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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 12, JUNE 15, 2009
2077
Finite-Difference Modeling of Dielectric WaveguidesWith Corners
and Slanted Facets
Yih-Peng Chiou, Member, IEEE, Yen-Chung Chiang, Member, IEEE,
Chih-Hsien Lai, Cheng-Han Du, andHung-Chun Chang, Senior Member,
IEEE, Member, OSA
AbstractWith the help of an improved finite-difference
(FD)formulation, we investigate the field behaviors near the
corners ofsimple dielectric waveguides and the propagation
characteristics ofa slant-faceted polarization converter. The
formulation is full-vec-torial, and it takes into consideration
discontinuities of fields andtheir derivatives across the abrupt
interfaces. Hence, the limita-tions in conventional FD formulation
are alleviated. In the firstinvestigation, each corner is replaced
with a tiny arc rather thana really sharp wedge, and nonuniform
grids are adopted. Singu-larity-like behavior of the electric
fields emerge as the arc becomessmaller without specific treatment
such as quasi-static approxi-mation. Convergent results are
obtained in the numerical analysisas compared with results from the
finite-element method. In thesecond investigation, field behaviors
across the slanted facet areincorporated in the formulation, and
hence the staircase approx-imation in conventional FD formulation
is removed to get bettermodeling of the full-vectorial
properties.
Index TermsCorners, dielectric waveguides,
finite-differencemethod (FDM), frequency-domain analysis,
full-vectorial, singu-larities, step index, tiny arcs.
I. INTRODUCTION
A MONG various structures for optical applications,
thestructures containing corners are almost inevitable
andsingularities of fields at corners are known as manifestationsof
the vector nature of electromagnetic waves ([1], and ref-erences
therein). Because of the simplicity of implement andsparsity of the
resultant matrix, the finite-difference method(FDM) is an
attractive numerical method to analyze the opticalwaveguides.
Although some improved finite-difference (FD)schemes [2][4] have
been proposed for full-vectorial modalanalysis, precise modeling of
field singularities near the cornerswith full-vectorial modal
analysis is still very difficult [5].
Manuscript received June 11, 2008; revised August 29, 2008.
First publishedApril 17, 2009; current version published June 24,
2009. This work was sup-ported in part by the Ministry of
Education, Taipei, Taiwan, under the ATU plan,by the National
Science Council of the Republic of China under Grant
NSC95-2221-E-005-127 and Grant NSC97-2221-E-005-091-MY2, and by the
Excel-lent Research Projects of National Taiwan University under
Grant 97R0062-07.
Y.-P. Chiou is with the Graduate Institute of Photonics and
Optoelectronicsand Department of Electrical Engineering, National
Taiwan University, Taipei106-17, Taiwan (e-mail:
[email protected]).
Y.-C. Chiang is with the Department of Electrical Engineering,
Na-tional Chung-Hsing University, Taichung 402-27, Taiwan (e-mail:
[email protected]; [email protected]).
C.-H. Lai and C.-H. Du are with the Graduate Institute of
Photonics andOptoelectronics, National Taiwan University, Taipei
106-17, Taiwan (e-mail:[email protected];
[email protected]).
H.-C. Chang is with the Department of Electrical Engineering,
the GraduateInstitute of Photonics and Optoelectronics, and the
Graduate Institute of Com-munication Engineering, National Taiwan
University, Taipei 106-17, Taiwan(e-mail:
[email protected]).
Digital Object Identifier 10.1109/JLT.2008.2006862
Fig. 1. Cross-sectional view of a square channel waveguide with
the rotatedcoordinate setup at the corner.
Since the field singularity is highly localized in nature,
mostanalysis methods focus on the behavior of fields very close
tothe corner. Within the vicinity of the corner, the spatial
varia-tions of the field is far more rapid than the temporal
variations,and the electromagnetic field near a corner can be
consideredquasi-static [1]. In such cases, the field may be
expanded asthe powers of the distance from the corner [6], [7],
i.e., inFig. 1 or more correctly with additional logarithmic terms
[8].Hadley [9] and Thomas et al. [10] utilized such expansionmethod
in their derivation of improved FD scheme regardingthe field near
the corner. Such treatments mostly focus on thevariation of the
fields in the radial direction, but they cannotproperly model the
behavior of fields in the rotational direc-tion, which is denoted
as the variable , as shown in Fig. 1. Wealso noticed that these
formulations are mostly based on themagnetic fields, which are
continuous at corners and experi-ence less singular difficulties in
their field behaviors, and thusobtaining a proper formulation for
electric fields is still noteasy due to the singularities. Lui et
al. [11] derived a simpleformula by expanding the E as the power of
, but this for-mula is not a thorough derivation as indicated by
Hadley [9].Besides, the applications of the above-mentioned
improved for-mulations are limited to those structures with
interfaces parallelto - or -axis. Finite-element method (FEM) is a
choice forsuch structures, since it can generally fit the structure
better.However, it still needs special treatment for the corner
cases.Efficient finite-element modal solvers with full-vectorial
prop-erties [12], [13] were proposed for the corner problems, but
themesh generation and the programming are relatively tedious.
