Design of integrated optical circuits using finiteelements

T.P. Young, BSc, PhD

Indexing terms: Electromagnetic theory, Optical waveguides, Computer-aided design, Integrated optics

Abstract: The finite-element method is describedfor modelling and characterising integratedoptical devices using an interactive CAD facility.The application of the method to componentdesign is illustrated by a series of examples,ranging from electrostatics to the optical fre-quency domain. The full vector H formulation ofthe optical problem is outlined and a new devel-opment described in which an electrostatic finite-element solver package is linked to the opticalsolver for a unified approach to the electro-opticeffect.

1 Introduction

The emergence of optical communications has stimulatedthe recent research and development interest in opticalsignal processing components and systems. For singlemode fibre systems, many signal processing functions canbe advantageously realised by monolithic integration ofthe optical waveguide components onto such crystallinesubstrates as LiNbO3, or semiconductor materials suchas InP, GaAs or Si.

Using many of the material growth and fabricationtechniques originally developed for integrated electroniccircuitry, it is possible to realise optical integrated circuitcomponents such as phase and amplitude modulators,wavelength division multiplexers, power splitters,switches and polarisation transformers. The basicelement from which these components are formed, andwhich links them together, is the optical waveguide.Essentially, an optical waveguide is a dielectric structureof higher refractive index than the surrounding medium,which provides two-dimensional confinement of the radi-ation to form a light-guiding channel a few micrometresdeep, several micrometres wide, and millimetres or evencentimetres long. The waveguides most commonly usedinclude the diffused channel guide and the etched rib, orridge guide. In the former, a strip of titanium is depositedonto a LiNbO3 substrate, and then diffused in at around1300 K for several hours to leave a graded region of ele-vated refractive index suitable for waveguiding. The latterare formed using etching techniques on layers of epi-taxially grown material to form the type of structure

Paper 5833A (S8), first received 22nd October 1986 and in revised form29th May 1987The author is with GEC Research Limited, Marconi Research Centre,West Hanningfield Road, Great Baddow, Chelmsford, EssexCM2 8HN, United Kingdom

shown in Fig. 1. Some waveguide analysis is described inSection 2.

A second structure fundamental to integrated optics isknown as the directional coupler [1], in which two wave-guides are kept in close proximity over a distance of

[-.-intrinsic semiconductorbuffer (doped)

buffer

guiding layer

substrate (doped)

Fig. 1 An etched rib waveguide on a semiconductor substrate

several millimetres. Light initially propagating down oneguide will transfer to the other guide, over a character-istic distance known as the coupling length Lc. An analo-gous system is that of a pair of coupled pendula, in whichpower transfer is periodic in time: here power transfer isspatially periodic. This effect can also be explained byconsidering that the composite structure has two eigen-modes which propagate at slightly different velocities,and the power transfer is a result of the constructive anddestructive interference taking place as they run in andout of phase along the coupler. The directional couplerproblem is examined in Section 3.

Clearly some control of the coupling characteristics isrequired to produce modulators, switches etc. Thiscontrol may be applied using the electro-optic effect, inwhich an electrostatic field alters the local refractiveindex. The effect is described using a well known tensorformulation [2], because the refractive index changes aredependent on the electrostatic field direction as well as itsmagnitude. Also, modes of orthogonal polarisations canbe affected differently by the same electrostatic field. Elec-trode design and electro-optic optimisation are, therefore,areas where a numerical technique such as the finite-element method becomes highly attractive. In Section 3, arecent electrode design study is described, and in Section4 an integrated approach to the electro-optic effect is pre-sented, in which the same finite-element mesh is used for

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988 135

an electrostatic solution to the electrode problem and forsolving the optical directional coupler.

In an industrial environment, a numerical analysistechnique is useful to the optical engineer in at least twoways. First, it represents the crucial stage in the designloop, providing a cost- and time-effective alternative todevice fabrication as means of assessing components andoptimising their performance. Secondly, as the dimen-sions of integrated optical devices are extremely small,and because the waveguides themselves are generallyinaccessible, it provides an insight into modal behaviourwithin the waveguide itself. The finite-element methodoffers a flexible way of modelling complex geometries anddistributed parameters within the same approach, and istherefore suited to the optical waveguide problem.

2 Modelling of optical waveguides

Although finite elements have only been applied tooptical waveguides relatively recently, electrostatic andmagnetostatic situations have been analysed over a muchlonger period, and a variety of preprocessing and postvie-wer packages have been developed. In our research, it hasbeen possible to use two such inhouse programs, SATANand DEMON, and to develop a two-dimensional wave-guide solver, OPTIC, compatible with existing finite-element software within GEC Research. The authorbelieves that together with an electrostatic solver,ELECST, this integrated software suite represents aunique facility, combining interactive flexibility with ana-lytical accuracy.

