September 18, 2002 Next-generation active integrated optic subsystems Information society technologies programme of the European Commission, project IST-2000-28018 Workpackage 3: Design — Confidential, NAIS internal document. — Coupled mode model for 3D directional couplers Contents 1 Adiabatic directional couplers, coupled mode viewpoint 1 2 Basis fields and supermodes 2 3 Mode amplitude evolution 5 4 Scattering matrices 6 Concepts and ideas in this draft originate from contributions from several participants in the NAIS project. Contact: Manfred Hammer, Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Phone: +31/53/489-3448, Fax: +31/53/489-4833, E-mail: [email protected]
1 Adiabatic dir ectional couplers,coupledmodeviewpoint
A commonmodel for optical ringresonators[1, 2] reliescrucially uponan adequatedescriptionof the cou-pling regionsbetweenthecavity andtheport channels.Figure1 shows a sketchof thatcouplingregion for acylindrical cavity.
WG1
WG2
� �a
A
b ��� � B
�������
Figure1: Top view of thedirectionalcouplerconfiguration:Asegmentof a cylindrical waveguidecore(WG2) with radius � ,evanescentlycoupledto a straightwaveguide(WG1). LettersA, B, a, b denotetheport positionsof thedevice, is thelocal� -displacementof thering rim at position � .
The � -axis indicatesthe dominantdirectionof light propagation,with the origin positionedsuchthat the ar-rangementis symmetricwith respectto theplane ����� . The lateral � -axis is orientedalonga radiusof thering; ����� markstheouterrim of thering at ����� . At position � , this outerrim is shiftedto ��������� �"! with��� �"!#�%$&�(' $*)+�,�-) . Notethat in generala threedimensionalconfigurationis considered,i.e. theport coremaybeplacedunderneaththering waveguide(seee.g.thecrosssectionof Figure2).
For a largering radius,thestructurecanbetreatedby meansof moreor lessstandardcoupledmodetheory[3]:. Theoptical electromagneticfield is representedby the two singleguidedmodesof properpolarizationthataresupportedby theisolatedcoresof theport andthering waveguide.For thepresentsimulations,thebendmodesof thecurvedcavity waveguidearereplacedby fieldscalculatedfor astraight waveguidewith analogouscrosssection.. After assigning— in a by no meansuniqueway — permittivity profilesto the individual waveguides,thebasisfieldsarecalculatednumerically[4, 5, 6]; for fixedcoredimensionsthis is necessaryonly once.. With thebasismodeprofilestranslatedover thecrosssectionplaneaccordingto theactual( � -dependent)coreposition,local couplingmatricescanbecomputedthatestablishthelocal coupledmodeequations.The implementedexpressions[7] arejustifiedasanapproximatedescriptionof longitudinally constantstructuresonly.. For (slowly) longitudinally varying structures,onecanexpect that thesecoupledmodeequationsstillyield reasonableapproximations,providedthatthelongitudinalvariationof thecouplingmatricesis takeninto account.The coupledmodeequationsconstitutea linear systemof ordinarydifferentialequationwith — dueto the varying waveguidedistance— nonconstantcoefficients. That systemis integratednumericallyby meansof astandardRunge-Kuttaalgorithm[8].. The solution of the coupledmodeequationsestablishthe scatteringmatrix of the directionalcouplerelement,thecoefficientsof whichserve asdirectinput to theexpressionsin Ref. [2].
As outlinedin thenext sections,thesestepsprovide a computationallyrelatively cheapmeansfor anapprox-imateassessmentof the couplerperformance,basedon existing implementationsof the modesolver andthecoupledmodeapproach.
1
2 Basisfields and supermodes
Specializeto a couplercrosssectionasshown in Figure2. Theport waveguidewith corewidth / , thickness0 , andrefractive index 1 g is embeddedin a mediumwith refractive index 1 b, buriedat a distance2 below thesurface34�5� of thatmedium.Thecavity stripof width 6 andthickness7 with refractive index 1 r is placedontopof thesurface,coveredby amaterialwith refractive index 1 a. Thehorizontalpositionof theportwaveguideis definedby the � -coordinate8 (positive or negative valuesareto beconsidered)of thecorecenter. � is thehorizontal,� -dependentdisplacementof thering rim asintroducedin Figure1.