Another limitation of conventional FDM is that grids in
thecomputation are normally parallel to the axes in the
discretiza-tion of field components. Staircase approximation is
often re-quired when the fields cross a slanted interface between
twodifferent materials. The convergence is slow due to the
staircaseapproximation as compared to other methods without
staircaseapproximation, e.g., FEM. In addition, the full-vectorial
proper-ties may not be accurately modeled under such
approximation.
0733-8724/$25.00 2009 IEEE
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2078 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 12, JUNE 15,
2009
Fig. 2. (a) Cross-sectional view of the stencil used near the
corner.
In our previous paper [14], we proposed an
improvedfull-vectorial FD scheme regarding dielectric waveguides
withpiecewise homogeneous structures. In this paper, we will
adoptthe scheme with nonuniform grids to demonstrate the
singu-larity-like behaviors of the electric fields here even
without anyfield expansion by powers of radius (or additional
logarithmicterms). In addition, our method will show that it can be
easilyextended to those structures with slanted facets.
Implemen-tation will be described in Section II, and some
numericalresults are given in Section III. A simple conclusion is
drawnin Section IV.
II. FORMULATIONIn this section, we will first introduce our
full-vectorial FD
scheme for structures with step-index interfaces. The
interfacecan be slanted to the - and -axes or even curved. Then,
wewill give a simple treatment for modeling a corner with a
smallarc.
A. Finite-Difference Schemes for Step-Index InterfaceThe cross
section of the problem under consideration is
shown in Fig. 2(a) in which a linear slanted interface or
acurved interface lies between the grid points. The basic ideais to
express the field quantities at the surrounding grid pointsas the
expansion of the field at the center point and itsderivatives.
Using the surrounding point atas an example, the derivation process
of the relation for gridpoints with an interface in between can be
summarized as thefollowing steps, and these steps are basically the
same as thoseintroduced in [14]:
1) Express the field as the 2-D Taylor seriesexpansion of the
field just right to the interface and
its derivatives. Similarly, we can also express and
itsderivatives as the expansion of and its derivatives.
2) As shown in Fig. 2(a), transform and its derivatives inthe
global coordinate system into corresponding termsin the local
rotated coordinate system for the linearslanted interface or into
the local cylindrical coordi-nate system for the curved interface
with effective radius
. Similarly, and its derivatives are transformed backto their
correspondings in the coordinates system.
3) Express and its derivatives as a linear combinationof the
field just left to the interface and its deriva-tives by matching
the boundary conditions. In addition tothose given in [14], some
detailed formulas are given in theAppendix.
In the steps, represents the electric field or the magneticfield
, and the subscript denotes the - or -component.Following the above
steps, we can express as thelinear combination of and its
derivatives. If there is nointerface between the grid points, such
expansion and boundarymatching is the same as normal Taylor series
expansion in a ho-mogeneous material. For the second-order scheme,
we need usenine grid points and corresponding derivative terms. We
collectall relation equations based on the nine points shown in
Fig. 2(a),including the point itself, and express them in a
matrixform:
(1)
whereis the vector of the fields
at the nine points, is the matrix of coeffi-cients derived with
the above steps, and
is thevector contains the field quantities at the point andits
derivatives with respect to or . We can obtain a finalset of FD
formulas by taking inverse operation of (1), andthe improved FD
formulas for the terms , ,
, , and so on in are then expressedas a linear combination of
the field values at the nine sampledpoints.
