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Propagation of light in quadratic index profile waveguides

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Title:
Propagation of light in quadratic index profile waveguides
Creator:
Rebolledo, Neil Aporongao ( author )
Language:
English
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1 electronic file (91 pages). : ;

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Light -- Scattering ( lcsh )
Optics ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Review:
Optical gradient index media have the property where the spatial variation of the index of refraction is continuous along the direction transverse to the optical axis. Many emerging technologies in optics require gradient index components, and a firm understanding of the physics of light propagation in these components is required. Presented here are the mathematical foundations used to analyze light propagation in planar quadratic index profile waveguides. One transverse direction is used in this analysis, and light propagation is seen to have periodic behavior. Also presented here is a comparison of ray bundles and wave intensities in quadratic index waveguides, with the intent to use this machinery to further ray chaos theory.
Thesis:
Thesis (M.I.S.)--University of Colorado Denver.
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Includes bibliographic references.
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System requirements: Adobe reader.
General Note:
Integrated Sciences Program
Statement of Responsibility:
by Neil Aporongao Rebolledo.

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University of Colorado Denver
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Auraria Library
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912926304 ( OCLC )
ocn912926304

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PROPAGATIONOFLIGHTINQUADRATICINDEXPROFILEWAVEGUIDES by NEILAPORONGAOREBOLLEDO B.S.,MetropolitanStateCollegeofDenver,2006 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofIntegratedSciences IntegratedSciencesProgram 2015

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2015 NEILA.REBOLLEDO ALLRIGHTSRESERVED ii

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ThisthesisfortheMasterofIntegratedSciencesdegreeby NeilAporongaoRebolledo hasbeenapprovedforthe IntegratedSciencesProgram by RandallP.Tagg,Chair MasoudAsadi-Zeydabadi BurtSimon Date:April24,2015 iii

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Rebolledo,NeilAporongaoM.I.S.,IntegratedSciences PropagationofLightinQuadraticIndexProleWaveguides ThesisdirectedbyAssociateProfessorRandallP.Tagg. ABSTRACT Opticalgradientindexmediahavethepropertywherethespatialvariationoftheindex ofrefractioniscontinuousalongthedirectiontransversetotheopticalaxis.Manyemerging technologiesinopticsrequiregradientindexcomponents,andarmunderstandingofthe physicsoflightpropagationinthesecomponentsisrequired.Presentedherearethemathematicalfoundationsusedtoanalyzelightpropagationinplanarquadraticindexprole waveguides.Onetransversedirectionisusedinthisanalysis,andlightpropagationisseen tohaveperiodicbehavior.Alsopresentedhereisacomparisonofraybundlesandwave intensitiesinquadraticindexwaveguides,withtheintenttousethismachinerytofurther raychaostheory. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:RandallP.Tagg iv

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ForCcile,Luna,BooBear,Liwan,Winnie,andColette. v

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ACKNOWLEDGEMENTS ManythanksandsinceregratitudetomythesiscommitteemembersDr.RandallTagg, Dr.MasoudAsadi-Zeydabadi,andDr.BurtSimon.Theyhavebeenextremelysupportive throughoutthisthesisproject,workingaroundmyworkscheduletosetupourmeetings, answeringquestionsoverthephone,respondingtolatenightandearlymorningemails, andhavegivenmeinvaluableadviceandguidancethatIwilltakewithmefortherest ofmycareer.ManymanythanksalsogoesouttoDr.MartinHuber,whoIrstmetas myundergraduateseniorlabprofessorthesametimeImetMasoud,whoatthetimewas assistingMartinwithhisupperdivisionlabcourses,andnowasmyprogramdirectorfor thisdegree.Yourtime,eort,andsupportasmyprogramadviseratUCDenverisvery muchappreciated,andIcannotthankyouenoughforhelpingmeasagraduatestudentin yourmastersdegreeprogram. AcknowledgementalsogoesouttothephysicsdepartmentattheUniversityofColorado ColoradoSprings,especiallytoDr.RobertCamley,Dr.ZbigniewCelinski,andDr.Anatoliy Glushchenko.Theysupportedmeasaphysicsgraduatestudent,andsupportedmydecision totransfertoUCDenver.SpecialthankstomyfriendsSethJohnson,ScottDavis,Juan Pino,GeorgeFarca,ScottRommel,andMichaelAnderson,whoIallmetwhileworkingat VescentPhotonics.Theygavememystartinopticalphysics,gavemetheboostIneeded tosucceedinscienceandengineering,andwereverysupportiveofmypursuitofamasters degree.AndaveryspecialthankyoutoCottonAndersonwhoImetwhileworkingat ResearchElectro-Optics.Hefurtheredmyprofessionalexposureinoptics,datacollection andanalysis,andshowedmesoftwarewritingtechniquesthatIlaterusedtowritethescripts forthisthesisproject. MysinceregratitudealsogoesouttoDr.RichardKrantzwhosupportedmypursuitof amastersdegreefromthestartbywritingaletterofrecommendationforadmissionintothe physicsgraduateprogramatUCCS.Dr.Krantzwasmyveryrstundergraduatephysics professor,andhealsosupervisedmyundergraduatecapstoneprojectinmathematicalmusic theory.ItwasduringthistimewhereIgotmyrstexposuretoscienticprogramming, collaboratingandtestingideas,andpresentingmyresults.Theseskillspreparedmefor vi

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bothmyprofessionalandacademicendeavors.Ithasalsobeenarealpleasurerunninginto himoncampusasIcompletemymaster'sdegree,catchingup,andtellinghimaboutmy thesisproject.ThankyouDr.K.! vii

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TABLEOFCONTENTS CHAPTER I.WHYSTUDYQUADRATICINDEXWAVEGUIDES?............1 HowDoWeSimulateLightPropagationInQuadraticIndexWaveguides?.3 PuttingItAllTogether,AndWhereToGoFromHere?...........4 II.ELECTROMAGNETICSANDGEOMETRICOPTICS...........6 ElectromagneticWaves..............................7 ReectionandTransmissionofWaves......................8 Polarization....................................8 PlaneWaves....................................9 EnergyandMomentuminElectromagneticWaves...............10 Maxwell'sEquations...............................11 Maxwell'sEquationsinLinearMedia......................12 BoundaryConditions...............................14 GeometricApproximationsofElectromagnetics................14 TotalInternalReection.............................16 TheEikonalEquation..............................18 TheVectorRayEquation............................20 FromTheEikonalEquation........................20 FromLeastActionPrinciples.......................21 III.THEHAMILTONIANFORMULATIONOFGEOMETRICOPTICS....23 DerivationofRayTrajectoriesinQuadraticIndexProleWaveguides....25 IV.COMPUTATIONALSIMULATIONSOFRAYBUNDLESINQUADRATIC INDEXWAVEGUIDES.............................28 NumericalSolutionsToHamilton'sEquations.................29 SimulationofaGaussianRayBundle......................33 viii

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V.WAVEOPTICSTREATMENTOFLIGHTPROPAGATIONINQUADRATIC INDEXWAVEGUIDES.............................35 DerivationofGaussianBeamsanditsProperties...............35 HermitePolynomials...............................39 HermitePolynomialExpansionofGaussianBeams..............40 DerivationofAnalyticSolutionsforFieldIntensitiesinQuadraticIndex Waveguides....................................42 VI.NUMERICALSOLUTIONSTOMAXWELL'SEQUATIONS........47 TheFiniteDierenceTimeDomainMethod..................47 MEEP.......................................49 MEEPSimulation.................................51 ComparisonofMEEPandAnalyticSimulations................53 VII.COMPARISONOFGAUSSIANRAYBUNDLESWITHTHENUMERICALSOLUTIONSTOMAXWELL'SEQUATIONSANDTHEANALYTICSOLUTIONSOFTHESCALARWAVEEQUATION........55 VIII.FUTUREWORK.................................58 REFERENCES......................................59 APPENDIX........................................61 A.Code:AnalyticSolutionforRayTrajectories.................61 B.Code:Hamilton'sEquationsSolvedUsing ode45 ...............62 C.Code:GaussianRayBundle...........................63 D.Code:AnalyticSolutiontotheScalarWaveEquation............65 E.Code:MEEPSimulation.............................69 F.TheQuantumHarmonicOscillator.......................70 G.HowToUseMEEP................................78 ix

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LISTOFFIGURES FIGURES 1Totalinternalreectioninsideaplanarwaveguide................17 2RaybundleusinggeneralsolutionfromHamilton'sEquations.........28 3Raybundleusingode45tosolveHamilton'sequations..............32 4Comparisonsofanalyticsolution,andode45solutionofHamilton'sequations.33 5GaussianRayBundle................................34 6Simulationofeldintensityprolefromtheanalyticsolution..........46 7Yeegrid,in2D....................................50 8IndexprolesimulationinMEEP.........................52 9MEEPsimulationofaGaussianbeaminaquadraticindexwaveguide.....52 10GrayscaleimageofMEEPsimulationforaGaussianbeaminaquadratic indexwaveguide...................................53 11ComparisonofMEEPsimulationwithanalyticsolutionexpansionupto2nd order.........................................54 12ComparisonofMEEPsimulationwithanalyticsolutionexpansionupto20th order.........................................54 13Time-averageeldintensityfortheanalyticsolutiontotheHelmholtzequationforordern=2..................................56 14Time-averageeldintensityfortheanalyticsolutiontotheHelmholtzequationforordern=20.................................56 15ComparisonofanalyticsolutionofGaussianraybundletrajectorieswithtimeaverageeldintensityofanalyticsolutionsimulationupto20thorder.....57 16Therst5Hermitepolynomials..........................76 17Therst5Hermitewavefunctions.........................77 18MEEPexample.ctlgeometryanddielectricconstantprole...........81 19MEEPexample.ctl E z eldintensityprole....................81 x

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CHAPTERI WHYSTUDYQUADRATICINDEXWAVEGUIDES? Thatquestionalonecontainssomuchinformationthatifonedoesn'tknowthemeaning ofanyofthewords quadratic,index, or waveguide ,thenthequestion "whystudyquadratic indexwaveguides?" onlyhasoneanswer, "I'drathernot." Butifonedoesunderstandthe question,andwishesforsomereasontoreadfurther,thenwhat'spresentedhereasan attempttoanswerthetitleofthissectionwillhopefullybeinformative,ifnotprofound. Quadraticindexwaveguidesaredevicesthathaveanindexofrefractionthatisspatially varyingperpendiculartotheopticalaxis.Thisvariationisquadraticallydecreasingtothe boundaryfromthecenterofthewaveguide.Waveguidesandopticaldevicesthathavethis featurearefundamentaltotheadvancementoftechnologiessuchasopticalcommunications, lasersystems,andphotonicintegratedcircuits.Athoroughunderstandingofquadraticindexwaveguidesisalsofundamentaltoproblemsinvolvingchaoticraybehavior,andlight propagationingeometrieswithregularinhomogeneitiesintheaxialdirection.Themathematicalformulationsusedtodescribelightpropagationinquadraticindexwaveguidescan alsobeusedtodescribeothertypesofwavepropagation,suchasacousticandoceanwaves. Therearetwoapproachestoanalyzinglightpropagationinquadraticindexwaveguides. Oneapproachistoassumethatthedimensionsofthewaveguidearemuchlargerthanthe wavelengthoftheincidentlight.Wethenonlyneedtoconsiderthepropagationdirection oftheenergyofthewavefronts,andwecanmodellightas rays .Thisiscalledthe geometricoptics treatmentoflightpropagation.Thisapproachtoanalyzinglightpropagationin quadraticindexwaveguideshasaremarkableconnectiontoaphysicallydierentsystem, theclassicalharmonicoscillator.Thatconnectionliesinthemathematicsbasedonthe Hamiltonianformulationofparticledynamics.TheHamiltonianformulationusesasetof rst-orderdierentialequations,calledHamilton'sEquations,todescribeparticletrajectoriessubjecttoaforce,andwhenappliedtotheclassicalharmonicoscillator,weobtain analyticsolutionsthatdescribeperiodic,sinusoidalparticletrajectories.WhenHamilton's equationsareappliedtolightpropagationinquadraticindexwaveguides,wecanthinkofthe indexproleasthe'force'.AftersolvingHamilton'sEquations,weobtainanalyticsolutions 1

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forraytrajectoriesinquadraticindexwaveguides,andtheraytrajectoriesareshowntobe sinusoidal[2],[3],[6],[10].Theexistenceoftheconnectionbetweenraypathsinquadratic indexwaveguides,andparticletrajectoriesfortheclassicalharmonicoscillatorshowsthat twophysicallydierentsystemsaredescribedbythesamemathematics. Theotherapproachtoanalyzinglightpropagationinquadraticindexwaveguidesisto assumethatthedimensionsofthewaveguideareofthesamelengthscaleasthewavelengthof theincidentlight.Thewavelengthnowmatters,somethingthatisignoredinthegeometric opticsapproach,andwenowneedtodenequantitiesthatwillhelpdescribethewaves interactionwiththegeometryofouranalysis.Thisapproachiscalledthe waveoptics treatmentoflightpropagation.Lightpropagationinquadraticindexwaveguidesisdescribed bysolvingtheHelmholtzwaveequation,whichisbaseduponMaxwell'sEquations.The waveopticstreatmentoflightpropagationofourproblemhasamathematicalconnectionto thequantumharmonicoscillator.Solutionstothequantumharmonicoscillatorarefoundby solvingtheSchrodingerEquation,andanexactsolutioncanbefoundthatwillspecifythe energy-levels.ThesolutionsfoundareacompletesetofsolutionsbasedupontheHermite polynomials.ThesolutionstotheHelmholtzwaveequationisalsoacompletesetofsolutions basedupontheHermitepolynomials.Againweseetwophysicallydierentsystemsrelated bythesamemathematics.Thisthesisaimstonotonlyshowthegeometricandwave opticstreatmentoflightpropagationinquadraticindexwaveguides,buttoalsoshowhow mathematics,thelanguageofphysics,connectssystemsthatarephysicallydierent. Sowhatisaquadraticindexprole?Inthisthesis,themathematicalexpressionforthe refractiveindexproleisoftheform n 2 x = n 2 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 a 2 0 where n o isthemaximumvalueatthecenterofthewaveguide,anddecreasesquadratically to n 1 attheboundary.Thequantity a o isthedistancefromcenterofthewaveguide,orthe waveguidehalf-width.Theindexprolewaschosensothat n 0 and n 1 aredimensionless, andunitsoflengtharecanceledbythefactor x 2 =a 2 0 .Equationisoneexampleofa quadraticorgradientrefractiveindexprole.Oneexampleofpracticaldevicesthatutilizes 2

