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Finite-Difference Modeling of Dielectric Waveguides With Corners and Slanted Facets Yih-Peng Chiou, Member, IEEE, Yen-Chung Chiang, Member, IEEE, Chih-Hsien Lai, Cheng-Han Du, and Hung-Chun Chang, Senior Member, IEEE, Member, OSA
Abstract—With the help of an improved finite-difference (FD) formulation, we investigate the field behaviors near the corners of simple dielectric waveguides and the propagation characteristics of a slant-faceted polarization converter. The formulation is full-vectorial, and it takes into consideration discontinuities of fields and their derivatives across the abrupt interfaces. Hence, the limitations in conventional FD formulation are alleviated. In the first investigation, each corner is replaced with a tiny arc rather than a really sharp wedge, and nonuniform grids are adopted. Singularity-like behavior of the electric fields emerge as the arc becomes smaller without specific treatment such as quasi-static approximation. Convergent results are obtained in the numerical analysis as compared with results from the finite-element method. In the second investigation, field behaviors across the slanted facet are incorporated in the formulation, and hence the staircase approximation in conventional FD formulation is removed to get better modeling of the full-vectorial properties. Index Terms—Corners, dielectric waveguides, finite-difference method (FDM), frequency-domain analysis, full-vectorial, singularities, step index, tiny arcs.
I. INTRODUCTION
A
MONG various structures for optical applications, the structures containing corners are almost inevitable and singularities of fields at corners are known as manifestations of the vector nature of electromagnetic waves ([1], and references therein). Because of the simplicity of implement and sparsity of the resultant matrix, the finite-difference method (FDM) is an attractive numerical method to analyze the optical waveguides. Although some improved finite-difference (FD) schemes [2]–[4] have been proposed for full-vectorial modal analysis, precise modeling of field singularities near the corners with full-vectorial modal analysis is still very difficult [5].
Manuscript received June 11, 2008; revised August 29, 2008. First published April 17, 2009; current version published June 24, 2009. This work was supported 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 NSC952221-E-005-127 and Grant NSC97-2221-E-005-091-MY2, and by the Excellent Research Projects of National Taiwan University under Grant 97R0062-07. Y.-P. Chiou is with the Graduate Institute of Photonics and Optoelectronics and Department of Electrical Engineering, National Taiwan University, Taipei 106-17, Taiwan (e-mail:
[email protected]). Y.-C. Chiang is with the Department of Electrical Engineering, National 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 and Optoelectronics, National Taiwan University, Taipei 106-17, Taiwan (e-mail:
[email protected];
[email protected]). H.-C. Chang is with the Department of Electrical Engineering, the Graduate Institute of Photonics and Optoelectronics, and the Graduate Institute of Communication 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 rotated coordinate setup at the corner.
Since the field singularity is highly localized in nature, most analysis methods focus on the behavior of fields very close to the corner. Within the vicinity of the corner, the spatial variations of the field is far more rapid than the temporal variations, and the electromagnetic field near a corner can be considered quasi-static [1]. In such cases, the field may be expanded as the powers of the distance from the corner [6], [7], i.e., in Fig. 1 or more correctly with additional logarithmic terms [8]. Hadley [9] and Thomas et al. [10] utilized such expansion method in their derivation of improved FD scheme regarding the field near the corner. Such treatments mostly focus on the variation of the fields in the radial direction, but they cannot properly model the behavior of fields in the rotational direction, which is denoted as the variable , as shown in Fig. 1. We also noticed that these formulations are mostly based on the magnetic fields, which are continuous at corners and experience less singular difficulties in their field behaviors, and thus obtaining a proper formulation for electric fields is still not easy due to the singularities. Lui et al. [11] derived a simple formula by expanding the E as the power of , but this formula is not a thorough derivation as indicated by Hadley [9]. Besides, the applications of the above-mentioned improved formulations are limited to those structures with interfaces parallel to - or -axis. Finite-element method (FEM) is a choice for such 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 properties [12], [13] were proposed for the corner problems, but the mesh generation and the programming are relatively tedious. Another limitation of conventional FDM is that grids in the computation are normally parallel to the axes in the discretization of field components. Staircase approximation is often required when the fields cross a slanted interface between two different materials. The convergence is slow due to the staircase approximation as compared to other methods without staircase approximation, e.g., FEM. In addition, the full-vectorial properties may not be accurately modeled under such approximation.
