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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
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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
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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.
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=
= 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 .
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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
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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.
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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.