H. Noro and T. Nakayama Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1451

Unusual molecular-dynamical method for vector-wave analysis of optical waveguides

Haruhito Noro

Department of Applied Physics, Hokkaido University, Sapporo 060, Japan, and Eniwa Laboratory, Showa ElectricWire and Cable Co., Ltd., Eniwa 061-14, Japan

Tsuneyoshi Nakayama

Department of Applied Physics, Hokkaido University, Sapporo 060, Japan

Received July 2, 1996; revised manuscript received January 2, 1997; accepted January 21, 1997

We propose an efficient numerical method, based on the unusual molecular-dynamical algorithm, for the modeanalysis in optical waveguides. This method can properly take into account the vector nature of light waves.The vector-wave analysis is treated as the eigenvalue problem of large-scale nonsymmetric matrices with realeigenvalues. The efficiency of the method is demonstrated by calculating dispersion relations of a rectangulardielectric waveguide and supermodes in a nonsymmetric rib waveguide directional coupler. The advantageslie in its being easily vectorized and parallelized for implementation on array-processing modern supercom-puters. © 1997 Optical Society of America [S0740-3232(97)00208-1]

1. INTRODUCTIONPrecise numerical analysis has become crucial to under-standing the propagation characteristics of light inwaveguides for the optimum design of optical devices.Although a lot of efficient methods for numerical analysisof optical waveguides have been proposed, most of themhave been performed under the scalar-wave approxima-tion. The scalar-mode analysis is valid only for weaklyguiding structures and is not accurate enough for wave-guide structures with a large variation of refractive indi-ces such as semiconductor waveguides. While the meth-ods incorporating the vector nature of light waves,1–8

namely, the polarization dependence on propagationcharacteristics and the hybrid coupling between polariza-tions of electromagnetic fields, are rigorous, these areusually complicated and require much CPU time for prac-tical calculations. This is because the vector-modeanalyses1–5 need to treat large-scale nonsymmetric (ornon-Hermitian) matrices or the Fourier transformtechniques6–8 require a long propagation distance forachieving highly accurate calculations. Among thesemethods for the vector-mode analysis, the method9 basedon the Lanczos recursion technique is effective with re-spect to computation time and memory requirements butis complicated and may not be efficient in determininghigher-order guided vector modes.Recently, a new approach,10 based on a molecular-

dynamical algorithm11 for extracting extreme eigenmodesof large systems, has been proposed for finding the lowestand a few of the higher-order guided scalar modes of op-tical waveguides. It has been demonstrated that thismethod is quite powerful for the scalar-mode analysis forweakly guiding systems. The feature of the method is itssimplicity compared with existing methods, in addition toefficient computation time (}N2) and memory require-

0740-3232/97/0701451-09$10.00 ©

ments (}N). This paper describes the extension of themethod10 to the analysis of guided vector modes. Forthis purpose it is necessary to treat the eigenvalue prob-lem of large-scale nonsymmetric matrices. The difficultyis mainly due to the lack of orthogonality among eigenvec-tors for nonsymmetric matrices. We extend the methodgiven in Refs. 10 and 11 for the vector-mode analysis ofoptical waveguides described by large nonsymmetric ma-trices with real eigenvalues.This paper is organized as follows: In Section 2 the

vector-wave equation is related to the eigenvalue problemof nonsymmetric matrices. Section 3 describes in detailthe mapping relationship between the wave equation andthe equation of motion for lattice vibrations. We extendthe method10,11 to be applicable to vector-wave equationsand also show that the method is efficient for calculatingnot only the lowest but also higher-order vector modes ofoptical waveguides. The effectiveness of the method isdemonstrated in Section 4, which presents calculated re-sults for an embedded rectangular dielectric waveguideoften used as a benchmark for comparison, and for a non-symmetric semiconductor rib waveguide directional cou-pler, whose analysis requires a large amount of computa-tional memory space. A summary is given in Section 5.

2. VECTOR-WAVE EQUATIONThe propagation characteristics of optical waveguideswith local refractive index n(r), which are source free andlinear, but lossless and isotropic, are determined by solv-ing the eigenvalue problem associated with a vector-waveequation. If we assume a monochromatic electromag-netic field with the angular frequency v and make use ofthe vector identity, the vector-wave equation for electricfields E(r) becomes

1997 Optical Society of America

1452 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 H. Noro and T. Nakayama

¹2E~r! 1 k2n2~r!E~r! 5 2¹$¹ ln@n2~r!# • E~r!%, (1)

where k 5 v/c 5 2p/l is the wave number of light invacuum and c and l are the velocity and the wavelengthof light in vacuum, respectively.If we take the z axis as the propagation direction and

assume a homogeneous structure along the z axis, the zcomponent of ¹ ln@n2(r)# on the right-hand side of Eq. (1)vanishes. Hence the components of the transverse elec-tric fields, Ex and Ey , are decoupled from the z compo-nent of the field, Ez . Two equations for the electric fieldsEx and Ey are given by