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Fig. 2. (a) Cross-sectional view of the stencil used near the corner.
In our previous paper [14], we proposed an improved full-vectorial FD scheme regarding dielectric waveguides with piecewise homogeneous structures. In this paper, we will adopt the scheme with nonuniform grids to demonstrate the singularity-like behaviors of the electric fields here even without any field expansion by powers of radius (or additional logarithmic terms). In addition, our method will show that it can be easily extended to those structures with slanted facets. Implementation will be described in Section II, and some numerical results are given in Section III. A simple conclusion is drawn in Section IV. II. FORMULATION In this section, we will first introduce our full-vectorial FD scheme for structures with step-index interfaces. The interface can be slanted to the - and -axes or even curved. Then, we will give a simple treatment for modeling a corner with a small arc. A. Finite-Difference Schemes for Step-Index Interface The cross section of the problem under consideration is shown in Fig. 2(a) in which a linear slanted interface or a curved interface lies between the grid points. The basic idea is to express the field quantities at the surrounding grid points and its as the expansion of the field at the center point derivatives. Using the surrounding point at as an example, the derivation process of the relation for grid points with an interface in between can be summarized as the following steps, and these steps are basically the same as those introduced in [14]: as the 2-D Taylor series 1) Express the field and expansion of the field just right to the interface
and its its derivatives. Similarly, we can also express and its derivatives. derivatives as the expansion of and its derivatives in 2) As shown in Fig. 2(a), transform the global – coordinate system into corresponding terms in the local rotated – coordinate system for the linear slanted interface or into the local cylindrical – coordinate system for the curved interface with effective radius . Similarly, and its derivatives are transformed back to their correspondings in the – coordinates system. and its derivatives as a linear combination 3) Express and its derivaof the field just left to the interface tives by matching the boundary conditions. In addition to those given in [14], some detailed formulas are given in the Appendix. In the steps, represents the electric field or the magnetic field , and the subscript denotes the - or -component. as the Following the above steps, we can express and its derivatives. If there is no linear combination of interface between the grid points, such expansion and boundary matching is the same as normal Taylor series expansion in a homogeneous material. For the second-order scheme, we need use nine grid points and corresponding derivative terms. We collect all relation equations based on the nine points shown in Fig. 2(a), itself, and express them in a matrix including the point form: (1) where is the vector of the fields is the matrix of coeffiat the nine points, cients derived with the above steps, and is the vector contains the field quantities at the point and its derivatives with respect to or . We can obtain a final set of FD formulas by taking inverse operation of (1), and , , the improved FD formulas for the terms , , and so on in are then expressed as a linear combination of the field values at the nine sampled points. Note that the interface between materials of refractive indexes and can be slanted or curved. No staircase approximation is required as that in common FD formulation. The boundary conditions across the slanted or curved interface in our formulation are satisfied through coordinate transformation of the fields. Noteworthily, the derivation process is the same for both E- and H-formulations. For waveguides made of nonmagnetic media, H is continuous across the interface between two media, while E may be discontinuous. Therefore, we generally expect the H-formulation converges faster. They do normally, but there is slight difference between two formulations, since derivative of H may also be discontinuous. Therefore, the singular behavior around a corner exists for E and H formulations. B. Treatment Near the Corner We may adopt another strategy in the analysis of the corner problems, since we have proposed an improved full-vectorial
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Fig. 3. Contours of the electric field distributions for the fundamental mode of the square channel waveguide. (a) E and (b) E .
Fig. 4. Contours of the magnetic field distributions for the fundamental mode of the square channel waveguide. (a) H and (b) H .
Fig. 5. 3-D plot of the electric field distributions for the fundamental mode of the square channel waveguide. (a) E and (b) E .
