Finite-Difference Modeling of Dielectric Waveguides With Corners and Slanted Facets

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.

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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


CHIOU et al.: FINITE-DIFFERENCE MODELING OF DIELECTRIC WAVEGUIDES WITH CORNERS AND SLANTED FACETS 2079

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.



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



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



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



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.



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)



(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.



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.


Finite-Difference Modeling of Dielectric Waveguides With Corners and Slanted Facets

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