¹2Ex 1 k2n2Ex 5 2]

]xF] ln n2

]xEx 1

] ln n2

]yEyG , (2a)

¹2Ey 1 k2n2Ey 5 2]

]yF] ln n2

]xEx 1

] ln n2

]yEyG . (2b)

where the differential operators are defined by

H xx 5]

]xF 1n2

]

]x~n2!G 1

]2

]y21 k2n2, (5a)

H xy 5]

]xF 1n2

]

]y~n2!G 2

]2

]x]y. (5b)

The differential operators H yy and H yx are obtained fromEqs. (5a) and (5b), respectively, means of the exchangex ↔ y.If we take the spatial position (x, y) of the system as

the grid point (i, j), the operators are transformed intodiscretized forms by using the finite-differencerepresentation.4,5 From this we have the following ex-plicit expressions of matrix elements:

^i, juH xxuex& 5G i11, jex~i 1 1, j ! 1 G i21, jex~i 2 1, j ! 2 ~4 2 G i11, j 2 G i21, j!ex~i, j !

~Dx !2

1ex~i, j 1 1 ! 1 ex~i, j 2 1 ! 2 2ex~i, j !

~Dy !21 ni, j

2 k2ex~i, j !, (6a)

^i, juH xyuey& 51

4DxDy@2L i21, j21ey~i 2 1, j 2 1 ! 1 L i11, j21ey~i 1 1, j 2 1 !

1 L i21, j11ey~i 2 1, j 1 1 ! 2 L i11, j11ey~i 1 1, j 1 1 !#, (6b)

^i, juH yxuex& 51

4DxDy@2Li21, j21ex~i 2 1, j 2 1 ! 1 Li11, j21ex~i 1 1, j 2 1 !

1 Li21, j11ex~i 2 1, j 1 1 ! 2 Li11, j11ex~i 1 1, j 1 1 !#, (6c)

^i, juH yyuey& 5Gi, j11ey~i, j 1 1 ! 1 Gi, j21ey~i, j 2 1 ! 2 ~4 2 Gi, j11 2 Gi, j21!ey~i, j !

~Dy !2

1ey~i 1 1, j ! 1 ey~i 2 1, j ! 2 2ey~i, j !

~Dx !21 ni, j

2 k2ey~i, j !. (6d)

For a homogeneous system along the z axis, the wave-forms for transverse electric fields should be expressed bythe following equation:

SEx

EyD 5 S exey D exp~ibz !, (3)

where ex and ey are functions of the spatial position in thex–y plane and b is the propagation constant along the zaxis. The substitution of Eq. (3) into Eqs. (2) yields thematrix representation of Eqs. (2):

Hue& 5 b2ue&. (4)

Here the matrix H and the ket vector ue& are given by

H 5 FH xx H xy

H yx H yyG , ue& 5 S uex&

uey&D ,

Here we have defined G, L, G, and L, introduced in theabove equations, as

G i61, j 52ni61, j

2

ni, j2 1 ni61, j

2 , L i61, j61 5ni61, j612

ni61, j2 2 1,

(7a)

Gi, j61 52ni, j61

2

ni, j2 1 ni, j61

2 , Li61, j61 5ni61, j612

ni, j612 2 1.

(7b)

We can see from Eqs. (6) that the resulting matrices de-scribed by H xx and H yy are nonsymmetric on account ofEqs. (7). In addition, the forms of matrices H xy andH yx are not identical, i.e.,H xy Þ H yx . Hence the matrixH is not symmetric, though the matrix elements are real.This nonsymmetricity arises from discontinuity of indexinterfaces. Discretized forms of Eqs. (6) take full accountof discontinuity for normal components of the electric field

H. Noro and T. Nakayama Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1453

across the index interfaces. Thus the vector-wave equa-tion (2) for electric fields is reduced to the eigenvalueproblem for nonsymmetric matrices. The vector-waveequations for magnetic fields are derived as in the case ofelectric fields. We analyze the propagation characteris-tics of vector modes for transverse electric fields by usingEq. (4).

3. NUMERICAL METHODIn this section the wave equation is related to the equa-tion of motion for lattice vibrations. The relationship en-ables us to treat the eigenvalue problem from themolecular-dynamical viewpoint. First, we treat thescalar-wave equation; then we extend the method10,11 tobe applicable to the vector-mode analysis by introducingthe transposed matrix and the adjoint eigenvectors for anonsymmetric matrix.12

A. Lattice Dynamical Equation of MotionThe equation of motion for lattice vibrations is givenby10,13

ml

d2ul~t !

dt25 (

l8

Kll8ul8~t ! 2 MlV02ul~t !, (8)

where ml and ul(t) are the mass and the displacement ofthe atom at the grid point l, respectively. Hereafter weuse the definition l 5 (i, j) for abbreviation. The massml is set to be unity without loss of generality. Kll8 is thespring constant between the grid points l and l8. Thespring constants Kll8 are nonzero for the nearest-neighborinteraction and zero otherwise. The spring constants inlattice systems should have the following relation, arisingfrom the invariance under uniform displacements13:

Kll 5 2(l8Þl

Kll8 .