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thistypeofindexprolearegradient-indexGRINoptics[13].GRINopticsarediscrete opticalcomponentsbehavinglikelensesthathave,similartoquadratic-indexwaveguides,a varyingindexofrefractionintheradialdirection.Theseopticshavebeenusedtoreduce aberrationsinimagingsystems,toreducecostandweightinopticalassemblies,andto producecollimatedopticalbers.Anothergradient-indexdeviceapplicationisthegradientindexber.Theindexofrefractionofthecoredecreaseswithradialdistancefromthe opticalaxis,andtheseberopticsareusedinopticalcommunicationsapplicationsfortheir capabilityofguidinglightoverlargedistances[5]. Theadvancementofintegratedphotonicsisincreasinglybecomingmorereliantonoptical componentsthathaveaspatiallyvaryingindexofrefraction.Itisbecomingincreasingly desiredtointegrateconventionalopticswithGRIN-likeopticstoreducecostandweightof integratedphotonicsystems[15]. HowDoWeSimulateLightPropagationInQuadraticIndexWaveguides? Letstakeacloserlookatthetwoapproachestoanalyzinglightpropagationinquadratic indexwaveguides.Firstthegeometricopticstreatment.ThisapproachusesHamilton's Equationswhicharerst-orderdierentialequationsthatcoupletheray'spositionandmomentum.WhenHamilton'sEquationsaresolvedforlightpropagationinquadraticindex waveguides,theyyieldexpressionsthatshowraypathspropagatesinusoidally.Thetechniquesof raytracing canbeusedtosimulatethislightpropagation[1],[8].Thesoftware toolusedinthisthesistosimulateraypathsisOctave,aMatLabclone.AnOctavescript canbewrittentodenethewaveguidedimensionsandrefractiveindices,andtheexact solutionsfromHamilton'sequationscanbecodedaswell.Built-inplottingfunctionsnative toOctavecanthenbeusedtocreateplotsshowingraypathsinoursystem.Wecansolve Hamilton'sEquationsdirectlybyusingadierentialequationsolverfunctionnativetoOctaveandMatlabcalled ode45 .ThisfunctionusesaniterativeRunge-Kuttaapproximation todirectlysolveHamilton'sequations.Thisnumericalapproachcanbecomparedtothe exactsolutionstoHamilton'sequations,anditwillbeseenheretobeinexcellentagreement witheachother. Thewaveopticstreatmentoflightpropagationinquadraticindexwaveguidesrequiresus 3

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tosolvetheHelmholtzwaveequation,whichisasecond-orderpartialdierentialequation whosesolutionrepresentsanycomponentofthewave.Fortheanalysisinthisthesis,a Gaussiandistributionisinitiallyattheleftboundaryofthewaveguide,andisexpanded intermsoftheHermitepolynomialstosolvetheHelmholtzwaveequation.Theanalytic solutionfortheeldproleisseentohaveaperiodicbehavior.Octavewasalsoused tosimulatewavepropagationinquadraticindexwaveguidesfromtheanalyticsolutionto theHelmholtzequation.Thenumericalsoftwaretoolusedtosimulatewavepropagation is MEEP [12],whichutilizesthe nitedierencetimedomain FDTDmethodtosolve Maxwell'sEquationsbydiscretizingspaceandtimeintoregularrectangulargrids[14].The electricandmagneticeldsarecalculatedatappropriategridlocations,andthestateof thewaveatanytimestepisbasedonthestateofthewaveattheprevioustimestep. Thismethodisapopularchoiceforcomputationalelectromagneticproblemsbecauseof thebalancebetweencomputingrequirementsandaccuracy.MEEPsimulationsarealso performedbyrstwritingascriptdeningthecomputationalcellsize,refractiveindices, andinitialGaussiandistributionattheleftboundary.Acomparisonofthewaveintensities fromtheanalyticsolutiontotheMEEPsimulationsisalsoshowninthisthesis. PuttingItAllTogether,AndWhereToGoFromHere? Onegoalofthisthesisistoshowthatifweknowhowrayspropagateinquadraticindex waveguides,wecanthenaccuratelypredictthecorrespondingwavebehavior,andvice-versa. Thevalidityofthenumericalapproachesusedforthegeometricandwaveopticstreatment isreinforcedbycomparingeachapproachtoknownanalyticalsolutions.Thiswillcreatea foundationtoextendthenumericalapproachestosituationswhereanalyticalsolutionsto therayandwavetreatmentsarenolongerviable.Ofspecialinterestistheintroduction ofaxialmodulationoftheindexofrefraction.Suchasystemhasbeenshowntogenerate chaoticraypaths,anditisofinteresttoknowhowthistranslatesintointensityvariationsof lightinactualsystems.Thus,anumericalschemetorelateintensityprolesobtainedfrom thewaveopticstreatmenttotheconcentrationofbundlesoftrajectoriesfromraytracing techniquesisstronglydesired. Suchavalidatedapproachwillbeusefulforfundamentalstudiesofchaoticpropagationin 4

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axiallymodulatedwaveguidesandwillalsobeofgreatpracticalvalueindescribingsystems inwhichlightpropagatesthroughperiodicstructureswhosescaleofvariationislargerthan thewavelengthoflightbutnotlargeenoughtoignorethewavenatureoflight.Anexample ismediacomposedofregularlyspacedlivingcellsinwhichthenucleicreatesaperiodic variationoftheindexofrefraction. 5

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CHAPTERII ELECTROMAGNETICSANDGEOMETRICOPTICS Inphysicswethinkoflightaselectromagneticwaves,thesamekindofwavesyouwould seepropagatinginwater.Theseelectromagneticwavesarewhatwebelievecausetheperceptionoflight.Ageneralmathematicalexpressionthatdescribesthemotionofthesewaves is f z;t = f z )]TJ/F23 10.9091 Tf 10.909 0 Td [(vt; 0= g z )]TJ/F23 10.9091 Tf 10.909 0 Td [(vt Thisequationdescribesthedisplacementofawaveadistance z attime t .Electromagnetic wavesareanalogoustowavesproducedonastringthatshakesupanddown.Thesekindsof wavesarecalledtransversewaves.ByusingNewton'ssecondlaw, F = m d 2 x dt 2 ,onasegment ofthestringwhichhasatension T onbothendsasitisdisplacedfromitsequilibrium position,thestringwillexperienceaforceinthetransversedirectionbetween z and z + z F = T @f @z j z + z )]TJ/F23 10.9091 Tf 12.104 7.38 Td [(@f @z j z = T @ 2 f @z 2 If isthemassperunitlength,Newton'ssecondlawforthissystemis F = z @ 2 f @t 2 andequatingthiswithgives @ 2 f @z 2 = T @ 2 f @t 2 : andforsmalldisturbances @ 2 f @z 2 = 1 v 2 @ 2 f @t 2 where v isthespeedofpropagationofthewave,andequalto q T .Equationisthe classicalwaveequationinonedimension,andhassolutionsoftheform f z;t = g z )]TJ/F23 10.9091 Tf 10.909 0 Td [(vt : 6

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ElectromagneticWaves Electromagneticwavesaremostoftenrepresentedassinusoidalwaves f z;t = A cos[ k z )]TJ/F23 10.9091 Tf 10.909 0 Td [(vt + ] : Theargumentof cos isthephaseofthewave, isthephaseconstantwhichhasvalues between 0 2 ,and k isthewavenumberofthewaveandisrelatedtothewavelength by k = 2 : Thewavenumberisthenumberofcyclesperunitdistance,ornumberofwavesperunit distance.Whenthewavetravelsadistanceofonewavelength = 2 k ,thecosineexecutes onecompletecycle.Atanyxedpoint z ,thestringvibratesupanddownandcompletes onecycleinaperiod T T = 2 kv : Theperiodisrelatedtothefrequency numberofoscillationsperunittimeby = 1 T = kv 2 = v : Sinusoidalwavescanbewrittenintermsoftheangularfrequency =2 = kv ,the numberofradianssweptoutperunittime.Theexpressionforsinusoidalwavesisofthe form f z;t = A cos kz )]TJ/F23 10.9091 Tf 10.909 0 Td [(!t + : UsingEuler'sformula, e i =cos + i sin ,sinusoidalwavescanbeexpressedincomplex notation ~ f z;t = ~ Ae i kz )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t wherethecomplexamplitudeis ~ A = Ae i 7

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Anywavecanbeexpressedasalinearcombinationofsinusoidalwaves ~ f z;t = Z + 1 ~ A k e i kz )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t dk: Equation3showsthatanywavecanbewrittenasalinearcombinationofsinusoidal waves,andifweknowthebehaviorofsinusoidalwaves,wethenknowthebehaviorofany typesofwaves. ReectionandTransmissionofWaves Considerasinusoidalwaveincidentonthe xy planearrivingfromtheleft.Immediately aftertheincidentwaveinteractswiththe xy plane,areectedwaveisproducedandtravels intheoppositedirectionoftheincidentwave.Atthesametime,atransmittedwaveis producedontherightsideofthe xy plane,andtravelsinthesamedirectionastheincident wave.Thereectedandtransmittedwavemayhavethesamespeedofpropagationasthe incidentwave,orthetransmittedwavemayhaveadierentspeedofpropagationifitisin adierentmedium.Taking z asthepropagationdirection,thesewavesaremathematically representedas ~ f I z;t = ~ A I e i k 1 z )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ,for z< 0 ~ f R z;t = ~ A R e i )]TJ/F24 7.9701 Tf 6.587 0 Td [(k 1 z )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ,for z< 0 ~ f T z;t = ~ A T e i k 2 z )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ,for z> 0 Thedisplacementofthewavejusttotheleftandrightofthe xy planemustbethesame, f )]TJ/F23 10.9091 Tf 7.085 -3.959 Td [(;t = f + ;t .Assumingthe xy planeinterfacehasnegligiblemass,thenthederivative of f isalsocontinuous, @f @z 0 )]TJ/F15 10.9091 Tf 9.783 1.991 Td [(= @f @z 0 + Polarization Asstatedearlier,electromagneticwavesaretransversewaves,andthedisplacementof thesetypesofwavesisperpendiculartothedirectionofpropagation.Electromagneticwaves havetwoquantitieswhosedisplacementisperpendiculartothepropagationdirection,and canthenhavetwostatesofpolarization.Analogoustoshakingastring,youcanshake 8

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ithorizontallyorvertically.Forexample,forawavepolarizedinthe ^ x directionwehave ~ f z;t = ~ Ae i kz )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ x ,andforawavepolarizedinthe ^ y direction ~ f z;t = ~ Ae i kz )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ^ y .In general ~ f z;t = ~ Ae i kz )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ^ n : Thepolarizationvector ^ n denestheplaneofvibration,andsincethisplaneisperpendicular tothedirectionofpropagation ^ n ^ z =0 ,and ^ n =cos ^ x +sin ^ y .Soawavecanbedescribed asasuperpositionoftwowaves,eachwaveinoneofthestatesofpolarization ~ f z;t = ~ A cos e i kz )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ x + ~ A sin e i kz )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ^ y : PlaneWaves Sinusoidalwavestravelinginthe z directionandhaveno x and y dependencearecalled planewaves.Planewaveshaveuniformeldsovereveryplaneperpendiculartothedirection ofpropagation.Theexpressionsforplanewavesforelectricandmagneticeldsinavacuum are ~ E z;t = ~ E 0 e i kz )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ~ B z;t = ~ B 0 e i kz )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t : ThesearealsosolutionstoMaxwell'sequationsinachargefree,andcurrentfreemedium. ~ E 0 and ~ B 0 arethecomplexamplitudes.Oversmallenoughregions,anywaveisaplane wave,aslongasitswavelengthismuchlessthantheradiusofthecurvatureofthewavefront. Wecangeneralizethepropagationdirection kz toanydirection r ,andbyintroducing thewavevector k ,whichpointsinthedirectionofpropagation. ~ E r ;t = ~ E 0 e i k r )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ^ n ~ B r ;t = ~ B 0 e i k r )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ n 9

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EnergyandMomentuminElectromagneticWaves Theenergyperunittimeinanelectromagneticeldis u = 1 2 0 E 2 + 1 0 B 2 where 0 isthepermitivityoffreespace, 0 isthepermeabilityoffreespace.Theenergy densitiesfortheelectriceldis u E = 1 2 0 E 2 ,and u B = B 2 2 0 forthemagneticeld.For monochromaticplanewaves, B 2 = 1 c 2 E 2 = 0 0 E 2 ,sotheelectricandmagneticeld contributionsareequal u = 0 E 2 = 0 E 2 0 cos 2 kz )]TJ/F23 10.9091 Tf 10.909 0 Td [(!t + : Theenergyuxdensity,orenergyperunitarea,perunittime,isgivenbythePoynting vector S = 1 0 E B andformonochromaticplanewavesinthe z direction S = c 0 E 2 cos 2 kz )]TJ/F23 10.9091 Tf 10.909 0 Td [(!t + ^ z = cu ^ z sothat S istheenergydensity u timesthevelocityofthewave c ^ z Electromagneticeldsalsocarrymomentum,andthemomentumdensitystoredinthe eldis } = 1 c 2 S andformonochromaticplanewaves } = 1 c 2 0 E 2 cos 2 kz )]TJ/F23 10.9091 Tf 10.91 0 Td [(!t + ^ z = 1 c u ^ z h u i = 1 2 0 E 2 h S i = 1 2 c 0 E 2 ^ z h } i = 1 2 c 0 E 2 0 ^ z 10

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Thebracketsrepresentthetimeaverageoveracompletecycle,ormanycycles.Theaverage powerperunitareatransportedbyanelectromagneticwaveistheintensity I h S i = 1 2 c 0 E 2 0 Lightcanimpartmomentumonasurfaceifthematerialisaperfectabsorberinatime interval t themomentumtransferis p = h } i Ac t ,sotheradiationpressureaverage forceperunitareais P = 1 A p t = 1 2 0 E 2 0 = 1 c Maxwell'sEquations Inachargefreeandcurrentfreespace,Maxwell'sequationsare r E =0 r B =0 r E = )]TJ/F23 10.9091 Tf 9.681 7.38 Td [(@ B @t r B = 0 0 @ E @t Theseequationsarecoupled,rst-orderpartialdierentialequationsfor E and B .They canbedecoupledbytakingthecurlofthecurlequations r r E = r r E )-222(r 2 E = r )]TJ/F23 10.9091 Tf 9.681 7.38 Td [(@ B @t = )]TJ/F23 10.9091 Tf 11.65 7.38 Td [(@ @t r B = )]TJ/F23 10.9091 Tf 8.485 0 Td [( 0 0 @ 2 E @t 2 r r B = r r B )-222(r 2 B = r )]TJ/F23 10.9091 Tf 9.68 7.38 Td [(@ E @t = )]TJ/F23 10.9091 Tf 11.65 7.38 Td [(@ @t r E = )]TJ/F23 10.9091 Tf 8.485 0 Td [( 0 0 @ 2 B @t 2 : Usingthedivergenceequationsand,andbecome r 2 E = 0 0 @ 2 E @t 2 r 2 B = 0 0 @ 2 B @t 2 11

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whicharethewaveequationsfortheelectricandmagneticeldsderivedfromMaxwell's equationsinavacuum.Whatstartedascoupledrst-orderpartialdierentialequationsare notseparatesecond-orderpartialdierentialequations. Faraday'sLawgivesarelationbetweentheamplitudesoftheelectricandmagnetic elds )]TJ/F23 10.9091 Tf 8.485 0 Td [(k ~ E 0 y = ~ B 0 x and k ~ E 0 x = ~ B 0 y : Theseresultscanbegeneralizedas ~ B 0 = k ^ z ~ E 0 : Wethenalsohavearelationbetweentheirrealamplitudes B 0 = k E 0 = 1 c E 0 : sothatbecomes ~ B r ;t = 1 c ^ k ~ E Maxwell'sEquationsinLinearMedia Considermatterthatallowsforelectromagneticwavepropagation,andisfreefrom chargeandcurrent.Theelectricdisplacementandauxiliarymagneticeldduetothematerialare D = E H = 1 B where isthepermitivityofthematerial,and isthepermeabilityofthematerial. Maxwell'sequationsforlinearmatterthenbecome r D =0 12

