R. Z. L. Ye and D. O. Yevick Vol. 18, No. 11 /November 2001 /J. Opt. Soc. Am. A 2819

Noniterative calculation of complex propagationconstants in planar waveguides

Raymond Z. L. Ye

Cisco Systems, Inc., 170 West Tasman Drive, San Jose, California 95134

David O. Yevick

Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Received November 20, 2000; revised manuscript received March 29, 2001; accepted May 7, 2001

We adapt an efficient finite-difference procedure for determining complex propagation constants to the analysisof modes in planar waveguides. The method requires solving a single rather than multiple eigenvalue equa-tions and does not require prior knowledge of either the nature of the solutions or the position of the modaleigenvalues in the complex plane. © 2001 Optical Society of America

OCIS codes: 130.2790, 230.7390, 230.7400.

1. INTRODUCTIONThe determination of the complex modal propagation con-stants and the associated modal field distributions of pla-nar waveguides that are composed of successive planarlayers with loss or gain is a frequently occurring problemin optoelectronic design, as noted in numerous overviewpapers such as Refs. 1–3. Many recent methods used toanalyze this problem are based on the transfer matrixtechniques for a multilayer structure.4–6 This leads to anequation of the form t(b) 5 0, where b is the complexpropagation constant, which is then solved numerically.Accordingly, a two-dimensional root-finding procedure inthe complex plane must be employed. The downhill6 andthe Newton–Raphson root-finding methods7 ensure first-order and second-order convergence, respectively, but re-quire an initial guess followed by multiple iterations todetermine each possible root. Modal field calculationsmay therefore be difficult if insufficient information isavailable to generate proper starting values.

Alternatively, Cauchy’s integral formula,3,8,9 which is anoniterative method, can be applied to find the roots oft(b) 5 0 within a specified contour of integration. Re-ferred to as the argument principal method, this proce-dure normally requires evaluating derivatives of the dis-persion equation. As well, the user must specify acontour in the complex plane such that the required nu-merical integration over the contour is convergent andthe propagation constants’ desired modes lie within thecontour. Recently, however, a more efficient root-findingtechnique that does not require the evaluation of deriva-tive functions in contour integral evaluation has beendeveloped,10 and, as well, two new procedures, namely,the reflection-pole and the wave-vector density methodshave been proposed for the determination of guided andleaky modes.11 The latter two techniques have the fur-ther advantage that the dispersion equation does nothave to be solved explicitly, but rather the propagationconstants are obtained respectively from the quasi-

0740-3232/2001/112819-04$15.00 ©

Lorentzian resonances in the reflection coefficient peaksand from the density of wave-vector states of planarmultilayer waveguide structures.

A second set of numerical methods are based on itera-tive finite-difference procedures. Here appropriateboundary conditions are imposed at the computationalwindow edges. The desired solutions are then generatediteratively by determining field distributions that satisfythese boundary conditions.12–14 If the problem is trans-formed into an eigenvalue system, the eigenvalue systemis generally solved repeatedly while the electric field dis-tribution is adjusted after each iteration to match theboundary conditions. This procedure can be complicatedif several complex propagation constants are required.

In this paper we apply an alternative finite-differencemethod to determine the guided modes of multilayerwaveguides subject to appropriate boundary conditionsthat should provide a useful complement to currently em-ployed methods. The modes are obtained by solving anonlinear eigenvalue system. No prior knowledge oftheir approximate positions in the complex plane is there-fore required. The technique is noniterative in the sensethat the boundary conditions are satisified after a singlesolution is found for an appropriate eigensystem. In con-trast, many other procedures require that matrix eigen-systems be solved multiple times while the field is ad-justed, to ensure that the boundary conditions aresatisified between each successive solution stage.

2. THEORYConsider a planar waveguide with y as the direction oftransverse confinement and z as the propagation direc-tion. Specializing to TE modes and writing

E~ y ! 5 xE~ y !exp@i~bz 2 vt !#, (1)

the Helmholtz equation yields

2001 Optical Society of America

2820 J. Opt. Soc. Am. A/Vol. 18, No. 11 /November 2001 R. Z. L. Ye and D. O. Yevick

d2

dy2 E~ y ! 1 @k2~ y ! 2 b2#E~ y ! 5 0, (2)

in which we have defined k( y) 5 (2p/l)n( y) and thepropagation constant can be written as b 5 (2p/l)neffwith l the light wavelength in vacuum, n( y) the complexrefractive index distribution and neff the effective modalindex.