2.1 Optical solver formulationSeveral formulations have been advanced for the opticalwaveguide problem in two and three dimenions [3-8],ranging from scalar to full vectorial approaches. To pre-serve the generality of the solver, a full vector approachhas been adopted, so that it is possible to analyse thenatural anisotropy of some waveguiding media and alsoto treat tensor effects, such as the electro-optic effectalready mentioned.

The functional which has been used throughout isbased on the //-field variational expression presented byBerk [9]:

J(V l (V x H)dQ(1)

where e and pi are the material permittivity and per-meability, described by Hermitian tensors of rank two,and * indicates a conjugate. Although most of thematerials used are dielectric rather than magnetic media,the full tensor form of each variable has been retained inview, partly, of the current interest in complementary for-mulations [10].

When a penalty function [11] is added, this can beexpressed as a functional:

-1F(H) = [(V x H)* • e'1 • (V x H)

+ (a/eo)(V x //)*(V x //)] dil

— co (2)

Uniformity of the waveguide is assumed and is intro-duced by integrating each term over a complete wave-length to remove one dimension, so that the volumeintegrals of eqns. 1 and 2 reduce to surface integrals.

These are then expressed algebraically for a singleelement using linear shape functions [8], assuming con-stant e and \i across the element. The field solutions willtherefore be linearly piecewise continuous, rather like afaceted diamond. The functional is then differentiatedwith respect to each nodal field component, which leadsto an eigenvalue equation:

Ax = k2Bx • (3)

A is a complex Hermitian matrix and B is real symmetricand positive definite, while x is a column vector contain-ing all the nodal field values. The square root of theeigenvalue k is the free-space wave number of the propa-gating radiation. Thus, with finite elements, the propaga-tion constant of a mode in the guide, /?, is specified toobtain k, whereas most other techniques start with thewavelength X (or with k), and /? is the computed variable.The only practical difference this makes is that an extrastage of iteration may be needed if the solution isrequired at a particular wavelength.

These elemental equations are then combined into aglobal matrix eigenvalue equation which is solved bymeans of a subspace iteration technique SSITER [12].With this routine, a region in &2-space is selected inwhich it converges on a specified number of eigenvalues.The number of solutions sought, along with all the otherinput data are entered as an auxilliary data file. Currentlymounted on an Apollo Domain DN 560 workstation, thealgorithm uses approximately 300 CPU seconds for a1000 node solution. Much larger meshes have, however,been used — extending above 2000 nodes, or 6000unknowns.

The waveguide situation differs from many otherproblems of interest in that the boundary is 'open'. Thatis, the fields decay, in the plane of the waveguide cross-section, to zero at infinity. This decay is approximatelyexponential, and so the boundary conditions have beenmet using infinite elements [13], in which a decay con-stant is specified for the optical field decay away fromeach of the four edges of the rectangular meshes used.The boundaries are normally placed far enough from thewaveguides for the decay constant to have only a smallperturbing effect on the solution. Where extreme accu-racy is required, however, it is possible to iterate to find astationary value for each decay constant.

2.2 Spurious modesThe problem of spurious solutions [14-16] remains.These modes are sometimes introduced because thedivergence of B is not explicitly forced to zero in thefunctional (unless introduced through the penalty para-meter a). Nonphysical modes can also occur in veryweakly guiding systems, such as diffused waveguides,because it is not always possible to disguise the boundaryof the field of analysis. Indeed, unless the precise fielddecay constants are known beforehand, modal powerreaching the edge of the field of analysis will effectivelysee another, infinitely extending medium of slightly differ-ent refractive index from that on the inside of the bound-ary. Such modes are not spurious in the accepted senseand may be termed 'box modes'. The surest means ofdetecting them is by inspecting the mode profiles,although the anomalously low effective index values cansometimes be used to identify them.

A number of ways of eliminating spurious modes havebeen suggested, and the problem does not occur at allwith the scalar formulation [3]. In that case, the vectorcomponents of the electric or magnetic field may be

136 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988

derived from some scalar potential function afterwardsby a stage of differentiation, which, in turn, reduces thepolynomial order of the solution. A piecewise linear solu-tion for the potential might therefore yield field com-ponent solutions which are piecewise constant. Thetechnique of working with scalars can reduce the amountof computation and eliminate spurious modes, but at theexpense of accuracy. A formulation in terms of the trans-verse //-field has been advanced [17] which eliminatesspurious modes. The Hx and Hy field components arerepresented to the order of the elemental shape functions,but Hz involves a stage of differentiation and is thus moreapproximately represented.