FD scheme to rigorously treat linear and curved step-index interfaces as in last section. As shown in Fig. 2(b), we model the rather than a recorner as a tiny arc with effective radius ally sharp wedge. In fact, this replacement may be even closer to the realistic engineering implementations. Outside the corner region, we adopt the linear slanted scheme to model the interface. The relation between the arc angle and the original corner angle is arc
corner
(2)
where corner is the angle of the corner and arc is the arc angle used to approximate the corner, as shown in Fig. 2(b). To
enhance the calculation efficiency, we use fine uniform meshes around the corner and nonuniform ones elsewhere. As indicated in 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 more and more smaller. After an iterative process of updating , , , we will obtain a convergent result. and Although we do not expand the field as the powers of the distance from the corner, i.e., in Fig. 1 or more correctly with additional logarithmic terms as others do, we will show that our treatment can still model the vectorial nature of the field via the curvature in our scheme. And this will be demonstrated in the following section.
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Fig. 6. 3-D plot of the electric field distributions for the fundamental mode of the square channel waveguide. (a)
Fig. 7. Electric field profiles along the diagonal of the waveguide near a corner obtained by using different
TABLE I CONVERGENCE OF THE COMPUTED REFRACTIVE INDEXES OF THE SQUARE CHANNEL WAVEGUIDE
III. NUMERICAL RESULTS A. Channel Waveguide Case Referring to the structure shown in Fig. 1, we first calculate the mode fields of a square channel waveguide with width m, the refractive indexes of the waveguide and the vacuum and , respectively, as shown in Fig. 1. being m. The The operating wavelength is assumed to be parameters used here are the same as those used by Sudbø [5] and Lui et al. [11]. This case can be calculated by some typical method, e.g., Goell’s approach [15], but they did not treat the corner problem well as indicated by Sudbø [5]. Because of the symmetry of the field, we only calculate one quarter of the whole region. We also apply transparent boundary condition (TBC) in this case and the calculation window is 1.0 m in both and directions. We use uniform mesh divisions at the vicinity of
H
r . (a) E
and (b)
H
and (b)
E
.
.
the corner and nonuniform mesh divisions elsewhere to save computation time and memory. The smallest grid size is m near the corner, and the largest grid size is m near the edge of the computational window. The effective radius of the arc at the corner is chosen to 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 surface plots of the corresponding transverse field components in Figs. 3 and 4. Although we do not add any singularity treatment near the corner, the resultant distributions of the electric fields still behave singularity-like near the corner. On the other hand, the resultant distributions of the magnetic fields behave smoothly around the corner as expected. Figs. 7 and 8 show the field profiles along the diagonal of the waveguide near a corner by using different values. It can be shown that both electric field components become more and more singularity-like as gets smaller. However, the field profiles near the corner and the computed effective index are found to converge uniformly. On the other hand, the H components near the corner converge faster due to their continuity nature. We calculate another case with the same waveguide width and refractive indexes as those used in the above case, except that the waveguide is operated at the normalized frequency and the waveguide is surrounded by a perfect electric conductor (PEC). The same structure has been analyzed by a vector FEM with inhomogeneous elements (VFEM-I) [13] and their converged effective indexes
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Fig. 8. Magnetic field profiles along the diagonal of the waveguide near a corner obtained by using different r . (a) H and (b) H .
EFFECTIVE INDEXES OF THE H
TABLE II MODE FOR THE RIB WAVEGUIDE FOR DIFFERENT D 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 and the E-VFEM, respectively. For further comparison, the result by Goell’s approach is 1.35617. Table I lists the convergent behavior of the computed effective refractive index in this case by using our improved full-vectorial - and -formulations, respectively. We can see that both the grid sizes and the effective radius of the arc at the corner influence the convergence of the results. When the grid sizes and become smaller, the results get more closer to those calculated using the VFEM-I by Li and Chang [13]. It also shows that the results using the -formulation converge faster than those using the -formulation. This is reasonable due to the more singular behavior in the electric fields. B. Rib Waveguide Case Although channel waveguides can provide better field confinement, the cost of fabricating the channel waveguides is relatively higher. In this section, we illustrate the capability of our formulations in treating the well-known rib waveguide involving the structure, as shown in Fig. 9. The slab-based structure provides the field confinement in the -direction and the rib region provides the field confinement in the -direction because
of the relatively higher equivalent refractive index in the rib region. Since this structure is relatively easier for semiconductor processing, it is one of the most popular structures in the design of integrated optic devices and systems. In our calculation, we use the following parameters: the opm, rib width m, and erating wavelength m. The outer slab depth varies from 0.1 to 0.9 m. The refractive indexes of the cover, the guiding layer, , , and , and the substrate are respectively. The parameters for the computational window are m, m, and m. We present in the last two columns of Table II the computed effective index mode obtained by our improved of the lowest order and -formulations. TBC is adopted. Table II also provides values obtained by previous authors using different methods: the VFEM with Aitken extrapolation [16], the VFEM with highorder mixed-interpolation-type elements (Edge-FEM) [17], and VFEM-I [13]. Figs. 10 and 11 show the contours of the com, , , and for puted transverse field components mode using our improved formulations with the lowest 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 Wall A typical photonic integrated system includes many components that are polarization sensitive, for example, integrated switches, interferometers, amplifiers, receivers, etc. Thus, it is often necessary to manipulate or convert polarization state in
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Fig. 10. Contours of the electric field distributions for the H
Fig. 11. Contours of the magnetic field distributions for the H
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mode of the rib waveguide. (a) E and (b) E .