On a discrete set of grid points y 5 jDy, j 5 1, 2,..., N,the finite-difference representation of Eq. (2) is

Ej11 2 2Ej 1 Ej21

~Dy !2 1 kj2Ej 5 b2Ej . (3)

Equation (3) then yields an eigenvalue system for thepropagation constant and field profiles once the ficticiousfield values E0 and EN11 are determined through an ap-propriate boundary condition. As the general solutionof Eq. (2) is E( y) 5 C1 exp(ay) 1 C2 exp(2ay) witha 5 @b2 2 k2( y)#1/2, for positive R(a) the field in thecover will be proportional to exp(ay) and the field in thesubstrate will be proportional to exp(2ay). The bound-ary conditions are then

E0 5 exp~2a0Dy !E1 , a0 5 ~b2 2 k02!1/2, (4)

EN11 5 exp~2aN11Dy !EN , aN11 5 ~b2 2 kN112 !1/2.

(5)

Equation (3) is thus transformed into the matrix equation

ME 5 b2E. (6)

While the above equation has the form of a standard ei-genvalue problem, the two matrix elements M11 andMNN are now functions of b2 through the quantities a0and aN11 of Eqs. (4) and (5). A time-consuming iterativeapproach is accordingly required to determine each of themodal field distributions and propagation constants, espe-cially for complex refractive-index distributions.

In this paper we propose a noniterative approach forthe modes of a waveguide with complex refractive indicesthat requires a single solution of Eq. (6). We first expressEq. (6) in terms of the variables9

u 5 ~b2 2 k02!1/2 1 ~b2 2 kN11

2 !1/2 (7)

and R 5 (kN112 2 k0

2)1/2, resulting in

a0 5u2 1 R2

2u, (8)

aN11 5u2 2 R2

2u, (9)

b2 5 kN112 1 aN11

2 5u4 1 2~2kN11

2 2 R2!u2 1 R4

4u2 .

(10)

In the case of guided modes that vanish at y → 6`,

Re$a0~u !% . 0, (11)

Re$aN11~u !% . 0. (12)

For a sufficiently small discretization step Dy we ap-proximate

exp~2a0Dy ! ' 1 2 a0Dy 112 a0

2~Dy !2, (13)

exp~2aN11Dy ! ' 1 2 aN11Dy 112 aN11

2 ~Dy !2 (14)

in the boundary conditions of Eqs. (4) and (5), respec-tively. Similarly, expressing a0 , aN11 and b2 and thematrix elements M11 and MNN as power series in uyields in place of Eq. (6)

~M4u4 1 M3u3 1 M2u2 1 M1u 1 M0!E 5 0, (15)

where the N 3 N matrices M0,..., M4 , are given in Ap-pendix A. The above nonlinear eigenvalue problem in ucan be rewritten as the 4N 3 4N linear eigensystem,15

Mf 5 uf, (16)

where

f 5 S f0

f1

f2

f3

D , f0 [ E, (17)

M 5 F 0 1 0 0

0 0 1 0

0 0 0 1

2M421M0 2M4

21M1 2M421M2 2M4

21M3

G .

(18)

The first N components of the eigenvector f then containthe modal field distributions, and the corresponding ei-genvalues u determine the propagation constants of themodes. The numerical solution of this eigenvalue systemmay lead to spurious modes in finite-difference proce-dures. To avoid these modes, we impose the physicalconstraints corresponding to Eqs. (11) and (12), as well asthe condition R(nsubstrate) , R(neff) , R(nguiding-layer) forguided modes when selecting eigenvalues.

3. EXAMPLES AND DISCUSSIONTo determine the accuracy of the numerical method devel-oped in Section 2, we now analyze waveguide structuresrepresented by Fig. 1. In our first calculation weconsider a high-loss multilayer waveguide corres-ponding to the parameters nC 5 1.0, n1 5 3.5912 i0.084, d1 5 1.502 mm, n2 5 3.211, d2 5 0.74 mm,n3 5 3.166, d3 5 3.5 mm, and nS 5 3.15.11 The light

Fig. 1. Multilayer planar waveguide structure. The computa-tional region is @ y0 , yN11#.

R. Z. L. Ye and D. O. Yevick Vol. 18, No. 11 /November 2001 /J. Opt. Soc. Am. A 2821

Table 1. TE Complex Effective Refractive Indices for a Multilayer Waveguide

Mode APM RPM This Work

TE0 3.5628 2 i0.8401(21) 3.5630 2 i0.909(21) 3.5627 2 i0.8401(21)TE1 3.4779 2 i0.8377(21) 3.4952 2 i0.831(21) 3.4776 2 i0.8374(21)TE2 3.3360 2 i0.8141(21) 3.3442 2 i0.687(21) 3.3359 2 i0.8135(21)TE3 3.1633 2 i0.5285(22) 3.1642 2 i0.512(22) 3.1631 2 i0.5281(22)TE4 3.1594 2 i0.3437(22) 3.1598 2 i0.298(22) 3.1594 2 i0.3513(22)

