dispersion relation of leaky modes in nonhomogeneous waveguides and its applications

3230 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 21, NOVEMBER 1, 2011

Dispersion Relation of Leaky Modes inNonhomogeneous Waveguides and Its Applications

Jianxin Zhu and Zheqi Shen

Abstract—For a nonhomogeneous waveguide, whose refractiveindex is not a constant, the problem is very complicated since thenonlinear eigenvalue problems are unable to reduce to algebraicequations yet. When the refractive index is varied, the dispersionrelation cannot be derived by using the analytic expressions of thesolutions in each layer. In this paper, this problem is solved by usingthe differential transfer matrix method, which is introduced to de-duce the dispersion relations of leaky modes for TE and TM cases,respectively.Moreover, for the waveguide whose refractive index isgradually varied, the dispersion relations can be approximated bysome simpler algebraic equations, which are close to the exact re-lations and very easy to analyze. Asymptotic solutions are used asinitial guesses, and followed by Newton’s method, to give very ac-curate solutions. This paper is a generalization of the asymptoticmethod of slab waveguides; all the results therein are consistentwith the analysis here.

Index Terms—Differential transfer matrix method (DTMM),dispersion relations, leaky modes, optical waveguide.

I. INTRODUCTION

I N computation of open waveguides, leaky modes [1]play an important role; they are useful to approximate

the continuum of radiation and evanescent waves. However,leaky modes are not members of the complete modes of anunbounded lossless waveguide. When the waveguide is un-bounded, the solutions can be decomposed into a few discretepropagation modes and a continuum of radiation modes [2].In practical computation, the infinite integration is not easy toevaluate, and leaky modes are therefore needed. For numericalmethods, the unbounded domain is always restricted to a finiteregion by means of absorbing boundary conditions or perfectlymatched layers (PML) [3]. Due to the fact that the originalopen problem is approximated by a bounded one, the propertyof orthogonality breaks, and the complete mode expansion isreplaced by the infinite sum of a set of discrete modes. As aresult, the integration of radiation modes is approximated bythe sum of leaky modes. Mathematically, they are essentiallytwo different problems [4] with different spectrums [5]. Butin numerical simulations, the leaky modes expansion is very

Manuscript received June 23, 2011; revised August 28, 2011; accepted Au-gust 29, 2011. Date of publication September 06, 2011; date of current versionOctober 19, 2011. This work was supported in part by Natural Science Founda-tion of China under Grant 11071217, in part by the Natural Science Foundationof Zhejiang Province, China, under Grant Y6100210, and in part by the Educa-tion Department of Zhejiang Province under Grant 200906635.The authors are with the Department of Mathematics, Zhejiang University,

Hangzhou 310027, China (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JLT.2011.2167129

efficient. In most cases, a small amount of leaky modes aresufficient for good approximation of the solutions [6].For the computation of leaky modes, there are two main di-

rections: first of all, the eigenvalue problem could be solved di-rectly by finite difference method [7] or finite-element method[8]. However, because of the absorbing boundary conditionsor the utilization of PMLs, the problem is usually nonlinear orcomplex, and very difficult to solve; these methods work wellonly in some certain circumstances. The other way is trans-forming the eigenvalue problem to an algebraic equation, ornamely the dispersion relation, whose roots correspond to theeigenvalues. There are lots of papers on this issue [9]–[12]; mostof them deal with the slab waveguide, in which the refractiveindices are piecewise-constant functions. For a slab waveguide,the solution can be represented analytically with only two un-determined coefficients in each layer. The dispersion relation is,therefore, derived exactly using the boundary conditions and thecontinuous conditions on each interfaces.However, when the refractive index is not a piecewise-con-

stant function, the situation is much more complicated. For ex-ample, if the refractive index in the core is a continuous func-tion, there is no analytical expression of the solutions here, andneither the explicit expression of dispersion relation. In [12], theWKB method [13] is used to approximate the solutions in thecore, and the dispersion relation is derived in the same way asthat used in the slab waveguide problems. However, this methodjust provides a relation for approximate solutions, and the accu-racy is unable to evaluate.The differential transfer matrix method (DTMM) is a useful