mode of the rib waveguide. (a) H and (b) H .
Fig. 12. The schematic representation of the single-section polarization converter. (a) The converter structure. (b) Cross-sectional view of the PRW section.
such guided wave structures, and polarization converters are key components in photonic integrated systems. Polarization rotation in optical devices can be achieved by induced material anisotropy. Previously, such polarization converters employing electrooptical [18] and photoelastic effects [19] had been reported. However, in many applications, a passive polarization converter is much preferred, and some very promising and simpler passive polarization converters have also been reported
[20]–[22]. Such passive components may be simpler to fabricate and an important characteristic of these converters is that the polarization rotation is achieved simply by adjusting the geometry of the devices. Most of these passive polarization converters employ a longitudinally periodic perturbation structure. Recently, it has been reported that it is possible to achieve polarization rotation in a single-section design [23]–[25], as mode is launched from a standard shown in Fig. 12(a). If an input waveguide (IW), this incident field excites both the first and second hybrid modes of nearly equal modal amplitudes. As these two hybrid modes propagate along the polarization rotating waveguide (PRW), they would become out of phase at the half-beat length and their combined modal fields produce mode in the following output waveguide (OW). mainly a The PRW is based on a rib waveguide with one side wall slanted at an angle around 45 , and the cross-sectional view of this structure is shown in Fig. 12(b). The parameters we use for an asymmetrical slanted-wall rib m, rib width waveguide are the operating wavelength m, m, and the outer slab depth 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. The slanted angle
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Fig. 13. 3-D plot of the electric field distributions for the first hybrid mode of the asymmetric slanted-wall rib waveguide. (a) E and (b) E .
Fig. 14. 3-D plot of the magnetic field distributions for the first hybrid mode of the asymmetric slanted-wall rib waveguide. (a) H and (b) H .
Fig. 15. 3-D plot of the electric field distributions for the second hybrid mode of the asymmetric slanted-wall rib waveguide. (a) E and (b) E .
is 52 . Figs. 13 and 14 show the 3-D plots of the E and M components for the first hybrid mode, respectively. Figs. 15 and 16 show the 3-D plots of the E and M components for the second hybrid mode, respectively. We use 317 by 251 grid meshes in calculation. The calculated effective refractive indexes for the first hybrid mode are 3.3273423 and 3.3272512 by using and -formulations, respectively. The calculated effective refractive indexes for the second hybrid mode are 3.3263567 and 3.3263383 by using - and -formulations, respectively. We can see that both hybrid modes have comparable field components in - and -directions, and their polarizations are no longer in the - or -direction but in the direction parallel or perpendicular to the slanted wall. Thus, they cannot be correctly obtained by using semivectorial formulation. To verify our simulation, 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 formulation of Yee-FD-BPM. Furthermore, we also use conventional FD scheme [2] with staircase approximation and index-average approximation to calculate the same problem. We find that the hybrid 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 facets that are not parallel to - or -axis.
