Novel finite element analysis of optical waveguide discontinuity problems

1420 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 22, NO. 5, MAY 2004

Novel Finite Element Analysis of OpticalWaveguide Discontinuity Problems

S. S. A. Obayya

Abstract—In this paper, a novel finite-element method (FEM) torigorously and efficiently solve the optical waveguide discontinuityproblems is presented. Instead of performing the time-consumingmodal solutions on both discontinuity sides, the square root of thecharacteristic matrix is efficiently approximated using Taylor’s se-ries expansion, and then the interface boundary conditions are en-forced at the discontinuity plane to solve for the reflected and trans-mitted fields. The high numerical precision and effectiveness of theproposed method is demonstrated through the analysis of variousdiscontinuity problems, and the excellent agreement of the resultsobtained using the present finite element method and those ob-tained using other rigorous approaches in the literature.

Index Terms—Discontinuity problems, finite-element method(FEM), laser-air facets, optical waveguide.

I. INTRODUCTION

OPTICAL waveguide discontinuities play very importantrole in the design of optical communication systems, and

are quite often faced in situations such as butt-coupled waveg-uides, waveguide ends, laser facets, and antireflection coatings[1]–[3]. Therefore, an accurate and efficient method for the so-lution of optical waveguide discontinuities is highly demanded.There are a number of methods proposed in the literature forthe analysis of optical waveguide discontinuity problems. Thesemethods can be classified, according to their solution strategy,into two groups. In the first group, the total field in either sideof the discontinuity is made up as a weighted summation of allguided and radiation modes. Then, the continuity of the tangen-tial electric and magnetic fields at the discontinuity plane is en-forced using, e.g., the least squares boundary residual (LSBR)[4], a combination of the finite elements and method of lines(FE-MoL) [5], the finite-element method (FEM) with analyt-ical techniques [6] or the free space radiation mode (FSRM)method [7], to solve for the reflection and transmission coef-ficients. In this class of methods, the solution accuracy is en-hanced as more guided and radiation modes are included in thefield summation. This would be very time-consuming especiallyin the case of three-dimensional (3-D) optical waveguide prob-lems where approximating the radiation modes with continuousspectrum turns to be tremendously complicated. Alternatively,in the second group of discontinuity analysis, the Pade approxi-mations, in the context of the finite difference method, are usedto efficiently approximate the square root of the characteristicmatrix of both discontinuity sides [3], [8], [9]. In this group

Manuscript received September 23, 2003; revised February 17, 2004.The author is with the Department of Design and Systems Engi-

neering, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K. (e-mail:[email protected]).

Digital Object Identifier 10.1109/JLT.2004.827671

of analyses, although the numerically intensive modal solutionstage in each side is totally avoided, the use of higher-order Padeapproximation requires matrix inversion that would be very de-manding in terms of computer resources especially in dealingwith 3-D problems.

In this paper, a novel finite element solution for the opticalwaveguide discontinuity problems is proposed. By “lumping”the mass matrix entries into the diagonal locations [10]–[12],this matrix renders itself to be diagonal, and, hence, its inver-sion and multiplication with the original characteristic matrixwill result in a modified characteristic matrix as sparse as theoriginal one. Then, upon the application of Talyor’s series ex-pansion to represent the square root of the modified character-istic matrix and satisfying the interface boundary condition atthe discontinuity plane, both the reflected and transmitted fieldscan be efficiently solved for. Therefore, the presented FEM notonly avoids the modal solution stage, but also, does not requireany matrix inversion to approximate the square root operator ofthe characteristic matrix using higher-order Talyor’s series ex-pansions.

The paper is organized as follows. Following this introductionsection, a brief mathematical treatment of the proposed finiteelement approach is described in Section II. The validity and thenumerical precision of the proposed method will be evaluated inSection III through the analysis of three discontinuity problems,including a junction between two different waveguide sections,laser-air interface and laser angled facet. Then, conclusions willbe drawn.

II. ANALYSIS

Consider a 2-D optical waveguide discontinuity problemwhose schematic diagram is depicted in Fig. 1. The transverseand propagation directions are assumed to be and , respec-tively, and the discontinuity plane is located at . Supposethat there is no variation in the -direction, the following 2-DHelmholtz equation for both sides 1 and 2 can be written as [13]

(1)

with

(2a)

(2b)

(3a)

(3b)

0733-8724/04$20.00 © 2004 IEEE

OBAYYA: OPTICAL WAVEGUIDE DISCONTINUITY PROBLEMS 1421

Fig. 1. Schematic diagram of a discontinuity between two sides each witharbitrary refractive index distribution n (x).