tool in computation of nonhomogeneous structures [14], [15].In this paper, we use DTMM to derive the dispersion relationsof leakymodes in nonhomogeneous waveguides, including bothtransverse electric (TE) and transverse magnetic (TM) polariza-tions. In addition, under the assumption that the refractive indexvaries gradually, the relations can be further reduced to simpleralgebraic equations, which are more solvable. The equation forTE polarization is just as the same as that in [12]. And we havealso given the equation for TM polarization, which has not yetproposed by other methods. In the special case when the refrac-tive index is a constant in the core, the resulting relations accordwith [11] totally. The same asymptotic analysis therein is usedafterward to give initial guesses of leaky modes, and for betterperformances, additional Newton’s method is applied.The organization of this paper is as follows. In Section II,

the formulations of both TE and TM modes are given, and theDTMM is introduced to derive the dispersion relations andthe approximate equations in each cases. In Section III, theleaky modes in gradually varied waveguides are derived via

0733-8724/$26.00 © 2011 IEEE

ZHU AND SHEN: DISPERSION RELATION OF LEAKY MODES 3231

Newton’s method, where the asymptotic solutions are used asinitial guesses. Numerical results and comparisons are given inSection IV. Finally, the conclusions are made in Section V.

II. FORMULATION FOR DISPERSION RELATIONS

A. Basic Equations

The equation for open waveguide problems is the unbounded2-D Helmholtz equation

(1)

in which for TE polarization and for TMpolarization. Let the refractive index function be

(2)

which has two discontinuous points at and . Andcan an arbitrary positive function of .Since the refractive index function is invariant along the

propagation direction , we can assume that the solutions aregiven by . Substitute into (1), the problemturns to be an unbounded eigenvalue problem

(3)

The waveguide modes are actually solutions of (3). The-oretically, the complete mode composition of (3) consists of afew discrete propagation modes and a continuum of radiationmodes. However, leaky modes turn up when the transverse di-rection is truncated by some artificial boundary conditions.The outgoing conditions are imposed at the positive and neg-

ative transverse directions by using the one-way approximationin the regions where the refractive indices are constants. Theyare

(4)

And the conditions at the discontinuous points are given by

(5)

(6)

Denote that ,the eigenvalue problem for TE case is

(7)

(8)

(9)

For TM case, due to the continuous conditions, the eigenvalueproblem is

(10)

(11)

(12)

Both eigenvalue problems are nonlinear and difficult to solvein a regular way. In the following sections, we are going toreduce these problems to some nonlinear algebraic equations,whose roots are corresponding to the leaky modes. For simplifi-cation, the following derivation is regardless of the dependenceof and on .

B. DTMM for the TE Case

We first consider the problem for TE case (7) and introducethe DTMM to reduce the problem. To show the definition ofDTMM, we introduce the transfer matrix method (TMM) inadvance.Suppose there are two horizontally connected homogenous

mediums whose refractive indices are and , respectively,and the interface point of which is . The solutions of (7) inthe regions and could be expressed analyticallyas

in which , andand are unknown coefficients, associated with the back-

ward and forward waves, to be determined. In this case, we have

(13)

with and . The matrixis the transfer matrix from the left layer to the right one,

it can be expressed analytically using the continuous conditionsin the interfaces (5), (6). The result is shown in (14) at the bottomof the page.

(14)


For slab waveguide, in which is constant, TMM is suffi-cient to derive the dispersion relation. The solutions to (7) areof the form

(15)

where . The transfer matrices across the in-terfaces are easy to know from (14) as

(16)

(17)

and in the homogeneous case. By the definitionof transfer matrices, we have the transfer relation

(18)

To satisfy the boundary conditions (8) and (9), the coefficientsmust have and . As a result, we have therelation

which is the same as that in [11].On the other hand, when varies continuously, the

transfer matrix is not identity any longer, since thesolutions to (7) are of the form