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r B =0 r E = )]TJ/F23 10.9091 Tf 9.681 7.38 Td [(@ B @t r H = @ D @t Inahomogeneousmedium, and donotvaryfrompointtopoint.Maxwell'sequations thenreduceto r E =0 r B =0 r E = )]TJ/F23 10.9091 Tf 9.681 7.38 Td [(@ B @t r B = @ E @t Thespeedofpropagationofanelectromagneticwaveinalinearhomogeneousmediumis v = 1 p = c n ,where n = q 0 0 istheindexofrefraction.Itistheratiobetweenthe speedofpropagationofelectromagneticwavesinthemediumandinavacuum.Formost materials, 0 ,therefore n p r where r isthedielectricconstantofthematerial. Since r isalwaysgreaterand1,lighttravelsmoreslowlythroughmatter. Waveequationsfor E and B inlinearmediacanbeobtainedbythesameprocedurethat yieldedthewaveequationsand r 2 E = @ 2 E @t 2 r 2 B = @ 2 B @t 2 : Lookingjustatthewaveequationfor E ,whichholdsforallcomponentsof E ,andis equivalenttosayingthatallcomponentsof E alsosatisfythescalarwaveequation r 2 = 1 v 2 @ 2 @t 2 : where v =1 = 1 = 2 .Thisdescribesinterferenceanddiractioneectsofthewave,and 13

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canrepresentanycomponentof E BoundaryConditions Consideranelectromagneticwaveinteractingwithaninterface.Thewaveobeysthefollowingboundaryconditions 1 E ? 1 = 2 E ? 2 B ? 1 = B ? 2 E k 1 = E k 2 1 1 B k 1 = 1 2 B k 2 Theseequationsrelatetheelectricandmagneticeldsjusttotheleftandrightofaninterface betweentwolinearmedia. GeometricApproximationsofElectromagnetics Considerawaveincidentonthe xy planefromtheleft,andthe xy planeseparatestwolinear mediathatsupportswavepropagation.Takingthepropagationdirectionoftheincident waveinthe + z direction,andpolarizedinthe + x direction ~ E I z;t = ~ E oI e i k 1 z )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ x ~ B I z;t = 1 v 1 ~ E oI e i k 1 z )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ^ y afterthewaveinteractswiththeinterface,areectedwaveisproducedwhichtravelstothe right ~ E R z;t = ~ E oR e i )]TJ/F24 7.9701 Tf 6.586 0 Td [(k 1 z )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ x ~ B R z;t = 1 v 1 ~ E oR e i )]TJ/F24 7.9701 Tf 6.586 0 Td [(k 1 z )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ y andatransmittedwaveisproducedontherightsideofthe xy plane,travelingtotheleft ~ E T z;t = ~ E oT e i k 2 z )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ x 14

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~ B T z;t = 1 v 1 ~ E oT e i k 2 z )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ^ y : Ingeneral,electromagneticwavesareincidentatasurfaceatmanydierentangles.For amonochromaticplanewave ~ E I r ;t = ~ E o I e i k I r )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ~ B I r ;t = 1 v 1 ^ k I ~ E I aretheincidentelectricandmagneticeldsapproachingfromtheleft.Afterinteracting withtheinterface,areectedwaveisproduced ~ E R r ;t = ~ E o R e i k R r )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ~ B R r ;t = 1 v 1 ^ k R ~ E R andatransmittedwaveisproduced ~ E T r ;t = ~ E o T e i k T r )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t ~ B T r ;t = 1 v 1 ^ k T ~ E T : Thefrequency isthesameforallthreewaves,andfromthedenitionofangularfrequency = kv k I v 1 = k R v 1 = k T v 2 = ,or k I = k R = v 2 v 1 k T = n 1 n 2 k T : Thecombinedelds ~ E I + ~ E R and ~ B I + ~ B R inmedium1mustbejoinedwiththetransmitted elds ~ E T and ~ B T usingtheboundaryconditions.Theeldssharethemathematicalform e i k I r )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t + e i k R r )]TJ/F24 7.9701 Tf 6.587 0 Td [(!t = e i k T r )]TJ/F24 7.9701 Tf 6.586 0 Td [(!t ,at z =0 : Sincetheboundaryconditionsmustholdforallpointsontheplane,foralltimes,thenthe exponentialfactorsmustbeequal.Thespatialtermsarethen k I r = k R r = k T r ,when z =0 : or x k I x + y k I y = x k R x + y k R y = x k T x + y k T y ; 15

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forall x andall y .Equationcanonlyholdifthecomponentsareseparatelyequal.If x =0 ,wehave k I y = k R y = k T y ; andif y =0 k I x = k R x = k T x : Ifweorientouraxessothat k I liesinthe xz plane k I y =0 ,then k R and k T alsoliein the xz plane.Wecanthengeneralizetoconclude Theplanewhichcontainstheincident,reected,andtransmitted waves,andthenormaltothesurfaceiscalledthe planeofincidence Equationimpliesthat k I sin I = k R sin R = k T sin T ; where I isthe angleofincidence R isthe angleofreection ,and T istheangleof transmission,orthe angleofrefraction .Alloftheseanglesaremeasuredwithrespectto thenormal.Wecanthengeneralizeandstatetwolaws.Therstisthe lawofreection Theangleofincidenceandangleofreectionareequal. Thesecondisthe lawofrefraction ,alsoknownas Snell'sLaw n 1 sin I = n 2 sin T whichgivesustheangleofthetransmittedanglewithrespecttothenormaloftheinterface. Thesetwolaws,alongwiththedenitionoftheplaneofincidence,arethethreefundamental lawsofgeometricoptics.Anyotherwaves,suchaswaterwavesorsoundwaves,willobey thesame"optical"lawswhentheypassfromonemediumtoanther. TotalInternalReection SolvingSnell'sLawfor T sin T = n 1 n 2 sin I 16

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Thisimpliesthatif n 1 n 2 then T > I .Thisimpliesthat inthecaseof n 1 >n 2 foracertainangleofincidence I = critical ,therefractionangle T is90 ,andthelightwillstayconnedinthematerialwithhigherindexofrefraction.From theseassumptions,wehave sin c = n 2 n 1 sin T = n 2 n 1 andthe criticalangle istherefore c =arcsin n 2 n 1 : Awaveguideisalightconningdevicewithahigherindexofrefractioninitscore,and alowerindexofrefractioninitscladding.Lightcanbecoupledintoopticalwaveguidesat anglesatwhichtotalinternalreectionhappensinsideofthewaveguide. Figure1:Totalinternalreectioninsideaplanarwaveguide. Tondthemaximumanglerelativetothenormalofthewaveguidewherewecanstill couplelightintothewaveguide,weusethe n =1 forair,andthe criticalangle derived above.FromSnell'slaw,usingtheentranceplaneofthewaveguideasourinterface,and n =1 forair,wehave 1 sin max = n 1 sin 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [( c = n 1 cos c 17

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usingthiswiththecriticalangle,wehave sin max = n 1 cos c = n 1 p 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(sin 2 c = n 1 s 1 )]TJ/F29 10.9091 Tf 10.909 15.382 Td [( n 2 n 1 2 sin max = q n 2 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 2 : Themaximumanglealsodenesthe numericalaperture NAofthewaveguide NA = q n 2 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 2 : TheEikonalEquation Allpropagationlawsforpencilsofraysoflightcanbederivedfromthe EikonalEquation TheequationgetsitsnamefromtheGreekword ~ meaningimage.Raysarethenormals tolightwavefronts,andalsorepresentthedirectionofenergypropagationofthesewaves. TheEikonalequationisanon-linearpartialdierentialequationthatcanbederivedfrom thescalarwaveequation,usingarst-order,planewave-likesolutionofthetime-harmonic electricandmagneticelds. Thescalarwaveequationdescribeslightpropagationinaopticalmediumwherethe indexofrefractionvariesslowlyasafunctionofposition.Using v = 1 p = c 0 n ,thescalar waveequationbecomes r 2 )]TJ/F23 10.9091 Tf 12.105 7.38 Td [(n 2 x c 2 0 @ 2 @t 2 =0 where c 0 isthevacuumspeedoflight.Asolutiontoisoftheformofamonochromatic wave = x e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i!t andaftertakingtheappropriatederivatives,thescalarwaveequationbecomes r 2 + k 2 0 n 2 x =0 where k 0 !=c 0 isthevacuumwavenumber,withunits 1 =length .Thesolutionto, 18

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alsoknownastheHelmholtzequation,isoftheform x = A x e ik 0 S x where S x istheeikonalfunction.Takingtheappropriatederivativesof r = fr A + A ik 0 r S g e ik 0 S x r 2 = n r 2 A + r A ik 0 r S + r A ik 0 r S + A ik 0 r 2 S )]TJ/F23 10.9091 Tf 10.909 0 Td [(Ak 2 0 jr S j 2 o e ik 0 S x @ @t = A )]TJ/F23 10.9091 Tf 8.485 0 Td [(i! e ik 0 S x @ 2 @t 2 = )]TJ/F23 10.9091 Tf 8.485 0 Td [(A! 2 e ik 0 S x : theHelmholtzequationbecomes r 2 A + r Aik 0 r S + r Aik 0 r S + Aik 0 r 2 S )]TJ/F23 10.9091 Tf 10.909 0 Td [(Ak 2 0 jr S j 2 )]TJ/F23 10.9091 Tf 12.105 7.38 Td [(n 2 x c 2 0 )]TJ/F23 10.9091 Tf 8.484 0 Td [(A! 2 e ik 0 S x =0 n r 2 A + r Aik 0 r S + r Aik 0 r S + Aik 0 r 2 S )]TJ/F23 10.9091 Tf 10.909 0 Td [(Ak 2 0 jr S j 2 + k 2 0 n 2 x A o e ik 0 S x =0 : Lookingatjusttherealpart r 2 A )]TJ/F23 10.9091 Tf 10.909 0 Td [(Ak 2 0 jr S j 2 + k 2 0 n 2 x A =0 : Dividingby k 2 0 A ,andinthegeometriclimit r 2 A k 2 0 A 0 [3][2],wehave jr S x j 2 = n 2 x whichistheeikonalequation. Wecandenealocalwavevectoras ~ k k 0 r S andwhen S x = constant aresurfacesofconstantphase.Thus ~ k pointsinthedirection 19

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normaltotheconstantphasesurfaces.Theeikonalequationimpliesthatthemagnitudeof ~ k isdeterminedbythelocalindexofrefraction k 2 = k 2 0 n 2 x : TheVectorRayEquation Thevectorrayequationisavectorialequationthatcandetermineraypathsinoptical media.Itwillrstbederivedherefromtheeikonalequation,thenfromleastactionprinciples. FromTheEikonalEquation Theeikonalequationisthebasicequationthatdescribeslightraypropagationinthe geometriclimit.Itwillbeseenherethattheeikonalequationcanbefurthersimplied toobtainthevectorrayequation.Takingthesquarerootof,anddening s asthe arc-lengthoftheray,wehave r S = n d~ r ds : Dierentiatingbothsidesof d ds r S = d ds n d ds andusingthedenitionofthegradoperator r ,wehave d~r ds r r S = d ds n d~r ds 1 n r S r r S = d ds n d~r ds 1 2 n r [ r S 2 ]= d ds n d~r ds 1 2 n r n 2 = d ds n d~r ds r n = d ds n d~r ds 20

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Equationistheeikonalequationforrayvectors,orthe paraxialrayequation whose solutionscorrespondtoraypaths. FromLeastActionPrinciples Fermat'sprincipleofleasttimestatesthataraypathbetweentwospeciedpointsis traversedintheleastamountoftimerequired Z x 2 x 1 nds = minimum where n istheindexofrefraction,and ds istheraypath'sarclength. Hamilton'sprincipleofleastactionisthebroadestofalldynamicalprinciples,andit yieldstheequationsofmotionforaclassicalparticle Z L q ; q ;t dt =0 where L istheLagrangianfunctionequalto T )]TJ/F23 10.9091 Tf 11.318 0 Td [(V kineticminuspotentialenergy,and isrepresentedingeneralizedcoordinates.Hamilton'sprinciplestatesthattheevolution ofadynamicalsystemwithinaspeciedtimeintervalisastationarypointoftheaction functional.Hamilton'sprincipleminimizesfunctionsoftime,whereasFermat'sprinciple minimizesfunctionsoflength.FromHamilton'sprinciplewecanderiveLagrange'sequations d dt @L @ q = @L @q whichyieldtheequationsofmotionforaparticleinanysystem,givenaknownLagrangian function.Lagrange'sequationsarethebasicrelationshipbetweenHamiltonianandNewtonianmechanics. FromFermat'sprinciple,the opticalLagrangian canbedetermined,andthenappliedto Lagrange'sequationstoobtaintherayequations.Fermat'sprinciplecanthenbewrittenas Z n x;y;z )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+_ x 2 +_ y 2 1 = 2 dz =0 21

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where L = L optical = n x;y;z )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1+_ x 2 +_ y 2 1 = 2 istheopticalLagrangian.Ifwechoosethe z -directionasthedirectionalongwhichrayspropagate,then z playsthesameroleastimeinHamiltonianmechanics.Lagrange'sequations thenbecome d dz @L @ x = @L @x d dz @L @ y = @L @y Fromthesetwoequations,wecanderivetherayequationinvectorform d ds n d~r ds = r n whichisthesameresultof,thevectorrayequation. 22

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CHAPTERIII THEHAMILTONIANFORMULATIONOFGEOMETRICOPTICS HeretheHamiltonianformulationofgeometricopticsispresented,andappliedtoour 2Dquadraticindexwaveguidesystem.Recallthatwearedeningraysasthedirectionof energypropagationoftheelectromagneticwavefronts.Inaninnitesimaltime dt ,apoint ontheraypathmovesadistance cdt inthedirectionoftheunitvector ~ k=k .Forthe x componentoftheraypath dx dt = c x k x k 0 : Sincethemediumiscontinuouslyvaryingwithpositioninthe x direction c x = c 0 n x : andsubstitutingandinto dx dt = c 0 n 2 x k x k 0 From, ~ k = ~ k x t ,andtheequationforthe x componentof ~ k is dk dt = @k dx dx dt : Since ~ k isproportionaltothegradientoftheeikonalfunction, @k x @x = @k @x andsubstitutingyields dk x dt = c 0 k 0 1 n 2 x 1 2 @k 2 dx andfrom dk x dt = c 0 k 0 1 n x @n x @x : Bynotingthat = c x k = c 0 n x k ,wecanobtainacorrespondencewithHamiltonian 23

