the Department of Microwaves and Optics, School of Electrical and Computer Engineering, University of Campinas (UNICAMP), 13083-970 Campinas, São Paulo, Brazil (e-mail: [email protected]. unicamp.br). Digital Object Identifier 10.1109/JLT.2005.843441 quencies and mode fields by using a supercell scheme with periodic boundary conditions. It assumes an infinite lattice with periodic defects, and the coupling of these defects leads to a defect band. The coupling effect between neighboring defects decays exponentially with the distance among defects, and a supercell of moderate size can give accurate information of the defect modes. Recently, the finite-element formulation using perfectly matched layers (PMLs) has been used to analyze finite-size cavities [5] where quadrilateral elements have been used to discretize the spatial domain and the monopole mode for the transverse-electric (TE) mode (electric field parallel to the rods) was analyzed. In the present approach, the second-order triangular element with curved sides, or so-called isoparametric curvilinear element, is used. It permits the accurate modeling of the geometry using a less number of elements. If compared with the linear ones, it can speed up the computer time by a factor of 2. Here, the TE and transverse-magnetic (TM) modes (electric and magnetic field parallel to the rods, respectively) are analyzed, and a strategy to compute the higher resonant modes is also presented. In the time domain, finite-difference time-domain (FDTD) algorithms [2], [4] and [6] are widely used and are the most commonly method used for PC cavities analysis. Finer discretization is required to avoid the staircase problem when curved geometries have to be modeled; additionally, short time steps have to be used because they simulated the total field propagation, and the time step size is given by the Courant stability criterion. On the other hand, the time-domain FEM has been applied to calculate the field distribution in microwave cavities [8]. However, the approach presented in [8] took into account the total field; then, the use of shorter time steps to attain the convergence is required. In order to overcome this limitation, a slowly varying envelope time-domain finite-element scheme for the analysis of PCs and optical devices has been introduced and successfully used in [9]–[11] where the narrow and wide bands have been treated in [9] and the full band has been treated in [10] and [11]. In this paper, the full-band slowly varying envelope finite-element time-domain method is used to analyze PC resonant cavities. In this approach, under the condition that the modulation frequency is much lower than the carrier frequency, the electromagnetic field is separated in the fast varying component and the slowly varying component (envelope); in this way only the field envelope is solved, and, consequently, larger time steps can be used. Here, the time step used is at least five times larger than that one necessary if the fast variation is taken into account. It is important when cavities with higher values are analyzed. Due to the intrinsic curvilinear nature of the PC devices (dielectric 0733-8724/$20.00 © 2005 IEEE RODRÍGUEZ-ESQUERRE et al.: FINITE-ELEMENT ANALYSIS OF PC CAVITIES 1515 TABLE I DEFINITION OF SYMBOLS IN WAVE EQUATION (1) TABLE II VALUES OF s AND s Fig. 1. Finite-size 2-D PC cavities with the rods/holes perpendicular to the y -z plane. The boundaries are surrounded by PMLs to simulate open boundaries: (a) square cavities and (b) hexagonal cavities. rods or circular holes), isoparametric triangular elements [12] were implemented to discretize efficiently the geometry. This favors the use of a moderated number of elements. Numerical integration based on seven-point formulas were applied to assemble the element matrices [13]. Only a quarter of the resonant cavity was discretized because of the symmetrical property of the cavities, and PMLs were used to simulate an open boundary, avoiding undesirable reflections from the computational window edges. All of these considerations considerably reduced the computational effort and time processing. The paper is organized as follows. In Section II, the finite-element time-domain approach is shown, highlighting the scheme used for time evolution, which allows wide-band pulse propagation to be considered. The frequency formulation, which starts from the wave equation, is also presented. In Section III, numerical results concerning the analysis of two-dimensional (2-D) square and hexagonal cavities are presented. Finally, in Section IV, the main conclusions are given. II. FINITE-ELEMENT FORMULATION We