(19)

in which and the coefficients dependon . In order to determine , it immediately comes tomind that the step approximation of can be used. Theproblem (7)–(9) is then transformed to the problem of multi-layered waveguides, which has been discussed in [8]: Stowelland Tausch have shown that the transfer matrices in this kind ofproblems are ill posed, and the resulting systems are too compli-cated to give the relations, especially when the amount of layersis large.Nevertheless, the idea of step approximation is still useful,

as soon as we make the number of layers infinity. There comesto the idea of DTMM. We first determine the transfer matrix

for an arbitrary small . Assume that the refractiveindices are still constants in the neighborhoods of and ,respectively. In this case, (14) is still valid. The transfer matrixin this circumstance is defined by

(20)

where . Equivalently

in which is the identity matrix. Taking results in

(21)

is exactly expressed using a similar formula of (14).Substitute into (21), we have

(22)

Or equivalently

(23)

in which the first term represents the phase variation along thetransverse direction, and the second term depends on the inter-action between the forward and backward waves. Under the as-sumption of gradual variation, the second term can be neglected.Anyway, based on (21), it can be solved that

(24)

in which is the matrix exponential defined as

It thus can be observed that

(25)

Bringing (16)–(18) and (25) together, we have

(26)

in which in (16) and (17) are replaced by and , re-spectively. The dispersion relation can be derived by imposingthe conditions and as well. However, theintegration and the matrix exponential in (24) make the relationvery complicated to solve. Fortunately, the varied refractive in-dices are usually assumed to be gradually varying in practicalapplications. Under this assumption, the matrix exponential iseasy to evaluate, since the off-diagonal elements of are ne-glected by only using the first term of (23) as

(27)

The resulting dispersion relation is


Using integration by parts, it equals to

(28)

which is equivalent to the relation derived by the WKB methodin [12].

C. DTMM for the TM Case

The equation for TM case is (10), and the conditions atand can be derived by (11), (12) and the continuous

conditions (5), (6). They are

which are no different from the boundary conditions of TE case.Before the derivation of differential transfer matrix, let

, the eigenvalue problem (10) becomes

(29)

The conditions at and for are the same, sinceand are constants. If we denote

the problem for is

(30)

(31)

(32)

which is almost the same as the TE problem. However, the con-tinuous conditions for are

(33)

(34)

The transfer matrix can be derived as (35), shown at thebottom of the page. Comparing to (14), (35) is much more com-plicated, and the related transfer matrices are very inconvenientto give the dispersion relation.Fortunately, when we assume that the refractive index func-

tion varies slowly, is much smaller than , the secondterm of (35) can, therefore, be neglected. The rest of the deriva-tion of dispersion relation is trivial; we just list the results analo-gous to (14), (16), and (17) as shown in (36)–(38) at the bottomof the page, in which andfor . Since the varied refractive index functionin the core is continuous, the differential transfer equation isas the same as (22) except that is replaced by . In the samemanner, we have the relation

(39)

for TM case, under the assumption of gradually varied.

III. ASYMPTOTIC FORMULA AND NEWTON’S METHOD

We have already transformed the eigenvalue problem (7)–(9)and (10)–(12) to the nonlinear equations (28) and (39) for TEand TM polarizations, respectively. To solve these equations,Newton–Raphson method is recommended as soon as appro-priate initial solutions are given. Fortunately, Zhu and Lu havealready given asymptotic expressions for slab waveguides in[11]. Since we are discussing the gradually varied waveguides,which are not very different from the slab waveguides numeri-cally, and the iteration process plays a more important role, theasymptotic solutions of slab waveguides are sufficient to be ourinitial guesses, especially when the index of modes is large.

(35)

(36)

(37)

(38)


The derivation of asymptotic solutions refers to [11]. The for-mula for th leaky mode in TE case is

(40)

in which the multivalued Lambert W functions [16] are usedwith , and .

the parameter is chosen arbitrarily in interval . It is notice-able that the case is meaningful in the graduallyvaried waveguides, so we have to shift the index by setting

. However, the leading order asymptoticformula is enough for the iteration will eventually be applied.When we have these , Newton’s method can be used to give

more accurate solutions. As mentioned in Section II, andare functions of , and is actually .We can denote

The iterative scheme reads

(41)

for each .Similarly, for the TM case, we have the asymptotic formula

(42)

for , in which,

and is still in . The iterative scheme is with

The procedures (41) converge quickly in finding the roots ofequation (28) and (39), which is to be shown in the next section.