Kobelansky and Webb [18] have suggested a two-stage procedure, solving first using the functional:

4G(H) = [(V • H)-kH* (4)

The solutions obtained will be divergence-free, and arethe only allowed trial functions for the second stage inwhich the usual functional is minimised. While this tech-nique eliminates spurious modes, it appears to havedrawbacks for very large problems, especially as theadvantage of matrix sparsity may be lost.

Angkaew et al. [19] have differentiated between realand spurious modes using a complex functional, differentfrom that given in eqn. 1. With this approach spuriouseigenvalues move off the real axis in the complex plane sothat all real eigenvalues are genuine. This gain is achievedat the expense of the increasing the order of the problemand therefore the computing effort. It is not clear whetherthis technique removes the spurious modes altogether, oracts as a filter to identify and remove them afterwards.Konrad [20] has also proposed a means of eliminatingspurious modes in three-dimensional cavities. Histechnique is particularly appealing because it reduces theorder of the problem and thus the number of eigenvalues.Spurious modes will therefore disappear, and not simplymove to a different region of the eigenvalue spectrum. Tothe author's knowledge, these ideas have yet to be imple-mented in two-dimensional open-boundary situations.

From the above discussion, it is clear that the spuriousmode problem continues to receive much attention, andthat most of the solutions represent a tradeoff of one sortor another. For instance, Kobelansky and Webb [18]and Angkaew et al. [19] have traded off the spuriousmodes for computational complexity. Reduced com-ponent formulations decrease the computation and caneliminate spurious modes but the cost is a poorer repre-sentation of one or more components of the mode, withpossible repercussions on the overall accuracy.

The experience here has been that a full vector solu-tion with a penalty function is certainly adequate, espe-cially when allied to a fast graphics capability foridentifying any doubtful modes. In practice, it is normallypossible to identify spurious modes simply by printingout the maximum and minimum values of each fieldcomponent. Also, because the penalty parameter is avariable, the user enjoys flexibility in applying it and can,where necessary, extrapolate back to discover the eigen-value of the unperturbed vector solution. This is useful incases where it is not possible to reach zero withoutdegrading the mode profile. Such degradation of theprofile has been attributed to a degree of mixing of thespurious solutions with the valid solution [21].

Hayata et al. [21] contend that extrapolation of a tozero is possible with just two solution runs using a =(1/n2) (where n is the substrate index) and a = 1. This

technique is particularly useful when the guiding isextremely weak and there is a very narrow window forthe allowed effective index. In such cases, the penaltyparameter required to produce a smooth profile can pullthe solution out of the allowed range of ne. However, theextrapolation procedure allows access to a good field rep-resentation, and also to high accuracy.

2.3 AccuracyAs the analysis is based on a variational approach, wewould expect it to be highly accurate. The question ofprecision is problematic, because we have only othernumerical or even semianalytical techniques with whichto compare the results. The absence of empirical data isdue to the extreme difficulty, first, in measuring refractiveindices of the thin material layers to sufficient precision,and also the practical problems of measuring the wave-guides' effective indices to better than three decimalplaces.

Because there is no generally recognised benchmarkstandard, the tendency has been to work with waveguidesof regular geometry and compare the answers with earlierwork. For such tests, the window of allowed effectiveindex values may be narrowed by choosing waveguidesystems with very small refractive index changes. Thiswill provide a useful test of the FEM, because analyticand semianalytic techniques become more accurate inthis limit, while the FEM which is not based on any suchallowed range of solutions, should work to its normalaccuracy. Clearly, factors such as the mesh quality andnode density will determine the accuracy achieved in anyset of circumstances. However, Rahman and Davies [11],Koshiba et al. [22] and Young and Smith [23] presentresults which support this claim to high accuracy. In thelatter reference, the accuracy of the FEM is shown toreach parts in 106 for a 250-node test example.