Note that the interface between materials of refractive
indexesand can be slanted or curved. No staircase approximation
is required as that in common FD formulation. The
boundaryconditions across the slanted or curved interface in our
formula-tion are satisfied through coordinate transformation of the
fields.Noteworthily, the derivation process is the same for both E-
andH-formulations. For waveguides made of nonmagnetic media,H is
continuous across the interface between two media, whileE may be
discontinuous. Therefore, we generally expect theH-formulation
converges faster. They do normally, but there isslight difference
between two formulations, since derivative ofH may also be
discontinuous. Therefore, the singular behavioraround a corner
exists for E and H formulations.
B. Treatment Near the Corner
We may adopt another strategy in the analysis of the
cornerproblems, since we have proposed an improved
full-vectorial
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CHIOU et al.: FINITE-DIFFERENCE MODELING OF DIELECTRIC
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Fig. 3. Contours of the electric field distributions for the
fundamental mode of the square channel waveguide. (a) and (b) .
Fig. 4. Contours of the magnetic field distributions for the
fundamental mode of the square channel waveguide. (a) and (b) .
Fig. 5. 3-D plot of the electric field distributions for the
fundamental mode of the square channel waveguide. (a) and (b) .
FD scheme to rigorously treat linear and curved step-index
in-terfaces as in last section. As shown in Fig. 2(b), we model
thecorner as a tiny arc with effective radius rather than a re-ally
sharp wedge. In fact, this replacement may be even closerto the
realistic engineering implementations. Outside the cornerregion, we
adopt the linear slanted scheme to model the inter-face. The
relation between the arc angle and the original cornerangle is
arc corner (2)
where corner is the angle of the corner and arc is the arcangle
used to approximate the corner, as shown in Fig. 2(b). To
enhance the calculation efficiency, we use fine uniform
meshesaround the corner and nonuniform ones elsewhere. As
indicatedin Fig. 2(a), the index of light-gray area was replaced
from to
, thus the index distribution differs from the real corner
case.However, the limit will approach the corner case as gets
moreand more smaller. After an iterative process of updating , ,and
, we will obtain a convergent result.
Although we do not expand the field as the powers of thedistance
from the corner, i.e., in Fig. 1 or more correctly withadditional
logarithmic terms as others do, we will show that ourtreatment can
still model the vectorial nature of the field via thecurvature in
our scheme. And this will be demonstrated in thefollowing
section.
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2080 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 12, JUNE 15,
2009
Fig. 6. 3-D plot of the electric field distributions for the
fundamental mode of the square channel waveguide. (a) and (b) .
Fig. 7. Electric field profiles along the diagonal of the
waveguide near a corner obtained by using different . (a) and (b)
.
TABLE ICONVERGENCE OF THE COMPUTED REFRACTIVE INDEXES OF THE
SQUARE CHANNEL WAVEGUIDE
III. NUMERICAL RESULTS
A. Channel Waveguide CaseReferring to the structure shown in
Fig. 1, we first calculate
the mode fields of a square channel waveguide with widthm, the
refractive indexes of the waveguide and the vacuum
being and , respectively, as shown in Fig. 1.The operating
wavelength is assumed to be m. Theparameters used here are the same
as those used by Sudb [5]and Lui et al. [11]. This case can be
calculated by some typicalmethod, e.g., Goells approach [15], but
they did not treat thecorner problem well as indicated by Sudb [5].
Because of thesymmetry of the field, we only calculate one quarter
of the wholeregion. We also apply transparent boundary condition
(TBC) inthis case and the calculation window is 1.0 m in both
and
directions. We use uniform mesh divisions at the vicinity of
the corner and nonuniform mesh divisions elsewhere to
savecomputation time and memory. The smallest grid size is
m near the corner, and the largest grid size ism near the edge
of the computational
window. The effective radius of the arc at the corner is
chosento be 0.0125 m.
The contours of the computed transverse field components, , ,
and for the fundamental mode are shown in
Figs. 3 and 4, respectively. Figs. 5 and 6 show the 3-D
surfaceplots of the corresponding transverse field components in
Figs. 3and 4. Although we do not add any singularity treatment
nearthe corner, the resultant distributions of the electric fields
stillbehave singularity-like near the corner. On the other hand,
theresultant distributions of the magnetic fields behave
smoothlyaround the corner as expected. Figs. 7 and 8 show the
fieldprofiles along the diagonal of the waveguide near a corner
byusing different values. It can be shown that both electric
fieldcomponents become more and more singularity-like as
getssmaller. However, the field profiles near the corner and the
com-puted effective index are found to converge uniformly. On
theother hand, the H components near the corner converge fasterdue
to their continuity nature.
