JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 12, DECEMBER 2000 2217

Application of Multiple Scales Analysis and theFundamental Matrix Method to Rugate Filters:Initial-Value and Two-Point Boundary Problem

FormulationsMohammed Bataineh, Member, IEEE,and Omar Rafik Asfar, Senior Member, IEEE

Abstract—In this paper, the filtering problem of apodizedrugates is solved by deriving first-order, as well as second-order,coupled-mode equations via the perturbation method of multiplescales. The first-order perturbation equations are the same asthose of coupled-mode theory. However, the second-order pertur-bation expansion is more accurate, and permits the use of largeramplitudes of the periodic index variation of the rugate. The cou-pled-mode equations are solved numerically by using two differentformulations. The first approach is a two-point boundary-valueproblem formulation, based on the fundamental matrix solution,that is essentially the exact solution for the unapodized rugate.The second approach is an initial-value problem formulation,that uses backward integration of the coupled-mode equations.Comparison with the characteristic matrix method is made forthe case of unapodized rugate in terms of speed and accuracy,and it is found that the fundamental matrix solution is the fastest.The accuracy of the multiple scales solution is measured in termsof the amplitude error and the phase error of the filter’s spectralresponse, taking the characteristic matrix solution as a referencefor the unapodized rugate. The proposed formulations are utilizedto calculate the spectral response of apodized rugates.

Index Terms—Apodization techniques, coupled-mode theory,multiple scales analysis, numerical solutions, optical waveguidefilters, periodic structures.

I. INTRODUCTION

WHEN the refractive index varies sinusoidally or, in gen-eral, periodically in the direction perpendicular to the

film plane of an optical thin film, the resulting structure is re-ferred to as a rugate filter. This inhomogeneous thin-film opticalcoating is assumed to be nonabsorbent, nonscattering, linear,isotropic, and nonmagnetic. The characteristic matrix method[1], used in the analysis of stack filters, also could be used toassess the spectral response of these periodic coatings. This re-quires a huge number of layers to represent faithfully the si-nusoidal profile of the rugate filter, resulting in a prohibitivelylarge number of matrix multiplications and, consequently, largecomputer times. This was realized by Southwell [2], who used

Manuscript received October 15, 1999; revised June 8, 2000.M. Bataineh is with the Hijjawi Faculty for Engineering Technology,

Yarmouk University, Irbid, Jordan.O. R. Asfar is with the Hijjawi Faculty for Engineering Technology, Yarmouk

University, Irbid, Jordan, on leave from Jordan University of Science and Tech-nology, Irbid, Jordan.

Publisher Item Identifier S 0733-8724(00)09102-7.

coupled-wave theory to reduce the computational time for thecase of a constant small amplitude sine-wave profile. Bovard[3] developed an admittance matrix approach for the treatmentof rugate filters. Villa et al. [4] used the same approach to calcu-late thin-film thickness of the rugate filter to obtain a specifiedreflectance.

This paper uses the perturbation method of multiple scales toderive coupled-mode equations for rugate filters. This methodhas been used successfully to treat propagation of waves in cor-rugated open and closed waveguides, optical fibers, and acousticwaveguides [5]–[7]. The method leads to a system of first-ordercoupled ordinary differential equations for the incident and re-flected fields, which are usually referred to as the coupled-modeequations. The filter response is obtained by considering twodifferent formulations for an essentially two-point boundary-value problem. Formulation, as an initial-value problem, is pos-sible, because the reflection coefficient at the substrate interfaceis known. However, the incident mode field is unknown, but theproblem can be circumvented by normalizing its amplitude, thusenhancing an initial-value formulation. On the other hand, thereflection coefficient at the end of a periodic waveguide sec-tion is zero, if the waveguide is matched or assumed to be in-finite in length. A two-point boundary-value problem formula-tion is necessary in the latter case. Both the initial-value problemand the two-point problem formulations are considered in thispaper, and found in agreement with each other, as well as withthe characteristic matrix method for small amplitude of the pe-riodic index perturbation. The two-point problem is solved nu-merically by using the fundamental matrix method [8], whichhas a distinct advantage in terms of speed over the character-istic matrix and the initial-value methods.

A new dimension is added to the filtering problem when theperturbation method of multiple scales is carried to higher orderin , where is the amplitude of the uniform periodic indexvariation. In this paper, both first-order and second-order expan-sions are derived and the range of validity of these expansionsis investigated. The first-order expansion accurately determinesthe spectral response for small, such as considered by South-well [2]. However, as increases, a second-order expansion isrequired, in order to overcome the phase error in the responseof the first-order expansion.