IV. CONCLUSION Replacing sharp wedges with tiny arcs, we have implemented full-vectorial FD scheme to investigate the field behavior near dielectric waveguide corners. Nonuniform grids are adopted to save computation. The electrical fields show singularity-like distribution due to abrupt field discontinuities around the corner, while the magnetic fields show smooth distribution due to field continuity. Numerical results are convergent and show excellent approximation to the real wedge structure. Five-digit accuracy or more is achieved as compared with the full-vectorial FEM. In addition, the formulation is applicable to slanted facets without staircase approximation. Numerical results from a passive polarization converter shows it can model well the full-vectorial properties of fields.
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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)
TABLE III EFFECTIVE INDEXES OF COMPUTED BY DIFFERENT METHODS
H
and (b)
H
.
(17) (18) for the electric field. For the curved interface case, the interface conditions required in addition to those in [14] are
APPENDIX For the linear slanted interface case, the interface conditions required in addition to those in [14] are
(19) (20)
(3)
(21)
(4)
(22)
(5) (6)
(23)
(7)
(24)
(8) (9)
(25)
(10) for the magnetic field and (11) (12)
(26) for the magnetic field and
(13)
(27)
(14)
(28)
(15) (16)
(29)
CHIOU et al.: FINITE-DIFFERENCE MODELING OF DIELECTRIC WAVEGUIDES WITH CORNERS AND SLANTED FACETS
(30) (31) (32)
(33) (34) for the electric field. REFERENCES [1] R. E. Collin, Field Theory of Guided Waves, ser. IEEE/OUP Series on Electromagnetic Wave Theory, 2nd ed. Oxford, U.K.: Oxford University Press, 1990, sec. 1.5. [2] C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full vectorial mode calculations by finite-difference method,” Proc. Inst. Electr. Eng., J, vol. 141, no. 5, pp. 281–286, Oct. 1994. [3] G. R. Hadley and R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightw. Technol., vol. 13, no. 3, pp. 465–469, Mar. 1995. [4] S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 1737–1739, 1999. [5] A. S. Sudbø, “Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?,” J. Lightw. Technol., vol. 10, no. 4, pp. 418–419, Apr. 1992. [6] J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag., vol. AP-20, no. 4, pp. 442–446, Jul. 1972. [7] J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag., vol. AP-26, no. 4, pp. 598–602, Jul. 1978. [8] G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radio-Phys. Quantum Electron., vol. 29, no. 6, pp. 544–549, Dec. 1986. [9] G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightw. Technol., vol. 20, no. 7, pp. 1219–1231, Jul. 2002. [10] N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightw. Technol., vol. 25, no. 9, pp. 2563–2570, Sep. 2007. [11] W. W. Lui, C.-L. Xu, W.-P. Huang, K. Yokoyama, and S. Seki, “Fullvectorial mode analysis with considerations of field singularities at corners of optical waveguides,” J. Lightw. Technol., vol. 17, no. 8, pp. 1509–1513, Aug. 1999. [12] M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightw. Technol., vol. 18, no. 5, pp. 737–743, May 2000. [13] D.-U. Li and H.-C. Chang, “An efficient full-vectorial finite element modal analysis of dielectric waveguides incorporating inhomogeneous elements across dielectric discontinuities,” IEEE J. Quantum Electron., vol. 36, no. 11, pp. 1251–1261, Nov. 2000. [14] Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightw. Technol., vol. 20, no. 8, pp. 1609–11618, Aug. 2002. [15] J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J., vol. 51, pp. 2133–2160, 1969. [16] B. M. A. Rahman and J. B. Davies, “Vector-H finite-element solution of GaAs/GaAlAs rib waveguides,” Proc.-J. Inst. Electr. Eng., vol. 132, pp. 349–353, 1985. [17] M. Koshiba, S. Maruyama, and K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightw. Technol., vol. 12, no. 3, pp. 495–502, Mar. 1994.