We calculate another case with the same waveguide widthand
refractive indexes as those used in the above case, exceptthat the
waveguide is operated at the normalized frequency
and the waveguide is surroundedby a perfect electric conductor
(PEC). The same structurehas been analyzed by a vector FEM with
inhomogeneouselements (VFEM-I) [13] and their converged effective
indexes
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CHIOU et al.: FINITE-DIFFERENCE MODELING OF DIELECTRIC
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Fig. 8. Magnetic field profiles along the diagonal of the
waveguide near a corner obtained by using different . (a) and (b)
.
TABLE IIEFFECTIVE INDEXES OF THE MODE FOR THE RIB WAVEGUIDE FOR
DIFFERENT s COMPUTED BY DIFFERENT AUTHORS
Fig. 9. Cross-sectional view of a rib waveguide.
are 1.35638307 and 1.35638381 obtained by the H-VFEM andthe
E-VFEM, respectively. For further comparison, the resultby Goells
approach is 1.35617. Table I lists the convergentbehavior of the
computed effective refractive index in this caseby using our
improved full-vectorial - and -formulations,respectively. We can
see that both the grid sizes and the effec-tive radius of the arc
at the corner influence the convergenceof the results. When the
grid sizes and become smaller, theresults get more closer to those
calculated using the VFEM-Iby Li and Chang [13]. It also shows that
the results using the
-formulation converge faster than those using the -formu-lation.
This is reasonable due to the more singular behavior inthe electric
fields.
B. Rib Waveguide CaseAlthough channel waveguides can provide
better field con-
finement, the cost of fabricating the channel waveguides is
rel-atively higher. In this section, we illustrate the capability
ofour formulations in treating the well-known rib waveguide
in-volving the structure, as shown in Fig. 9. The slab-based
struc-ture provides the field confinement in the -direction and the
ribregion provides the field confinement in the -direction
because
of the relatively higher equivalent refractive index in the rib
re-gion. Since this structure is relatively easier for
semiconductorprocessing, it is one of the most popular structures
in the designof integrated optic devices and systems.
In our calculation, we use the following parameters: the
op-erating wavelength m, rib width m, and
m. The outer slab depth varies from 0.1 to0.9 m. The refractive
indexes of the cover, the guiding layer,and the substrate are , ,
and ,respectively. The parameters for the computational window
are
m, m, and m. We presentin the last two columns of Table II the
computed effective indexof the lowest order mode obtained by our
improved -and -formulations. TBC is adopted. Table II also
providesvalues obtained by previous authors using different
methods:the VFEM with Aitken extrapolation [16], the VFEM with
high-order mixed-interpolation-type elements (Edge-FEM) [17],
andVFEM-I [13]. Figs. 10 and 11 show the contours of the com-puted
transverse field components , , , and forthe lowest mode using our
improved formulations with
m. Note that the field confinement is not very good in-direction
for and 0.9, and using PEC instead of TBC
may affect the sixth and fifth significant digits,
respectively.
C. Rib Waveguides With One Slanted Side WallA typical photonic
integrated system includes many com-
ponents that are polarization sensitive, for example,
integratedswitches, interferometers, amplifiers, receivers, etc.
Thus, it isoften necessary to manipulate or convert polarization
state in
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2082 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 12, JUNE 15,
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Fig. 10. Contours of the electric field distributions for the
mode of the rib waveguide. (a) and (b) .
Fig. 11. Contours of the magnetic field distributions for the
mode of the rib waveguide. (a) and (b) .
Fig. 12. The schematic representation of the single-section
polarization con-verter. (a) The converter structure. (b)
Cross-sectional view of the PRW section.
such guided wave structures, and polarization converters arekey
components in photonic integrated systems. Polarizationrotation in
optical devices can be achieved by induced materialanisotropy.