9:
;<>=@?A
b <BA
g
Ar
Aa
CD
EFG H
WG2
WG1 Figure2: Crosssectionview of thecouplerstructure. JIK�@Lis thelocal lateraldisplacementof theouterflankof theringwaveguidefrom theposition MION>L�P,N at �QPRN .
Omittingtheportcore(WG1)to definetherefractive index profilefor theisolatedring waveguide(WG2)seemsto bea naturalchoice. Hencethebasismodeassociatedwith the ring is calculatedfor a raisedstrip of width6 , thickness7 , refractive index 1 r, supportedby a materialwith refractive index 1 b, andcoveredby a mediumwith refractive index 1 a.
Whendefiningtheisolatedportwaveguide,thecavity coreis to beremoved.Mainly for reasonsof conveniencewe alsoremove thecladdingmedium,extendingthelower backgroundmaterialto theregion 3RS�� . To somedegreethis is justified,becausefor the interestingconfigurationsof coupled,overlappingfieldstheport modeprofilewill sensemainly thepresenceof thering corefor 3�ST� , not thelower refractive index claddingregion.Definingtheportwaveguideprofileasarectangularcoreof width / , thickness0 , refractive index 1 g, embeddedin amediumwith refractive index 1 b, allows to usetheport modeprofileasabasisfield for arbitraryvaluesoftheburyingdepth 2 .Thefirst rowsof Figures3,4show exampleprofilesfor thebasisfieldswith theparametersof Table1,calculatedby thesemivectorialversionof asemianalytictechniqueasdescribedin Refs.[4, 5, 6]. For TE polarizedlight,thetwo isolatedwaveguidesarealmostphasematchedwith aneffective refractive index of UWVYXW�� at a vacuumwavelength ZR�[UWVYXX�\ m. Thefieldsareinternally representedby analyticfunctionswhich arenot restrictedto a spatialcomputationalwindow, hencethey arewell suitedfor furtherprocessingin the framework of thecoupledmodetheory.] ^_ Na` mb N _dcfe ` mg ^_ihkj ` ml cm_ Na` mn cm_ojmj ` m� c N>Na` m
pb
c>_ eWj (SiOq )pg
c>_ r>s (Sit N u )pr
c>_ jmrmv (PMMA–DR1 @ c>_ j>j ` m)pa
c>_ Nw x4y N _ j-z{^_oj}| ` m~ x4yY��� _ N z�e�_ N | ` m
Table1: Simulationparametersthat leadto the re-sultsof Figures3–7,for couplerstructuresaccordingto Figures1, 2.
for thevector� �����Q�k��� ) ! T of � -dependentmodeamplitudes,whereindices U and � refer to theport- and
cavity modes,respectively. Seethereferencefor theprecisedefinitionof the ����� couplingmatrices�, � , �
andfor their specificproperties.
2
−4
−2
0
2
−3−2
−10
12
0
1
2
3
y [µm]x [µm]
Ey [a
.u.]
−4
−2
0
2
−3−2
−10
12
−1
0
1
2
3
y [µm]x [µm]
Ey [a
.u.]
−4
−2
0
2
−3−2
−10
12
−1
0
1
2
y [µm]x [µm]
Ey [a
.u.]
−4
−2
0
2
−3−2
−10
12
−1
0
1
2
y [µm]x [µm]
Ey [a
.u.]
Figure3: Basismodeprofilesof the ring andof the port waveguide(top row), andthe local CMT-supermodeprofiles(bottom)at �QP,N ( �PRN ), for parametersasgivenin Table1, with ~ P,N _oj ` m and
w P cm_ Na` m. Theplotsshow the(real)lateralelectriccomponent��� of theTE polarizedmodes.The relative amplitudesof thebasisfields in thesupermodesaregivenby vectorsION _ e@s-z N _ s@^ L (left, symmetric-likefield) and ION _ s>s-z � N _ojks L (right, antisymmetric-likefield).
Assumingfor themomentthatthestructureis longitudinallyinvariant,Eq.(1) is readilysolvedby anansatzofso-calledsupermodes,asoutlinedin Ref. [7]. Thecorrespondingfield profiles,shown in thebottomrows ofFigures3, 4, canbeviewedasapproximationsto thetwo guidedTE modesof thetotal, compositewaveguidethat includesbothcores.Thecorrespondingpropagationconstants� s and � a definea couplinglengthor halfbeatlength � c �5�¡ ¢��� s �£� a ! thatcangiveafirst hint whetherasignificantpower transfercanbeexpectedfroma limited couplingregionof thisshape.Specifically, for theconfigurationof Figure3 with largercoredistance,thuswith weaker coupling,this lengthevaluatesto � c �¤U¥�-��\ m. Theconfigurationwith alignedcorecentersof Figure4 leadsto strongercouplingwith ashorterbeatlength � c ��¦¦�\ m.