Table 2. TE0 Complex Effective Refractive Index for a Slab Waveguide with @ y0 , yN11#Ä@21 mm, 3 mm#

Transverse Grid Points TE0 Mode

N 5 40 1.51874 2 i0.45852(22)N 5 80 1.51875 2 i0.45861(22)N 5 120 1.51875 2 i0.45863(22)Benchmark 1.51875 2 i0.45866(22)

Table 3. TE0 Effective Refractive Index for a Slab Waveguide, Discretization Step Fixed with DyÄ0.03 mm

@ y0 , yN11# N Zero Boundary Conditions This Work

[20.5 mm, 2.5 mm] 99 1.51720 2 i0.48976(22) 1.51875 2 i0.45867(22)[21.0 mm, 3.0 mm] 132 1.51843 2 i0.47063(22) 1.51875 2 i0.45863(22)[21.5 mm, 3.5 mm] 166 1.51868 2 i0.46236(22) 1.51875 2 i0.45864(22)[22.0 mm, 4.0 mm] 199 1.51875 2 i0.45974(22) 1.51875 2 i0.45870(22)

wavelength is l51.55 mm, and N5200 transverse gridpoints and t 5 1 mm are used. Further, the refractive in-dex at each grid point is taken as the average of the re-fractive index over the region closest to the given point.In Table 1 the resulting values of the TE propagation con-stants of the first five lowest-order modes are comparedwith those of the argument principle method and thereflection-pole method as presented in Ref. 11. Clearly,the procedure of this paper is equivalent in accuracy to es-tablished techniques.

Next, in Table 2 we examine the convergence of thenoniterative method with respect to the number of trans-verse grid points. Here we consider a slab wave-guide with l51.0 mm, nC 5 1.0, n1 5 1.53 2 i0.005,d1 5 2 mm, and nS 5 1.5. The benchmark value sup-plied in the table is obtained by the iterative procedure ofRef. 6.

Finally, in Table 3 we display the propagation-constantchange induced by displacements of the computationalwindow boundary. Since the boundary condition is di-rectly incorporated into a matrix equation that is thensolved numerically, the calculated TE0 propagation con-stants for the preceeding slab waveguide are found to be

effectively independent of the boundary position for a suf-ficiently large number of computational grid points. As acomparison, we have presented equivalent results calcu-lated by setting the electric field to zero at the boundary.While zero boundary conditions are generally not em-ployed in practice, the results of the table clearly illus-trate the dependence of the propagation constants on theboundary location for nonexact boundary conditions andinsufficiently large computational window widths.

4. CONCLUSIONSWe have applied a noniterative finite-difference method tothe determination of the complex propagation constant ofplanar waveguides. This procedure transforms the equa-tion into a trival eigenvalue system that is easily solvedwith standard matrix procedures. Further, the boundaryconditions are adapted to the proper field behavior in theexternal region so that the calculated propagation con-stants are insensitive to the computational window width.As a result, the procedure should be a useful complementto current search algorithms for complex propagation con-stants.

APPENDIX A: MATRIX ELEMENTSBelow we present formulas for the elements of the matrices M0 ... M4 of Eq. (15):

@M0# ij 5 H R4~Dy !2 i 5 j 5 1 or i 5 j 5 N

2R4~Dy !2 i 5 j 5 2 ,..., N 2 1

0 otherwise,

@M1# ij 5 H 4R2Dy i 5 j 5 1

24R2Dy i 5 j 5 N

0 otherwise,

@M2# ij 5

~8kN112 2 8ki

2 2 6R2!~Dy !2 1 8 i 5 j 5 1

~8kN112 2 8ki

2 2 2R2!~Dy !2 1 8 i 5 j 5 N

~8kN112 2 8ki

2 2 4R2!~Dy !2 1 16 i 5 j 5 2,..., N 2 1 ,

2822 J. Opt. Soc. Am. A/Vol. 18, No. 11 /November 2001 R. Z. L. Ye and D. O. Yevick

5 28 i 5 j 1 1 5 2,..., N or i 1 1 5 j 5 2 ,..., N

0 otherwise

@M3# ij 5 H 4Dy i 5 j 5 1 or i 5 j 5 N

0 otherwise,

@M4# ij 5 H ~Dy !2 i 5 j 5 1 or i 5 j 5 N

2~Dy !2 i 5 j 5 2,..., N 2 1

0 otherwise.

ACKNOWLEDGMENTThe authors thank Jun Yu for helpful discussions. Aswell, David Yevick acknowledges financial support re-ceived from the Natural Sciences and Engineering Re-search Council (Canada), Canadian Institute for PhotonicInnovations, and Nortel Networks.

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