This relation is obtained by putting ul 5 constant andV0 5 0 in Eq. (8). The prefactor of the second term ofthe right-hand side of Eq. (8), MlV0

2, represents the forceconstant of the spring connected to the atom at the gridpoint l.14 We rearrange Eq. (8) as follows:

d2ul~t !

dt25 2(

l8

f ll8ul8~t !. (9)

Here the matrix element is defined as f ll8 5 MlV02d ll8

2 Kll8 , where d ll8 is the Kronecker delta. A time depen-dence of ul(t) is taken as exp(2imt). The parameter mrepresents the eigenfrequency of the lattice system; thatis, the eigenvalue becomes m2 for the vibrational system.

B. Mapping of the Wave Equation to theMolecular-Dynamical EquationThe wave equation (2) is reduced to the following eigen-value equation for the case of weakly guided structures:

Due& 5 b2ue&, (10)

where ue& represents the ket vector of the scalar field andthe discretized form of the operator D(5]2/]x2 1 ]2/]y2

1 k2n2) becomes a real symmetric matrix. We rewriteEq. (10) as

D8ue& 5 ~A 2 b2!ue&. (11)

The meaning of the parameter A will be described below.The discretized form of the operator D8(5A 2 k2n2

2 ]2/]x2 2 ]2/]y2) is expressed by a real symmetric ma-trix. The parameter A in Eq. (11) is a constant intro-duced so that the one-to-one correspondence between thediscretized form of the scalar-wave equation [Eq. (10)]and the equation of motion for lattice vibrations [Eq. (9)]is satisfied. The corresponding relationships betweenthese equations are shown in Table 1. The constant Ashould always satisfy A 2 b2 > 0 and A 2 k2nmax

2 > 0because of the condition that eigenvalues of the lattice dy-namical equation of motion should always be positive.Hence we have taken A 5 k2nmax

2 , since the propagationconstant b2 of the guided mode is estimated as b2

< k2nmax2 .

Since the minimum of A 2 b2 corresponds to the maxi-mum of b2, we must transform the matrix D8 of Eq. (11)into the following form10:

D8→ BI 2 D8, (12)

where B is a constant that satisfies B > (A 2 b2)max .The reason for making the transformation (12) is the factthat eigenvalues are computed one by one from the high-est eigenvalue by the algorithm.11 By replacing 2D8with a new matrix D defined below, we rewrite Eq. (11)again as follows:

2Due& 5 2~B 2 A 1 b2!ue&. (13)

From a comparison of Eqs. (9) and (13), we can obtain thekey equation in the present work,

2(l8

Dll8el8~z ! 5]2el~z !

]z2, (14)

where Dll8 is an element of matrix D, where l or l8 rep-resents the grid point of the spatial position (x, y). Itshould be noted that the z axis plays the role of time inEq. (14); i.e., one can presume time (z) dependence of theform exp@2i(B 2 A 1 b 2)1/2z# for el(z). If we discretize

Table 1. Corresponding Relations between theScalar-Wave Equation and the Equation of Motion

for Lattice Vibrations

Scalar-Wave EquationLattice DynamicalEquation of Motion

A 2 b2 (propagation constant)a m2 (eigenfrequency)el (electromagnetic field) ul(t) (displacement)1/D l

2 b Kll8 (spring constant)A 2 nl

2k2 (wave number)c MlV02 (Einstein frequency)

aThe parameter A is chosen as A 2 b2 . 0 because of the conditionm2 . 0.

bD l is spatial distance at the grid point l.cThe parameter A is a constant that satisfies A 2 nl

2k2 . 0.

1454 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 H. Noro and T. Nakayama

with respect to time z by a time step t, Eq. (14) yields themolecular-dynamical equations with positive eigenvalues(B 2 A 1 b2)1/2 as follows:

vl~n 1 1 ! 5 vl~n ! 2 t(l8

Dll8el8~n !, (15a)

el~n 1 1 ! 5 el~n ! 1 tvl~n 1 1 !, (15b)

where vl(n) denotes the velocity of the grid point l at timez (5nt), where the integer n denotes the time develop-ment. Thus we see that the mapping relation betweenthe scalar-wave equation and the equation of motion forlattice vibrations enables us to treat Eq. (14) from the lat-tice dynamical viewpoint.