Apodization techniques, to modulate the refractive indexsine-wave profile, were used by various investigators (e.g.

0733–8724/00$10.00 © 2000 IEEE

2218 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 12, DECEMBER 2000

Fig. 1. Schematic model of rugate index distribution and adjacent media.

[9]–[11]) in order to eliminate or reduce the undesiredside-ripple level near to and far from the stopband. In thispaper, we employ a truncated Gaussian apodization functionused by Erdogan [12], Wei et al. [13] for fiber Bragg gratingsas an example.

II. DERIVATION OF COUPLEDMODE EQUATIONS VIA MULTIPLE

SCALES ANALYSIS

The perturbation method of multiple scales is employed toanalyze the interaction of the incident and reflected waves in-side the filter’s structure. The problem considered is that of awave incident from a homogenous medium of refractive index

upon a filter with a refractive index varying according to

(1)

whereis the average refractive index,is a slowly varying function measuring the amplitudemodulation of the sine-wave or the apodization func-tion, andis the wavenumber of the refractive index.

Here, denotes the relative fluctuation of the filter’s refractiveindex about the average value. The filter is deposited on atransparent substrate of index. The geometry is shown inFig. 1, for the unapodized case.

A normal -directed plane wave with is incidenton the interface at . Due to mismatch of refractive indexesas the wave travels down the structure, multiple-reflection willtake place. This scenario is similar to that encountered in mul-tisectional transmission lines [14]. The governing equation for

is Helmholtz’s equation

(2)

where the wavenumber is given by

(3)

with the free-space wavenumber. Substituting for from(1), we can express in the following form

(4)

Here, is the average wavenumber in the rugate.The governing Helmholtz equation (2) is a homogenous second-order differential equation with variable coefficients, whose so-lution is sought via the perturbation method of multiple scales.

We seek a second-order perturbation expansion forinpowers of in the form

(5)

where is a fast varying scale, while , andare slowly-varying scales. Expressing the derivatives

in terms of , and by using the chain rule, substituting in(2), and equating coefficients of equal powers of, we obtain

(6)

(7)

(8)

Note that (6) corresponds to the unperturbed system, whose gen-eral solution is given by

(9)

In this solution, the functions and areslowly-varying functions, representing the amplitudes of theincident and reflected waves, respectively. They are determinedfrom the solvability conditions of the higher order problems.

A. First-Order Coupled-Mode Equations

Substituting (9) into (7), we obtain

(10)

Had we attempted a straightforward expansion, correspondingto , we would have found that it breaksdown when the following resonance condition is satisfied

(11)

This is usually referred to as the Bragg condition, which is aresonance condition in the jargon of theory of waves in periodicstructures. This condition ensures that maximum response oc-curs at . However, it is not necessary that this conditionbe satisfied exactly; it is sufficient for the phase to

BATAINEH AND ASFAR: MULTIPLE SCALES ANALYSIS AND THE FUNDAMENTAL MATRIX METHOD TO RUGATE FILTERS 2219

have a slow spatial variation. The nearness to resonance is mea-sured by a detuning parameterdefined by

(12)

Here is the order of unity. Substituting (12) into (10), we get

(13)

The particular solution of (13) contains secular terms; i.e.,solutions that are proportional to . This means that

would soon become greater than , and, consequently,the perturbation expansion in (5) for is not uniform, and itbreaks down. To eliminate the secular producing terms fromthe right-hand side of (13), we set the coefficients ofequal to zero, and arrive at the following solvability conditionsfor the first-order problem

(14)

(15)

Equations (14) and (15) are first-order coupled equations withcoupling coefficient equal to .

B. Second-Order Coupled-Mode Equations

For the second-order problem, we seek a particular solutionfor (13) in the form

(16)

Substituting (16) into (8), and following the same procedure asin the first-order problem, the second-order solvability condi-tions are given by

(17)

(18)

Instead of solving the first order equations, (14) and (15), andthen the second-order equations, (17) and (18), we combine

them into one set of equations, making use of the chain rule, thus obtaining

(19)

(20)

where the coefficients are given in Appendix A.Thus, (19) and (20) are the second-order coupled-mode equa-tions.