where the subscript is 1 or 2 and corresponds to the quan-tities in the side, and are the -component of theelectric and magnetic fields, respectively, is the free spacewavenumber, is the refractive index distribution, is the op-erating wavelength, is the PML parameter, is the distanceinside the PML region from the beginning of PML and isthe theoretical reflection coefficient placed at the boundary be-tween the PML region and the computational window. Dividingthe cross section into a number of first-order linear element andapplying the standard Galerkin’s finite element procedure re-sults in [13]

(4)

where is the nodal field vector, is the null vector,and are the global mass and characteristic matrices whoseexpressions are

(5a)

(5b)

In (5) represent the shape functions vector over each ele-ment, is the transpose and the summation is performedover all the elements. Conventionally, both the guided and radi-ation modes of each side are found using a modal solution tech-nique for (4), and then upon matching the transverse electricand magnetic field components across the discontinuity plane,the solutions for both reflected and transmitted fields are ob-tained [5]. However, the modal solution analysis is very timeconsuming and turns to be complicated in dealing with radi-ation modes with continuous spectrum. Alternatively, a novelfinite element procedure completely avoids the modal solutionstage is presented as follows. The global mass matrix ofeach side is lumped into a diagonal matrix using the row sumprocedure [10]–[12]. In this procedure, the diagonal entry of the

new lumped mass matrix at a particular row “ ” is the sum ofall the entries in that row of the original mass matrix as

(6)

where is the diagonal entry of the new lumped mass ma-trix at the row, and the summation in (6) is performed overall entries in that row. Having lumped the mass matrix,its inverse is merely the reciprocals of its diagonal elements. So,by multiplying both sides of (4) with the inverse of the “lumped”mass matrix leads to

(7)

with

(8)

It should be noted that the numerical advantages of lumpingthe mass matrix is the elimination of the need for any matrixinversion and also keeping the new characteristic matrix assparse as the original matrix . This would lead to massivesaving in the use of computer resources especially in dealingwith 3-D problems. The formal solution of (7) in each side canbe written as

(9)where and are the forward and backward propa-gating fields in each side at the discontinuity plane . As-sume that the incident field is launched from side 1 and applyingthe interface boundary conditions of the continuity of transverseelectric and magnetic fields at results in

(10)

(11)

where , and are the incident, reflectedand transmitted fields, respectively. So, upon the solution of thelinear system of equations in (10) and (11), quantities such asthe reflection and transmission coefficients can be determinedonce the incident field is known. Usually, the square root ofthe characteristic matrix is evaluated using Pade approximation,which is numerically very accurate and efficient [3], [8], [9].However, the use of higher-order Pade approximants requiresmatrix inversion, which may be computationally intensive es-pecially in dealing with 3-D problems. On the other hand, theTaylor’s series expansion of the square root of the characteristicmatrix around a reference wavenumber has been employedhere, and its -order approximation is given as

(12)


Fig. 2. Schematic diagram of a junction between two waveguides.

with , , , and is the identityunity matrix.

As may be seen from (12), the use of Taylor’s series expan-sion of the square root of the characteristic matrix turns out to bemerely a series of matrix multiplications that can be efficientlyperformed by exploiting the matrix sparisty. In order to properlydeal with the “evanescent” modes generated at the interface be-tween the two sides, the reference wavenumber has to be a com-plex valued [9] which can be straightforwardly implemented ifa phasor term has been included so that

(13)

where is the original real-valued reference wavenumbernormally taken as the propagation constant of the fundamentalmode, and is a phase angle in the range . Thisapproach is equivalent to the use of Pade approximation withreal coefficients and the branch cut of the square root of thecharacteristic matrix [8]. It will be shown in the following sec-tion that introducing is mandatory to ensure the satisfactionof the power balance condition, however, it has a very littleeffect on the accuracy of the solution.

III. NUMERICAL EXAMPLES

The first example considered is a junction between two op-tical waveguides shown in Fig. 2. For the range of waveguideheights considered, both waveguides are monomoded. The samestructure has been considered in [4], [6]. In this example, theoperating wavelength, , is 1.0 , and the waveguide heightof side 1, , is fixed at 0.6 . For an incident modeand the waveguide height of side 2, , is 0.4 , shown inFig. 3 is the variation of the total power, , normalized to theincident power, with the phasor angle . The total power is cal-culated as the sum of both the reflected and transmitted powers.As may be observed from this figure, the power balance con-dition is fulfilled for , either for second- or third-orderTaylor’s approximations. On the other hand, Fig. 4 illustratesthe effect of the phasor angle on the values of the reflected

and transmitted powers of the fundamental mode insides 1 and 2, respectively. It may be noted from Fig. 4 that for

, both and are almost unchangeable with thechange in , and they converge to values of 0.16 and 0.61, re-spectively, which are in excellent agreement with the results ob-tained in [4] using the LSBR method employing an expansion

Fig. 3. Variation of the total power P with the phasor angle � for launchedTM mode with H = 0:4 �m.