consider a finite-size PC cavity on a 2-D spatial domain on the - plane with the axis of the rods/holes parallel to the axis, as shown in Fig. 1. The cavities are formed by introducing point defects, in this case by removing or filling the central rod/hole in a periodic array. The cavity is surrounded by PMLs to simulate open boundaries, and the variation in the direction is neglected . As we can see from the geometry, the cavity has two-folded symmetry, and only a quarter of the cavity needs to be computed. A. Time Domain for TE modes; , , , and for TM modes; is the refractive index; is the free-space permeability; and is the unit vector in the direction (see Table I). are parameters related to the absorbing Here, , , and boundary conditions of the PML type, and the parameter is given by in PML regions in other regions (2) where , is the angular frequency, is the thickness of the PML layer, is the distance from the beginning of PML, and is the theoretical reflection coefficient. The other parameters and take the values described in Table II. Since the wave is considered to be centered at the frequency , the field is given by , represents the wave’s slowly varying envewhere lope, and the current density is given in the same way, i.e., , with being the slowly varying current density. Substituting these expressions into (1), dividing the spatial domain into curvilinear triangular elements, and applying the conventional Galerkin/FEM procedure, we obtain the following equation for the slowly varying envelope: (3) where the matrices and are given by (4a) The TE and TM fields are compactly described by the Helmhothz equation considering an external current density , as follows: (4b) where is the speed of light in free space; , current (external excitation); (1) is the density of , , and where is the shape function vector, denotes a transpose, extends over all different elements, and represents the external excitation and is given by (5), shown at the bottom of 1516 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 is the energy at an arbitrary time position, is where the energy after one cycle, and the cycle corresponds to the resonant frequency. A more general way to compute is (8b) where and are the amplitudes of the electromagnetic fields at the arbitrary times and , respectively. B. Frequency-Domain Analysis Fig. 2. Six-node isoparametric second-order triangular element. the page. Here, the vector is nonzero only at the positions corresponding to the nodal points, where it is applied. We use isoparametric curvilinear six-node elements for the spatial discretization [12] (see Fig. 2). The discretization in the time domain is based on Newmark–Beta formulation [14], and following [10], we obtain The frequency-domain scalar equation governing the transverse TE and TM modes, over a 2-D spatial domain - in, is obtained by replacing cluding PMLs, free of charges with the factor in (1) as follows: the operator (9) The parameters in (9) are the same given in (1). Applying the Galerkin method to (9), we obtain (10) (6) where given by is the vector field, and the matrices and are where is the time step, the subscripts , , and denote the th, th, and the th time steps, respectively, controls the stability of the method. The and marching relation is given as (11a) (11b) (7) We solved (7) by lower and upper triangular matrices (LU) decomposition at the first time step and by forward and backward substitutions at each time step to obtain the subsequent field. The initial conditions are . The factor is an important parameter of the cavity and tells us the number of oscillations for which the energy decays to of its initial value. From energy time variation, we can obtain the factor as (8a) The resulting sparse complex eigenvalues system is efficiently solved by the subspace iteration method [15]. We obtain then the field distribution and its complex resonant frequency . The factor of the mode associated to the complex frequency is given by (12) where and stand for the real and imaginary parts, respectively. The modal area is computed from the field distribution as in [16] (13) where is the or field. for TE modes for TM modes. (5) RODRÍGUEZ-ESQUERRE et al.: FINITE-ELEMENT ANALYSIS OF PC CAVITIES 2 Fig. 3. Mesh used in the simulation of the square n n cavity, where only a quarter of the cavity is analyzed, and the PML thickness is 1 m. 1517 = = 0 Fig. 4. Variation of the amplitude of the electric field at y z corresponding to the 5 5 cavity, during the excitation (region A) and after turning off the excitation (region B). 