finite-difference-based mode-matching method for 3-d waveguide structures under semivectorial...

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 12, DECEMBER 2005 4239

Finite-Difference-Based Mode-Matching Methodfor 3-D Waveguide Structures Under

Semivectorial ApproximationKai Jiang and Wei-Ping Huang, Senior Member, IEEE

Abstract—A finite-difference-based mode-matching method(FD-MMM) is developed and demonstrated for three-dimensional(3-D) optical waveguide structures with arbitrary index profilesand multiple scattering interfaces along the waveguide axis. Thecomputation domain is enclosed by a perfectly conducting boxcoated by a perfectly matched layer (PML). Semivectorial approx-imation is employed to simplify the formulations and calculations.Tradeoff and guidelines for choice of the PML parameters in themode-matching analysis are discussed by comparison with knownbenchmark solutions for a two-dimensional (2-D) structure. Themethod is subsequently applied to and validated for 3-D struc-tures in which both single and multiple scattering interfaces aresimulated.

Index Terms—Mode-matching methods, optical waveguides,perfectly matched layer (PML), waveguide junctions.

I. INTRODUCTION

MODE-MATCHING method is a powerful technique forthe analysis of waveguide structures with longitudinal

discontinuities such as junctions and facets. The method hasbeen extremely successful in treating closed waveguides forwhich the mode spectra are discrete. For open waveguidescommonly used in guided-wave optical devices and photonicintegrated circuits, however, the mode-matching method hasbeen limited to applications in which the modal solutions areknown analytically such as one-dimensional (1-D) multilayerwaveguides [1], [2]. In this respect, the inclusion of the con-tinuous radiation modes in the field expansion constitutes sig-nificant challenges for the applications of the mode-matchingmethod in the context of open waveguide structures. In orderto avoid the problems associated with the radiation modes, onemay enclose the waveguide structure with a large box made ofperfect conductors so that all modes become discrete. Thesemodes can be divided into two categories: the guided modes,which are confined to the core of the waveguide, and the boxmodes, which are related to the enclosing box. If the box issufficiently large, then the original problem can be accuratelysimulated by the computation model. On the other hand, themode spectral spacing is inversely proportional to the size ofthe box, and a large number of the modes in the box need to be

Manuscript received March 16, 2005; revised July 15, 2005.The authors are with the Photonic Group, Department of Electrical and

Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/JLT.2005.858227

included in the field expansion to ensure adequate accuracy. Thesituation becomes much more acute when dealing with three-dimensional (3-D) structures of arbitrary index profiles wherethe modes need to be computed numerically. Mathematically,the field expansion in terms of the guided and box modes isnot very effective, especially for accurate representation of theradiation fields.

The radiation field in an open waveguide can also be approx-imated by the summation of leaky modes. Strictly speaking,leaky modes are not the proper solutions of Maxwell’s equa-tions, but an analytical extension of the guided-mode solutionsto the complex domain. It can be shown that the leaky modescan be used to approximate the radiation fields not far from thewaveguide axis. The imaginary parts of the modal constants forthe leaky modes stand for the leakage loss and the modal-fielddistributions within the critical points represent the radiationfields [3]. Beyond the critical points, the modal fields of theleaky modes diverge. For these reasons, the use of the leakymodes as part of the modal expansion in the mode-matchingmethod appears to be tricky when dealing with such problemsas modal orthogonality and normalization [4].

A major advance in the mode-matching method for analysisof open optical waveguides is the recent work by Derudderand co-workers [5]–[7]. By enclosing the optical waveguidestructure by a perfectly conducting box with the inner perfectlymatched layer (PML), the entire mode spectrum is discretizedinto the guided and complex modes, all of which are welldefined and possess the normal mode characteristics such asmodal orthogonality and normalization. The complex modesare formed as a result of the PML sandwiched between theguiding region and the perfectly conducting box enclosing thecomputation domain. If the parameters of the PML and the sizeof the box are properly chosen, the guided modes of the originalwaveguide structure will not be affected significantly, whereasthe complex modes represent the radiation fields or leakymodes. Due to the presence of the PML, the closed waveguidestructure can be seen as an open one in the sense that the waveswould not be reflected at the boundaries. It was demonstratedthat the approach of expansion of the field in terms of the guidedand complex modes is more effective than the conventionalapproach of expansion in terms of the guided and radiation orleaky modes. So far, this approach has been applied to two-dimensional (2-D) waveguide structures with 1-D modes aswell as 3-D structures in which analytical modal solutions forthe 2-D modes exist such as circular step-index fibers. It is