2.4 Optical waveguidesBecause most practical guides do not possess the simplerectangular geometry tackled by semianalytic methods[24, 25], interactive CAD facilities are important if amodelling technique such as finite elements is to makethe same impact in optical waveguide engineering as ithas in other areas. As an example, we consider the type ofdesign questions which must be addressed and the way inwhich they are tackled with the present facility. Fig. 2shows the ^-component of the Hy mode of the waveguideof Fig. 1, for a wavelength X around 1.55 /im. This guideis fabricated on a semiconducting substrate. A 2 /mi thickbuffer layer of intrinsic semiconductor is grown over this,to separate the mode from the optically lossy substrate.This is overgrown with a 0.3 nm guiding layer and thenanother buffer layer. The top layer is semiconductor onceagain and is open to the air. The etched surfaces areorthogonal to the planes of layer interfaces, becausedirectionally preferential etching techniques can be used,although sidewalls at any angle can be generated by thepreprocessor and therefore analysed by OPTIC. Table 1lists the various layer properties.

Knowledge of the power distribution inside the guideis extremely useful, especially in a case such as thepresent, where we require as little power as possible toreach the lossy substrate. A vertical section graph takenthrough the mode (Fig. 3) shows us that the intensityalong the substrate boundary is attenuated by about50 dB with respect to its peak value, indicating that thebuffer layer is sufficiently thick. As all three vector com-ponents are computed, one can also see what the Hz field,

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988 137

for instance, looks like. A contour map of this componentis shown in Fig. 4A.

When discussing optical waveguides, a useful para-meter is the effective refractive index no, which is the

2.5 Directional couplersElectro-optically controlled directional couplers are thebasic element of most integrated optical switches, andcan also be used for amplitude modulators and tunable

Fig. 2 Y-component ofHy mode of the waveguide shown in Fig. 1

Table 1: Layer dimensions and refractive indices for thewaveguide of Fig. 1

Layer

1. Substrate(doped)

2. Buffer3. Guiding layer4. Buffer5. Doped

semiconductor6. Air

Thickness,//m

-v450

2.00.31.5*0.5

Refractiveindex, n

3.16

3.173.383.173.16

1.00

* This layer has been etched to 0.2 //m over the outer slab region.

modal propagation constant divided by the free-spacewave number (/?/&). Fig. 5 is a plot of the effective indexagainst waveguide width for the fundamental Hy mode.Such graphs can obviously be calculated with respect towavelength to obtain a dispersion curve, or with respectto any other useful design parameter. Rahman andDavies [26], for example, have looked in some detail atthe behaviour of regular rib structures using finite ele-ments.

wavelength division multiplexers. The simplest imple-mentation is a coupler designed to be about 1.5LC long.Surface electrodes are used to apply an electrostatic fieldwhich induces switching action through the refractiveindex changes [27]. Precise prediction of the couplinglength enables device design parameters to be obtaineddirectly.

The directional coupler problem is a rather sensitiveapplication of any analysis technique, because the normalmodes are so closely spaced. The symmetric and anti-symmetric modes are shown in Figs. 6 and 7 for acoupler based on the structure already illustrated, using awaveguide width of 1.65 pan and separation of 2 pan. Thecoupling strength K is defined as

to o \ / ^ fc\

where /?1(2) is the propagation constant of the symmetric(antisymmetric) mode. The coupling length Lc is given by

Lc = K/2K (6)

To obtain the coupling length, therefore, both the sym-metric and antisymmetric modes must be found at thesame wavelength, which involves a stage of iteration asalready mentioned. Fig. 8 shows the natural logarithm ofcoupling strength plotted against waveguide spacing, fora directional coupler using the waveguide structure

138 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988

described above. A fundamental plot such as this cannotusually be generated from normalised curves, and mustbe produced for each waveguide geometry.

tional couplers can be limited by this effect, unless specialtapering is implemented [28]. The fact that the modes incoupled waveguides are not power orthogonal is clearly

1.00r

0.80-

0.60-

* 0.40-

0.20-

Fig. 3Fig. 2

2.5 . 5.0distance, |jm

Vertical section through the y-component of that shown in

Some physical insight can be gained by noting that thefunctions will not cancel completely over the whole of theguide, where their amplitudes have opposite sign. Thecancellation improves as the waveguides are separated,but the crosstalk performance of closely spaced direc-

1.0

0.6

- 0.2Q.

eo? -0.2

-0.6

-1.0L

Fig. 4B Vertical section through component shown in Fig. 4 A

illustrated here, and it is at this point that simple coupledmode theory breaks down. A reformulation of coupledmode theory to cater for residual crosstalk is receiving alot of attention at present [29, 30].