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mechanics.Fromand dx dt = @! @k x dk x dt = )]TJ/F23 10.9091 Tf 9.68 7.38 Td [(@! @x whichrepresentHamilton'sequationsforparticledynamics. Forraypropagationinawaveguide,weassumethattherayneverturnsbackuponitself, andtaking z astheopticalaxis dx dz 6 =0 : Thisallowsustodescribetheraypathsbytheone-wayequations.Dividingequation by dz=dt ,wehave dx dt dt dz = c 0 n 2 x k x k 0 dt dz : Theraypathinthe z directionis dz dt = c x k z k 0 andaftersolvingfor k z ,becomes dx dz = k x k z : Dividingby dz=dt dk x dt dt dz = c 0 k 0 1 n x @n x @x dt dz dk x dz = k 2 0 k z n @n x @x Ifwedene p k x =k 0 asthenewconjugatevariableto x ,thedynamicalsystemfortheray trajectorybecomes dx dz = p p n 2 )]TJ/F23 10.9091 Tf 10.91 0 Td [(p 2 dp dz = n @n @x p n 2 )]TJ/F23 10.9091 Tf 10.91 0 Td [(p 2 : 24

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TheequationsforthisdynamicalsystemcanberewrittenintheformofHamilton'sequations dx dz = @H @p dp dz = )]TJ/F23 10.9091 Tf 9.68 7.38 Td [(@H @x wheretheHamiltonianisgivenby H = )]TJ/F29 10.9091 Tf 8.485 9.913 Td [(p n 2 x;y )]TJ/F23 10.9091 Tf 10.909 0 Td [(p 2 : Theanglebetweentherayandtheopticalaxis z isgivenby tan = dx dz : Intermsoftheindexofrefraction n andtherayangle ,theHamiltonianandmomentum canbewrittenas H = )]TJ/F23 10.9091 Tf 8.485 0 Td [(n cos p = n sin : Iftheindexofrefractionisindependentoftheaxialcoordinate,thentheHamiltonian H = )]TJ/F23 10.9091 Tf 8.485 0 Td [(n cos isaconstantalongtheraypath.ThisisageneralizationofSnell'sLaw foramediumwithcontinuouslyvaryingindexofrefractionintransverse x -direction.The presenceofcosineratherthansineisduetothedenitionoftheangle relativetothe opticalaxis. DerivationofRayTrajectoriesinQuadraticIndexProleWaveguides Themostcommonindexproleforgradient-indexopticalwaveguidesisaquadraticfunctionofthetransversecoordinate x .Forarectangularopticalwaveguideofxedthickness a ,theproleis n 2 x = n 2 0 )]TJ/F29 10.9091 Tf 10.909 8.836 Td [()]TJ/F23 10.9091 Tf 5 -8.836 Td [(n 2 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 a 2 25

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forthisindexprole,theHamiltonianis H = )]TJ/F29 10.9091 Tf 8.485 9.913 Td [(p n 2 x )]TJ/F23 10.9091 Tf 10.909 0 Td [(p 2 : Hamilton'sequationsandthenbecome dx dz = p p n 2 )]TJ/F23 10.9091 Tf 10.91 0 Td [(p 2 dp dz = )]TJ/F15 10.9091 Tf 8.485 0 Td [( n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x a 2 p n 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(p 2 : Usingthechainrule dp dx = )]TJ/F15 10.9091 Tf 8.485 0 Td [( n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x a 2 p : Separatingvariables,wehave pdp = )]TJ/F15 10.9091 Tf 8.485 0 Td [( n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 xdx p 2 = p 2 o + n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 o a 2 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 a 2 : Wenowdene N 2 p 2 o + n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 o a 2 and n 2 n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 : Themomentumthenbecomes p 2 = N 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 p = p N 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 : Substitutionof p and n 2 x = n 2 o )]TJ/F21 7.9701 Tf 12.105 5.856 Td [( n 2 o )]TJ/F24 7.9701 Tf 6.586 0 Td [(n 2 1 x 2 a 2 = n 2 o )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 intogives dx dz = p N 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 p n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(N 2 26

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rearrangingas dx q )]TJ/F24 7.9701 Tf 8.508 -4.541 Td [(N n 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(x 2 = n p n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(N 2 dz thenintegratinggives arcsin x N n )]TJ/F15 10.9091 Tf 10.909 0 Td [(arcsin x o N n = n p n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(N 2 z )]TJ/F23 10.9091 Tf 10.909 0 Td [(z o x = N n sin arcsin x o N n + n p n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(N 2 z )]TJ/F23 10.9091 Tf 10.909 0 Td [(z o # : Lookingjustat N q p 2 o + n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 a 2 = p p 2 o + n 2 x 2 o ,andusing p = nsin = p n 2 o )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 sin ,wehaveafterrearranging N = p n 2 o sin 2 o + cos 2 o n 2 x 2 o : Iftheinitialangleis o =0 at z o =0 ,then cos o =1 and N = p n 2 x 2 o and N n = p n 2 x 2 o n = x o ,becomes x = x o sin arcsin x o x o + n p n 2 o )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 o z # x = x o sin 2 + n p n 2 o )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 o z # x = x o cos n p n 2 o )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 o z # : From k =2 = ,wedene 2 p n 2 o )]TJ/F15 10.9091 Tf 10.909 0 Td [( n 2 x 2 o n ; thenthegeneralexpressionforraytrajectoriesis x = x o cos 2 z 27

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CHAPTERIV COMPUTATIONALSIMULATIONSOFRAYBUNDLESINQUADRATIC INDEXWAVEGUIDES TheanalyticsolutionforraytrajectoriesinquadraticindexwaveguideswassimulatedinOctave.First,refractiveindicesaredenedtobe n o =4 alongtheopticalaxis,and n 1 =1 attheboundary.Thecharacteristiclengthscaleofthesimulationandallsimulation inthisthesisisinmicrons.Thewaveguidehalf-widthdistancetotheboundaryfromthe centerofthewaveguideis5microns,andthewaveguidelengthis32microns.Theray's initialpositionsontheleftboundaryweregivenequalspacings,andeachwithaninitial angleof 0 =0 degrees.Allofthisinformationispassedintoafunctionthatcalculates equation,andthenequation.Theresultofthisfunctionisacolumnvectorfor eachraythatgivesthetransversepositionatallaxiallocations.Figure2istheplotfrom thesimulationresultsseeAppendixAforcode Figure2:RaybundleusinggeneralsolutionfromHamilton'sEquations NotethatfromFigure2,weseethewavelengthdependenceontheinitialpositionofthe raypredictedin.Thisismoreclearlyseenatthesecondnode.Thisisuniquefrom theothersystemswithoscillatorysolutionssuchastheclassicalharmonicoscillator. 28

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NumericalSolutionsToHamilton'sEquations Theproblemoflightpropagationinquadraticindexmediainvolvesolvingdierential equationsforsomeinitialvalues.ThedierentialequationsareHamilton'sequations,and theinitialvaluesthatmustbespeciedaretheinitialtransversepositionsofeachray,andthe initialaxialposition,whichinallofoursimulationsisat z =0 .Giventheseinitialvalues,we canpropagateHamilton'sequationsforwardastheaxialpositionincreases.Euler'smethod isanumericaltechniquetosolvingordinarydierentialequations.Hamilton'sequations andareoftheform dx dt = f x;t : TakingtheTaylorseriesexpansionof x about t gives x t + t = x t + dx dt t + 1 2! d 2 x dt 2 t 2 + : where x t istheinitialvalueofthefunction.Ifwetake t tobesmallenoughsothatthe termssecondorderandhigherarenegligible,thenwecanignorethoseterms,andweare leftwithanapproximationforthevalueofthefunctionattimestep t + t x t + t x t + dx dt t: Thisapproachiscalledthe Eulermethod .Euler'smethodcanbeexpressedbyusingthe indices i x t i +1 x t i + f x t i ;t i t where x t i istheinitialvalueofthefunction.Thelocalorderoftheapproximationis determinedbytheorderin t towhichtheapproximationagreeswiththeexactsolution. Euler'smethodisarstorderapproximationat t i +1 Foranimprovedapproximationmethod,wecankeephigherordersoftheTaylorexpansion.Wecanthinkoftheproblemofsolvingdierentialequationsoftheformofequation asintegrating dx=dt from t to t + t .Then,fromthemeanvaluetheorem,thereexists avalue t m intheintervalwheretheexactsolutionisfoundwhilestoppingatrstorderin 29

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t x t + t = x t + dx dt j t m t: Theslope dx=dt j t m containsinformationofthecurvaturesecondandhigherorderterms. Byproperlyestimating t m and dx=dt j t m ,approximationsofhigherorderthantheEuler methodcanbeobtained.Oneofthoseapproximationsisthesecond-orderRunge-Kutta approximation x t + t = x t + f x 0 ;t 0 t where x 0 = x t + 1 2 f x t ;t t t 0 = t + 1 2 t: Theexpression t 0 isthemidpointoftheinterval,and x 0 istheEulerapproximatedvalueof x at t 0 ,andthisapproximationisaccuratetosecondorder,andrstorderaccurateglobally. TheRunge-Kuttaapproximationestimatestheslope dx=dt j t m byaweightedaverageof severaltermsoftheform f x 0 ;t 0 where t 0 i aresuitablychosenvaluesintheinterval [ t;t + t ] ,and x 0 i areobtainedbyan Euler-like approximation.Antherpopularhigherorder approximationisthefourth-orderRunge-Kuttamethod x t + t x t + 1 6 f x 0 1 ;t 0 1 +2 f x 0 2 ;t 0 2 +2 f x 0 3 ;t 0 3 + f x 0 4 ;t 0 4 t where x 0 1 = x t x 0 2 = x t + 1 2 f x 0 1 ;t 0 1 t x 0 3 = x t + 1 2 f x 0 2 ;t 0 2 t x 0 4 = x t + f x 0 3 ;t 0 3 t and t 0 1 = t 30

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t 0 2 = t + 1 2 t t 0 3 = t + 1 2 t t 0 4 = t + t: Thisfourth-orderRunge-Kuttamethodrequiresthecalculationof f x 0 ;t 0 fourtimesand Eulerapproximationsfourtimes,andthusrequiresroughlyfourtimesthecomputingrequirementsperstepoftheEulermethodatthesameaccuracy.Butwecanchoselarger valuesof t thanwiththeEulermethodatthesameaccuracy.ThismakesthefourthorderRunge-Kuttamethodtherstchoiceforapproximationmethodsforcomputational problemsrequiringhighaccuracies. Matlab's ode45 functioncanbeusedtosolveHamilton'sequations.Thisfunctionsolves non-stidierentialequations.ItisbasedonanexplicitRunge-Kuttamethod.Thesyntax forusing ode45 is [Z,X]=ode45@funcname,t,x0,options,param where Z isacolumnvectorofaxialpointsonthewaveguide,and X isthesolutionarrayfor thetransversepositionoftherayattherespectiveaxialposition.Foroursimulation, t is theintervalofintegrationoveradenedaxialregion.The ode45 solverwillusetherst elementofthisvector,andintegratefromtherstelementtothelast.Arowinthesolution array X containstwoelements,whichcorrespondtothesolutionsofHamilton'sequations for x z and p z atthecorrespondingaxiallocation. 31

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Figure3:Raybundleusingode45tosolveHamilton'sequations. Theinitialcondition y0 isarowvectoroftwoelements,therstbeingtheinitialposition,andthesecondelementistheinitialmomentum p z =0= p 0 = n 0 sin 0 .Theinitial angle 0 ,indices'sofrefraction n 0 and n 1 ,and a 0 ,themaximumtransversedistancefrom thecenterofthewaveguideareallgiven. Acomparisonofthetwosimulationscanbemadebyoverlayingthe ode45 plotoverthe analyticsolutionsimulation.Figure44istheoverlayplotofbothsimulations,andbothare inexactagreementwitheachother. 32

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Figure4:Comparisonsofanalyticsolution,andode45solutionofHamilton'sequations. SimulationofaGaussianRayBundle Inordertogetinsightsofhowareallaserbeamwillpropagateinquadraticindex waveguidesfromtheraytrajectoriespresented,wehavetocreateaninitialraydistribution thatrepresentsaGaussianfunction.TocreateaGaussianraybundlewedenerayspacings thatareproportionaltoaGaussianfunctionas x i +1 = x i + e x 2 i = 2 where istheinitialspacingfromtheon-axisraytobothofitsadjacentraysoneachside. TheGaussianrayspacingdenedby59isimplementedtotheanalyticsolutionforray trajectories,andtheresultisseeninFigure5. 33

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Figure5:GaussianRayBundle. Againweseethewavelengthdependenceontheinitialpositionofeachray,andwecan predictthatthiswillcorrespondtofocusingregionsofareallaserbeampropagatingina quadraticindexwaveguide. 34

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CHAPTERV WAVEOPTICSTREATMENTOFLIGHTPROPAGATIONIN QUADRATICINDEXWAVEGUIDES DerivationofGaussianBeamsanditsProperties StartingwiththeHelmholtzscalarwaveequation,andtaking n x =1 r 2 + k 2 =0 where isthecomplexeldamplitudeforanypolarizationoftheelectriceld,and k = 2 = = !=v = p isthevacuumwavenumber.Taking inthe z directionas = u x;y;z e )]TJ/F24 7.9701 Tf 6.587 0 Td [(ikz thederivativesare r = r ue )]TJ/F24 7.9701 Tf 6.587 0 Td [(ikz )]TJ/F23 10.9091 Tf 10.909 0 Td [(ikue )]TJ/F24 7.9701 Tf 6.587 0 Td [(ikz r 2 = r 2 ue )]TJ/F24 7.9701 Tf 6.587 0 Td [(ikz )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ik r ue )]TJ/F24 7.9701 Tf 6.587 0 Td [(ikz )]TJ/F23 10.9091 Tf 10.909 0 Td [(k 2 ue )]TJ/F24 7.9701 Tf 6.587 0 Td [(ikz andsubstitutingbackintothe,wehave )]TJ/F26 10.9091 Tf 5 -8.836 Td [(r 2 u )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 ik r u e )]TJ/F24 7.9701 Tf 6.586 0 Td [(ikz =0 : Atthispointweimposethe paraxialapproximation inwaveanalysis.Therstapproximationisthatthevariationofpropagationisslowonthescaleofthewavelength Mathematicallythisisexpressedas @ 2 u @z 2 2 k @u @z : Thenextapproximationisthatthevariationofpropagationisslowinthetransversedirection @ 2 u @z 2 @ 2 u @x 2 @ 2 u @y 2 : 35