IV. NUMERICAL RESULTS

We have given the dispersion relations for TE and TM cases,respectively, in Section II as (28) and (39), and the asymptoticformulas and Newton’s method are used to solve them as in(40)–(42).We are going to use them to compute the leakymodesof some waveguides with gradually varied indices, and to showthey relate to the solutions of our dispersion relations as weexpected.For example, let m, , and

is gradually varied as . The free spacewavelength is assumed to be m and, therefore, thewavenumber . For TE and TM cases, we use theasymptotic formulas with to give initial solutions ofdispersion relations (28) and (39) which are denoted as , while

Fig. 1. Asymptotic solutions and leaky modes of example 1: the asymptoticsolutions are marked by “ ,” the solutions of (28) and (39) are marked by “ ,”and the exact solutions are marked by “ .”

TABLE ISOLUTIONS AND RELATIVE ERRORS OF EXAMPLE 1—THE TE CASE

TABLE IISOLUTIONS AND RELATIVE ERRORS OF EXAMPLE 1—THE TM CASE

the solutions using Newton’s method are given by . stepsof iterations are used to achieve the accuracy of for eachmodes. The solutions are derived in Fig. 1. It is noticeable thatthe performance of asymptotic formula (40) is not so good, andmore steps of iterations are needed. In the TM case, the firstthree asymptotic solutions direct to the same propagation mode.Based on the derivation of differential matrices and the

transfer relation (26), the exact solutions, i.e., leaky modes, canbe found when the residues defined by

(43)

are zero, where

and . How-ever, due to the small modification in (35), such solutions arenot actually the exact solutions for TM cases. But in graduallyvaried waveguide, the difference is negligible. The solutions ofour dispersion relations and relative errors of TE and TM casesare given in Tables I and II, respectively. We have derived theexact solutions , which make the residues less than


Fig. 2. Leaky loss of TE1 mode in example 2 in which and. (Solid line) Solution by current method.

(Dashed line) Exact solution.

Fig. 3. Relative errors of TE modes by multilayer approximations and currentmethod of example 3.

. And the relative errors for each cases are tiny in Tables Iand II, which means the solution of our dispersion relationsare good approximations to the leaky modes. Moreover, it canbe observed that the relative errors in the TM case is muchsmaller, and thus the solutions of relation (39) correspond toleaky modes better.In addition, we consider another example in which the leaky

structure has outer cladding layers with refractive indices higherthan the core, our dispersion relations are still valid. We use

and keep other parameters unchanged. Theleaky loss of the mode TE1 which is defined by

dB

like that in [17] is plotted in Fig. 2.The third example is computed when m,

. This wave-guide is asymmetric when . The free-space wavelengthis still m. TE and TM modes together with the rela-tive errors are given in Table III. The omitted case TE1 is not amode, while the situations of the TM case directto a same propagation mode. The results are similar to the firstexample, the performances of both dispersion relations are ex-cellent. Fig. 3 compares the present method and the mentionedmethod in which the waveguide is approximated by a few piece-

TABLE IIIMODES AND RELATIVE ERRORS OF EXAMPLE 3

wise-constant layers. TE case is considered, and the relative er-rors by are plotted. The picture shows that thedispersion relations derived by differential transfer matrices arebetter than those by multilayer approximation, as we expected.

V. CONCLUSION

We have derived dispersion relations for leaky modes of TEand TM cases in nonhomogeneous waveguides. The transferrelations using DTMM are exact, but very complicated to solve.Nevertheless, (28) and (39) are formally simple for both TEand TM cases when the refractive index varies gradually. Themethod used to solve these equations is not difficult at all. Weuse Newton’s method here, in which the initial solutions areexpressed through asymptotic analysis; the performance is verydelightful when the index of modes is large. The derivationof dispersion relations for nonhomogeneous waveguides ismeaningful, for it transforms the nonlinear eigenvalue prob-lems (7)–(9) or (10)–(12) into nonlinear equations whose rootscorrespond to the eigenvalues. Obviously, the latter is mucheasier to solve.