2 III. NUMERICAL RESULTS To show the usefulness and validity of the approaches presented in this paper, several cavities are analyzed. Comparisons between results in frequency and time domains are given through several numerical examples. The cavities analyzed in the present paper are formed by removing the central rod at the center of a square [5], and the hexagonal lattice of dielectric rods [2], [4] which have resonance for the TE modes and by filling the central hole in a hexagonal lattice of air holes in a dielectric substrate [1], which presents resonance for the TM modes. Localized modes will appear into the bandgap as a consequence of the defect introduced [1]–[6]. In time domain, we assume a current density with a Gaussian variation in the time of the form (14) Fig. 5. Energy spectra in the cavity, showing a resonance at the normalized : . frequency !a= c 2 = 0 378 where and were defined to have a sufficiently wide band1.0 fs, and width; in addition, the time step size used was in (7). we used A. Square Cavity This cavity is formed by removing the central rod in an array of dielectric rods with refractive index , in air [5] [see Fig. 1(a)]. Only a quarter of the cavity needs to be simulated because of its symmetry. Here, is taken as the values 5, 7, 9, and 11. 0.586 52 m to From [5], we chose the lattice constant obtain the resonance frequency at 1.55 m. We used a computational domain of 4.25 m 4.25 m discretized using 26 869 nodal points, as shown in Fig. 3. With this number of nodal points, we obtained solutions, which are not affected by increasing the number of them (convergence). The PML thickness was 1.0 m. The central wavelength used 1.5 m, 30 fs, and 70 fs. was , corresponding The time variation of the field at to the 5 5 cavity is shown in Fig. 4. When the excitation is present, the field grows and, after turning off the excitation, the resonance effect is present, and we can see an exponential decay of the amplitude of the electric field. The factor is determined from this time variation as the ratio of the stored power divided Fig. 6. Electric-field distribution for the resonant mode after 1024 fs. This : . mode corresponds to !a= c 2 = 0 378 by the lost power after one cycle, using (8a) or (8b). For 1.55 m, the cycle is 5.16 fs. The spectra distribution of the energy in the 5 5 cavity, which is determined by Fourier transform of the complex elec, is shown in Fig. 5. In tric field monitored at 1024 fs. Fig. 6, we show the electric-field distribution at From this, the mode area was computed as in [16], obtaining 0.32 . factor for different sizes of the cavity is shown in The Fig. 7. We can see an excellent agreement between the two methods, time and frequency domains, in the calculation of . 1518 Fig. 7. JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 Quality factor and dependence of the resonant frequency on the size of the cavity. TABLE III VALUES OF FACTOR AND RESONANT FREQUENCIES OBTAINED FOR DIFFERENT CAVITY SIZES IN TIME AND FREQUENCY DOMAINS Q The relative error between both approaches is about 1%, and the results agree very well with previously published ones [5]. We also observed a variation of the resonant frequency as a function of the size of the cavity. It decreases as the number of rods increases. This variation is observed in the frequency- and time-domain analyses, and it is shown in Fig. 7. This variation corresponds to a maximum of 2 nm in the optical communication region. The numerical values for the factor and resonant frequencies for different cavity size are shown in Table III. Fig. 8. Mesh used in the simulation of the hexagonal resonant cavity showing the computational domain used in the simulations (a quarter of cavity). TABLE IV VALUES OF FACTOR AND RESONANT FREQUENCIES OBTAINED FOR DIFFERENT CAVITY SIZES IN TIME AND FREQUENCY DOMAINS Q B. Hexagonal Cavities As a second example, we analyzed a hexagonal cavity; these cavities are formed by removing the central rod in four-ring and five-ring hexagonal lattices of dielectric rods with refrac, 0.378 in air [2], [4] [see tive index Fig. 1(b)]. From [2] and [4], we chose the lattice constant 0.7254 m to obtain the resonance frequency at 1.55 m. In Fig. 5, from the symmetry of the problem, only a quarter of the cavity was discretized using 16 805 nodal points. The resulting mesh is shown in Fig. 8. The PML thickness was 1.0 m. 1.5 m, 30 fs, and The central wavelength used was 70 fs. The factor in time domain was determined using (8), since 5.16 fs for 1.55 m, the factor computed in the time and frequency domains for the four-ring and the five-ring , and cavities are shown in Table IV. , respectively. The spectra distribution of the energy in the cavity shown in Fig. 