3 Electrostatic problems

3.1 FormulationAs with the optical formulation, uniformity of the wave-guide properties is assumed along the propagation axis

Fig. 4A Z-component ofHy mode of the waveguide shown in Fig. 1

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988 139

to reduce the problem to two dimensions. The voltagedistribution can be calculated by solving Laplace's equa-tion:

V 2 • <f> = 0 (7)

3.236

> 3.226

3.216

Fig. 5width

gu ide wid th , u m

Variation of Hy fundamental mode effective index with guide

Fig. 6 Symmetric mode of a directional coupler with inset graphshowing a horizontal section through the guiding layer

where <j> is the electrostatic potential. The analysis isbased on the following two-dimensional functional:

JA| d(f>ldx dx • dy (8)

3r

Fig. 7 Antisymmetric mode of a directional coupler with inset graphshowing a horizontal section through the guiding layer

distance, urn

Fig. 8 Natural logarithm of coupling strength as a function of wave-guide separation

This expression can also be minimised with respect toeach nodal <f) value subject to a series of fixed nodal volt-ages [31], or constraints, representing the electrode volt-ages with which the electrostatic fields are applied. Thisprocedure leads to a system of linear equations of theform:

Sty = b (9)

where the right-hand terms of this expression originatefrom imposed voltages, or constraints, and <)> is a vectorcontaining all the nodal values of electrostatic potential.The local electrostatic field components Ex and Ey canthen be calculated from the potential distribution.

ELECST handles the open boundary condition usingballooning [32] instead of the infinite elements used byOPTIC, but the mesh is bounded by line elements ineach case, and a given mesh is therefore acceptable toeither solver programme. To ensure full compatibility,the mesh nodes must be properly numbered for ELECSTin order to optimise the matrix bandwidth and to placethe constrained nodes last in the numbering system. Suchnumbering is not required for the subspace iterationroutine in OPTIC which does not rely on the bandednature of the matrices.

3.2 Electrode designAn interesting electrode design problem occurred recent-ly, when designing high-speed Mach-Zehnder interfero-meters for use as high-frequency amplitude modulators.The integrated optical implementation of a travelling-wave interferometer is shown in Fig. 9, where the electricfield has been applied using a microwave coplanar strip-guide over X-cut LiNbO3. The horizontal field acrossone waveguide is in the opposite direction to that in theother waveguide, and so the refractive index changeinduced by the r33 coefficient is also reversed. The modein one guide is therefore accelerated, while the mode inthe other is retarded, so that a phase difference accumu-lates along the device. When the waveguides are recom-bined, the modes interfere to give either the fundamentalmode or else a higher order mode which radiates into thesubstrate. The voltage required to switch from one state

140 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988

to the other is known as Vn, because it induces a differen-tial phase shift of n rad. A travelling-wave configurationwill offer the highest bandwidth, because ideally the elec-

CPWelectrodes"

asymmetricinterferometer

Fig. 9 Top view of an integrated optical travelling-wave Mach-Zehnder interferometer

trie wave and the optical wave would remain in phasethrough the device to give maximum efficiency at highfrequencies. However, the permittivity of LiNbO3 isroughly a factor of 7 greater at microwave frequenciesthan it is at optical frequencies. The situation is compli-cated by the anisotropy of the substrate and the fact thatthe thin buffer layer, which isolates the waveguide modefrom the lossy electrodes, is not anisotropic.

The radio-frequency wavelength in the coplanar wave-guide decreases as the modulation frequency increases, sothat the efficiency is reduced, reaching zero when theoptical wave gains a complete cycle over the modulatingwave. One method of raising the modulation bandwidth

is to reduce the velocity mismatch between the opticaland electrical signals by separating the coplanar wave-guide from the substrate of high dielectric constant, byusing a much thicker buffer layer of SiO2 [33]. The cross-section of the equipotentials has been computed by finiteelements using ELECST, for the coplanar waveguidestructure, and an example is given in Fig. 10. It showshow the field strength is reduced across the waveguide,although refraction at the buffer/substrate interface some-what offsets this decrease. The data obtained from thisstudy, on the variation with buffer layer thickness of linecapacitance and electric field strength in the waveguideregion, have enabled a detailed assessment of the trade-offs between modulation bandwidth and drive voltage.

The reduction in capacity is, to first approximation,accompanied by a proportional increase in Vn. Fig. 11shows the modulator efficiency and bandwidth as a func-tion of buffer layer thickness for an 8 mm long device.The bandwidth increases rapidly until the interaction isvelocity matched, at which point the frequency responseis limited by the microwave attenuation through thedevice. The results of this study have shown how toincrease the modulation bandwidth from around 8 GHzto at least 30 GHz, with minimal reduction in electro-optic efficiency.