C. Vector-Mode Analysis for the System Described byNonsymmetric MatricesThere is no direct relation between the vector-wave equa-tion (4), which is reduced to the eigenvalue problem ofnonsymmetric matrices, and the equation of motion forlattice vibrations [Eq. (9)], unlike the case for the scalar-wave equation (14). This subsection is devoted to de-scribing the numerical method, which is extended to beapplicable to the vector-wave equation.In the same way as in the case of the scalar equation,

the matrix H of Eq. (4) is transformed as H→ 2H 2 (B2 A)I. From this transformation one sees that thehighest eigenvalue of B 2 A 1 b2 corresponds to thelowest-order mode. A nonsymmetric matrix has two setsof eigenvectors, namely, the eigenvector u f(l)&, of thematrix H defined by14–16

vl2u f~l!& 5 Hu f~l!&, (16)

and the adjoint eigenvector ^g(l)u defined by the trans-posed matrix,15–17

vl2ug~l!& 5 H†ug~l!&, (17)

where H† is the transposed matrix of H. These eigenvec-tors belong to the same eigenvalue vl

2 corresponding toB 2 A 1 b2. Though the eigenvectors u f(l)& [or^ g(l)u# do not form orthonormal sets, on account of thenonsymmetricity of the matrixH, the biorthogonality con-dition holds between u f(l)& and ^g(l)u.15,16 The bior-thogonality conditions are written as

(l

u f~l!&^g~l!u 5 I, (18a)

^g~l!u f~l8!& 5 dll8 . (18b)

In the case of the vector-wave equation (4) described bya nonsymmetric matrix H, we need to treat two equationsfor the electric field el(z) 5 ^lue(z)& and its adjoint e l(z)5 ^e(z)ul&, which correspond to Eq. (14) for scalar waves:

]2el~z !

]z25 2(

l8

Hll8el8~z !, (19a)

]2e l~z !

]z25 2(

l8

Hl8le l8~z !, (19b)

where el(z) [or e l(z)] is the strength of the electric field atthe grid point l. Note here that z plays the role of time.

We define the x component of el(z) by l 5 1 ; N and they component of el(z) by l 5 N 1 1 ; 2N. Then H(H†) takes the form of a 2N 3 2N matrix, where N de-notes the total number of grid points. Since u f(l)& formsa complete set of vectors (but does not form an orthonor-mal set, though they are linearly independent), the elec-tric field el(z) can be decomposed into a set of eigenvec-tors u f(l)& as

el~z ! 5 (l

Ql~z !fl~l!, (20)

where Ql(z) is the amplitude of the eigenvector u f(l)&and varies as ;exp(2ivlz) (vl

2 5 B 2 A 1 b2), as seenfrom substituting Eq. (20) into Eq. (19a). In the sameway, e l(z) can be expanded in a set of adjoint eigenvectors^ g(l)u as

e l~z ! 5 (l

Sl~z !gl~l!, (21)

where Sl(z) is the amplitude of the adjoint eigenvector^g(l)u and varies as ;exp(2ivlz) as well.Equation (19a), discretized with respect to z by a time

step t, is given by

vl~n 1 1 ! 5 vl~n ! 2 t(l8

Hll8el8~n !, (22a)

el~n 1 1 ! 5 el~n ! 1 tvl~n 1 1 !, (22b)

where vl(n) denotes the time derivative of the x compo-nent for l 5 1 ; N and the y component for l 5 N 1 1; 2N at time z (5nt). Similarly to the case for theelectric field el(n), vl(n) can be decomposed into a set ofeigenvectors u f(l)& as

vl~n ! 5 (l

Pl~n !fl~l!, (23)

where Pl(n) is the amplitude of the eigenvector u f(l)&and varies as ;exp(2ivlz). Analogous to Eqs. (22), themolecular-dynamical equations for Eq. (19b) are given by

v l~n 1 1 ! 5 v l~n ! 2 t (l8

Hl8le l8~n !, (24a)

e l~n 1 1 ! 5 e l~n ! 1 t v l~n 1 1 !, (24b)

where v l(n) corresponds to the adjoint of vl(n) defined byEqs. (22). The velocity v l(n) can be decomposed into aset of eigenvectors ^g(l)u as

v l~n ! 5 (l

Rl~n !gl~l!, (25)

where Rl(n) is the amplitude of the eigenvector ^g(l)uand varies as ;exp(2ivlz).Using Eqs. (16), (17), and (20)–(25), we have

Ql~n 1 1 ! 2 ~2 2 vl2t 2!Ql~n ! 1 Ql~n 2 1 ! 5 0, (26a)

Sl~n 1 1 ! 2 ~2 2 vl2t 2!Sl~n ! 1 Sl~n 2 1 ! 5 0. (26b)

By assuming a solution of the form Ql(n) [or Sl(n)]5 al

n with a constant al , one has

H. Noro and T. Nakayama Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1455

al6 5

2 2 vl2t 2 6 @vl

2t 2~vl2t 2 2 4 !#1/2

2. (27)