REFERENCES[1] A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Elec-

tron., vol. 17, no. 5, pp. 311–321, 1985.[2] A. Snyder and J. Love, Optical Waveguide Theory. Berlin, Germany:

Springer, 1983.[3] J. Berenger, “A perfectly matched layer for the absorption of electro-

magnetic waves,” J. Comput. Phys., vol. 114, no. 2, pp. 185–200, 1994.[4] J. Hu and C. Menyuk, “Understanding leaky modes: Slab waveguide

revisited,” Adv. Opt. Photon., vol. 1, no. 1, pp. 58–106, 2009.[5] F. Olyslager, “Discretization of continuous spectra based on perfectly

matched layers,” SIAM J. Appl. Math., vol. 64, pp. 1408–1433, 2004.[6] S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approxi-

mations for modal expansion in multilayer open waveguides,” IEEE J.Quantum Electron., vol. 31, no. 10, pp. 1790–1802, Oct. 1995.

[7] R. Ye and D. Yevick, “Noniterative calculation of complex propagationconstants in planar waveguides,” J. Opt. Soc. Amer. A, vol. 18, no. 11,pp. 2819–2822, 2001.

[8] D. Stowell and J. Tausch, “Variational formulation for guided and leakymodes in multilayer dielectric waveguides,” Commun. Comput. Phys.,vol. 7, no. 3, pp. 564–579, 2010.

[9] H. Rogier and D. De Zutter, “Berenger and leaky modes in microstripsubstrates terminated by a perfectly matched layer,” IEEE Trans. Mi-crow. Theory, vol. 49, no. 4, pp. 712–715, Apr. 2001.

[10] L. Knockaert, H. Rogier, and D. De Zutter, “An FFT-based signal iden-tification approach for obtaining the propagation constants of the leakymodes in layered media,” AEU-Int. J. Electron. C., vol. 59, no. 4, pp.230–238, 2005.

[11] J. Zhu and Y. Lu, “Leaky modes of slab waveguides—Asymptotic so-lutions,” J. Lightw. Technol., vol. 24, no. 3, pp. 1619–1623, Mar. 2006.

[12] J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguideswith varied refractive index for microchip optical interconnect appli-cations—Asymptotic solutions,”Microelectron. Reliab., vol. 48, no. 4,pp. 555–562, 2008.


[13] C. Bender and S. Orszag, Advanced Mathematical Methods for Scien-tists and Engineers: Asymptotic Methods and Perturbation Theory.Berlin, Germany: Springer-Verlag, 1999.

[14] S. Khorasani and K. Mehrany, “Differential transfer-matrix methodfor solution of one-dimensional linear nonhomogeneous optical struc-tures,” J. Opt. Soc. Amer. B, vol. 20, no. 1, pp. 91–96, 2003.

[15] M. Eghlidi, K. Mehrany, and B. Rashidian, “Modified differen-tial-transfer-matrix method for solution of one-dimensional linearinhomogeneous optical structures,” J. Opt. Soc. Amer. B, vol. 22, no.7, pp. 1521–1528, 2005.

[16] R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On theLambertW function,” Adv. Comput. Math., vol. 5, no. 1, pp. 329–359,1996.

[17] M. Fedin, A. Popov, and A. Vinogradov, “On the waveguide leakylosses induced by the outer cladding,” J. Opt. A—Pure Appl. Op., vol.10, p. 085003, 2008.

Jianxin Zhu received the B.S. degree in mathematics, the M.S. degree in com-putational mathematics, and the Ph.D. degree in applied mathematics from Zhe-jiang University, Hangzhou, China, in 1984, 1991, and 1998, respectively.Since 1984, he has been with the Department of Mathematics, Zhejiang Uni-

versity, where he is currently a Professor. His current research interests includescientific computation and mathematical modeling.

Zheqi Shen received the B.S. degree in applied mathematics from ZhejiangUniversity, Hangzhou, China, in 2007, where he is currently working towardthe Ph.D. degree in the Department of Mathematics.His current research interests include computational science and engi-

neering, numerical partial differential equations, and simulation of unboundedwaveguides.

dispersion relation of leaky modes in nonhomogeneous waveguides and its applications

Documents