Many of the design tradeoffs for electro-optically con-trolled devices can be determined by an electrostaticfinite-element solver. The above illustration serves toshow the way in which the electrode dimensions can befinely tuned to reach optimum performance, and also toshow how those parameters which are most importantcan be identified once the designer has a clear picture ofthe electrostatic field behaviour.

Fig. 10 Equipotential plot in 5% intervals for Mach-Zehnder velocity matching structure

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988 141

4 The electro-optic effect

Having shown how an optical waveguide solver and anelectrostatic solver can be used independently in inte-grated optical circuit design, it is also possible to demon-

12r

10

o 6

q60

50

30 S

20

10

" 0 1 2 3buffer layer thickness, urn

Fig. 11 Modulator bandwidth (solid) and drive voltage (dashed) as afunction of buffer layer thickness

strate the combined operation of the two solvers for anintegrated analysis of the electro-optic interaction. Thishas been implemented so that the potential distributiongenerated by ELECST, is used to calculate the electro-static field at the centre of each element, and thence todetermine the changes in refractive index at the centroidof each element. Using the same mesh, OPTIC goes on tosolve for the waveguide propagation.

The line elements represent ballooning for the electro-static case, and infinite elements for the optical analysis.Once this condition is met, a common mesh can be usedwhich is central to information transfer from oneprogram to the other.

continuous, the refractive index is stepwise constant.Here, too, a phase mismatch builds up along the couplerand the coupling characteristics are affected. With suchelectrode structures, phase mismatch, or A/?-switching isachieved [34].

Coupled-mode theory states that it is impossible torealise full power transfer using a uniform phase velocitymismatch along a coupler. This is reflected in the normalmode picture presented by the FEM. The main feature torecognise is that the larger lobe of the symmetric mode ison the opposite side to that of the antisymmetric mode.The situation in which light is input to one of the guidesis solved by finding the relative weighting of the sym-metric and antisymmetric modes, which largely cancelsthe field in the other guide. Once selected, these weight-ings can only be multiplied by propagators of the forme~PiZ, where /?, is the propagation constant of the ith (1stor 2nd) mode to represent the changing interferencealong the directional coupler. This relative weightingdetermines the maximum power which can transfer intothe other waveguide. The deformation shown means thatone mode will always require a higher weighting than theother, no matter which waveguide the light is originallyput into, and so light transfer will never be complete.Furthermore, the greater the deformation, the greater therelative weighting disparity, and the less power is trans-ferred. This link between the coupled and normal modepictures is a simple example of the insight afforded by thephysical pictures provided by the FEM.

This approach does not assume an approximate form,for either the electrostatic or the optical aspects of theproblem, nor does it seek to impose one effect as a per-turbation on the other. Such approximations must beadopted in almost any other waveguide analysis methodin order to treat a complex interaction such as theelectro-optic effect. Here, however, the electro-optic mod-elling flows naturally and directly from the vectorialimplementation of finite elements outlined above. Theresults, therefore, will be more accurate and tend to offer

Fig. 12A Symmetric mode of channel waveguide directional coupler under uniform A/? action

Figs. 12 and 13 show a rectangular channel guidecoupler, with the symmetric and antisymmetric modesdrawn in contours over it, for 10 V applied between theelectrodes. The electro-optic tensor used in this exampleis that of Z-cut LiNbO3, so that it is the vertical com-ponent of the electrostatic field which will increase therefractive index in one guide and decrease it in the other.As these changes in refractive index are small, a verylarge voltage must be applied before the refractive indexchanges can be observed on a graph, although devicesnormally operate at much lower voltages.

Fig. 14 shows a cross-section of (1/M)2 (x 1000) takenabout 1.5 /im below the surface when 100 V is applied tothe electrodes, and shows the actual change of extraordi-nary refractive index in the channels and the substrate.Note that, because the electric field is piecewise linearly

a better insight into device operation, especially wherecomplex geometries are involved.

5 Conclusions

The full vector //-field formulation of the waveguideproblem has been developed into a general solver andlinked with preprocessing and postviewer packages toprovide a highly felxible and accurate solution to opticalwaveguide analysis. The interactive input of data has leadto convenient and efficient geometry definition ofcomplex waveguide and directional coupler structuresinvolving very large meshes, and examples have beengiven to illustrate the potential of such a facility. Thepostviewer has been especially useful for rapid identifica-tion of spurious modes generated by the finite-element

142 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988

solver, and also in providing a clear picture of modalbehaviour within the waveguide.