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theHelmholtzequationthembecomes @ 2 u @x 2 + @ 2 u @y 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 ik @u @z =0 : Equationiscalledthe paraxialwaveequation Tosolvetheparaxialwaveequation,weassumethetrialsolutionhastheform[11] u = Aexp )]TJ/F23 10.9091 Tf 8.485 0 Td [(i P z + r 2 q z where r 2 = x 2 + y 2 .Wecanrewrite u using k =2 = ,and A =1 [9],[11], u = exp )]TJ/F23 10.9091 Tf 8.484 0 Td [(i P z + k x 2 + y 2 2 q z : Takingtheappropriatederivatives @u @x = )]TJ/F23 10.9091 Tf 11.323 7.38 Td [(ikx q z u @ 2 u @x 2 = )]TJ/F23 10.9091 Tf 11.322 7.38 Td [(ikx q z )]TJ/F23 10.9091 Tf 11.322 7.38 Td [(ikx q z u )]TJ/F23 10.9091 Tf 16.864 7.38 Td [(ik q z u = )]TJ/F23 10.9091 Tf 10.835 7.38 Td [(k 2 x 2 q 2 z u )]TJ/F23 10.9091 Tf 16.864 7.38 Td [(ik q z u andsimilarlyfor @ 2 u=@y 2 @ 2 u @y 2 = )]TJ/F23 10.9091 Tf 11.083 7.38 Td [(k 2 y 2 q 2 z u )]TJ/F23 10.9091 Tf 16.864 7.38 Td [(ik q z u and @u @z = )]TJ/F23 10.9091 Tf 8.485 0 Td [(i P 0 z )]TJ/F23 10.9091 Tf 16.13 7.38 Td [(kr 2 q 2 z q 0 z u theHelmholtzequationthenbecomes )]TJ/F23 10.9091 Tf 10.835 7.38 Td [(k 2 x 2 q 2 z )]TJ/F23 10.9091 Tf 16.864 7.38 Td [(ik q z )]TJ/F23 10.9091 Tf 13.506 7.38 Td [(k 2 y 2 q 2 z )]TJ/F23 10.9091 Tf 10.909 0 Td [(i k q z )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ik )]TJ/F23 10.9091 Tf 8.485 0 Td [(iP 0 z + i kr 2 2 q 2 z q 0 z =0 )]TJ/F23 10.9091 Tf 11.34 7.38 Td [(k 2 r 2 q 2 z )]TJ/F23 10.9091 Tf 10.909 0 Td [(q 0 z )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 k i q z + P 0 z =0 : 36

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Eachtermmustvanishindependently,so dq z dz =1 and dP z dz = )]TJ/F23 10.9091 Tf 17.451 7.38 Td [(i q z : Thesolutiontois q z = q o + z Nextweintroducethe complexbeamparameter ,whichrelatestworealbeamparameters R and w [9] 1 q = 1 R )]TJ/F23 10.9091 Tf 10.909 0 Td [(i w 2 : Whenissubstitutedintothetrialsolution,thephysicalmeaningoftheparameters R and w isseen.Theparameter R istheradiusofcurvatureofthewavefrontat z ,and w isthebeamradius,whichisdenedasthedistancefromthemaximumamplitudeofthe eld,tothepointwhereitdecreasestothevalue 1 =e .The spotsize ofthebeamis 2 w ,and isalsocalledthe beamdiameter .Theminimumdiameterofthebeamis 2 w 0 ,andiscalled the beamwaist .Thecomplexbeamparameteratthewaistispurelyimaginary[9] q 0 = i w 2 0 : sothatthesolutiontothenbecomes q = i w 2 0 + z: Substitutinginto,wehave 1 i w 2 0 + z = 1 R )]TJ/F23 10.9091 Tf 10.909 0 Td [(i w 2 )]TJ/F23 10.9091 Tf 8.485 0 Td [(i w 2 0 + z 2 w 4 0 2 + z 2 = 1 R )]TJ/F23 10.9091 Tf 10.909 0 Td [(i w 2 37

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)]TJ/F23 10.9091 Tf 8.485 0 Td [(i w 2 0 + z = 1 R 2 w 4 0 2 + z 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(i w 2 2 w 4 0 2 + z 2 Equatingtheimaginarypartsof,weobtaintheexpressionforthesquareofthebeam radius )]TJ/F23 10.9091 Tf 8.485 0 Td [(i w 2 0 = )]TJ/F23 10.9091 Tf 8.485 0 Td [(i w 2 2 w 4 0 2 + z 2 w 2 z = 2 2 w 2 0 2 w 4 0 2 + z 2 w 2 z = w 2 0 1+ z w 2 0 2 # : Equatingtherealpartsof,weobtaintheexpressionfortheradiusofcurvature z = 1 R 2 w 4 0 2 + z 2 R z = 1 z 2 w 4 0 2 + z 2 R z = z w 2 0 z 2 +1 # : Thebeamcontourof w z isahyperbola,withasymptotesthatmakeananglewiththe opticalaxis z = w 0 : Dividingby,weobtaintherelation z w 2 0 = w 2 R whichcanbeusedtoexpress w 0 and z intermsof w and R w 2 0 = w 2 w 2 R 2 +1 z = R )]TJ/F24 7.9701 Tf 12.853 -4.541 Td [(R w 2 +1 : 38

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Nextwesolvethedierentialequationfor P z ,equation dP z dz = )]TJ/F23 10.9091 Tf 17.451 7.38 Td [(i q z = )]TJ/F23 10.9091 Tf 28.114 7.38 Td [(i i w 2 0 + z andintegratingyields iP z =ln 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(i z w 2 0 iP z =ln s 1+ z w 2 0 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(i tan )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 z w 2 0 : Therealpartof P z isthephaseshiftdierencebetweentheGaussianbeamandanideal planewave.Theimaginarypartrepresentstheexpectedintensitydecreaseofthebeamdue totheexpansionofthebeam.ThetrialsolutionoftheHelmholtzwaveequationis now = w 0 w exp )]TJ/F23 10.9091 Tf 8.485 0 Td [(i kz )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F23 10.9091 Tf 10.909 0 Td [(r 2 1 w 2 + ik 2 R where =tan )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F23 10.9091 Tf 5 -8.837 Td [(z=w 2 0 HermitePolynomials InCartesiancoordinates,theparaxialwaveequationcanbesatisedby u x;y;z = g x w h y w exp )]TJ/F23 10.9091 Tf 8.484 0 Td [(i P + k 2 q x 2 + y 2 were g isafunctionof x and z ,and h isafunctionof y and z .Lookingatonlythetransverse dimension x ,is u x;z = g x w exp )]TJ/F23 10.9091 Tf 8.484 0 Td [(i P + kx 2 2 q Usingintheparaxialwaveequation,weobtainHermite'sdierentialequation forthefunction g d 2 H n dx 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x dH n dx +2 nH n =0 : where H n arethesolutionsto,aretheHermitepolynomialswhere n denesitsorder. TheHermitepolynomialsareacompletesetoforthogonalfunctionsintheinterval 39

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[ ; 1 ] ,andhavetheorthogonalityrelation Z 1 dxe )]TJ/F24 7.9701 Tf 6.586 0 Td [(x 2 H n x H m x =2 n n p nm : Thisrelationcanbeusedtodeneasetofpolynomials n x n x = 1 p 2 n n p e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(x 2 2 H n x ;n =0 ; 1 ; 2 ;::: sothattheorthogonalityrelationbecomes Z 1 dx n x m x = nm : Sincethisdenesacompleteset,thenanywell-behavedfunctionintheinterval [ ; 1 ] canbeexpandedintermsof n x f x = 1 X n =0 C n n x wherethecoecientscanbefoundby C n = Z 1 dx 0 f x 0 n x 0 = constant 6 =0 : HermitePolynomialExpansionofGaussianBeams AGaussianfunctionisdenedas f x = e )]TJ/F25 5.9776 Tf 9.772 3.258 Td [(x 2 2 2 = e )]TJ/F25 5.9776 Tf 10.22 3.258 Td [(b 2 2 2 x 2 b 2 = e )]TJ/F25 5.9776 Tf 10.22 3.258 Td [(b 2 2 2 x 0 2 whichwecanexpandintermsofHermitepolynomials[4]equation4.51ontheleftboundary z =0 as f x = 1 X n =0 C n n x : 40

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Using,theexpansionbecomes f x = 1 X n =0 Z 1 dx 0 f x 0 n x 0 n x andthensubstituting,wehave f x = 1 X n =0 Z 1 dx 0 f x 0 p 2 n n p e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(x 0 2 2 H n x 0 # 1 p 2 n n p e )]TJ/F25 5.9776 Tf 5.756 0 Td [(x 2 2 H n x f x = 1 X n =0 1 2 n n p Z 1 dx 0 f x 0 e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(x 0 2 2 H n x 0 e )]TJ/F25 5.9776 Tf 5.756 0 Td [(x 2 2 H n x Justlookingattheintegralinthesquarebracket Z 1 dx 0 f x 0 e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(b 2 x 2 2 2 e )]TJ/F25 5.9776 Tf 7.782 3.258 Td [(x 0 2 2 H n x 0 = Z 1 dx 0 e )]TJ/F22 5.9776 Tf 7.782 3.258 Td [(1 2 b 2 2 +1 x 2 H n x 0 : SincewearemodelingaGaussianbeam,whichbysymmetryisanevenfunction,weonly usetheevenordersoftheHermitepolynomials, n =2 m .Usingthesubstitution 2 = )]TJ/F21 7.9701 Tf 9.681 4.295 Td [(1 2 b 2 2 +1 fortheexponentialontherightsideof,andusingtheresultfrom Bayin[4]problem4.10,becomes Z 1 dx 0 f x 0 e )]TJ/F25 5.9776 Tf 7.782 3.258 Td [(b 2 x 2 2 2 e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(x 0 2 2 H n x 0 = m m p 1 )]TJ/F23 10.9091 Tf 10.91 0 Td [( 2 2 m : TheexpansionoftheGaussianfunctionisnow f x = 1 X n =0 1 2 2 m m p m m p 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 2 m e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(x 2 2 H 2 m x f x = 1 X n =0 1 4 m m 1 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 2 m e )]TJ/F25 5.9776 Tf 7.782 3.259 Td [(x 2 2 H 2 m x 41

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DerivationofAnalyticSolutionsforFieldIntensitiesinQuadraticIndex Waveguides Theanalyticsolutionformodesinaplanarquadraticindexwaveguideisderivedhere fromthereducedscalarwaveequation,ortheHelmholtzequation r 2 + n 2 x k 2 0 =0 where n 2 x isdenedby,and k 0 =2 = isthevacuumwavenumber.Wavesare assumedtohaveaharmonictimedependence e i!t .TheHelmholtzequationissolved usingthetrialsolution = f x e )]TJ/F24 7.9701 Tf 6.586 0 Td [(iz whereweareonlyconsideringonetransversedimension x .Asolutionoftheformis amodeofthequadraticindexwaveguide,andthefunctions f and areundeterminedat thispoint.Takingtheappropriatederivativesof @ @z = )]TJ/F23 10.9091 Tf 8.485 0 Td [(f i e )]TJ/F24 7.9701 Tf 6.587 0 Td [(iz @ 2 @z 2 = )]TJ/F23 10.9091 Tf 8.485 0 Td [(f 2 e )]TJ/F24 7.9701 Tf 6.586 0 Td [(iz andthesubstitutingthesederivativeandtheindexofrefractioninto,wehave d 2 f dx 2 + n 2 o k 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(k 2 o n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 a 2 o f =0 : f 00 x e )]TJ/F24 7.9701 Tf 6.587 0 Td [(iz )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 f x e iz + k 2 0 n 2 0 )]TJ/F29 10.9091 Tf 10.909 8.836 Td [()]TJ/F23 10.9091 Tf 5 -8.836 Td [(n 2 0 )]TJ/F23 10.9091 Tf 10.91 0 Td [(n 2 1 x 2 a 2 0 f x e )]TJ/F24 7.9701 Tf 6.587 0 Td [(iz =0 : Theexponentialtermsvanish,andwehave f 00 x )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 f x + k 2 0 n 2 0 )]TJ/F29 10.9091 Tf 10.909 8.836 Td [()]TJ/F23 10.9091 Tf 5 -8.836 Td [(n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 x 2 a 2 0 f x =0 ThisequationcanbetransformedintotheformofthequantumharmonicoscillatorAp42

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pendixFbymakingthefollowingsubstitution = p k o n 2 0 )]TJ/F23 10.9091 Tf 10.91 0 Td [(n 2 1 a 2 0 1 = 4 x: Solvingfor x x = p k o n 2 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [(n 2 1 a 2 0 1 = 4 : Usingthechainrule df dx = df d d dx wehave df dx = df d p k o n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 0 1 = 4 d 2 f dx 2 = d d df d p k o n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 0 1 = 4 # d dx d 2 f dx 2 = d 2 f d 2 k 0 n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 0 1 = 2 Equationbecomes d 2 f d 2 k 0 n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 0 1 = 2 + )]TJ/F23 10.9091 Tf 5 -8.836 Td [(n 2 0 k 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 f )]TJ/F23 10.9091 Tf 10.909 0 Td [(k 2 0 n 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 a 2 0 2 k 0 n 2 0 )]TJ/F24 7.9701 Tf 6.586 0 Td [(n 2 1 a 2 0 1 = 2 f =0 d 2 f d 2 + n 2 0 k 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 k 0 n 2 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [(n 2 1 a 2 0 1 = 2 f )]TJ/F23 10.9091 Tf 12.104 11.345 Td [(k 0 n 2 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [(n 2 1 a 2 0 1 = 2 k 0 n 2 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [(n 2 1 a 2 0 1 = 2 2 f =0 : Dening = n 2 0 k 2 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 k 0 n 2 0 )]TJ/F24 7.9701 Tf 6.586 0 Td [(n 2 1 a 2 0 1 = 2 wehave d 2 f d 2 + )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 f =0 whichisthesamedierentialequationoftheformofthequantumharmonicoscillator.The modesofthesquarelawmediamustbeguidedneartheaxisofthewaveguide,whichmeans 43

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werequirethefunction f tovanishas x !1 .Thesamerequirementisimposedforthe wavefunctionsofthequantumharmonicoscillator.Boundsolutionsaremetifweimpose thecondition =2 p +1 : Thiscondition,togetherwith3,andwithassigningthevariable b = s a o k o n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 1 2 determinethepossiblevaluesofthepropagationconstant 2 p +1= n 2 o k 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 a o k o n 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [(n 2 1 1 2 2 p +1= n 2 o k 2 o )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 b 2 p = r n 2 o k 2 o )]TJ/F15 10.9091 Tf 12.104 7.38 Td [( p +1 b 2 : Returningto,thegeneralsolutionis f = H p e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 takingtheappropriatederivatives df d = dH p d e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 + H p )]TJ/F23 10.9091 Tf 8.485 0 Td [( e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 d 2 f d 2 = d 2 H p d 2 e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 + dH p d )]TJ/F23 10.9091 Tf 8.485 0 Td [( e 0 2 = 2 + dH p d )]TJ/F23 10.9091 Tf 8.485 0 Td [( e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 + H p n )]TJ/F23 10.9091 Tf 8.485 0 Td [( e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(e 2 = 2 o d 2 f d 2 = d 2 H p d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 dH p d + 2 H p e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(H p e )]TJ/F24 7.9701 Tf 6.587 0 Td [(= 2 : Usingthe,becomes d 2 H p d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 dH p d +2 pH p =0 44

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whichofthesameformof,Hermite'sDierentialequation.Thus,themodesolutions forthequadraticindexwaveguidearetheHermiteGaussianfunctions,andthesemodes formacompletesetoforthogonalfunctions.Wecanthendescribeeverywaveasaseries expansionintermsofthesemodes. From x 2 = 2 k o n 2 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [(n 2 1 a 2 0 1 = 2 x 2 = 2 k o n 2 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [(n 2 1 a 2 0 1 = 2 x 2 = 2 w 2 2 andsolvingfor 2 2 = 2 x 2 w 2 and w 2 is w 2 = 2 k 0 n 2 0 )]TJ/F24 7.9701 Tf 6.586 0 Td [(n 2 1 a 2 0 1 = 2 Using,thegeneralsolutionto14canbeexpressedas f = H p x e )]TJ/F24 7.9701 Tf 6.587 0 Td [(x 2 = 2 andthiscanbeexpandedinaseriesexpansionfollowingthesameprocedurethatderived .Thetrialsolutionbecomes x;z = N X m = even 1 r 1 2 b 2 2 +1 4 m m 1 )]TJ/F23 10.9091 Tf 10.909 0 Td [( 2 2 m H m x cos m z e )]TJ/F24 7.9701 Tf 6.587 0 Td [(x 2 = 2 whichgivestheeldsintensityproles,ormodes,ofourplanarquadraticindexwaveguide system.Figure5istheOctavesimulationusing,wherethewaveguidelengthis 32 m andthewaveguidewidthis 10 m 45