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4240 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 12, DECEMBER 2005

nevertheless desirable to extend this approach to 3-D structuresin which the analytical expressions for the 2-D modes are notavailable, and therefore, make the mode-matching method aneffective and efficient simulation technique for 3-D structuresmade of open dielectric waveguides.

Given an index profile, the complex modal solutions in awaveguide structure enclosed by a perfectly conducting boxwith the PML can be calculated numerically by the finite-difference-based eigenvalue solvers. In general, the modalfields are hybrid in nature, as obtained from the solutions ofthe full-vectorial modal equations. For most practical opticalwaveguides, the modal fields are predominantly linearly polar-ized so that only one major field component needs to be con-sidered. Under this circumstance, the simplified semivectorialformulations [8] can be used to calculate the modal-propagationconstants and fields with sufficient accuracy yet much lesscomputation time and memory storage.

We write this paper with the following objectives. First ofall, we carry out an investigation for the dependence of theaccuracy and efficiency on the key PML parameters for themode-matching method in the context of the 2-D structure.Secondly, we extend the method to the 3-D waveguide junc-tions of arbitrary index profiles based on the expansion of theincident, reflected, and transmitted fields in terms of semivec-torial guided and complex modes. The modes are computed byusing a semivectorial finite-difference mode solver so that itis applicable to optical waveguide structures of arbitrary indexprofiles. Section II presents the main theoretical formulationsin which the governing equations under the semivectorial ap-proximation in the real and PML media are solved by using thefinite-difference-based mode-matching method (FD-MMM).The guideline for the proper choice of the PML parameters isproposed and demonstrated in Section III. The semivectorialapproximation is briefly discussed in Section IV. Several ex-amples of 3-D structures are studied in Section V, where thesimulation results are shown to be in excellent agreement withthe results in the literature.

II. THEORETICAL FORMULATIONS

For simplicity, we make the following assumptions.1) The medium in the waveguide structure is lossless, lin-

ear, and isotropic. The permittivity and permeability ofvacuum are denoted as ε0 and µ0, respectively. Thepermeability µ in the medium is equal to the free-spacevalue µ0 throughout this paper.

2) The time dependence is expressed as exp(jωt). The waveis propagating along z, and z dependence is expressed asexp(−jβz), which refers to the propagation in the posi-tive z direction, or exp(jβz) in the negative z direction.ω and β are the angular frequency and the propagationconstant, respectively.

A. Modal Governing Equations and Solutions

We consider a waveguide structure where the transverseindex profile n(x, y) is arbitrary and defined in the Cartesiancoordinate system. The waveguide structure is surrounded by

an anisotropic PML medium terminated by a perfectly conduct-ing boundary enclosing the entire computation domain. Theboundary is far away from the waveguide-guiding region so thatthe guided modes are not affected. The Maxwell’s equations inthe real and PML media can be written as

∇× �E = − jωµ0[Λ] �H

∇× �H = jωε0n2[Λ] �E (1)

where the general form for [Λ] in 2-D cases (x and y) is [9]

[Λ] =

sy

sx0 0

0 sx

sy0

0 0 sxsy

. (2)

The parameter sk, also called the coordinate-stretching factor[10], is the function of k only, where k = x, y. In the realmedium, sx = sy = 1; in the PML medium, sx and sy aregiven by

sx =κx − jσx

ωε0n2(3a)

sy =κy − jσy

ωε0n2(3b)

which are complex. The vector wave equation for the electricfield is

∇×([Λ]−1∇× �E

)= ω2µ0ε0n

2[Λ] �E. (4)