Previously, such polarization converters employingelectrooptical
[18] and photoelastic effects [19] had been re-ported. However, in
many applications, a passive polarizationconverter is much
preferred, and some very promising andsimpler passive polarization
converters have also been reported
[20][22]. Such passive components may be simpler to fabri-cate
and an important characteristic of these converters is thatthe
polarization rotation is achieved simply by adjusting thegeometry
of the devices. Most of these passive polarizationconverters employ
a longitudinally periodic perturbation struc-ture. Recently, it has
been reported that it is possible to achievepolarization rotation
in a single-section design [23][25], asshown in Fig. 12(a). If an
mode is launched from a standardinput waveguide (IW), this incident
field excites both the firstand second hybrid modes of nearly equal
modal amplitudes.As these two hybrid modes propagate along the
polarizationrotating waveguide (PRW), they would become out of
phase atthe half-beat length and their combined modal fields
producemainly a mode in the following output waveguide (OW).The PRW
is based on a rib waveguide with one side wall slantedat an angle
around 45 , and the cross-sectional view of thisstructure is shown
in Fig. 12(b).
The parameters we use for an asymmetrical slanted-wall
ribwaveguide are the operating wavelength m, rib width
m, m, and the outer slab depthm. The refractive indexes of the
cover, the guiding layer,
and the substrate are , , and ,respectively. The parameters for
the computational window are
m, m, and m. The slanted angle
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CHIOU et al.: FINITE-DIFFERENCE MODELING OF DIELECTRIC
WAVEGUIDES WITH CORNERS AND SLANTED FACETS 2083
Fig. 13. 3-D plot of the electric field distributions for the
first hybrid mode of the asymmetric slanted-wall rib waveguide. (a)
and (b) .
Fig. 14. 3-D plot of the magnetic field distributions for the
first hybrid mode of the asymmetric slanted-wall rib waveguide. (a)
and (b) .
Fig. 15. 3-D plot of the electric field distributions for the
second hybrid mode of the asymmetric slanted-wall rib waveguide.
(a) and (b) .
is 52 . Figs. 13 and 14 show the 3-D plots of the E and M
com-ponents for the first hybrid mode, respectively. Figs. 15 and
16show the 3-D plots of the E and M components for the secondhybrid
mode, respectively. We use 317 by 251 grid meshes incalculation.
The calculated effective refractive indexes for thefirst hybrid
mode are 3.3273423 and 3.3272512 by using -and -formulations,
respectively. The calculated effective re-fractive indexes for the
second hybrid mode are 3.3263567 and3.3263383 by using - and
-formulations, respectively. Wecan see that both hybrid modes have
comparable field compo-nents in - and -directions, and their
polarizations are no longerin the - or -direction but in the
direction parallel or perpen-dicular to the slanted wall. Thus,
they cannot be correctly ob-tained by using semivectorial
formulation. To verify our simu-lation, FEM and Yee-mesh-based FD
beam propagation method(Yee-FD-BPM) [26] are adopted, as shown in
Table III. Both
and fields are used at the same time in the formulationof
Yee-FD-BPM. Furthermore, we also use conventional FDscheme [2] with
staircase approximation and index-average ap-proximation to
calculate the same problem. We find that the hy-brid mode cannot be
correctly obtained either by our codes or
by commercial software. The fundamental modes may become- or
-dominant modes, not as expected. It is shown that our
improved FD scheme can easily handle this structure with
facetsthat are not parallel to - or -axis.
IV. CONCLUSION
Replacing sharp wedges with tiny arcs, we have
implementedfull-vectorial FD scheme to investigate the field
behavior neardielectric waveguide corners. Nonuniform grids are
adopted tosave computation. The electrical fields show
singularity-likedistribution due to abrupt field discontinuities
around thecorner, while the magnetic fields show smooth
distribution dueto field continuity. Numerical results are
convergent and showexcellent approximation to the real wedge
structure. Five-digitaccuracy or more is achieved as compared with
the full-vecto-rial FEM. In addition, the formulation is applicable
to slantedfacets without staircase approximation. Numerical results
froma passive polarization converter shows it can model well
thefull-vectorial properties of fields.
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2084 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 12, JUNE 15,
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Fig. 16. 3-D plot of the magnetic field distributions for the
second hybrid mode of the asymmetric slanted-wall rib waveguide.
(a) and (b) .
TABLE IIIEFFECTIVE INDEXES OF COMPUTED BY DIFFERENT METHODS
APPENDIXFor the linear slanted interface case, the interface
conditions
required in addition to those in [14] are
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
for the magnetic field and
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
for the electric field.For the curved interface case, the
interface conditions re-
quired in addition to those in [14] are
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
for the magnetic field and
(27)
(28)
(29)
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(30)
(31)
(32)
(33)
(34)
for the electric field.