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Figure6:Simulationofeldintensityprolefromtheanalyticsolution. 46

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CHAPTERVI NUMERICALSOLUTIONSTOMAXWELL'SEQUATIONS TheFiniteDierenceTimeDomainMethod ThenitedierencetimedomainFDTDmethodsolvesMaxwell'scurlequationsFaraday'sLawandAmpere'sLawbydiscretizingspaceandtimeintoniterectangular grids.Itusescentraldierenceapproximations,whichhas2ndorderaccuracy,to'propagate'equationsandintime[14].Toobtainthecentraldierenceequations,rst considertheTaylorseriesexpansionofafunction f z expandedaboutthepoint z 0 with aninitialoset = 2 f z 0 + = 2= f z + 2 df dz j z = z 0 + 2 2 d 2 f dz 2 j z = z 0 + 1 3 2 3 d 3 f dz 3 j z = z 0 + f z 0 )]TJ/F23 10.9091 Tf 10.91 0 Td [(= 2= f z )]TJ/F23 10.9091 Tf 12.201 7.38 Td [( 2 df dz j z = z 0 + 2 2 d 2 f dz 2 j z = z 0 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 3 2 3 d 3 f dz 3 j z = z 0 + : Subtractingfrom f z 0 + = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(f z 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(= 2= df dz j z = z 0 + 2 3! 2 3 d 3 f dz 3 + thendividingbythestepsize wehave f z 0 + = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(f z 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(= 2 = f 0 z 0 + 1 3! 2 2 f 000 z 0 + ::: wheretheprimeindicatedierentiationwithrespectto z .Rearranging,wesolvefor f 0 z df dz j z = z 0 = f z 0 + = 2 )]TJ/F23 10.9091 Tf 10.91 0 Td [(f z 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(= 2 + O 2 : andnotethatthattheterm O 2 containsallthesecond-orderandhigherterms.If wechooseastepsize thatissucientlysmallenoughsothatalltermsin O 2 are negligiblysmallerthanthersttermontherighthandside,thenwecanmakeareasonable 47

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approximationfor df=dz j z = z 0 bydropping O 2 .Wehaveforourapproximation df dz j z = z 0 f z 0 + = 2 )]TJ/F23 10.9091 Tf 10.91 0 Td [(f z 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(= 2 + O 2 : Thisisthecentraldierenceapproximationforthederivativeofthefunctionat z 0 .Butthe functionisnotsampledat z 0 ,butinsteadissampledattheneighboringpoints z 0 + = 2 and z 0 )]TJ/F23 10.9091 Tf 10.909 0 Td [(= 2 Forthesimulationsinthisthesis,weconsiderthecaseforawavetravelinginthezdirectionpolarizedinthe x -direction.Faraday'slawisthen )]TJ/F23 10.9091 Tf 8.485 0 Td [( @ ~ H @t = r ~ E =^ e y @E x dz )]TJ/F23 10.9091 Tf 8.485 0 Td [( @H y @t = @E x dz andAmpere'slawis @ ~ E @t = r ~ B = )]TJ/F15 10.9091 Tf 8.903 0 Td [(^ e x @H y dz @E x @t = @H y dz : Wenowintroducenotationtoidentifythespatialandtemporalstepoftheelds.For E x wewilluse E x z;t = E p x q ,andfor H y wewilluse H y z;t = H p y q wherepindicatesthe temporalstep,andqindicatesthespatialstep. Tondthevalueof H y ,athenextone-halfspatialstep q 1 = 2 z;p t weuseFaraday's law )]TJ/F23 10.9091 Tf 8.485 0 Td [( H p +1 = 2 y q +1 = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(H p )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = 2 y q +1 = 2 t = E p x q +1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(E p x q z : Solvingfor H p +1 = 2 y q +1 = 2 ,weobtaintheupdateequationfor H y H p +1 = 2 y q +1 = 2= H p )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = 2 y q +1 = 2 )]TJ/F15 10.9091 Tf 16.198 7.38 Td [( t z [ E p x q +1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(E p x q ] : Thecorrespondingupdateequationfor E x isobtainfromAmpere'slawusingthesame 48

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procedure.Thecentral-dierenceapproximationis E p +1 x q )]TJ/F23 10.9091 Tf 10.909 0 Td [(E p x q t = H +1 = 2 y q +1 = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(H p +1 = 2 y q )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 z : Solvingfor E p +1 x q ,wehave E p +1 x q = t z h H p +1 = 2 y q +1 = 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(H p +1 = 2 y q )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 i Theseequationscalculatetheelectricandmagneticeldsin 1 = 2 increments,knownasthe leapfrogmethod ,inwhichtheelectriceldiscalculatedinaunitgridvolume,thenthe magneticeldiscalculatedinthesameunitgridvolume,butatthenexttimestep.The FDTDrstcalculatestheelectriceldfortheentirespaceatthersttimestep,thenthe magneticeldsfortheentirespaceinthenexttimestep.Inotherwords,itsolvestheinitial valueproblemwheretheeldsandcurrentsarezerofor t< 0 ,thennon-zerovaluesevolve inresponsetosomecurrents,orsources. MEEP MEEPisaFDTDsolverthatsimulatesMaxwell'scurlequations.MEEPisusedto calculateeldintensityprolesincomputationalelectromagneticproblems[12].Itusesthe Yeegrid discretizationtodividespaceandtimeintoregularniterectangulargrids.Inthis grid,eacheldcomponentissampledatdierencespatiallocationsosetbyahalf-pixel usingthecenter-dierencecalculationofspaceandtime.A2DillustrationoftheYeegrid isshowninFigure7.Thetimederivativeofthevectoreld H producesanelectriceld arounditaspredictedbyFaraday'sLaw.AndaspredictedbyAmpere'sLaw,thetime derivativeoftheelectriceldproducesthecurlofthevectoreld H aroundit.InMEEP, theelectriceldandvectoreld H areinitiallyzeroat t =0 .Theeldsarecalculatedas timeincreases,andtheeldsarefoundtobenon-zerobecauseoftheirinteractionwitha source,andinoursimulation,aGaussianlaserbeam. 49

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Figure7:Yeegrid,in2D. MEEPusesthescriptinglanguageScheme.SchemediersinsyntaxfromMatLaband Octaveinuniqueways.Forexample,inOctavetoperformtheoperation4minus5,one wouldimplementthisas 4 )]TJ/F16 10.9091 Tf 9.167 0 Td [(5 butinScheme,thisisimplementedas )]TJ/F16 10.9091 Tf 15.614 0 Td [(45. AGaussiandistribution f y = y 2 2 2 canbeimplementedinMEEPas exp/ )]TJ/F35 5.9776 Tf 5.595 0 Td [(1vector3 )]TJ/F35 5.9776 Tf 5.338 0 Td [(yspointvector3 )]TJ/F35 5.9776 Tf 5.338 0 Td [(yspoint 2sourcesigmasourcesigma where isthevarianceorspreadofthedistribution. Userscanwritescriptstodeneparameters,computationalcellsizes,geometries,and sourcesofthesimulation.Accesstothesourcecodeallowsexibilitytotheusernotseen inGUI-basedorCAD-basedFDTDpackages. MEEPusesdimensionlessunits, o = o = c =1 .Thisemphasizesthescaleinvariance ofMaxwell'sequations,andthefactthatmostmeaningfulquantitiesarealmostalways dimensionlessratiossuchasscatteredpoweroverincidentpower.MEEPallowstheuser tochoosealengthscale a ,thenalldistancesaregiveninunitsof a .Alltimesareinunits a=c 50

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andallfrequenciesareinunits c=a ,orequivalently a= ,where isthevacuumwavelength. RelativepermitivityandpermeabilityconstantscanbedenedinMEEP. BoundaryconditionscanbedenedandMEEP,andintheMEEPsimulationusedinthis thesisthe perfectlymatchedlayers PMLboundaryconditionisused.Thissimulatesopen boundaryconditionsbyabsorbingallwavesincidentonit,anddoesnotallowreections. Strictlyspeaking,thisisnota'boundarycondition'.Rather,itisactitiousabsorbing material.ThePMLisgivensomenitethicknessthatcausesthectitiousmaterialto graduallyturnon.Thisisbecauseinanactualdiscretizedsystem,thePMLmaterialhas somesmallreections. MEEPSimulation TosimulatewavepropagationinquadraticindexwaveguidesusingMEEP,werstdene thecomputationalcell,whichcanalsobeinterpretedasthedimensionsofourwaveguide. Thenwedenetherefractiveindicesalongtheopticalaxisandboundary,anddenea continuousGaussianbeamsource.Dening x asthetransversedimension,aGaussian distributionis f x = e )]TJ/F24 7.9701 Tf 6.587 0 Td [(x 2 = 2 2 where isthevarianceofthedistribution,andinthissimulationisequalto1. Thefrequencywasdenedtoequal =1 : 5 .Ifalengthscaleof a =1 m isused,thenthe wavelengthis = a= =1 m= 1 : 5=0 : 667 m or 667 nm .Ahalf-waveguidewidth a o was denedtoequal 5 m ,andthelong-axiswaveguidedimensionis 32 m .Thesedimensions werechosentomatchtheGaussianraybundlesimulation. TheMEEPsimulationofaGaussianbeaminaplanarquadraticindexwaveguidewas performedusingPMLboundary's,andaninitialbeamwaistsizeof 10 m .Theindexprole isoftheformofequation. 51

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Figure8:IndexprolesimulationinMEEP. Thesimulationyieldsthefollowingintensityprole Figure9:MEEPsimulationofaGaussianbeaminaquadraticindexwaveguide. Thecorrespondinggrayscaleimageis 52

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Figure10:GrayscaleimageofMEEPsimulationforaGaussianbeaminaquadraticindex waveguide. ComparisonofMEEPandAnalyticSimulations ThissectionsshowsplotsthatcomparetheanalyticsolutiontotheHelmholtzwave equationtothesimulationresultsobtainfromMEEP.Thesearegraphicalcomparisons comparingthegrayscalepixelintensitiesofthetwosimulation.Twoplotsareshown,one comparingtheMEEPsimulationwiththeanalyticsolutionupto2ndorder,andtheother usingtheanalyticsolutionup20thorder.Currently,resultsareseentoagreeonlythrough therstcycle.Afterthat,thetwosimulationsareoutofphase. 53

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Figure11:ComparisonofMEEPsimulationwithanalyticsolutionexpansionupto2nd order. Figure12:ComparisonofMEEPsimulationwithanalyticsolutionexpansionupto20th order. 54

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CHAPTERVII COMPARISONOFGAUSSIANRAYBUNDLESWITHTHENUMERICAL SOLUTIONSTOMAXWELL'SEQUATIONSANDTHEANALYTIC SOLUTIONSOFTHESCALARWAVEEQUATION HeretheanalyticsolutiontoHamilton'sequationsforaGaussianbundleofraytrajectorieswillbegraphicallycomparedtotheanalyticsolutiontotheHelmholtzwaveequation. Insteadofusingthesignedeldintensitysolution,wewanttousethetime-averageeld intensity.Tondthetime-averageeldintensity I = e )]TJ/F24 7.9701 Tf 6.586 0 Td [(x 2 = 2 T Z T 0 x;z x;z dt Goingoutto2ndorder I = e )]TJ/F24 7.9701 Tf 6.586 0 Td [(x 2 = 2 T Z T 0 C 0 H 0 e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i 0 z + C 2 H 2 e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i 2 z e i!t C 0 H 0 e i 0 z + C 2 H 2 e i 2 z e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i!t dt I = e )]TJ/F24 7.9701 Tf 6.586 0 Td [(x 2 = 2 T C 2 0 H 2 0 e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i 0 )]TJ/F24 7.9701 Tf 6.587 0 Td [( 0 z +2 C 0 C 2 H 0 H 2 e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i 0 )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 z + C 2 2 H 2 2 e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 z Z T 0 e )]TJ/F21 7.9701 Tf 6.586 0 Td [( )]TJ/F24 7.9701 Tf 6.587 0 Td [(! t dt: Thetimedependentintegralequals T ,andtheexponentialfactorsofthesquaredterms vanish,andwe'releftwith I = e )]TJ/F24 7.9701 Tf 6.587 0 Td [(x 2 = 2 n C 2 0 H 2 0 +2 C 0 C 2 H 0 H 2 e )]TJ/F24 7.9701 Tf 6.587 0 Td [(i 0 )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 z + C 2 2 H 2 2 o : Thegeneralizedexpressionforthetime-averageeldintensityforevenorders n is I = e )]TJ/F24 7.9701 Tf 6.587 0 Td [(x 2 = 2 N X n =0 2 n +2 N X n =0 N X m = n +2 n m cos n )]TJ/F23 10.9091 Tf 10.909 0 Td [( m z : ThegeneralizedeldintensitywassimulatedinOctave.Figure13isthesimulationup to2ndorder,andFigure14isthesimulationupto20thorder.Weseethattakingthe expansiontohigherordersincreasestheaccuracyofthesimulation,producingamore real lookingbeamthatwewouldexpecttoseeinexperiment 55

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Figure13:Time-averageeldintensityfortheanalyticsolutiontotheHelmholtzequation forordern=2. Figure14:Time-averageeldintensityfortheanalyticsolutiontotheHelmholtzequation forordern=20. 56

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NextwewishtocomparetheGaussianraybundlewiththeaverageeldintensity.Figure 15isaplotoverlayingtheGaussianraybundlestrajectoriesovertheaverageeldintensity plot. Figure15:ComparisonofanalyticsolutionofGaussianraybundletrajectorieswithtimeaverageeldintensityofanalyticsolutionsimulationupto20thorder. 57

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CHAPTERVIII FUTUREWORK Thischapterdescribesfutureworkontheanalysispresentedinthisthesis,andproblems thatwilluseallofthetheoryandsimulationtechniquesusedinthisthesis.It'ssuggested thatwecanmakepredictionsabouttheeldintensityfromtheGaussianraybundleanalysis. Anexactagreementwasn'tseengraphically,butfutureworkwouldincludequantifyinghow thefocusregionsoftheeldintensitiescorrespondwiththenodescreatedbytheGaussian raybundletrajectoriesontheopticalaxis. SimulatingaGaussianbeaminplanarquadraticindexwaveguideswasperformedby expandingaGaussianfunctionintermsofHermitepolynomials.Butitisalsosuggestedby Macuse[11]thatthereexistsaclosed-formsolutionforthemodesinasquarelawmedium. Itsproposedthatthisclosedformsolutioncanbeveriedbythesamemethodsusedin thisthesis,andacomparisontotheseriesexpansioncanbemadetorevealanypotential disagreementsbetweenthetwosolutions. Thecomparisonsbetweensolutionofscalarwaveequationandthenumericalsolutions toMaxwell'sequationsfromMEEPalsorequiredmoredetailinvestigation.Theanalyticsolutiontothescalarwaveequationwasobtainbyimposingparaxialapproximations,whereas thesimulationsperformedinMEEPisafullvectorialwavesimulationofMaxwell'sEquations.Howdothesedier,andhowcanthesedierencesbequantied?Ifthesedierences areunderstood,wecanthenmakeaccuratepredictionofraypropagationfromtheeld intensityproles. Thetheoreticaldevelopmentandsimulationspresentedinthisthesiscanalsobeapplied tosystemsinvolvingelectromagneticlightpropagationwithspatiallyvaryingmediumwhere ananalyticsolutiondoesnotexist.Onesuchsystemisaplanarquadraticindexwaveguide withperiodicaxialperturbations.Thenumericalsimulations,veriedbycomparisonwith analyticsolutions,canbeusedtosimulatelightpropagationinthissystem,andtodetermine theconditionsthatpredictchaoticraybehavior. 58