Further, we have

∇t ×(

1sxsy

∇t × �Et

)+ z ×

[[Λ]−1

T ∇t × (−jβEz z)]

−β2z ×([Λ]−1

T z × �Et

)= ω2µ0ε0n

2[Λ]T �Et (5)

where

[Λ]T =[ sy

sx0

0 sx

sy

]. (6)

Utilizing ∇ · �D = ∇ · (ε0n2[Λ] �E) = 0, we obtain

Ez =∇t ·

(n2[Λ]T �Et

)jβn2sxsy

. (7)

Substituting (7) into (5), we obtain the full-vectorial waveequation for the transverse electric field, shown in (8) at thebottom of the next page.

Under semivectorial approximation, we assume that only onedominant transverse electric component exists for the quasi-TEand quasi-TM modes, respectively. For quasi-TE modes whereEy = 0, we can obtain the following semivectorial governing

JIANG AND HUANG: FINITE-DIFFERENCE-BASED MODE-MATCHING METHOD FOR 3-D WAVEGUIDE STRUCTURES 4241

equation for the nonzero transverse electric-field component Ex

by using the above full-vectorial wave equation

1sx

∂

∂x

[1n2

1sx

∂

∂x

(n2 1

sxEx

)]+

1sy

∂

∂y

[1sy

∂

∂y

(1sx

Ex

)]

+k20n

2 1sx

Ex = β2 1sx

Ex (9)

where k0 = ω√µ0ε0. The transverse magnetic-field compo-

nent Hy can be readily obtained by the following expressionin terms of the transverse electric-field component:

Hy =β

ωµ0

sy

sxEx − 1

βωµ0

sy

sx

∂

∂x

[1n2

1sx

∂

∂x

(n2 1

sxEx

)].

(10)

Similarly, for the quasi-TM modes, the transverse electric-field component Ey is dominant, whereas Ex is zero. Thesemivectorial governing equation for Ey can be obtained as

1sy

∂

∂y

[1n2

1sy

∂

∂y

(n2 1

syEy

)]+

1sx

∂

∂x

[1sx

∂

∂x

(1sy

Ey

)]

+k20n

2 1sy

Ey = β2 1sy

Ey. (11)

The transverse magnetic-field component Hx is relatedto the transverse electric-field component by the followingrelationship:

Hx = − β

ωµ0

sx

syEy +

1βωµ0

sx

sy

∂

∂y

[1n2

1sy

∂

∂y

(n2 1

syEy

)].

(12)

The finite-difference method [8], [11] is utilized to discretizethe semivectorial modal governing equations (9) and (11) andto perform mode calculation. By imposing the proper boundaryconditions, we derive a matrix equation as follows:

[A]N×N [E]N×1 = β2[E]N×1 (13)

where N is the total number of unknown nodes. [E] represents[Ex] for quasi-TE modes or [Ey] for quasi-TM modes, andβ2 is the eigenvalue, which stands for the modal-propagationconstants and might be complex. The numerically calculatedmodes are used in the mode-matching method.

B. Mode-Matching Solutions

In a waveguide structure surrounded only by a perfectlyconducting box, the mode spectrum includes guided and boxmodes. With the introduction of the PML, the box modesmigrate into the complex domain with complex propagationconstants and field patterns. These complex modes that depend

on the PML parameters, together with guided modes, are tobe utilized for expanding any field in the closed waveguidestructure. Consider a typical waveguide junction where twowaveguides (A and B) of different transverse configurationare joined at the position z = 0. The transverse electric andmagnetic field ( �Et and �Ht) in waveguide A in the Cartesiancoordinate system can be written as

�EAt (x, y, z) =

N∑n=1

(a+

n e−jβAn z + a−nejβA

n z)�eA

tn(x, y) (14a)