III. FORMULATION AND SOLUTION OF FILTERING PROBLEM

Equations (14) and (15) may be solved to determine afirst-order approximation for the amplitude reflection co-efficient , which is the ratio of the backward-propagatingamplitude to the forward-propagating amplitude. Similarly,(19) and (20) may be solved to determine a second-orderapproximation for the amplitude reflection coefficient. In thefollowing, we solve them as an initial-value problem, then as atwo-point boundary-value problem, when subjected to appro-priate initial and boundary conditions. Before undertaking thisstep, we write the coupled-mode equations for the first-orderproblem, (14) and (15), and for the second-order problem, (19)and (20), in the following autonomous matrix form

(21)

the coefficient matrix is given in Appendix B, for boththe first-order and second-order problems. In deriving (21), weused the following substitutions

(22)

(23)

The filtering problem for a rugate section of lengthconsists ofcoupled-mode equations (21) subject to appropriate boundary orinitial conditions. Imposing initial conditions at , we mayintegrate (21) backward as an initial-value problem. Or, we canimpose conditions at both ends of the rugate section, and solvea two-point boundary-value problem. Both formulations indeedare feasible, as explained in the sequel. In either formulation,we make use of the amplitude reflection coefficient at

(24)

We also need to make use of the recursive Fresnel reflectionformula [15], in order to find the overall reflection coefficientat ; i.e.,

(25)

Here is the reflection coefficient at the outer interfacegiven by

(26)

2220 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 12, DECEMBER 2000

The power reflection coefficient, or reflectance,is calculatedthen from

(27)

where is the complex conjugate of.

A. Boundary-Value Problem

We first consider the formulation of the filtering problem,as a two-point boundary-value problem. One of the boundaryconditions is known at the substrate interface throughthe relation between electric and magnetic fields; i.e.,

(28)

where is the intrinsic wave impedance of free space. In termsof the incident and reflected amplitudes, this condition may bewritten in the form

(29)

where is the reflectivity at the rugate-substrate interface givenby

(30)

If an incident mode at is given; i.e.,

(31)

where is an arbitrary constant, then the reflected mode atis found by solving the two-point boundary-value problem

consisting of (21) subject to the boundary conditions given by(29) and (31).

B. Fundamental Matrix Method

An efficient numerical technique, the fundamental matrixmethod, developed by Asfar and Hussein [8], is chosen tosolve the two-point boundary-value problem. Besides itscomputational speed, the method can handle systems exhibitingstiffness [16], that is encountered near transitions from stop-band to passband and vice versa in the spectral response ofrugate filters. Numerical solutions using this method may besummarized in the following steps:

1) The integration interval is divided intosubintervals each of width, given by

(32)

2) The matrix is approximated over the subintervalby its value at the midpoint of that subin-

terval.3) The fundamental matrix solution is constructed

by computing the matrix of eigenvalues and the matrixof eigenvectors ; i.e.,

(33)

4) A coordinate transformation

(34)

is introduced to eliminate the ill-conditioning problemsassociated with stiff systems. The solution in each subin-terval then may be written as

(35)

where is a constant column vector.5) The solutions given by (35) are matched at mesh points

starting from to , in order to arrive at thefollowing system of algebraic equations

(36)

whereand matrices;

number of missing boundary conditions;number of equations ( , and inour case). In this particular problem ,and . This system hasband structure, and may be solved by Gaussianelimination.

C. Initial-Value Problem

An initial-value formulation is possible by normalizing theamplitude of the incident mode; i.e., . Thus, (21) maybe solved, subject to two-initial conditions

(37)

(38)

Backward integration may be employed to find the amplitudereflection coefficient. In this work, the fifth-order Runge-Kuttamethod was used to solve the initial-value problem.

IV. DISCUSSION

We illustrate the applicability of both formulations presentedin this paper by giving some examples. The spectral responses,obtained using the present analysis, were compared with thoseobtained by using the characteristic matrix method [1]. The pa-rameters used are 100 cycles of rugate with m,

, and in the range 0.025 to 0.224.The comparison is made in terms of speed and accuracy. Consid-ering accuracy first, we note that this depends on the value ofand the order of approximation being used. For small values of

up to , the first-order coupled-mode approximation(whether solved as initial-value or boundary-value problem) isin good agreement with the characteristic matrix method. Fig. 2shows an example of this comparison.