The solution for the amplitude is then given by

Ql~n ! 5 cl1~al

1!n 1 cl2~al

2!n, (28a)

Sl~n ! 5 dl1~al

1!n 1 dl2~al

2!n, (28b)

where cl6 and dl

6 are to be determined for the initial con-dition. The solutions describe the time development ofthe system under the molecular-dynamical equations (22)and (24).The solution has two remarkable features, which are

ualu . 1 for vl2t 2 . 4 and ualu 5 1 for 0 < vl

2t 2 < 4.This indicates that, for vl

2t 2 . 4, the amplitude of themode l grows exponentially as a function of time develop-ment n, while, for 0 < vl

2t 2 < 4, the amplitude oscil-lates. The present method chooses the time step t sothat only the maximum eigenvalue satisfies vl

2t 2 . 4,and the rest of the eigenvalues satisfy0 < vl

2t 2 < 4.11,12 We have obtained vlmax

2 and the cor-responding eigenvector by iterating Eqs. (22a) and (22b)[Eqs. (24a) and (24b)] with respect to time z (5nt) fromarbitrary initial conditions el(0) and vl(0) @ e l(0) andv l(0)]. The present calculations have taken the Gauss-ian distribution function as the initial condition. This isunique and unusual compared with ordinary molecular-dynamical calculations.To determine the optimum time step t, we introduce

the quantity U(n) defined by

U~n ! 518 (

l,l8$ e l~n !Hll8@el8~n 1 1 ! 1 2el8~n !

1 el8~n 2 1 !# 1 el~n !Hl8l@ e l8~n 1 1 ! 1 2 e l8~n !

1 e l8~n 2 1 !#%. (29)

The quantity U(n) given by Eq. (29) corresponds to thetotal potential energy of the system. This yields

U~n ! 518 (

lvl2~4 2 vl

2t 2!@Sl* ~n !Ql~n !

1 Ql* ~n !Sl~n !#, (30)

where the biorthogonality condition (19b) is used. Pro-vided that the initial values for el(0) and e l(0) are takenas the same, S (l)(n) equals Ql(n) from Eqs. (28), sincethese eigenvectors belong to the same eigenvalue vl

2 .Hence the quantity Sl* (n)Ql(n) 1 Ql* (n)Sl(n) takes apositive real value. The time step t, chosen as U(n) ofEq. (30) to be negative but as close as possible to zero, sat-isfies the condition vl

2t 2 . 4 for obtaining only the high-est eigenvalue. The highest eigenvalue in our system,which corresponds to the lowest-order mode for lightwaves, can be obtained from the final set of el(n) [ore l(n)] by the Rayleigh quotient11:

vlmax

2 5

(l,l8

elHll8el8~n !

(l

@el~n !#2. (31)

This follows from the requirement that the final set ofel(n) satisfy the relation Hue(n)& 5 vlmax

2 ue(n)&. We haveused the bisection method10 for finding the optimum timestep t.The next highest eigenvalue and the corresponding ei-

genvector can be obtained as follows: The extraction ofthe eigenvector u f(lmax)& from a given set of field ue0(0)&and velocity uv0(0)& creates the system in which the nexthighest eigenvalue is the new maximum eigenvaluevlmax21

2 . Then we choose for a given set of field and veloc-ity the original sets used for calculations of the eigenvaluevlmax

2 . For this situation the initial field ue(0)& and thevelocity uv(0)& are chosen to be orthogonal to the normal-ized adjoint vector ^g(lmax)u according to the following re-lations [see Eq. (18b)]:

ue~0 !& 5 ue0~0 !& 2 u f~lmax!&^g~lmax!ue0~0 !&, (32a)

uv~0 !& 5 uv0~0 !& 2 u f~lmax!&^g~lmax!uv0~0 !&. (32b)

Using these initial sets ue(0)& and uv(0)&, we calculatetheir time development ue(n)& and uv(n)&. In calculatinghigher-order modes, in order to reduce round-off errorsthat are due to the loss of biorthogonality between^g(lmax)u and u f(lmax)& among the successive iterations,we have reorthogonalized the vectors ue(n)& and uv(n)&@^e(n)u and ^v(n)u# with respect to ^g(lmax)u @ u f(lmax)&] byusing the biorthogonality condition (18b) again after somenumber of iterations.We use the following formula for judging the accuracy

of calculated eigenvalues and the correspondingeigenvectors17,18:

d2~v ! [

(l

@ql 2 v2el~n !#2

(lql2

, (33)

where v is a parameter and ql [ ( l8Hll8el8(n). It is evi-dent that, if only a single mode l1 is included in ue(n)&,the value of d2(vl1

) becomes zero. If ue(n)& consists of afew eigenmodes with eigenfrequencies close to the model1 , d becomes small but finite. Therefore d(v) is an in-dex for the degree of accuracy.