�HAt (x, y, z) =

N∑n=1

(a+

n e−jβAn z − a−nejβA

n z)�hA

tn(x, y) (14b)

where t denotes the transverse component, βAn is the propaga-

tion constant of the nth mode, �eAtn and �hA

tn are the transverseelectric and magnetic vectors of the nth mode, respectively, anda+

n and a−n are the amplitudes of forward and backward wavesof the nth mode, respectively. The sum is over N guided andcomplex modes. Similar equations can be written for waveguideB with the corresponding notations. With the use of boundaryconditions at the junction plane formed by waveguide A andwaveguide B and orthogonality relations between the modes,we have

b+m =N∑

n=1

a+n

⟨�eA

tn,�hB

tm

⟩+

⟨�eB

tm,�hAtn

⟩

2⟨�eB

tm,�hBtm

⟩

+N∑

n=1

a−n

⟨�eA

tn,�hB

tm

⟩−

⟨�eB

tm,�hAtn

⟩

2⟨�eB

tm,�hBtm

⟩ (15a)

b−m =N∑

n=1

a+n

⟨�eA

tn,�hB

tm

⟩−

⟨�eB

tm,�hAtn

⟩

2⟨�eB

tm,�hBtm

⟩

+N∑

n=1

a−n

⟨�eA

tn,�hB

tm

⟩+

⟨�eB

tm,�hAtn

⟩

2⟨�eB

tm,�hBtm

⟩ (15b)

where the coefficients b+m and b−m are the amplitudes of forwardand backward waves of the mth mode in waveguide B, respec-tively, and the inner product of the field vectors are defined ingeneral as

⟨�e,�h

⟩=

12

∫∫S

(�e× �h) · zds. (16)

The modal orthogonality of the guided and complex modesfor waveguides A and B is defined in terms of the above inner

∇t ×(

1sxsy

∇t × �Et

)− z ×

[Λ]−1

T ∇t ×

∇t ·

(n2[Λ]T �Et

)n2sxsy

z

− β2z ×

([Λ]−1

T z × �Et

)= ω2µ0ε0n

2[Λ]T �Et. (8)


Fig. 1. Planar step discontinuity surrounded by the perfectly conducting plateswith the PML. The waveguide core is located at the center of the structure. His the distance between the plate and the center.

product. Note that the complex modes are orthogonal in thesense that

⟨�etm,�htn

⟩=

12

∫∫S

(�etm × �htn) · zds =⟨�etm,�htm

⟩δmn

(17)

where m,n = 1, 2, . . . , N . For multiple discontinuities alongthe waveguide, the transfer or scattering-matrix formulationscan be utilized to connect the modal amplitudes of the differentsections so that the overall transfer or scattering matrix can beobtained [12], [13].

It should be mentioned that the semivectorial modes maynot be, in general, orthogonal in the sense of (17), due to theapproximations made in the derivation of the governing equa-tions (9) and (10) for the quasi-TE modes and (11) and (12) forthe quasi-TM modes, respectively. For many practical opticalwaveguides, however, the orthogonality relations are usuallyvalid, since the errors caused by the nonorthogonality may beignored, as illustrated by way of an example in Section IV.

III. PML PARAMETERS

In the case that the waveguide structure is enclosed witha perfectly conducting box that facilitates the zero-boundarycondition, the accuracy of the solution simply depends on thesize of the box and also the number of modes required forconvergence. The farther away from the guiding region of thewaveguide structure the boundary is, the larger the number ofmodes used in the modal expansion is, the higher the achievableaccuracy is. The introduction of the PML brings additional pa-rameters, and thus justifies the need to investigate their effectson the accuracy and computation efficiency of the numericalsolutions. To do so, we consider the example of the step discon-tinuities taken from [1] (see Fig. 1). The coordinate-stretchingfactors sx and sy defined in (3a) and (3b) are used to describethe PML. While the parameters σx and σy in the imaginary part

control the decay of the propagating waves in the PML region,the real part κx and κy act to cause additional attenuation ofthe evanescent waves. Although a properly chosen PML canabsorb both propagating and evanescent waves effectively [14],the position of the boundary still plays an important role onthe accuracy of the solution. In general, the boundary can belocated where the amplitudes of the fields of the guided wavesare less than some prescribed value. The real part κx and κy

of the coordinate-stretching factors can be simply assigned 1.The PML thickness can be chosen arbitrarily, as long as it hasno significant effects on the guided waves. However, the PMLshould not be too thin; otherwise, the parameters σx and σy

must take a large value to produce enough attenuation and thushave sharp variation within the PML, which is not preferred inthe numerical implementation.