The maximumamountof power that canbe transferredbetweenthe participatingcoresis indicatedby thebalanceof therelativecontributionsof thebasismodesto thesupermodefields.For theconfigurationwith mis-alignedcoresonefindssupermodeswheretheabsolutevaluesof theport- andcavity modeamplitudesdeviateconsiderably(thoughonecanstill recognizea symmetryproperty, accordingto thesignsof theindividual am-plitudes),in contrastto thestronglycoupledstructure,wherethe individual amplitudesareproperlybalancedin bothsupermodevectors.Hencefor a (still longitudinallyinvariant)couplerwith theparametersof Figure4of length � c onecanexpectan almostcompletepower transfer, while for theshiftedcoresoneobservesnotonly a couplinglengththat is threetimesaslong, but alsoa considerablyreducedupperlimit for the relativeamountof opticalpower thatis exchangedbetweenthecores.
3
−4
−2
0
2
−3−2
−10
12
0
1
2
3
y [µm]x [µm]
Ey [a
.u.]
−4
−2
0
2
−3−2
−10
12
−1
0
1
2
3
y [µm]x [µm]
Ey [a
.u.]
−4
−2
0
2
−3−2
−10
12
−2
−1
0
1
y [µm]x [µm]
Ey [a
.u.]
−4
−2
0
2
−3−2
−10
12
−2
−1
0
1
y [µm]x [µm]
Ey [a
.u.]
Figure4: Basismodeprofilesof thering andof theport waveguide(top row), andthelocal CMT-supermodeprofilesat��P5N ( §P5N ), for parametersasgiven in Table1, with ~ P � g+¨k^ andw P c>_ N�` m. Theplots show the (real) lateral
electriccomponent��� of theTE polarizedmodes.Therelativeamplitudesof thebasisfieldsin thesupermodesaregivenby vectorsION _ v@^z N _ v>^ L (left, symmetric-likefield) and I � N _ s>jz N _ s>j L (right, antisymmetric-likefield).
Thisfindingis differentfrom whatis usuallyobservedfor symmetricalcouplerswith two identicalcores,whereenlarging thecoredistancemerelyleadsto anincreasedbeatlength,while still completepower transfercanberealized.Closerexaminationof thecoupledmodeequationrevealsthattheupperlimit of thepowerconversionis determinedby aphasematchingconditionthatinvolvesthepropagationconstantsof thebasismodes,shiftedby a small amountdueto the presenceof the secondcore. While for a symmetricalcouplerboth individualeffective indicesareperturbedby the sameamount,this is no longertrue for a nonsymmetricalarrangementwith different corecrosssections. Aiming at a nonsymmetriccouplerdesignwith a high maximumpowertransfer, onehasto achieve phasematchingof thebasisfieldsincludingthephaseshiftsthatareinducedby thecompositerefractive index profile.
For theconfigurationof Figure3, thelargerphasemismatchis mainlycausedby thepresenceof thelow indexcladdingmediumthatlowersthepropagationconstantof theportwaveguidemode.For themodesof Figure4,the port waveguide modesensesthe cavity core ratherthan the cladding,hencethe effective modeindicesremainbalanced.
4
3 Mode amplitude evolution
If appliedto themoderatelylongitudinallyvaryingstructureof Figure1 (with large $ ), thematricesin Eq. (1)mustbeassumedto dependon � aswell asthemodeamplitudes
Figure5showsplotsof amplitudeevolutionsthatarecomputedin thisway, uniformly for integrationparameters½Ô��Õ-� and ¯5��Õ-��\ m. Onthedisplayedscale,thecurvesarestablewith respectto both ½ and ¯ . Modelinganexcitationin theportwaveguide,thecomputationsstartwith
� � ��!Ç�Ö�®UW�Å��! T.