9, was determined by Fourier transform of the complex . In Fig. 10, we show the electric field monitored at 1024 fs. From the distribution electric-field distribution at Fig. 9. Energy spectra in the cavity, showing the resonance at the normalized 2 = 0 468. frequency !a= c : of the electric field, the mode area 0.51 . obtaining was computed as in [16], RODRÍGUEZ-ESQUERRE et al.: FINITE-ELEMENT ANALYSIS OF PC CAVITIES 1519 Fig. 10. Electric-field distribution of the resonant mode for the five-ring cavity after 1024 fs, corresponding to !a=2c = 0:468. Fig. 11. Mesh used in the simulation of the hexagonal resonant cavity formed by holes in a dielectric substrate showing the computational domain used in the simulations (a quarter of cavity). As a third example, we analyzed a hexagonal cavity formed by filling the central hole in a five-ring hexagonal lattice of air 0.45 , in a dielectric substrate [1]. The characholes with teristics of the cavity are and . This cavity exhibits resonance for several TM modes, including degenerated ones; for simplicity and without loss of generality, we as1 m. The computational sume the constant lattice to be domain was discretized using 16 793 nodal points, and the PML thickness was 1 m (see Fig. 11). The parameters for the current source and time step were the same used in the previous simulations. In time-domain analysis, to obtain the factor and the magnetic-field distribution corresponding to the several modes, we used different initial boundary condition and current source arrangement, as shown in Fig. 12. The first quadrupole mode direction, was computed using one source of current in the direction and the boundary condibut it can also be in the at and . The second quadrupole tions mode was computed using two sources of current in the and directions, respectively, with the boundary conditions at and , respectively. The monopole mode was computed using one source of current, and with the same boundary condition it was placed at than the latter, and the hexapole mode was computed using two and directions, respectively. sources of current in the and The boundary conditions in this case were at and , respectively. Fig. 12. Boundary condition and placement of the resultant J in order to obtain the correspondent (a) quadrupole 1, (b) quadrupole 2, (c) monopole, and (d) hexapole mode. represents the current entering the plane of the paper, ^ is the vector and represents the current leaving the plane of the paper and n normal to the boundary. In the frequency-domain analysis, we impose the same boundary conditions used in time domain to compute a specific mode. The resonant frequencies and magnetic-field pattern determined by time domain are shown in Fig. 13. The same results are obtained in frequency domain. factor The results obtained for resonant frequency and agree well in both the approaches and are shown in Table V. The computation of higher modes are more elaborated because we have to impose appropriate boundary conditions. Besides, in the time domain, the placement of the sources has to be done in a strategic way in order to excite the desired mode, and in the frequency domain we have to choose the initial input guess to start the iterative method for solving the eigenvalues equation to get the convergence to the appropriate eigenvalue. The results obtained in time and frequency domains using the FEM are in good agreement with those previously published obtained in the frequency domain [8], FDTD [4], and the plane-wave expansion method [1]. Although the results obtained here were for specific values of the constant lattice , they are scalable and are valid for any cavity with the same parameand refractive index with other values of . ters IV. CONCLUSION An efficient time-domain FEM for the analysis of PC resonant cavities has been presented, and the results are in good agreement with those obtained in the frequency domain and those previously published. The method proved to be very effective in this calculation because it provides a wealth of data from which 1520 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 3, MARCH 2005 REFERENCES Fig. 13. Magnetic-field pattern of three resonant modes in the 2-D five-rings 0.45 : (a) the quadrupole 1 and hexagonal single-defect cavity with r quadrupole 2; (b) the monopole; and (c) the hexapole mode. = a TABLE V VALUES OF Q FACTOR AND RESONANT FREQUENCIES OBTAINED DIFFERENT MODES IN TIME AND FREQUENCY DOMAINS FOR it is possible to derive all the quantities of interest for the design of nanocavities ( factor, resonant frequency, field pattern, and mode area) analyzing only quarter of the cavity. This approach