Increasing beyond 0.05, a phase shift is introduced in thespectral response. This effect is evident by examining the re-sponse shown in Fig. 3 for , where we focus ourattention at the region of transition from stopband to passbandto show the fine detail in the passband near the transition. Usingthe second-order coupled-mode equations may eliminate this

BATAINEH AND ASFAR: MULTIPLE SCALES ANALYSIS AND THE FUNDAMENTAL MATRIX METHOD TO RUGATE FILTERS 2221

Fig. 2. Spectral response of the first-order expansion and the characteristicmatrix method for� = 0:05.

Fig. 3. Spectral response near the transition from stopband to passband(first-order multiple scales expansion and the characteristic matrix method) for� = 0:224.

phase shift in the response of the first-order approximation. Thiscorrection is evident from examination of Fig. 4, wherein thesecond-order response is in remarkable phase agreement, withthe characteristic matrix solution.

To gain insight into the multiple scales solution and how thephase correction is achieved, let us examine the analytical formof the propagating modes in the unapodized rugate for both thefirst-order and the second-order expansions.The solution of (14) and (15) may be taken in the form

(39a)

(39b)

where and are constants andis a solution of; i.e.,

(40)

Fig. 4. Spectral response near the transition from stopband to passband(second-order multiple scales expansion and the characteristic matrix method)for � = 0:224.

For the second-order equations, (19) and (20), the solution hasthe form

(41a)

(41b)

The eigenvalues are the solutions of , whereand . Thus, to second order in

(42)

Equations (40) and (42) may be used to determine the width ofthe stopband between transitions from passband to stopband andvice versa. This is determined whenever to firstorder in or whenever to second order in .When the form of the amplitudes in (39) or (41) is substitutedin (9), we see that the wavenumber of any propagating mode ismodified to a new value . To first order in , we have

(43)

While to second-order in , we have

(44)

For the first-order solution to be uniformly valid throughout thespectral band of interest, we require that the maximum valueof the normalized phase error in the band to bewhen . This limits the spectral range of validity of thefirst-order expansion when increases. We found that an upperbound of 0.05 on could conveniently be chosen withoutintroducing phase error in the response. Imposing this limit onthe second-order normalized phase error , weexpect to obtain a response having good phase agreement upto . The variation of the normalized phase error tofirst- and second-order is shown in Fig. 5 for , fromwhich it is obvious that the second-order normalized phase erroris throughout the band. The linear portion of this curvecorresponds to the width of the stopband.

2222 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 12, DECEMBER 2000

Fig. 5. Normalized phase error for first- and second-order multiple scalesexpansions.

We point out here that the characteristic matrix method isnot an exact solution to the rugate filter. It is an approximationthat approaches the exact solution, as the number of subdivi-sions per period of the sine wave approaches infinity. However,one should exercise caution, when increasing the number ofsubdivisions, because roundoff errors are present when a largenumber of matrix multiplications is involved. We also note thatfor the uniform rugate, the fundamental matrix solution is anexact solution of the coupled-mode equations throughout theperiodic structure without subdividing the interval ( in(32)). Subdividing the interval may be necessary, if the cou-pled-mode equations are strongly stiff. The initial-value solu-tion of the coupled-mode equations is in full agreement withthe boundary-value solution, except that numerical integrationis demanding on computer time.

Although speed of numerical methods is not an issue withpresent desktop computers, we nevertheless compared the speedof the fundamental matrix method with that of the characteristicmatrix method for an apodized rugate having and aGaussian apodization function of the form

(45)

In executing the respective numerical programs, it took 32.63S of CPU time for the characteristic matrix program with 40 di-visions per period, while it took 9.94 S of CPU time for the fun-damental matrix program with subintervals on a 300MHz Pentium II machine. The comparison was made for thesame number of spectral points in both programs (1000 points),yielding the same accuracy. The effect of the Gaussian apodiza-tion is seen to reduce the ripples on both sides of the stopband,as suggested by the spectral response shown in Fig. 6.

V. CONCLUSION

The method of multiple scales was used to derive the cou-pled-mode equations, governing the filtering behavior of rugateoptical filters. Two formulations are given to assess the spectral

Fig. 6. Spectral response of a Gaussian apodized rugate filter.

response of the rugate filter. Though both of them are in agree-ment, the two-point boundary-value problem numerical solution(the fundamental matrix method) is faster from the computa-tional point of view. For apodized rugates, this method is alsofaster than the characteristic matrix method.