4. NUMERICAL RESULTSTo demonstrate the efficiency of our method for vector-mode analysis, we have computed mode indices and modeprofiles for the following two examples. One is the modeanalysis of an embedded rectangular dielectric waveguidewith a large refractive-index variation. The propagationcharacteristics of this waveguide have been presented byusing the circular-harmonic analysis of Goell.19 The re-sults have often been used as a benchmark for compari-son with other methods. Another example is the super-mode analysis of a nonsymmetric rib waveguidedirectional coupler consisting of two nonidentical closelyseparated semiconductor rib waveguides.20 This type ofnonsymmetric directional coupler has many practical ap-plications in optoelectronics, such as tunable filters andoptical switches. For the modeling of such devices, the

1456 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 H. Noro and T. Nakayama

propagation constants and the mode profiles of the twolowest-order quasi-symmetric and quasi-antisymmetricsupermodes are important quantities. This is becauseone can calculate the coupling length of the power trans-fer between separated semiconductor rib waveguides.For these two examples, the calculations have been

performed until d, defined in Eq. (33), converges to avalue less than 1027. We have divided the systems intomesh grids with internal grid points so that the refractiveindex in each cell becomes uniform. We have applied thefixed boundary condition by which field values on theouter boundary of the calculated region are set to zero.Taking into account the substantial field penetration intosubstrate or cladding, we took sufficiently large calcu-lated regions for these two examples. Our calculations

Fig. 1. Dispersion relations for the lowest-order quasi-TE andquasi-TM modes in an embedded rectangular dielectric wave-guide. The closed circles are our results, and the open squaresare the results produced by Goell’s circular-harmonic analysis.

Fig. 2. Contour plot of the lowest-order quasi-TE mode: (a) xcomponent (dominant), (b) y component (minor).

have treated quasi-TE modes with the dominant x com-ponent and quasi-TM modes with the dominant y compo-nent.

A. Rectangular Dielectric WaveguideThe calculated region for the rectangular dielectric wave-guide is taken as 13.0 mm 3 6.5 mm, which we have di-vided into 260 3 130 mesh grids. Figure 1 shows calcu-lated results of the dispersion relations for our routine.The waveguide structure is shown in the inset to Fig. 1,and the relevant parameters are set to W 5 3.0 mm, L5 1.5 mm, n1 5 1.5, and n2 5 1.0. The dispersioncurves are drawn in terms of the normalized index n andthe normalized frequency n, which are defined byn 5 @(b/k)22 n1

2]/(n12 2 n2

2) and n 5 kLAn12 2 n2

2/p,respectively. Owing to the symmetry of the structure,only half of the cross section is often used. In the presentcalculation, we have taken an overall cross section.Goell’s solutions19 are given by open squares in Fig. 1 to-gether with our numerical results, given by closed circles.Our results are in good agreement with Goell’ssolutions.19

The calculated contour plots of mode profiles are givenin Figs. 2 and 3. The operating wavelength l is set to 1.3mm, which corresponds to the normalized frequency n5 2.58. Parts (a) and (b) of Figs. 2 and 3 represent thedominant and the minor component, respectively, for eachmode; that is, the dominant component corresponds to thex component of quasi-TE modes and the y component ofquasi-TM modes as shown in Figs. 2(a) and 3(a), respec-tively. The level of the minor component is 2 orders ofmagnitude smaller than that of the dominant component.As shown in Figs. 2(b) and 3(b), the minor component ap-pears at the corners of the structure. In addition, thediscontinuities of a normal electric field across the indexinterfaces are clearly observed. This is also observed in

Fig. 3. Contour plot of the lowest-order quasi-TM mode: (a) ycomponent (dominant), (b) x component (minor).

H. Noro and T. Nakayama Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1457

Figs. 2(a) and 3(a). These results demonstrate quite wellthe vector nature of modes. The normalized index n cal-culated for the lowest quasi-TE and quasi-TM modes is0.8707 and 0.8500, respectively. The CPU time neededto calculate both of the mode indices and the correspond-ing mode profiles for the lowest-order mode is 214 on aSPARC station 20. We should note that the mode indicesand the corresponding mode profiles of 15 modes havebeen calculated by turn. For this the memory space re-

Fig. 4. Nonsymmetric rib waveguide directional coupler.

Fig. 5. Contour plot of the lowest-order symmetric supermodefor the quasi-TE mode: (a) x component (dominant), (b) y com-ponent (minor).

Fig. 6. Same as Fig. 5, but for the lowest-order antisymmetricsupermode.

quired is 20 Mbytes. We have demonstrated the effi-ciency of the present method in Ref. 10 by comparing itwith that of the Lanczos method in the case of scalaranalysis.