seems to be highly efficient because it uses curvilinear elements and the slowly varying wave approximation. With these considerations, coarse meshes and larger time step can be used. The computational time can be sped up by a factor of two because of the former condition if it is compared with the use of the linear element, and the latter one can speed up the time step by at least five times if the fast variation is taken into account. It is important when cavities with higher values are analyzed. Consequently, significant improvement in computational efficiency can be attained without sacrificing calculation accuracy. A 3-D approach is now under consideration ACKNOWLEDGMENT The authors would like to thank Dr. Y. Tsuji from Hokkaido University, Sapporo, Japan, for valuable discussions. [1] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton Univ. Press, 1995, ch. 7. [2] K. Sakoda, Optical Properties of Photonic Crystals. New York: Springer-Verlag, 2001, ch. 6. [3] “Photonic Crystal,” IEEE J. Quantum Electron, vol. 38, pp. 724–963, Jul. 2002. [4] K. Sakoda, “Numerical study on localized defect modes in two-dimensional triangular photonic crystals,” J. Appl. Phys., vol. 84, pp. 1210–1214, Aug. 1998. [5] J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Y. H. Lee, “Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 15, pp. 2316–2324, Aug. 1998. [6] J. Huh, J. K. Hwang, H. Y. Ryu, and Y. H. Lee, “Nondegenerate monopole mode of single defect two-dimensional triangular photonic band-gap cavity,” J. Appl. Phys., vol. 92, pp. 654–659, Jul. 2002. [7] M. Marrone, V. F. Rodriguez-Esquerre, and H. E. Hernandez-Figueroa, “Novel numerical method for the analysis of 2D photonic crystals: The cell method,” Opt. Express, vol. 10, pp. 1299–1304, Nov. 2002. [8] D. C. Dibben and R. Metaxas, “Frequency domain vs. time domain finite element methods for calculation of fields in multimode cavities,” IEEE Trans. Magn., vol. 33, no. 2, pp. 1468–1471, Mar. 1997. [9] M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuit components,” J. Lightw. Technol., vol. 18, no. 1, pp. 102–110, Jan. 2000. [10] V. F. Rodríguez-Esquerre and H. E. Hernández-Figueroa, “Novel timedomain step-by-step scheme for integrated optical applications,” IEEE Photon. Technol. Lett., vol. 13, no. 4, pp. 311–313, Apr. 2001. [11] A. Cucinotta, S. Selleri, L. Vincetti, and M. Zooboli, “Impact of the cell geometry on the spectral properties of photonic crystal structures,” Appl. Phys. B, Photophys. Laser Chem., vol. 73, pp. 595–600, Oct. 2001. [12] A. Niiyama, M. Koshiba, and Y. Tsuji, “An efficient scalar finite element formulation for nonlinear optical channel waveguides,” J. Lightw. Technol., vol. 13, no. 9, pp. 1919–1925, Sep. 1995. [13] G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng., vol. 7, pp. 405–408, 1973. [14] N. M. Newmark, “A method of computation for structural dynamics,” J. Eng. Mechan. Divis., ASCE, vol. 85, pp. 67–94, July 1959. [15] B. N. Parlet, The Symmetric Eigenvalue Problem. Englewood Cliffs, NJ: Prentice-Hall, 1990. [16] M. R. Watts, S. G. Johnson, H. A. Haus, and J. D. Joannopoulos, “Electromagnetic cavity with arbitrary Q and small modal volume without a complete photonic bandgap,” Opt. Lett., vol. 27, pp. 1785–1787, Oct. 2002. Vitaly Félix Rodríguez-Esquerre was born in Peru, on February 4, 1973. He received the B.S. degree in electronic engineering from the University Antenor Orrego UPAO, Trujillo, Peru, in 1994 and the M.Sc. and Ph.D. degrees in electrical engineering in 1999 and 2003, respectively, from the University of Campinas, São Paulo, Brazil. He is a Postdoctoral Research Fellow at the Division of Media and Network Technologies, Hokkaido University, Sapporo, Japan. His current research interest includes numerical methods for modal and propagation analysis of electromagnetic fields in conventional and photonic-crystal integrated optics waveguides, integrated optics, and optical fibers. RODRÍGUEZ-ESQUERRE et al.: FINITE-ELEMENT ANALYSIS OF PC CAVITIES Masanori Koshiba (SM’84–F’03) was born in Sapporo, Japan, on November 23, 1948. He received the B.S., M.S., and Ph.D. degrees in electronic engineering from Hokkaido University, Sapporo, Japan, in 1971, 1973, and 1976, respectively. In 1976, he joined the Department of Electronic Engineering, Kitami Institute of Technology, Kitami, Japan. From 1979 to 1987, he was an Associate Professor of Electronic Engineering at Hokkaido University and became a Professor in 1987. He has been engaged in research on wave electronics, including microwaves, millimeter waves, lightwaves, surface acoustic waves (SAW), magnetostatic waves (MSW), electron