Practically, the parameter σ (σx or σy) usually takes thefollowing form [15]:

σ = σmax

(ρ

dPML

)m

, m = 1, 2, 3, . . . (18)

where dPML is the PML thickness and ρ is the distance from thestarting point of the PML. There may be other spatial profilesof σ, but we do not expect that the results would be differentin any significant way. In our case, we have chosen the profileto be parabolic, i.e., m = 2. In addition, the maximum valueσmax can be obtained from the PML reflection coefficient R atthe interface between the real and PML media.

Considering a plane wave perpendicularly incident to thePML, we can write the PML reflection coefficient R as

R = exp

−2σmax

cε0n

dPML∫0

(ρ

dPML

)m

dρ

(19)

where c is the speed of light in free space. Therefore, we canwrite the coordinate-stretching factor s (sx or sy) as follows:

s(ρ) = 1 − jσ

ωε0n2

=1 − jλ

4πndPML

[(m + 1) ln

1R

] (ρ

dPML

)m

. (20)

It can be seen that the PML reflection coefficient R repre-sents the attenuation level within the PML, and hence, playsa key role in the choice of the PML parameters in the mode-matching method. As the PML reflection coefficient R reflectsthe collective effects of all PML parameters such as σmax

and PML thickness, we will choose R as the measure of thePML. In general, the smaller the PML reflection coefficient is,the more effective the PML is. In practice, it is sufficient tochoose the magnitude of the PML reflection coefficient belowa certain level, e.g., R = 0.01, to ensure the accuracy of themode-matching calculation, as demonstrated below. It is notedfrom (19) that many combinations of m, σmax, and dPML inthe PML parameters may lead to the same magnitude of thePML reflection coefficient. Therefore, there appears to be adegree of freedom in the choice of the PML parameters. We


Fig. 2. Amplitude of reflection coefficient (TE) for incidence from waveguide A versus the PML reflection coefficient R for the planar step discontinuity.ncore = 2.236, ncladding = 1.0, D = 0.2387 µm, d = 0.2D, H = 3.2387 µm, and λ = 1.5 µm.

TABLE ICOMPARISON (QUASI-TE) BETWEEN FD-MMM AND ROZZI’S METHOD [1] FOR THE PLANAR STEP DISCONTINUITY. PML REFLECTION

COEFFICIENT IS 0.01. r1 AND r2 ARE THE REFLECTION COEFFICIENTS FOR INCIDENCE FROM WAVEGUIDE A AND WAVEGUIDE B,RESPECTIVELY. t IS THE TRANSMISSION COEFFICIENT. THE LOSSES REFER TO INCIDENCE FROM WAVEGUIDE A

have investigated some of these combinations such as σmax

and dPML and found that they make a negligible impact on theaccuracy of the mode-matching calculation for the same PMLreflection coefficient. We have chosen m = 2 in our analysis,which has been commonly used in the literature.