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REFERENCES [1]SSAbdullaevandGMZaslavski.Classicalnonlineardynamicsandchaosofrays inproblemsofwavepropagationininhomogeneousmedia. SovietPhysicsUspekhi 34:645,1991. [2]M.Asadi-ZeydabadiandR.P.Tagg.Raychaosinmediawithtransversalandaxial variationofindexofrefraction:Linearstability. [3]M.Asadi-ZeydabadiandR.P.Tagg.Parametricinstabilityinwaveguideswithaxialmodulationoftheindexofrefraction. PhDDissertation,UniversityofColorado Boulder ,2004. [4]SSBayin.Mathematicalmethodsinscienceandengineering.2006. JohnWiley&Sons, NewYork ,pageChapter4. [5]JosselinGarnier.Lightpropagationinsquarelawmediawithrandomimperfections. WaveMotion ,31:1,2000. [6]AKGhatakandEGSauter.Theharmonicoscillatorproblemandtheparabolicindex opticalwaveguide:I.classicalandrayopticanalysis. EuropeanJournalofPhysics 10:136,1989. [7]DavidJereyGrithsandEdwardGHarris. Introductiontoquantummechanics volume2.PrenticeHallNewJersey,1995. [8]MHashimoto.Geometricalopticsofguidedwavesinwaveguides. ProgressInElectromagneticsResearch ,13:115,1996. [9]HerwigKogelnik.Onthepropagationofgaussianbeamsoflightthroughlenslikemedia includingthosewithalossorgainvariation. AppliedOptics ,4:1562,1965. [10]SLonghi,GDellaValle,andDJanner.Rayandwaveinstabilitiesintwistedgradedindexopticalbers. PhysicalReviewE ,69:056608,2004. 59

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[11]DietrichMarcuse. LightTransmissionOptics .1982. [12]ArdavanFOskooi,DavidRoundy,MihaiIbanescu,PeterBermel,JohnDJoannopoulos,andStevenGJohnson.Meep:Aexiblefree-softwarepackageforelectromagnetic simulationsbythefdtdmethod. ComputerPhysicsCommunications ,181:687, 2010. [13]JSShirk,MSandrock,DScribner,EFleet,RStroman,EBaer,andAHiltner. Biomimeticgradientindexgrinlenses.Technicalreport,DTICDocument,2006. [14]ATaoveandKRUmashankar.Thenite-dierencetime-domainfd-tdmethodfor electromagneticscatteringandinteractionproblems. JournalofElectromagneticWaves andApplications ,1:243,1987. [15]QiWu,JeremiahPTurpin,andDouglasHWerner.Integratedphotonicsystemsbased ontransformationopticsenabledgradientindexdevices. Light:Science&Applications 1:e38,2012. 60

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APPENDIXA Code:AnalyticSolutionforRayTrajectories %%%%%%%%%%EXACT_SOLN_MAIN.M %usesfunction'qdinxexact.m' % % n0=4;%centerlineindexopticalaxis n1=1;%indexattheboundary a0=5;%waveguidedistancefromcenter z0=0; zf=32;%waveguidelength theta0_degs=0;%initialangle,indegrees xstart=1.25;%initialtransverseposition whilexstart>= )]TJ/F35 5.9776 Tf 5.833 0 Td [(1.25 theta0_rads=theta0_degs 3.14159/180;%convertinitialangletoradians theta0=theta0_rads; [x_z]=qdinxexactn0,n1,a0,z0,zf,xstart; z=linspacez0,zf,2000; xlabel'z'; ylabel'xz'; plotz,x_z,'r )]TJ/F35 5.9776 Tf 7.123 0 Td [(','LineWidth',2; legendexact=legend'ExactSoln'; holdon xstart=xstart )]TJ/F35 5.9776 Tf 6.531 0 Td [(.25; end function[x_z]=qdinxexactn0,n1,a0,z0,zf,xstart z=linspacez0,zf,2000; lambda=2 pi a0 sqrtn0^2/n0^2 )]TJ/F35 5.9776 Tf 5.976 0 Td [(n1^2 )]TJ/F35 5.9776 Tf 6.088 0 Td [(xstart/a0^2; x_z=xstart cos2 pi z/lambda;%generalequationforraytrajectories end 61

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APPENDIXB Code:Hamilton'sEquationsSolvedUsing ode45 %%%%%%%%%%%%%%HAMILTONS_EQNS_RAYPATHS.M %Usesode45tosolveHamilton'sequation %todetermineraytrajectoriesinaplanar %quadraticindexwaveguide. % % % n0=4;%centerlineindexopticalaxis n1=1;%indexattheboundary a0=5;%waveguidedistancefromcenter z0=0; zf=32;%waveguidelength theta0_degs=0;%initialangle,indegrees theta0_rads=theta0_degs 3.14159/180;%convertinitialangletoradians theta0=theta0_rads; p0=n0 sintheta0; options=[n0,n1,a0,theta0]; xinit=1.25;%initialtransverseposition whilexinit>= )]TJ/F35 5.9776 Tf 5.825 0 Td [(1.25 x0=[xinit,p0]; t=z0:0.02:zf; [z,x]=ode45@raypath,t,x0,options; xx=x:,1; p=x:,2; plotz,xx,'b )]TJ/F35 5.9776 Tf 7.122 0 Td [(','LineWidth',2 legend2=legend'HamiltonsEquationssolvedbyode45'; holdon xinit=xinit )]TJ/F35 5.9776 Tf 6.525 0 Td [(0.25; end functiondy=raypatht,x0,options n0=options1; n1=options2; a0=options3; theta0=options4; ifisrealx0 xx=x01; p=x02; else return end n=sqrtn0^2 )]TJ/F35 5.9776 Tf 5.892 0 Td [(n0^2 )]TJ/F35 5.9776 Tf 5.976 0 Td [(n1^2 xx^2/a0^2; dn= )]TJ/F35 5.9776 Tf 5.828 0 Td [(n0^2 )]TJ/F35 5.9776 Tf 5.977 0 Td [(n1^2 xx/a0^2; dx=p/sqrtn^2 )]TJ/F35 5.9776 Tf 5.79 0 Td [(p^2; dp=dn/sqrtn^2 )]TJ/F35 5.9776 Tf 5.79 0 Td [(p^2; dy=[dx;dp]; p0=n sintheta0; end 62

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APPENDIXC Code:GaussianRayBundle %%%%%%%%%%GAUSSIAN_RAY_BUNDLE.M %SimulatesthetrajectoriesofaGaussianraybundle %Usesfunction'qdinxexact.m' % % n0=4;%centerlineindexopticalaxis n1=1;%indexattheboundary a0=5;%waveguidedistancefromcenter z0=0; zf=32;%waveguidelength theta0_degs=0;%initialangle,indegrees spacing=0.12;%spacingbetweenon )]TJ/F35 5.9776 Tf 6.072 0 Td [(axisrayandadjacentrays.Determines#ofrays raydist1=0; alpha1=spacing; raydist2=alpha1; raydist3=raydist2+alpha1 e^raydist2^2/2; irays=3; rayflag=0; whilerayflag==0 raydistirays+1=raydistirays+alpha1 e^raydistirays^2/2; irays=irays+1; ifraydistirays>a0 raydistirays=[]; rayflag=1; end end raydistlengthraydist=[]; raydist; maxnumrays=lengthraydist foriray=1:maxnumrays xstart=raydistiray; theta0_rads=theta0_degs 3.14159/180;%convertinitialangletoradians theta0=theta0_rads; [x_z]=qdinxexactn0,n1,a0,z0,zf,xstart; z=linspacez0,zf,2000; xlabel'z'; ylabel'xz'; plotz,x_z,'r )]TJ/F35 5.9776 Tf 7.123 0 Td [(','LineWidth',1; legendexact=legend'ExactSoln,RayTrajectory'; holdon end %nowswitchsignofinitalraypostions,forother'half'ofgaussianraybundle foriray=1:maxnumrays xstart= )]TJ/F35 5.9776 Tf 6.157 0 Td [(raydistiray; theta0_rads=theta0_degs 3.14159/180;%convertinitialangletoradians theta0=theta0_rads; 63

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[x_z]=qdinxexactn0,n1,a0,z0,zf,xstart; z=linspacez0,zf,2000; xlabel'z'; ylabel'xz'; plotz,x_z,'r )]TJ/F35 5.9776 Tf 7.123 0 Td [(','LineWidth',1; legendexact=legend'ExactSoln,RayTrajectory'; holdon end 64

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APPENDIXD Code:AnalyticSolutiontotheScalarWaveEquation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%MAINSCRIPTFORANALYTICSOLN,WAVEEQUATIONINSQUARELAWMEDIUM %%Requiredfunctions:bbeta.m,hermitecoeffs.m,mfac.m,EE004.m, %%calcexpancoeffs.m,evalinten.m %% % % % % % % % clearall nn=20%Hermitepolynomialorder,mustbeeven iorder=nn; iorderdiv=iorder/2; m=0:1:iorderdiv; %%%%%%%%%%%%%%Waveguidedims %ForcomparisonwithMEEPsimulation y= )]TJ/F35 5.9776 Tf 6.54 0 Td [(9:0.03:9;%changestepsizetoincreaseresolutionofimages z=0:0.02:32; %Parameters a0=5.0;%Waveguidehalfwidth %ForcomparisonwithMEEPsimulation n0=4.0;%centerindexofrefraction n1=1.0;%boudaryindexofrefraction sourcesigma=1.0; frequency=1.5;%Thisismeep'sdimensionlessfrequency wavelength=1/frequency; k0=2 pi/wavelength; b=sqrta0/k0 sqrtn0 )]TJ/F35 5.9776 Tf 5.427 0 Td [(n1;%Thenaturalscale ggamma=sqrt0.5 b/sourcesigma^2+1; ggugga=1 )]TJ/F35 5.9776 Tf 5.661 0 Td [(ggamma^2/ggamma^2; propconst=bbetam,n0,b,k0,iorder; fac=mfacm,ggamma,ggugga;%expansioncoefficients eeta=y/b;%ydimensiondividedbynaturalscale x=eeta';%UsedtoevaluateeachHermitepolynomial functionH_0=H0x H_0=x; H_0:=1;%lowestorderHemitepolynomialis1singlerow end 65

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ihermitepolys=1; whileiorder>0 coeffs=hermitecoeffsiorder; numofcoeffs=lengthcoeffs; istep=1; whileistep
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functionfac=mfacm,ggamma,ggugga fac=ggugga.^m./ggamma 4.^m. gammam+1; endfunction functionhk=hermitecoeffsn ifn==0 hk=1; elseifn==1 hk=[20]; else hkm2=zeros1,n+1; hkm2n+1=1; hkm1=zeros1,n+1; hkm1n=2; fork=2:n hk=zeros1,n+1; fore=n )]TJ/F35 5.9776 Tf 5.338 0 Td [(k+1:2:n hke=2 hkm1e+1 )]TJ/F35 5.9776 Tf 10.434 0 Td [(k )]TJ/F35 5.9776 Tf 6.074 0 Td [(1 hkm2e; end hkn+1= )]TJ/F35 5.9776 Tf 6.022 0 Td [(2 k )]TJ/F35 5.9776 Tf 6.074 0 Td [(1 hkm2n+1; ifk0 Hn=H.num2striherm; icols=lengthHn1,:;%numberofcolumnsinthestructfield whileicols>=2 ifremicols,2==0%ifnumberofcolumnsiseven hn=hn+Hn:,icols+Hn:,icols )]TJ/F35 5.9776 Tf 6.384 0 Td [(1;%addlasttwocolumns icols=icols )]TJ/F35 5.9776 Tf 6.165 0 Td [(2;%subtractlasttwocolumns elseifremicols,2==1%&&icols>1%ifnumberofcolumnsisodd hn=hn+Hn:,icols; icols=icols )]TJ/F35 5.9776 Tf 6.165 0 Td [(1; endif endwhile%HavenowevaulatedeachtermforthecurrentHermitepolynomialorder inten.num2striinten=facifac hn; yy=yy+facifac hn cospropconstibeta z; 67

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hn=0;%resetvariable iinten=iinten+1; ifac=ifac+1; ibeta=ibeta+1; iherm=iherm )]TJ/F35 5.9776 Tf 6.166 0 Td [(1; endwhile moo=z; moo:=1; yy=exp )]TJ/F35 5.9776 Tf 6.332 0 Td [(0.5 x.^2. moo. yy; endfunction function[TimeAvgIntensity,crossfield,fieldsqrd]=evalinteninten,iorder,propconst,z,x,fac moo=z; moo:=1; intenfieldnames=fieldnamesinten; N=lengthintenfieldnames fieldsqrd=0; ifieldsqrd=1; whileifieldsqrd<=N fieldsqrd=fieldsqrd+inten.num2strifieldsqrd.^2;%Sumsallofthesquardterms ifieldsqrd=ifieldsqrd+1; end %fieldsqrd=exp )]TJ/F35 5.9776 Tf 6.332 0 Td [(0.5 x.^2. moo. fieldsqrd; field=0; ifield=1; whileifield<=N ifieldnxt=ifield+1; whileifieldnxt<=N betadiff=propconstifield )]TJ/F35 5.9776 Tf 6.598 0 Td [(propconstifieldnxt; crossfield=2 inten.num2strifield. inten.num2strifieldnxt cosbetadiff z;%Sumsallofthecross )]TJ/F35 5.9776 Tf 5.736 0 Td [(terms field=field+crossfield; ifieldnxt=ifieldnxt+1; end ifield=ifield+1; end %field; intensum=exp )]TJ/F35 5.9776 Tf 6.332 0 Td [(0.5 x.^2. moo. fieldsqrd+exp )]TJ/F35 5.9776 Tf 6.332 0 Td [(0.5 x.^2. moo. field; TimeAvgIntensity=exp )]TJ/F35 5.9776 Tf 6.332 0 Td [(0.5 x.^2. moo. intensum; end 68