waves, and computer-aided design and modeling of guided-wave devices using the finite-element method, the boundary-element method, and the beam-propagation method. He is an author or coauthor of more than 260 research papers in English and more than 130 research papers in Japanese, published in refereed journals. He is an author of books Optical Waveguide Analysis (New York: McGraw-Hill, 1992) and Optical Waveguide Theory by the Finite Element Method (Tokuo, Japan: KTK Scientific; Dordrecht, The Netherlands: Kluwer Academic, 1992) and is coauthor of the books Analysis Methods for Electromagnetic Wave Problems (Boston, MA: Artech House, 1990), Analysis Methods for Electromagnetic Wave Problems, Vol. Two (Boston, MA: Artech House, 1996), Ultrafast and Ultra-Parallel Optoelectronics (Chichester, U.K.: Wiley, 1995), and Finite Element Software for Microwave Engineering (New York: Wiley, 1996). Dr. Koshiba is a Fellow of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan and is a Member of the Institute of Electrical Engineers (IEE) of Japan, the Institute of Image Information and Television Engineers of Japan, the Japan Society for Simulation Technology, the Japan Society for Computational Methods in Engineering, and the Optical Society of America (OSA). In 1987, 1997, and 1999, he was awarded the Excellent Paper Awards from the IEICE; in 1998, he received the Electronics Award from the IEICE-Electronics Society; and in 2004, he received the Achievement Award from the IEICE. From 1999 to 2000, he served as a President of the IEICE Electronics Society, and in 2002, he served as a Chair of the IEEE Lasers & Electro-Optics Society (LEOS) Japan Chapter. Since 2003, he has served on the Board of Directors of the Applied Computational Electromagnetics Sociery (ACES). 1521 Hugo E. Hernández-Figueroa (M’94–SM’96) received the B.Sc. in electrical engineering from the Federal University of Rio Grande do Sul (UFRGS), Porto Alegre, Brazil, in 1983; the M.Sc. degree in electrical engineering and the M.Sc. degree in applied mathematics and computer science from the Pontifical Catholic University of Rio de Janeiro (PUC/RJ), Rio de Janeiro, Brazil, in 1986 and 1988, respectively; and the Ph.D. degree in physics from the Imperial College of Science, Technology and Medicine, University of London, London, U. K., in 1992. From 1984 to 1986, he was a Research Assistant at the Computer Science Department, PUC/RJ. From 1985 to 1989, he was an Assistant Professor at the Military Institute of Engineering, Department of Electrical Engineering, Rio de Janeiro, Brazil. From 1992 to 1995, he was a Senior Research Fellow at the Department of Electronic and Electrical Engineering, University College London (UCL), University of London. In March 1995, he joined the Department of Microwaves and Optics (DMO), School of Electrical and Computer Engineering (FEEC), University of Campinas (UNICAMP), São Paulo, Brazil, where he is a Professor. From January to March 1998, he was a Visiting Professor at the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque. He has published more than 120 papers in international indexed journals and conferences. His research interests concentrate on a wide variety of modeling and computational aspects of wave-electromagnetics-based devices and phenomena. He is also involved in research projects dealing with information technology applied to technology-based education. Dr. Hernández-Figueroa served as the IEEE Microwave Theory and Techniques Society (MTT-S) Representative for Latin America (1998–2002), the IEEE MTT-S Chapter Chair linked to the IEEE South Brazil Section (1998–2002), and Administrative Committee (AdCom) (Board of Directors) Member of the IEEE Education Society (1998–2001). In 2000, he was awarded the IEEE Third Millennium Medal. In 2002, he was elected Member of the Electromagnetics Academy, held at the Massachusetts Institute of Technology (MIT), Cambridge, and was also listed in the Who’s Who in Electromagnetics (The Electromagnetics Academt, 2002). Between 1996 and 1998, he was Vice-President of the Brazilian Electromagnetics Society (SBMAG), and between 1998 and 2000, he was Vice-President of the Brazilian Microwave and Optoelectronics Society (SBMO). He has served as general chairman and organizer of several international conferences and editor of several special issues in top technical journals related to photonics, microwaves, and education. Since January 2001, he has been a Member of the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE, THEORY AND TECHNIQUES and has served as an Associate Editor (Theory) of the JOURNAL OF LIGHTWAVE TECHNOLOGY since January 2004.