B. Nonsymmetric Directional Coupler

We have calculated the propagation constants and thecorresponding mode profiles of two lowest-order super-modes (symmetric and antisymmetric supermodes) of thenonsymmetric rib waveguide directional coupler shown inFig. 4. The same structure has been analyzed in terms ofa scalar approximation.20 The parameters employed forthe calculations are W1 5 3.0 mm, W2 5 5.0 mm, L5 0.9 mm, h 5 0.1 mm, S 5 2.0 mm, nC 5 1.0, nG5 3.44, and nS 5 3.36. The operating wavelength is1.55 mm. We have taken the region as 22.0 mm 3 5.0mm, which is divided into 220 3 100 mesh grids, and cal-culated six supermodes for the structure. The memoryspace required is 7 Mbytes.The contour plots of supermodes for the nonsymmetric

rib waveguide directional coupler are presented in Figs.5–8, and the corresponding propagation constants aresummarized in Table 2. Figures 5 and 6 give symmetricand antisymmetric supermodes for quasi-TE modes, re-spectively. In addition, Figs. 7 and 8 represent symmet-ric and antisymmetric supermodes for quasi-TM modes,respectively. Parts (a) and (b) of Figs. 5–8 show thedominant and the minor component, respectively. Thelevel of the minor component is 1 order of magnitudesmaller than that of the dominant component.The minor component for the quasi-TE mode of Figs.

5(b) and 6(b) is localized very close to the corners of therib waveguide. This is due to the fact that both deriva-tives of the refractive index in Eq. (5b) with respect to thedirections x and y are nonzero close to the corners of therib waveguide. On the other hand, the minor componentfor the quasi-TM mode of Figs. 7(b) and 8(b) is locatedclose to the corners of the rib waveguide but the maxi-mum part is not close to the corners. The minor compo-nent is not well confined, indicating that the minor com-ponent of the quasi-TM mode is leaky compared with thatof the quasi-TE mode. The dominant component of thesymmetric supermode of Fig. 5(a) is unbalanced towardthe larger rib waveguide (see Fig. 4), while that of the an-tisymmetric supermode of Fig. 6(a) is unbalanced towardthe smaller rib waveguide. These supermode profilesrepresent clearly the degree of the interaction betweenthe two rib waveguides. The complete power transfer be-tween the two rib waveguides does not occur. This is alsothe case in the quasi-TM mode.The numerical results for supermodes enable us to cal-

culate the degree of power transfer and the couplinglength. In addition, if the mode dispersion of two iso-lated rib waveguides can be arranged so that propagationconstants of the two rib waveguides become the same at awavelength by controlling refractive indices of the two ribwaveguides, the nonsymmetric rib waveguide couplershould show a complete power transfer. Thus the modeanalysis will be directly related to the design of deviceswith nonsymmetric structures.

1458 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 H. Noro and T. Nakayama

5. SUMMARYThis paper has presented an efficient method for thevector-wave analysis in optical waveguides. The methodpresented here is based on the unusual molecular-dynamical calculation and applicable to eigenvalue prob-lems for large nonsymmetric (or non-Hermitian) matricesassociated with vector-wave equations. The efficiency ofthe present method has been demonstrated by applying itto the vector-mode analysis of optical waveguides,namely, by calculating mode dispersion relations of arectangular dielectric waveguide and supermodes of a

Table 2. Propagation Constants of theNonsymmetric Rib Waveguide Coupler

Mode Quasi-TE Mode Quasi-TM Mode

Symmetric 13.769546 13.750483Antisymmetric 13.764359 13.745666

Fig. 7. Contour plot of the lowest-order symmetric supermodefor the quasi-TM mode: (a) y component (dominant), (b) x com-ponent (minor).

Fig. 8. Same as Fig. 7, but for the lowest-order antisymmetricsupermode.

nonsymmetric rib waveguide directional coupler. Thecalculated results of dispersion relations for an embeddedrectangular dielectric waveguide are in good agreementwith the solutions obtained by Goell.19 The calculatedmode profiles for the transverse electric field clearly showthe discontinuity of the normal electric field across indexinterfaces and the hybrid nature of modes, surely reflect-ing the vector nature of electric fields. In addition, su-permodes of the nonsymmetric rib waveguide directionalcoupler have been calculated by the present method. Itis difficult to analyze such a nonsymmetric structure interms of conventional numerical methods because of therequirement of a large amount of computational memoryspace.We should stress that our method is useful for perform-

ing large-scale calculations of the vector analysis of opti-cal waveguides, since the memory space scales linearlywith grid points N, even for nonsymmetric structures.This is due to the fact that the time-consuming part incomputations is to solve the time-development equationsof motion [Eqs. (22) and (24)]. The additional advantagesof the method lie in its being easily vectorized and paral-lelized for implementation on array-processing modernsupercomputers. We conclude that the present methodis powerful in the analysis, the modeling, and the designof optical waveguides. In addition, it is possible to applyour method to the analysis of photonic band structures,taking into account the vector nature of electromagneticwaves.