At the extreme case of R = 1, the PML is absent. Underthis circumstance, the solution, for instance, the reflectioncoefficient in the planar step discontinuity, may oscillate withrespect to the location of the boundary. The convergence rate,i.e., the level of accuracy as a function of distance between theboundary and the center of the waveguide as well as the numberof modes in the modal expansion, is low. Therefore, the bound-ary has to be located sufficiently far away from the guidingregion, and hence, a large number of box modes are neededin the modal expansion. The magnitude of such oscillation isrelated to the excitation of the box modes. On the other hand,as we introduce the PML to reduce the reflection, the amplitudeof the oscillation is reduced accordingly. It appears that we canset a prescribed value for the amplitude. When the amplitude issmaller than the value, the corresponding R is selected, and thesolution is expected to be accurate. Fig. 2 shows the dependenceof the amplitude of the reflection coefficient (TE) for incidencefrom waveguide A on the PML reflection coefficient R inthe planar step discontinuity where the waveguide cores arelocated at the center of the structure. Note that the transversecomponents of the TE modes are Ey and Hx in the case. The

number of modes is 120. As a benchmark, the result obtainedby Rozzi in [1] is also shown for comparison. It can be seenthat the accuracy can be improved significantly, even with aweak attenuation within the PML. Good agreement is achievedwhen R ≤ 0.01. The comparison with Rozzi’s results for thescattering matrix and loss is shown in Table I. No significantimprovements (with respect to Rozzi’s results) are observedwhen R is reduced further. In fact, a very small R is notexpected to result in a more accurate solution.

For a strongly absorbing PML (very small R), the modespectrum is clearly divided into three groups [6], [16], [17]. Thefirst group includes guided modes with modal effective indicesstill on the real axis. The modes in the second and third groupsare quasi-leaky and PML (Berenger) modes, respectively. Forthe quasi-leaky modes, the modal fields grow exponentiallyinto the cladding and get damped in the PML region beforereaching zero at the boundary. For the PML modes, the modalfield is mainly concentrated in the PML region. In this case,the guided and quasi-leaky modes are expected to play majorroles in the modal expansion since the incident guided waves,which have a substantial field in the core and, to some extent,in the cladding region not too far from the core, excite moreguided and quasi-leaky modes at the waveguide discontinuity.The relative contribution of the PML modes is small, as theoverlapping integrals of the PML modes with the guided modesare small.


Fig. 3. Two equal dielectric rectangular waveguides with an air gap.

IV. SEMIVECTORIAL APPROXIMATION

In derivation of the semivectorial governing equations (9)and (11) for the 3-D waveguide structures, the polarization-coupled terms

1sx

∂

∂x

[1n2

1sy

∂

∂y

(n2 1

syEy

)]− 1

sy

∂

∂y

[1sx

∂

∂x

(1sy

Ey

)]

and

1sy

∂

∂y

[1n2

1sx

∂

∂x

(n2 1

sxEx

)]− 1

sx

∂

∂x

[1sy

∂

∂y

(1sx

Ex

)]

are ignored. In other words, the x polarized field and y po-larized field are assumed to be decoupled. For many practicaloptical devices, these coupling terms are usually small andcan be neglected. For this reason, the semivectorial methodyields sufficiently accurate results [18], [19]. Take a losslessweakly guided waveguide structure with a rectangular coreas an example. The waveguide core width is 1.0 µm, and itsthickness is 0.5 µm. The indices of refraction of the core andthe cladding are 3.44 and 3.39, respectively. The wavelength λis 0.86 µm. The PML reflection coefficient R is 10−7. Theerrors in the real or imaginary part of the propagation constantsand the field distributions of the major components of the firstfour computed quasi-TE modes between the full-vectorial andsemivectorial methods were found to be less than 1.5 × 10−4.

An orthogonality relationship is highly desirable for thesimplification of the mode-matching formulations and for theeffective evaluation of expansion coefficients in the mode-matching method. Under the semivectorial approximation, theorthogonality is still valid, provided that the following criterionis satisfied:

Error =N∑

n=1(n �=m)

∣∣∣∣∣〈�etm,�htn〉〈�etm,�htm〉

∣∣∣∣∣ < ε (21)

where N is the number of modes, and ε is a small number(e.g., ε = 10−5). The modal orthogonality can be examined

numerically. For the first four calculated quasi-TE modes, theerrors are smaller than 10−5. Therefore, the nonorthogonalityerror between the normalized semivectorial modes is negligiblysmall, and the conventional mode-matching method can still beutilized.