Thepresentapproachdoesnot includeany lossmechanism,hencethemodalpowers ×o�Q�@× ) and ×o� ) × ) addal-waysto unity, at leastwith reasonableaccuracy. As expected,thepower transferbetweenthetwo waveguidesstartsonly after a certainpropagationdistance,whenthe coresaresufficiently close,andit levels at the endof whatmaythusbecalledthecouplingregion, with thestrongestinteractionaround���Ö� . Theplotscorre-spondto a configuration,wherethecorecentersarealignedat thatposition.For a largeburying depth,hardlyany power transferis observed. When 2 is reduced,i.e. the coresarebroughtvertically closertogether, theinteractionincreases,up to almosttotal couplingfor 2°�ØUWVi��\ m. Reducing2 furtherleadsto anevenstrongerinteraction,i.e. to morerapidoscillationsof theopticalpower, but the relative amountof power found in thecavity waveguideat theoutputport is reducedagain,dueto thebackcouplinginto theport core.Thusincreas-ing the interactionstrengthby reducingthedistancebetweenthecoupledcoresmaywell leadto a lower totalamountof power transfer.
5
−30 −20 −10 0 10 20 300
0.2
0.4
0.6
0.8
1
z [µm]
|C1|2 ,
|C2|2
b = 1.0 µm
|C2|2
|C1|2
−30 −20 −10 0 10 20 300
0.2
0.4
0.6
0.8
1
z [µm]
|C1|2 ,
|C2|2
b = 2.0 µm
|C2|2
|C1|2
−30 −20 −10 0 10 20 300
0.2
0.4
0.6
0.8
1
z [µm]
|C1|2 ,
|C2|2
b = 0.8 µm
|C2|2
|C1|2
−30 −20 −10 0 10 20 300
0.2
0.4
0.6
0.8
1
z [µm]
|C1|2 ,
|C2|2
b = 1.2 µm
|C2|2
|C1|2
Figure5: Evolution of thebasismodeamplitudesaccordingto theCMT equations,for couplerstructureswith differentburyingdepth
wandparametersasin Table1, with ~ P � g+¨m^ (centeredcoresat �QP,N ).
4 Scattering matrices
In theframework of themicroresonatormodel,thecoupleroperationis describedin termsof a ���º� scatteringmatrix Ë thatrelatestheoutputamplitudes
� ��¯±!��¤� Ù*É��f2}É�! T to thetheinitial values� �®�°¯±!��¤� Ú±É��ÅÛ¢É�! T.
By virtue of abstractsymmetryarguments[9, 2], oneexpectsa relationin theformÞ Ù*É2}É�ß � Þ+à áá â ß Þ Ú±ÉÛ¢ÉÓß (7)
with a symmetricscatteringmatrix,wherelosslesscoupleroperationrequires× â × ) �¤× à × ) �ØUQ�5× á × ) . Thoughnotexplicitly incorporated,thesepropertiesprovide ameansto checktheconsistency of thenumericalresults.For thegivenexamples,thisholdsreasonablywell, at leaston thescaleof theshown plots.
Apart from thecavity attenuationconstant,themicroresonatorperformancedependson theabsolutevaluesofthecouplingconstant
áandof thecouplertransfercoefficient
â(wherefor thepresentmodel × â ×ã�ä× à × ). Fig-
ures6 and7 show examplesfor thesometimesquiteirregulardependencesof thesequantitiesontheparameters8 and 2 thatcontrol thehorizontalandverticaldistanceof thecoreswithin thecouplerelement.Thesecurvesshouldbemoreor lessexplainableby a reasoningaccordingto Sections2, 3.
of thecoreof thestraightwaveguide,for differentvaluesof its centerposition ~ . Otherparametersareasgivenin Table1.
8
References
[1] A. Yariv. Universalrelationsfor couplingof opticalpower betweenmiroresonatorsanddielectricwaveguide. Elec-tronic Letters, 36(4):321–322,2000.
[2] M. Hammer. Standardmodelfor opticalringresonantors.NAIS internalmanuscript,09.2002.
[3] D. G. Hall andB. J.Thompson,editors.Selected Papers on Coupled-Mode Theory in Guided-Wave Optics, volumeMS 84 of SPIE Milestone Series. SPIEOpticalEngineeringPress,Bellingham,WashingtonUSA, 1993.
[4] M. Lohmeyer. Wave-matchingmethodfor modeanalysisof dielectricwaveguides.Optical and Quantum Electronics,29:907–922,1997.
[5] M. Lohmeyer. Vectorial wave-matchingmodeanalysisof integratedoptical waveguides. Optical and QuantumElectronics, 30:385–396,1998.