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Yih-Peng Chiou (M03) was born in Taoyuan,Taiwan, in 1969. He
received the B.S. and Ph.D.degrees from the National Taiwan
University, Taipei,Taiwan, in 1992 and 1998, respectively, both
inelectrical engineering.
From 1999 to 2000, he was with the TaiwanSemiconductor
Manufacturing Company, where hewas engaged in research on the
plasma enhancedchemical vapor deposition of dielectric films.
From2001 to 2003, he was with the RSoft Design Group,New York,
where he was engaged in research on
the modeling of simulation techniques and developing of photonic
com-puter-aided-design tools. In 2003, he joined the faculty of the
Graduate Instituteof Electro-Optical Engineering (now Institute of
Photonics and Electronics)and Department of Electrical Engineering,
National Taiwan University, wherehe is currently an Assistant
Professor. His current research interests includemodeling and
computer aided design (CAD) of optoelectronics, which
includesphotonic crystals, nano structures, waveguide devices,
optical fiber devices,light extraction enhancement in LED, display,
solar cell devices, and thedevelopment and improvement of numerical
techniques in optoelectronics.
Yen-Chung Chiang (M06) was born in Hualien,Taiwan, on March 10,
1970. He received the B.S.,M.S., and Ph.D. degrees in electrical
engineeringfrom the National Taiwan University, Taipei, Taiwan,in
1992, 1994, and 2002, respectively.
From 2002 to 2005, he was with the Very Innova-tive Architecture
(VIA) Technologies Inc., Taiwan,where he was engaged in research on
the designof radio-frequency integrated circuits. In 2005, hejoined
the faculty of the Electrical EngineeringDepartment, National
Chung-Hsing University,
Taichung, Taiwan, where he is currently an Assistant Professor.
His currentresearch interests include the numerical analysis
techniques for optical ormicrowave devices and the design of
radio-frequency integrated circuits.
Chih-Hsien Lai, photograph and biography not available at the
time of publi-cation.
Cheng-Han Du, photograph and biography not available at the time
of publi-cation.
Authorized licensed use limited to: Svetlana Boriskina.
Downloaded on September 6, 2009 at 14:50 from IEEE Xplore.
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-
2086 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 12, JUNE 15,
2009
Hung-Chun Chang (S78M83SM00) was bornin Taipei, Taiwan, on
February 8, 1954. He receivedthe B.S. degree from the National
Taiwan University,Taipei, Taiwan, in 1976, the M.S. and Ph.D.
degreesfrom the Stanford University, Stanford, CA, in 1980and 1983,
respectively, all in electrical engineering.
From 1978 to 1984, he was with the Space,Telecommunications, and
Radioscience Laboratoryof Stanford University. In August 1984, he
joined thefaculty of the Department of Electrical
Engineering,National Taiwan University, where he is currently
a Professor. From 1989 to 1991, he served as the Vice-Chairman
of theDepartment of Electrical Engineering and, from 1992 to 1998,
as the Chairmanof the newly-established Graduate Institute of
Electro-Optical Engineeringat the National Taiwan University. He is
also with the Graduate Institute ofCommunication Engineering,
National Taiwan University. His current research
interests include the theory, design, and application of
guided-wave structuresand devices for fiber optics, integrated
optics, optoelectronics, and microwave-and millimeter-wave
circuits.
Dr. Chang is a member of Sigma Xi, the Phi Tan Phi Scholastic
Honor Society,the Chinese Institute of Engineers, the Photonics
Society of Chinese-Ameri-cans, the Optical Society of America, the
Electromagnetics Academy, and theChina/SRS (Taipei) National
Committee (a Standing Committee member during19881993 and since
2006) of the International Union of Radio Science (URSI).He has
been serving as the Institute of Electronics, Information, and
Commu-nication Engineers (Japan) Overseas Area Representative in
Taipei. In 1987, hewas among the recipients of the Young Scientists
Award at the URSI XXIIndGeneral Assembly. In 1993, he was one of
the recipients of the DistinguishedTeaching Award sponsored by the
Republic of China, Ministry of Educationand in 2004, he received
the Merit National Science Council (NSC) ResearchFellow Award
sponsored by the Republic of China, NSC.
Authorized licensed use limited to: Svetlana Boriskina.
Downloaded on September 6, 2009 at 14:50 from IEEE Xplore.
Restrictions apply.