The semivectorial approximation is accurate only when theindex contrast over the waveguide cross section is small and/orthe hybrid nature of the modal fields is negligible. When theindex contrast over the transverse cross section becomes large,and/or the polarization coupling between the two semivectorialmodes is not negligible (e.g., for the waveguide structures to beused for polarization rotation), the semivectorial approximationmay be subject to considerable errors and is no longer valid.Under this circumstance, the mode-matching method based onsemivectorial modes would suffer from the poor representationof the fields, so that the rigorous full-vectorial method based onthe full-vectorial modes must be used. The full vectorial mode-matching method is beyond the scope of this paper and will bethe subject of our future work.

V. NUMERICAL RESULTS FOR 3-DWAVEGUIDE STRUCTURES

With a guideline we have developed for the proper choiceof the position of the boundary and the PML parameters, weare now in a position to simulate 3-D waveguide structuresby using FD-MMM based on the semivectorial modes. Asthe first example, we consider two equal dielectric rectangularwaveguides with an air gap, as shown in Fig. 3 [20]. Thewaveguide is the same as that in the previous section with corewidth Lx = 1.0 µm, core thickness Ly = 0.5 µm, core indexncore = 3.44 and cladding index ncladding = 3.39. The wave-length λ is 0.86 µm. The waveguide structure is surroundedby a rectangular perfectly conducting box with the PML, andthe waveguide core is located at the center of the computationdomain. The cross section for the waveguide structure is shownin Fig. 4. The entire PML consists of eight regions. There isattenuation in both the x and y directions in regions PML1,


Fig. 4. Cross section for the rectangular dielectric waveguide enclosed by the perfectly conducting box with the PML. The waveguide core is located at thecenter of the structure.

Fig. 5. Power-transmission coefficients (quasi-TE) for the 3-D waveguide air gap with box 1 (Wx = 3.0 µm; Wy = 2.0 µm) and box 2 (Wx = 4.0 µm;Wy = 3.0 µm), respectively. Lx = 1.0 µm; Ly = 0.5 µm; ncore = 3.44; ncladding = 3.39; λ = 0.86 µm.

PML3, PML6, and PML8 (sx and sy are all complex), only inthe x direction in regions PML4 and PML5 (sx is complex andsy = 1), and only in the y direction in regions PML2 and PML7(sy is complex and sx = 1). The PML reflection coefficientR is 0.01. As discussed, the box should be located where theguided waves are sufficiently damped. Beyond the point, theposition of the box becomes less important and would notaffect the solutions significantly when the PML with enoughattenuation is introduced. For this reason, the box backed withthe PML may be located closer to the waveguide structurethan the box without the PML. Although the accuracy can be

higher with the larger box, the computation effort would beincreased as more mesh points for mode calculation, and agreater number of modes for mode matching are required. Forthis typical example, we aim to calculate the dependence of thepower transmission, reflection, and loss coefficients (quasi-TE)for an incident guided wave that is launched from one of thewaveguides on the length L of the air gap. The matrix cascadeis used to deal with two junctions. Fig. 5 shows the power-transmission coefficients for two different boxes (Wx = 3.0 µmand Wy = 2.0 µm; Wx = 4.0 µm and Wy = 3.0 µm). Thosecalculated by setting R = 1.0, which corresponds to the case


Fig. 6. Power-transmission coefficients (quasi-TE) versus the number of modes for the 3-D waveguide air gap. Wx = 3.0 µm; Wy = 2.0 µm; Lx = 1.0 µm;Ly = 0.5 µm; ncore = 3.44; ncladding = 3.39; λ = 0.86 µm.

Fig. 7. Power-reflection coefficients (quasi-TE) for the 3-D waveguide air gap. Wx = 3.0 µm; Wy = 2.0 µm; Lx = 1.0 µm; Ly = 0.5 µm; ncore = 3.44;ncladding = 3.39; λ = 0.86 µm.

without the PML, are also shown for comparison. It was foundthat the results for the two different boxes are very close to eachother when the proper PML is used. The results without thePML suffer from the reflection from the box. We also checkedthe number of modes used for the mode-matching method.Fig. 6 presents the effects of the number of modes on the power-transmission coefficients (Wx = 3.0 µm and Wy = 2.0 µm).In this case, a total of 90 modes are required for the mode-matching analysis. The calculated power-reflection coefficientsand loss are shown in Figs. 7 and 8, respectively.