For small values of , a first-order coupled-mode analysisis sufficiently accurate. By extending the analysis to second-order in , it is possible to obtain filter responses with widerstopbands, yet with negligible phase error. The amplitude errorin the response observed in the passbands for largeis of theorder of far from the transition. If one desires to reduce thiserror, a third-order analysis would be necessary. Thiswas not done in this paper, in order to contain the mathematicaldetail, but is a systematic and straightforward procedure.

APPENDIX A

(A1)

(A2)

(A3)

(A4)

APPENDIX B

For the first-order coupled mode equations the coefficientsmatrix is given by

(B1)

while for the second-order problem it is given by

(B2)

The coefficients are given in Appendix A.

BATAINEH AND ASFAR: MULTIPLE SCALES ANALYSIS AND THE FUNDAMENTAL MATRIX METHOD TO RUGATE FILTERS 2223

REFERENCES

[1] H. A. Macleod,Thin-Film Optical Filters: McGraw-Hill, 1989.[2] W. Southwell, “Spectral response calculations of rugate filters using cou-

pled-wave theory,”J. Optical Soc. Amer., vol. 5, pp. 1518–1564, 1988.[3] B. Bovard, “Derivation of a matrix describing a rugate dielectric thin

film,” Appl. Optics, vol. 27, pp. 1998–2004, 1988.[4] F. Villa, R. Machorro, J. Siqueiros, and L. Regalado, “Admittance of

rugate filters derived from a2�2 inhomogeneous matrix,”Appl. Optics,vol. 33, pp. 2672–2676, 1994.

[5] O. Asfar, “Perturbation Theory of modes in nonuniform waveguides,”Ph.D. dissertation, Virginia Polytechnic Inst. State Univ., Blacksburg,VA, 1975.

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[7] M. Hawwa and O. Asfar, “Mechanical-wave filtering in a periodicallycorrugated elastic plate,”Trans. ASME, vol. 118, pp. 16–20, 1996.

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[9] W. Southwell, “Using apodization functions to reduce sidelobes in ru-gate filters,”Appl. Optics, vol. 28, pp. 5091–5093, 1989.

[10] O. Asfar, “Calculation of filter response of a dielectric slab waveguidehaving multiperiodic interface corrugations via the fundamental matrixmethod,”J. Electromagn. Waves Applicat., vol. 3, pp. 697–709, 1989.

[11] M. Muriel, A. Carballar, and J. Azaña, “Field distributions inside fibergratings,”IEEE J. Quantum Electron., vol. 35, pp. 548–558, 1999.

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Mohammed Bataineh (S’94–M’97) received theB.Sc. and M.Sc. degrees from Jordan Universityof Science and Technology in 1990 and 1993,respectively, and the Ph.D. degree from WalesUniversity, U.K. in 1996.

He is an Assistant Professor of Electrical Engi-neering at Yarmouk University, Irbid, Jordan. Hisresearch interests are in tropospheric radiowavepropagation, and the application of perturbationmethods to general periodic structures.

Omar Rafik Asfar (M’75–SM’89) is Professor ofElectrical Engineering at Jordan University of Sci-ence and Technology (JUST), Irbid, Jordan.

He joined King Saud University, Riyadh, SaudiArabia as an Assistant Professor in 1975, andYarmouk University, Irbid, Jordan, as an Asso-ciate Professor in 1981. He held the positions ofChairman of the Electrical Engineering Departmentat Yarmouk University (November 1982–August1986), Dean of Engineering at JUST (September1986–August 1987), Dean of Graduate Studies at

JUST (September 1987–August 1988), Dean of Engineering at JUST (October1992–January 1996). He was a Fulbright Visiting Scholar at Virginia Poly-technic Institute and State University, Blacksburg, VA, during the 1988–1989academic year. He spent two years teaching at the Department of ElectronicEngineering, Princess Summaya University College for Technology, RoyalScientific Society, Amman, Jordan. Since September 1998, he has beenserving as Acting Dean of the Hijjawi College for Engineering Technology,Yarmouk University. His teaching and research interests are in the areas ofapplied mathematics, electromagnetics, and wave propagation in periodicstructures, perturbation techniques, and numerical methods. He has authoredand coauthored more than twenty papers published in refereed internationaljournals and conferences and has one patent.

Prof. Asfar is a member of Sigma Xi, The Scientific Research Society, andthe Electromagnetics Academy.