ACKNOWLEDGMENTThis work was sponsored in part by a Grant-in-Aid forScientific Research from the Japan Ministry of Education,Science and Culture.

REFERENCES1. B. M. A. Rahman and J. B. Davies, ‘‘Vector-H finite element

solution of GaAs/GaAlAs rib waveguides,’’ IEE Proc. Part J132, 349–353 (1985).

2. K. Biewirth, N. Schulz, and F. Arndt, ‘‘Finite-differenceanalysis of rectangular dielectric waveguide structure,’’IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114(1986).

3. A. P. Ansbro and I. Montrosset, ‘‘Vectorial finite differencescheme for isotropic dielectric waveguides: transverseelectric field representation,’’ IEE Proc. Part J 140, 253–259 (1993).

4. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri,‘‘Full-vectorial mode calculations by finite differencemethod,’’ IEE Proc. Optoelectron. 141, 281–286 (1994).

5. M. S. Stern, ‘‘Semivectorial polarized finite differencemethod for optical waveguides with arbitrary index pro-files,’’ IEE Proc. Part J 135, 56–63 (1988); M. S. Stern,‘‘Semivectorial polarized H field solutions for dielectricwaveguides with arbitrary index profiles,’’ IEE Proc. Part J135, 333–338 (1988).

6. W. P. Huang and C. L. Xu, ‘‘Simulation of three-dimensional optical waveguides by a full-vector beampropagation method,’’ IEEE J. Quantum Electron. 29,2639–2649 (1993).

7. C. L. Xu, W. P. Huang, and S. K. Chaudhuri, ‘‘Efficient andaccurate vector mode calculations by beam propagationmethod,’’ J. Lightwave Technol. 11, 1209–1215 (1993).

8. Em. E. Kriezis and A. G. Papagiannakis, ‘‘A joint finite-

H. Noro and T. Nakayama Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1459

difference and FFT full vectorial beam propagationscheme,’’ J. Lightwave Technol. 13, 692–700 (1995).

9. P. L. Liu and B. J. Li, ‘‘Full-vector mode analysis of ribwaveguides by iterative Lanczos reduction,’’ IEEE J. Quan-tum Electron. 29, 2859–2863 (1993).

10. H. Noro and T. Nakayama, ‘‘Mode analysis of opticalwaveguides: a new approach,’’ Opt. Lett. 20, 1227–1229(1995); H. Noro and T. Nakayama, ‘‘A new approach to sca-lar and semivector mode analysis of optical waveguides,’’ J.Lightwave Technol. 14, 1546–1556 (1996).

11. Y. Okamoto and H. J. Marris, ‘‘A novel algorithm for calcu-lation of the extreme eigenvalues of large Hermitian matri-ces,’’ Comput. Phys. Commun. 76, 191–202 (1993).

12. D. M. Young, Iterative Solution of Large Linear Systems(Academic, New York, 1971).

13. M. Born and K. Huang, Dynamical Theory of Crystal Lat-tices (Clarendon, Oxford, 1954).

14. T. Nakayama, M. Takano, K. Yakubo, and T. Yamanaka,‘‘Numerical method for the analysis of optical waveguides,’’Opt. Lett. 17, 326–328 (1992).

15. C. Lanczos, ‘‘An iteration method for the solution of the ei-

genvalue problem of linear differential and integral opera-tors,’’ J. Res. Natl. Bur. Stand. 50, 255–283 (1950).

16. T. Terao, K. Yakubo, and T. Nakayama, ‘‘Numericalmethod for large-scale non-Hermitian matrices and its ap-plication to percolating Heisenberg antiferromagnets,’’Phys. Rev. E 50, 566–576 (1994).

17. T. Nakayama, ‘‘Computing very large matrices: the forcedoscillator method,’’ in Computational Physics as a NewFrontier in Condensed Matter Research, H. Takayama, M.Tsukada, H. Shiba, F. Yonezawa, M. Imada, and Y. Okabe,eds. (The Physical Society of Japan, Tokyo, 1995), pp. 21–33.

18. K. Yakubo, T. Nakayama, and H. J. Maris, ‘‘Analysis of anew method for finding eigenmodes of very large lattice sys-tems,’’ J. Phys. Soc. Jpn. 60, 3249–3259 (1991).

19. J. E. Goell, ‘‘A circular-harmonic computer analysis of rect-angular dielectric waveguides,’’ Bell Syst. Tech. J. 48,2133–2160 (1969).

20. T. M. Benson and P. C. Kendall, ‘‘Equipartition method forthe analysis of two nonidentical closely separated ribwaveguides,’’ IEE Proc. Part J 140, 62–65 (1993).