A particularly powerful facet of this analysis technique

Finally, the present solftware suite which resides on itsown workstation has allowed electro-optical finite-element analysis to move successfully into an industrial

?2 500

1.250Eo

~ 012 15 18

distance, pm

21 27 30

Fig. 12B Horizontal section through mode

Fig. 13A Antisymmetric mode of channel waveguide directional coupler under uniform A/? action

- 200

a -25o

^ - 2 5 0 L

Fig. 13B Horizontal section through mode

100

5 50

10 15 20distance.m

25 30

Fig. 14A Potential distribution along horizontal cross-section throughsubstrate

208.0

206.0

20405 10 15 20 25 30

distance, pm

Fig. 14B Resultant change in (l/ny)2 (x 1000)

has been developed by cascading two finite-elementsolver programs which operate on the same mesh, inorder to analyse electro-optically controlled devicesdirectly.

environment where it is contributing to the design andmodelling of practical integrated optical circuitry.

6 Acknowledgments

The author wishes to acknowledge the sponsorship ofESPRIT for the development of the solver, OPTIC,under its programme InP-based optoelectronics. He alsoacknowledges the use of the Harwell subroutine library,and the sub-space iteration package, SSITER, from Uni-versity College London. Thanks are also due to A.J.Davies, and P. Smith for development of SATAN,DEMON, OPTIC and ELECST, and for some numericalassistance.

The author wishes also to thank Prof. J.B. Davies ofthe Department of Electrical and Electronic Engineering,UCL, Dr. R. Ferrari and Dr. F. Payne of the EngineeringDepartment, Cambridge University, for useful dis-cussions, and especially, M.Y. Bourbin of Thomson-CSF,for information on the semiconductor waveguides anddirectional couplers.

Finally, I wish to acknowledge the helpful support ofMr. N.J. Parsons throughout this work, and for details ofthe travelling-wave modulator. Other assistance fromA.C. O'Donnell and J.R. Wilcox and Miss H. Senior hasalso been greatly appreciated.

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988 143

7 References

1 HUNG-CHIA, H.: 'Coupled mode theory as applied to microwaveand optical transmission' (VNU Science Press, Utrecht, The Nether-lands, 1984)

2 NYE, J.F.: 'Physical properties of crystals' (Clarendon Press,Oxford, 1967), Chap. 13, pp. 235-259

3 MABAYA, N., LAGASSE, P.E., and VANDENBULCKE, P.:'Finite element analysis of optical waveguides', IEEE Trans., 1981,MTT-29, (6), pp. 600-605

4 WELT, D., and WEBB, J.: 'Finite element analysis of dielectricwaveguides with curved boundaries', ibid., 1985, MTT-33, (7),pp. 576-585

5 YEH, C, HA, K., DONG, S.G., and BROWN, W.P.: 'Single modeoptical waveguides', Appl. Opt., 1979,18, (10), pp. 1490-1504

6 KONRAD, A.: 'Vector variational formulation of electromagneticfields in anisotropic media', IEEE Trans., 1976, MTT-24, (9),pp. 553-559

7 WEBB, J.P, MAILE, G.L, and FERRARI, R.L.: 'Finite-elementsolution of three-dimensional electromagnetic problems', IEE Proc.H, Microwaves, Opt. & Antennas, 1983,130, (2), pp. 153-159

8 KONRAD, A.: 'High-order triangular finite elements for electro-magnetic waves in anisotropic media', IEEE Trans., 1977, MTT-25,(5), pp. 353-360

9 BERK, A.D.: 'Variational principles for electromagnetic resonatorsand waveguides', IRE Trans., 1956, AP-4, pp. 104-111

10 KOSHIBA, M., HAY AT A, K., and SUZUKI, M.: 'Vector £-fieldfinite element analysis of dielectric optical waveguides', Appl. Opt.,1986, 25, (1), pp. 10-11

11 RAHMAN, B.M.A., and DA VIES, J.B.: 'Finite element solution ofintegrated optical waveguides', IEEE J. Light Tech., 1984, 2, (5),pp. 682-688

12 FERNANDEZ, F.A.F.: 'Finite element analysis of rotationally sym-metric electromagnetic cavities'. PhD thesis, Department of Electri-cal & Electronic Engineering, University College, London, 1981

13 RAHMAN, B.M.A., and DA VIES, J.B.: 'Finite element analysis ofoptical and microwave waveguide problems', IEEE Trans., 1984,MTT-32, pp. 20-28