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[18] R. C. Alferness and L. L. Buhl, “High-speed waveguide electro-optic polarization modulator,” Opt. Lett., vol. 7, pp. 500–502, 1982. [19] K. Yamanouchi, K. Wakazono, and K. Shibayama, “High-speed waveguide polarization modulator using the photoelastic effect,” IEEE J. Quantum Electron., vol. QE-16, no. 6, pp. 628–630, Jun. 1980. [20] Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett., vol. 59, pp. 1278–1280, 1991. [21] C. M. Weinert and H. Heidrich, “Vectorial simulation of passive TE/TM mode converter devices on InP,” IEEE Photon. Technol. Lett., vol. 5, no. 3, pp. 324–326, Mar. 1993. [22] J. J. G. M. van der Tol, F. Hakimzadeh, J. W. Pedersen, D. Li, and H. van Brug, “A new short and low-loss passive polarization converter on InP,” IEEE Photon. Technol. Lett., vol. 7, no. 1, pp. 32–34, Jan. 1995. [23] V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun., vol. 127, pp. 7–13, 1996. [24] J. Z. Huang, R. Scarmozzino, G. Nagy, M. J. Steel, and R. M. Osgood, Jr., “Realization of a compact and single-mode optical passive polarization converter,” IEEE Photon. Technol. Lett., vol. 12, pp. 317–319, Mar. 2000. [25] B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and characterization of compact single-section passive polarization rotator,” J. Lightw. Technol., vol. 19, no. 4, pp. 512–519, Apr. 2001. [26] J. Yamauchi, T. Mugita, and H. Nakano, “Implicit Yee-mesh-based finite-difference full-vectorial beam-propagation method,” J. Lightw. Technol., vol. 23, no. 5, pp. 1947–1955, May 2005. Yih-Peng Chiou (M’03) 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 in electrical engineering. From 1999 to 2000, he was with the Taiwan Semiconductor Manufacturing Company, where he was engaged in research on the plasma enhanced chemical vapor deposition of dielectric films. From 2001 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 computer-aided-design tools. In 2003, he joined the faculty of the Graduate Institute of Electro-Optical Engineering (now Institute of Photonics and Electronics) and Department of Electrical Engineering, National Taiwan University, where he is currently an Assistant Professor. His current research interests include modeling and computer aided design (CAD) of optoelectronics, which includes photonic crystals, nano structures, waveguide devices, optical fiber devices, light extraction enhancement in LED, display, solar cell devices, and the development and improvement of numerical techniques in optoelectronics.
Yen-Chung Chiang (M’06) was born in Hualien, Taiwan, on March 10, 1970. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the National Taiwan University, Taipei, Taiwan, in 1992, 1994, and 2002, respectively. From 2002 to 2005, he was with the Very Innovative Architecture (VIA) Technologies Inc., Taiwan, where he was engaged in research on the design of radio-frequency integrated circuits. In 2005, he joined the faculty of the Electrical Engineering Department, National Chung-Hsing University, Taichung, Taiwan, where he is currently an Assistant Professor. His current research interests include the numerical analysis techniques for optical or microwave devices and the design of radio-frequency integrated circuits.
Chih-Hsien Lai, photograph and biography not available at the time of publication.
Cheng-Han Du, photograph and biography not available at the time of publication.
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Hung-Chun Chang (S’78–M’83–SM’00) was born in Taipei, Taiwan, on February 8, 1954. He received the B.S. degree from the National Taiwan University, Taipei, Taiwan, in 1976, the M.S. and Ph.D. degrees from the Stanford University, Stanford, CA, in 1980 and 1983, respectively, all in electrical engineering. From 1978 to 1984, he was with the Space, Telecommunications, and Radioscience Laboratory of Stanford University. In August 1984, he joined the faculty 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 the Department of Electrical Engineering and, from 1992 to 1998, as the Chairman of the newly-established Graduate Institute of Electro-Optical Engineering at the National Taiwan University. He is also with the Graduate Institute of Communication Engineering, National Taiwan University. His current research
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interests include the theory, design, and application of guided-wave structures and devices for fiber optics, integrated optics, optoelectronics, and microwaveand 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-Americans, the Optical Society of America, the Electromagnetics Academy, and the China/SRS (Taipei) National Committee (a Standing Committee member during 1988–1993 and since 2006) of the International Union of Radio Science (URSI). He has been serving as the Institute of Electronics, Information, and Communication Engineers (Japan) Overseas Area Representative in Taipei. In 1987, he was among the recipients of the Young Scientists Award at the URSI XXIInd General Assembly. In 1993, he was one of the recipients of the Distinguished Teaching Award sponsored by the Republic of China, Ministry of Education and in 2004, he received the Merit National Science Council (NSC) Research Fellow Award sponsored by the Republic of China, NSC.