Fig. 4. Variation of the transmitted power jT j and the reflected power jRjof the fundamental TM modes in sides 2 and 1, respectively, with the phasorangle � for H = 0:4 �m.

of 25 modes in each side. Next, the effect of varying the wave-guide height in side 2, , on the reflected, and transmitted

powers of the fundamental mode in sides 1 and 2, respec-tively, and also the radiated power is shown in Fig. 5. Asmay be observed from this figure, the transmitted power of thefundamental TM mode in side 2 is increasing as increasesand reaches its peak value of 0.755 when is 1.2 . Conse-quently, as the transmission efficiency is enhanced with the in-crease in , the radiated power is continuously reducingand settles down to a value of nearly 0.1 as . Onthe other hand, the reflected power of the fundamental modeof side 1 is almost constant with the change in . The resultsshown in Fig. 5 have been obtained using third-order Taylor’sexpansion and they are almost indistinguishable form those ob-tained using the LSBR method [4]. For a excitation from


Fig. 5. Variation of the transmitted power jT j the reflected power jRj of thefundamental TM modes in sides 2 and 1, respectively, and the radiated powerP with the waveguide height in side 2, H .

Fig. 6. Variation of the transmitted power jT j the reflected power jRj of thefundamental TE modes in sides 2 and 1, respectively, and the radiated powerP , with the waveguide height in side 2, H .

side 1, Fig. 6 shows the variation of the reflected powerand transmitted power of the fundamental modes in sides 1and 2, respectively, and the radiation power with the wave-guide height in side 2, . As may be seen from this figure, thebehavior of , , and is quite similar to the TM exci-tation case, except for the maximum transmission efficiency isslightly enhanced to 0.776, while the minimum radiated poweris dropped to 0.079. In all examples considered in this paper, thecomputational window is enclosed by two PML regions each ofwidth 1.0 , eight elements and the value of the theoreticalreflection coefficient at the interface between the PML and thecomputational window is . In this example, a computa-tional window of 10.0 is nonuniformly represented by 200

Fig. 7. Schematic diagram of a laser-air facet.

Fig. 8. Variation of the fundamental TE mode reflected power with thewaveguide height, H , using the present finite element approach with first-,second-, and third-order Taylor’s series expansions, and the FSRM.

line elements. With this discretization, it took less than 0.5 s tocalculate the reflected and transmitted power pair on a PC (Pen-tium IV, 2.66 GHz). On the other hand, the LSBR with the FEMmethod [4] employing 25 modes for each side would require 25times the execution time and storage requirement compared tothe proposed method.

Next, the more challenging discontinuity problem of laser-airfacet is considered. The schematic diagram of the structure isshown in Fig. 7. The refractive indexes of the core and sub-strate are 3.60 and 3.42, respectively, and the operating wave-length is taken as 0.86 . For a mode launched, thevariation of the reflected power of the fundamental modewith the waveguide height is shown in Fig. 8. It may be ob-served from this figure that the results of the present approachare converging very rapidly with those obtained using the FSRM[7]. As the index difference between the two discontinuity sidesis large, it can be seen from Fig. 8 that the first-order Taylor’sexpansion, which is equivalent to the paraxial approximation, isunderestimating the values of the reflected power. On the otherhand, the results obtained using the third-order Taylor’s expan-sion are in excellent agreement with those obtained using theFSRM [7]. For a mode excitation, the variation of the re-flected power of the fundamental mode with obtainedusing the present finite element approach employing the first-,second-, and third-order Taylor’s expansion and the FSRM [7]are shown in Fig. 9. Again, this figure shows clearly that the use


Fig. 9. Variation of the fundamental TM mode reflected power with thewaveguide height H using the present finite element approach with first-,second-, and third-order Taylor’s series expansions, and the FSRM.

Fig. 10. Schematic diagram of a laser-air with facet angle �.

of the present finite element approach with third-order Taylor’sexpansion gives reflected power results that are almost indis-tinguishable from those obtained using the FSRM [7]. In thisexample, a computational window of 6.0 is nonuniformlyrepresented by 180 line elements.