Next, we calculate the reflection coefficient of a waveguidefacet [21]. The waveguide is composed of a rectangularlyshaped core whose width Lx is twice the thickness Ly . Thefacet is as cleaved without any coating. The indices of refrac-tion of the waveguide core and the cladding are ncore = 3.6and ncladding = 3.492, respectively. The wavelength is λ =0.86 µm. The normalized core thickness is defined as h =2Ly

√n2

core − n2cladding/λ. The PML reflection coefficient R

is 0.08 for the case. We choose Wx = 2.56 µm and


Fig. 8. Loss (quasi-TE) for the 3-D waveguide air gap. Wx = 3.0 µm; Wy = 2.0 µm; Lx = 1.0 µm; Ly = 0.5 µm; ncore = 3.44; ncladding = 3.39;λ = 0.86 µm.

Fig. 9. Power-reflection coefficient (quasi-TE) versus normalized core thickness for the 3-D waveguide facet. ncore = 3.6; ncladding = 3.492; Lx = 2Ly ;λ = 0.86 µm.

Wy = 1.56 µm. The number of modes for mode matching is200. Fig. 9 shows the dependence of the calculated power-reflection coefficient (quasi-TE) on the normalized core thick-ness. The results from [21] (from reading) are also shown forcomparison. Very good agreement is achieved.

VI. CONCLUSION

The finite-difference-based mode-matching method (FD-MMM) has been applied for solving three-dimensional (3-D)waveguide-junction problems. It was shown that the discretesemivectorial complex modes with the guided modes may be

utilized to expand the fields. With the method, the difficultiesarising from the inclusion of radiation modes can be avoided.To show the validity of the method, the numerical results for a3-D waveguide air gap and facet were presented. In addition,the effects of the key PML parameters were discussed anddemonstrated.

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Kai Jiang, photograph and biography not available at the time of publication.

Wei-Ping Huang (SM’00) received the B.S. degree in electrical engineeringfrom Shandong University, Jinan, China, in 1982 (with provincial and nationalhonors), the M.S. degree from University of Science and Technology of China,Hefei, China, in 1984, and the Ph.D. degree from Massachusetts Institute ofTechnology (MIT), Cambridge, in 1989, conducting research on fiber andintegrated optics, as well as studying international business management atSloan School of Management.

He has held a variety of faculty positions in the Department of Electrical andComputer Engineering, University of Waterloo, ON, Canada, and McMasterUniversity, Hamilton, ON, including Assistant Professor (1989–1992), Asso-ciate Professor with tenure (1992–1996), and Full Professor (1996–present). Hehas had visiting, adjunct, consulting, and/or managerial positions with severalacademic and industrial institutions in North America and Asia. He was aVisiting Researcher with Nortel from January to August 1992 and from Mayto August 1993 and a Visiting Professor with Nippon Telephone & Telegraph(NTT) Optoelectronics Laboratory from September 1995 to August 1996. Asthe leader of the Photonic Research Group at University of Waterloo, and laterat McMaster University, he has carried out a number of major research projectsin a wide range of areas of fiber and integrated optics, sponsored by Canadianfederal/provincial governments and the private sector. He is internationallyknown for his contributions and expertise on photonic devices and integratedcircuits, and he has authored and coauthored more than 100 journal papers and70 conference papers and holds seven U.S. patents.

Dr. Huang is a Member of the Optical Society of America and the Inter-national Society for Optical Engineers (SPIE). He was elected to the MITElectromagnetics Academy and received the Distinguished Leadership Awardfrom the American Biographical Institute in 1996.

finite-difference-based mode-matching method for 3-d waveguide structures under semivectorial...

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