14 KOSHIBA, M., HAY AT A, K., and SUZUKI, M.: 'Study of Spur-ious solutions of finite element method in the three-component mag-netic field formulation for dielectric waveguide problems', Elec. &Comm. Jpn. Part 1, 1985, 68, p. 114

15 CHING-CHUAN SU: 'Origin of spurious modes in the analysis ofoptical fibre using the finite-element or finite-difference technique',Electron. Lett., 1985, 21, (19), pp. 858-860

16 RAHMAN, B.M.A., and DAVIES, J.B.: 'Penalty function improve-ment of waveguide solution by finite elements', 1984, MTT-32, (8),pp. 922-928

17 HAYATA, K., KOSHIBA, M., EGUCHI, M, and SUZUKI, M.:'Novel finite-element formulation without any spurious solutions fordielectric waveguides', Electron. Lett., 1986, 22, (6), pp. 295-296

18 KOBELANSKY, A.J., and WEBB, J.P.: 'Eliminating spuriousmodes in finite-element waveguide problems by using divergence-free fields', ibid., 1986, 22, (11), pp. 569-570

19 ANGKAEW, T., MATSUHARA, M., and KUMAGAI, N.: 'Finiteelement analysis of waveguide modes: a novel approach that elimi-nates spurious modes', IEEE Trans., 1987, MTT-35, (2), pp. 117-123

20 KONRAD, A.: 'On the reduction of the number of spurious modesin the vectorial finite-element solution of three-dimensional cavitiesand waveguides', ibid., 1986, MTT-34, (2), pp. 224-227

21 HAYATA, K., EGUCHI, M., KOSHIBA, M., and SUZUKI, M.:'Vectorial wave analysis of side-tunnelling type polarization-maintaining optical fibres by variational finite elements', IEEE J.Light. Tech., 1986,4, (18), pp. 1090-1096

22 KOSHIBA, M, HAYATA, K., and SUZUKI, M.: 'Improved finite-element formulation in terms of the magnetic field vector for dielec-tric waveguides', IEEE Trans., 1985, MTT-33, (3), pp. 227-233

23 YOUNG, T.P., and SMITH, P.: 'Finite element modelling of inte-grated optical waveguides', GEC J. Res., 1986, 4, (4), pp. 249-255

24 KNOX, R.M., and TOULIOS, P.P.: 'Integrated circuits for themillimetre wave through optical frequency range'. Proc. Symp. Sub-millimetre Waves (Polytechnic Press, Brooklyn, New York, 1970),pp. 497-516

25 MARCATILI, E.A.J.: 'Dielectric rectangular waveguide and direc-tional coupler for integrated optics', Bell Syst. Tech. J., 1969, 48,pp. 2097-2102

26 RAHMAN, B.M.A., and DAVIES, J.B.: 'Vector-H finite elementsolution of GaAs/GaAlAs rib waveguides', IEE Proc. J., Opto-electron., 1986,132, (6), pp. 349-353

27 SCHLAAK, H.F.: 'Modulation behaviour of integrated opticaldirectional couplers', J. Opt. Commun., 1984, 5, (4), pp. 122-131

28 HAUS, H.A., and WHITAKER, N.A.: 'Elimination of cross talk indirectional couplers', Appl. Phys. Lett., 1985, 46, (1), pp. 1-3

29 HAUS, H.A., HUANG, W.P, KAWAKAMI, S., and WHITAKER,N.A.: 'Coupled-mode theory of waveguides', IEEE J. Light. Tech.,1987, 5, (1), pp. 16-23

30 STREIFER, W., OSINSKI, M., and HARDY, A.: 'Reformulation ofthe coupled-mode theory of multiwaveguide systems', ibid.,19%1, 5,(1), PP- 1-4

31 SILVESTER, P.P., and FERRARI, R.L.: 'Finite elements for electri-cal engineers' (Cambridge University Press, 1983), Chap. 1

32 SILVESTER, P.P., LOWTHER, D.A., CARPENTER, C.J., andWYATT, E.A.: 'Exterior finite elements for 2-dimensional fieldproblems with open boundaries', Proc. IEE, 1977, 124, (12),pp. 1267-1270

33 PARSONS, N.J., O'DONNELL, AC, and WONG, K.-K.: 'Designof efficient and wideband travelling wave modulators'. Proc. SPIEConf. 651, Third Integrated Optical Circuit Engineering, Innsbruck,Austria, 1986

34 KOGELINIK, H., and SCHMIDT, R.V.: 'Switched directionalcouplers with alternating A/T, IEEE J. Quantum Electron., 1976,QE-12, (7), pp. 396-401

144 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988