Finally, the design of laser-air facet with a tilt angle to min-imize the reflected power is considered using the present finiteelement approach. The schematic diagram of the structure isshown in Fig. 10 where the wavelength and the other param-eters are the same as in the previous example. In order to applythe present finite element approach, a virtual interface perpen-dicular to the waveguide cross section replaces the original tiltedinterface, as shown in Fig. 10. Equation (10) is solved for the re-flected field at the normal “virtual” interface, and then its phasefront is modified to be , where is thereal-valued propagation constant of the launched fundamentalmode, is the facet angle and is the center y-coordinate of thewaveguide core, in order to obtain the reflected field at the realtilted interface. Then, as in the previous examples, the reflectedpower is calculated as the power carried by the fundamentalmode normalized to the incident mode power. For a waveguideheight of 0.4 , shown in Fig. 11 is the variation of thereflected power of the fundamental mode, in decibels, with thefacet angle, , for both TE and TM excitations. As may be notedfrom this figure, the increase of the facet angle has a significant

Fig. 11. Variation of the fundamental TE and TM mode reflected powers,in decibels, with the facet angle � using the present finite element approach andthe BBPM. The waveguide height H is 0.4 �m.

Fig. 12. Variation of the fundamentalTE mode reflected powers, in decibels,with the waveguide height H and the facet angle � as a parameter.

effect on reducing the reflected power. In particular, for TE exci-tation, the reflected power is reduced from 4.1 dB to 11.9 dBas the facet angle increases from 0 to 12 . Also, for TM exci-tation, a reduction in the reflected power from 5.95 dB to 14dB occurs as increases from 0 to 12 . As may be observedalso from Fig. 11, the results obtained using the present finite el-ement approach with third-order Taylor’s expansion are in closeproximity to those obtained using the bidirectional beam propa-gation method (BBPM) [14]. Next, the variation of the re-flected power, in decibels, with the waveguide height, , withthe facet angle, , as a parameter is shown in Fig. 12. It can benoted from this figure that for , a significant reduction inthe reflected power occurs for the whole range of the waveguideheights considered. In particular, for and ,


Fig. 13. Variation of the fundamental TM mode reflected powers, indecibels, with the waveguide height H and the facet angle � as a parameter.

reflected power as low as 30 dB can be obtained. On the otherhand, Fig. 13 shows that the TM reflected power behaves in avery similar way to the TE case, e.g, the TM reflected power for

and is nearly 34 dB.

IV. CONCLUSION

A novel finite element procedure has been presented inorder to accurately and efficiently analyze optical waveguidediscontinuity problems. The proposed method completelyavoids the computationally intensive modal solution stage, andrather lumping the global mass matrix and using the Taylor’sseries expansion to accurately represent the square root of theresulting sparse characteristic matrix. The flexibility, efficiencyand accuracy of the proposed finite element approach hasbeen demonstrated through the excellent agreement of thereflected, transmitted and radiated power results obtained herewith their counterparts in the literature and obtained usingother rigorous approaches. The extension of the present finiteelement approach to a full vectorial analysis of single andmultiple 3-D optical waveguide discontinuity problems is nowunder consideration.

ACKNOWLEDGMENT

The author would like to acknowledge the long and fruitfulcollaboration with Prof. B. M. A. Rahman, City University,London, U.K.

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[7] M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” IEE Proc.—Optoelectron., vol. 143, no. 4,pp. 214–220, 1996.

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S. S. A. Obayya was born in Shirbeen, Egypt, onMay 27, 1969. He received the B.Sc. degree (with“excellent” grade), the M.Sc. degree in electronicsand communications engineering, as well as thePh.D. degree from Mansoura University, Egypt, in1991, 1994, and 1999, respectively. In his Ph.D. dis-sertation, he developed a novel finite element-basedfull vectorial beam propagation algorithm for theanalysis of various photonic devices.

In September 1995, he joined the Department ofElectronics and Communications, Mansoura Univer-

sity, as a Teaching Assistant. From September 1997 to September 1999, he waswith the Department of Electrical, Electronic and Information Engineering, CityUniversity London, as a Visiting Fellow carrying out research as part of hisPh.D. degree under the “joint supervision scheme” between City UniversityLondon and Mansoura University, Egypt. From January 2000 to July 2000, hewas with Mansoura University as an Assistant Professor, and from July 2000 toJune 2003, he worked as a Research Fellow with the School of Engineering, CityUniversity London. During his stay at City University, he has carried out confi-dential research to QinetiQ, Malvern, U.K., and also conducted research on thedesign of semiconductor electrooptic modulators funded by the EPSRC, U.K.Since June 2003, he joined the Department of Design and Systems Engineering,Brunel University, U.K., to work as a Lecturer. His main research interests arefocused on the areas of frequency selective surfaces, linear, nonlinear, passiveand active photonic devices, microstrip antennas for mobile phone networks,and radio over fiber systems. He has published more than 50 papers in the bestinternational journals and conferences in the areas of optics and microwaves.

Dr. Obayya received the “The Best Ph.D. Thesis” prize from Mansoura Uni-versity, Egypt, for the years 2001–2002.

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Novel finite element analysis of optical waveguide discontinuity problems