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APPENDIXE. Code:MEEPSimulation definemypointvector3000 definespointvector3000 definea05.0;waveguidehalf )]TJ/F35 5.9776 Tf 5.678 0 Td [(width defineeps04;center )]TJ/F35 5.9776 Tf 6.274 0 Td [(linedielectricconstant defineeps11;edgedielectricconstant definesourcesize10.0;sizeofinitialfullbeamwaist definesourcesigma1.0;varianceofGaussiandistribution set!geometry )]TJ/F35 5.9776 Tf 6.386 0 Td [(latticemakelatticesize3218no )]TJ/F35 5.9776 Tf 6.266 0 Td [(size ;;;;;;;;;;;;;;; defineepsfuncmypointmakedielectricepsilonif>vector3 )]TJ/F35 5.9776 Tf 5.338 0 Td [(ymypointa0eps1 )]TJ/F35 5.9776 Tf 10.903 0 Td [(eps0 )]TJ/F35 5.9776 Tf 10.903 0 Td [(eps0eps1/ vector3 )]TJ/F35 5.9776 Tf 5.338 0 Td [(ymypointvector3 )]TJ/F35 5.9776 Tf 5.337 0 Td [(ymypoint a0a0 set!geometrylist makeblockcenter00sizeinfinity10infinity materialmakematerial )]TJ/F35 5.9776 Tf 6.042 0 Td [(functionmaterial )]TJ/F35 5.9776 Tf 5.848 0 Td [(funcepsfunc definegaussprofspoint exp/ )]TJ/F35 5.9776 Tf 5.595 0 Td [(1vector3 )]TJ/F35 5.9776 Tf 5.338 0 Td [(yspointvector3 )]TJ/F35 5.9776 Tf 5.338 0 Td [(yspoint 2sourcesigmasourcesigma set!sourceslist makesource srcmakecontinuous )]TJ/F35 5.9776 Tf 6.092 0 Td [(srcfrequency1.5 componentEz center )]TJ/F35 5.9776 Tf 5.774 0 Td [(150 size0sourcesize amp )]TJ/F35 5.9776 Tf 5.848 0 Td [(funcgaussprof ;;;;;;;;;;;;;;;;;;;;;; set!pml )]TJ/F35 5.9776 Tf 6.16 0 Td [(layerslistmakepmlthickness1.0 set!resolution20 run )]TJ/F35 5.9776 Tf 6.195 0 Td [(until200 at )]TJ/F35 5.9776 Tf 5.92 0 Td [(beginningoutput )]TJ/F35 5.9776 Tf 6.14 0 Td [(epsilon at )]TJ/F35 5.9776 Tf 5.52 0 Td [(endoutput )]TJ/F35 5.9776 Tf 6.223 0 Td [(efield )]TJ/F35 5.9776 Tf 5.628 0 Td [(z 69

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APPENDIXF TheQuantumHarmonicOscillator Introduction Thequantumharmonicoscillatorisasystemwhichcanbesolvedanalyticallyandhas signicantimportanceinphysics.Theanalyticalsolutionofthequantumharmonicoscillator alsodescribesotherquantummechanicalsystemssuchaswavepropagationinanenclosure. Thequantummechanicalbehaviorofthewavefunctionatinnityfromdeterminedeigenvaluesandtheircorrespondingeigenfunctionsandforangularmomentumoperatorswill providethebasistowhichsuchsystemscanbeanalyzed[7]. TheSchrodingerWaveEquation Inquantummechanics,thebehaviorofaparticleisdescribedbyawavefunction x;t herein1D,andthiswavefunctionisasolutiontotheSchrodingerwaveequation i h @ @t = )]TJ/F15 10.9091 Tf 11.688 7.38 Td [( h 2 2 m @ 2 @x 2 + V where V isthepotentialenergyofthesystemand h isPlank'sconstant h = h 2 =1 : 054573 x 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(34 Js: TheSchrodingerequationallowsustond x;t forallfuturetimesifwearegivenappropriateinitialconditions. TheStatisticalInterpretationoftheWaveFunction Thewavefunctiongivestheprobabilityoflocatingaparticleatsomespeciedtime. Butitdoesnottelluswhereexactlytheparticleislocatedin x ,butratherthattheparticle'slocationisafunctionof x .MaxBornproposedastatisticalinterpretationofthewave functiontoallowustodescribethestateoftheparticleinspaceandtime.Bornproposed that j x;t j 2 givestheprobabilityofndingtheparticleatalocation x attime t 70

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TheTime-IndependentSchrodingerEquation TheSchrodingerequationcanbeusedtondthewavefunction x;t usingseparation ofvariables,assumingthepotential V isindependentoftime.Thesolutionsareoftheform x;t = x f t where isafunctionof x alone,and f isafunctionof t alone.Takingthederivatives @ @t = df dt @ 2 @x 2 = d 2 dx 2 f theSchrodingerequationbecomes i h 1 f df dt = )]TJ/F15 10.9091 Tf 11.688 7.38 Td [( h 2 2 m 1 d 2 dx 2 + V: Theleftsideofisafunctionof t ofalone,andtherightsideisafunctionof x alone. Thisholdsonlyifbothsidesareconstant.Wethensettheleftandrighthandsidesequal toaseparationconstant E df dt = )]TJ/F23 10.9091 Tf 9.68 7.38 Td [(iE h f; and )]TJ/F15 10.9091 Tf 11.688 7.38 Td [( h 2 2 m d 2 dx 2 + V = E: Separationofvariableshasturnedapartialdierentialequationintotwoordinarydifferentialequationsand.Equationisthetime-independentSchrodinger equationanditsgeneralsolutionis C exp )]TJ/F23 10.9091 Tf 8.484 0 Td [(iEt= h .Theconstant C canbeabsorbedinto sincethequantityofinterestistheproduct f .Then f t = e )]TJ/F24 7.9701 Tf 6.586 0 Td [(iEt= h : 71

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TheQuantumHarmonicOscillator Thepotentialforthequantumharmonicoscillatoris V x = 1 2 m! 2 x 2 andthetime-independentSchrodingerequationbecomes )]TJ/F15 10.9091 Tf 11.688 7.38 Td [( h 2 2 m d 2 dx 2 + 1 2 m! 2 x 2 = E: AnalyticSolutionbasedonHermitePolynomials Tosolve,werewriteitintermsofadimensionlessvariable r m! h x: TheSchrodingerequationthenbecomes d 2 d 2 = )]TJ/F23 10.9091 Tf 5 -8.836 Td [( 2 )]TJ/F23 10.9091 Tf 10.909 0 Td [(K where K istheenergyinunitsof = 2 h! andisequalto K 2 E h! .Solvinggivesthe allowedvaluesof K ,andthecorrespondingenergyvalues E Forverylarge and x ,theconstant K dominates 2 d 2 d 2 2 whichhastheapproximatesolutions Ae )]TJ/F25 5.9776 Tf 7.782 3.693 Td [( 2 2 + Be + 2 2 : For x !1 ,The B termblowsupandthephysicallyacceptablesolutionshaveanasymptotic formgivenby 72

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! De )]TJ/F25 5.9776 Tf 7.782 3.693 Td [( 2 2 : Dierentiating d d = dh d )]TJ/F23 10.9091 Tf 10.909 0 Td [(h e )]TJ/F25 5.9776 Tf 7.782 3.692 Td [( 2 2 andtakingthesecondderivative d 2 d 2 = d 2 h d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 dh d + 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 h e )]TJ/F25 5.9776 Tf 7.782 3.693 Td [( 2 2 theSchrodingerequationisthen d 2 h d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 dh d + K )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 h =0 : Tondsolutionsto6,weexpand h inapowerseries h = a 0 + a 1 + a 2 2 + = 1 X j =0 a j j : Takingtherstderivativeoftheseriestermbytermwehave dh d = a 1 =2 a 2 +3 a 3 2 + = 1 X j =0 ja j j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 : Takingthesecondderivativeoftheseriestermbytermwehave d 2 h d 2 =2 a 2 +2 3 a 3 +3 4 a 4 2 + = 1 X j =0 j +1 j +2 a j +2 j : PuttingthesederivativeintotheSchrodingerequationwehave 1 X j =0 [ j +1 j +2 a j +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ja j + K )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 a j ] j =0 : Fromtheuniquenessofpowerseriesexpansions,thecoecientofeachpowerof must vanish 73

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j +1 j +2 a j +2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ja j + K )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 a j =0 : Solvingfor a j +2 ,wehave a j +2 = j +1 )]TJ/F23 10.9091 Tf 10.909 0 Td [(K j +1 j +2 a j : TherecursionformulaisequivalenttotheSchrodingerequation. Writing h as h = h even + h odd where h even a 0 + a 2 2 + a 4 4 + isanevenfunctionof builton a 0 ,and h odd a 1 + a 3 3 + a 5 5 + isanoddfunctionbuilton a 1 Theconditionfor K toyieldphysicallyacceptablesolutionsis K =2 n +1 forsomepositiveintegern.Thisisequivalentto E n = n + 1 2 h!; forn=0,1,2,.... whichisthefundamentalquantizationconditionfortheenergylevelsofthequantumharmonicoscillator. Fortheallowedvaluesof K ,therecursionformulais a j +2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 n )]TJ/F23 10.9091 Tf 10.909 0 Td [(j j +1 j +2 a j : For n =0 ,onlyonetermintheenergyseriesexist,and a 1 =0 toeliminate h odd .In ,when j =0 a 2 =0 h 0 = a 0 74

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andthestatefunctionis 0 = a 0 e )]TJ/F25 5.9776 Tf 7.782 3.693 Td [( 2 2 : For n =1 ,choose a 0 =0 ,andinwhen j =1 wendthat a 3 =0 .Wethenhave h 1 = a 1 andthestatefunctionis 1 = a 1 e )]TJ/F25 5.9776 Tf 7.782 3.693 Td [( 2 2 : For n =2 ;j =0 yields a 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 a 0 ,and j =2 gives a 4 =0 ,so h 2 = a 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 andthestatefunctionis 2 = a 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 e )]TJ/F25 5.9776 Tf 7.782 3.693 Td [( 2 2 : Wecangeneralizetheseresults,andseethat h n isapolynomialofdegree n in involvingonlyevenpowersif n isaneveninteger,andoddpowersonif n isanoddinteger. Apartfromtheoverallfactor a 0 or a 1 ,thesearepolynomialsare Hermitepolynomials H n .Therst5Hermitepolynomialsare H 0 =1 H 1 =2 x H 2 =4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 H 3 =8 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x H 4 =16 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(48 x 2 +12 75

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Figure16:Therst5Hermitepolynomials. Anarbitrarymultiplicativefactorischosensothatthecoecientofthehighestpower of is 2 n .Thenormalizedstationarystatesfortheharmonicoscillatorarethen n x = m! h 1 = 4 1 p 2 n n H n e )]TJ/F25 5.9776 Tf 7.782 3.692 Td [( 2 2 : 76

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Figure17:Therst5Hermitewavefunctions. Thedierencesbetweentheclassicalharmonicoscillatorandquantumharmonicoscillatorarenotonlytheenergiesthequantumharmonicoscillatorhavingquantizedenergies, buttherearealsodierencesintheirpositiondistributions.Forexample,theprobability ofndingtheparticleoutsidetheclassicallyallowedrange x isgreaterthantheclassical amplitudefortheenergyinquestionisnotzero,andinalloddstatestheprobabilityof ndingtheparticleatthecenterofthepotentialwelliszero.Onlyforlarge n dowesee somesimilaritiestotheclassicalcase. 77

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APPENDIXG HowToUseMEEP ThissectiondescribesoperationsinMEEPsuchasinstallingMEEP,deningparameter,setingupgeometries,howrunMEEPsimulations,andcreatevisuals.MEEPcanbe installedonaUbuntucomputerfromtheterminalbytyping apt-getinstallmeeph5utils TherestofthissectionassumesMEEPisusedonaUbuntuGNU/Linuxoperatingsystem. ScriptswrittenforMEEPmustbesavedasa .ctl le.Userscanusetexteditors, suchas gedit ,tocreatethesescripts.Fortheexamplepresentedhereadoptedfromthe MEEPtutorialfoundat http://ab-initio.mit.edu/wiki/index.php ,thelewillbe called example.ctl ,andsavedinthelocationDocuments/MeepFolder Beforethegeometryofthesimulationcanbedened,thecomputationalcellmustbe dened.Thisisthetotalareaofthesimulation.Thecomputationalcellcanbedenedby thefollowingcommand set!geometry-latticemakelatticesize168no-size Thiscommanddenesthecomputationalareaas16unitsinthehorizontal x direction,and 8unitsinthevertical y direction.Thecommand no-size meansthatthe z dimension isnotdenedasize,thereforewehavedeneda2Dgeometry. Forthisexample,letscalculatetheeldintensityproleinaplanarhomogeneouswaveguide.Thegeometryofthewaveguidecanbedenedbythefollowingcommand set!geometrylistmakeblockcenter00sizeinfinity1infinity materialmakedielectricepsilon12 78

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Thecommand makeblock createsarectangulargeometry, center00 denesthe rectangletobecenteredatthepoint,0,and sizeinfinity1no-size denesa waveguideinnitivelylonginthe x direction, 1 unitinthe y direction,andthe z direction doesnothaveadeneddimension.Othertypesofshapesthatcanbedenedarespheres, cylinders,cones,andellipsoids.Morecomplexgeometriesarecreatedbyoverlayingthese shapesinasystematicwayuntilthedesiredgeometryiscreated. Thematerialisdenedasadielectricmaterialby materialmakedielectric andthe dielectricconstantof =12 isdenedby epsilon12 .Othermaterialpropertiessuch asrelativepermeability,electricandmagneticconductivitycanalsobedened. Sourcesaredenedbythecommand set!sourceslist makesource srcmakecontinuous-srcfrequency0.15 componentEz center-70 Thisdenesacontinuouswavesourceproportionalto exp )]TJ/F23 10.9091 Tf 8.485 0 Td [(i!t ,withafrequency= 0 : 15 whichisinunitsof c=distance ,and isinunitsof 2 c=distance .Thesourceislocatedat thepoint )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 ; 0 ,whichisoneunitawayfromtheleftboundary.ThisissothatthePML boundaryconditiondoesnotinterferewiththesource.Thecomponent E z ischosenasthe eldcomponentwewishtosee.Thedatawritteninthecorresponding .h5 lewillonlybe forthe E z componentoftheeld. Wewishtohavereectionlessboundaries.Weusethe perfectlymatchedlayers command set!pml-layerslistmakepmlthickness1.0 withathicknessof1unit.Thefollowingcommanddiscretizesthegeometry 79

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set!resolution10 Thiswillcreate160gridsinthe x dimension,and80gridsinthe y dimension.Lastly, wedenehowlongtorunthesimulation,andwhattooutputwiththefollowingcommand run-until200at-beginningoutput-epsilonat-endoutput-efield-z Thismeansthesimulationwillrunfor200timesteps,andwillcreatetwooutputles.The command at-beginningoutput-epsilon willwritealethatwillshowtheindexprole,orthedielectricproleofthegeometry.Thecommand at-endoutput-efield-z willcreatealethatcontainsthe E z intensityproledata. ToruntheMEEPsimulation,fromtheterminalgotothedirectorywherethesimulation wassaved unix:~$cdDocuments/MeepFolder thentypethecommand unix:~/Documents/MeepFolder$meepexample.ctl MEEPoutputssimulationresultsintheHDF5format.Theselescanbepost-processed byanyimageprocessingsoftware.Anexampleofacommandthatwillcreateavisualfrom the .h5 leis unix:~/Documents/MeepFolder$h5topng-S3example-eps-000000.00.h5 Thisusesthe h5topng function,andwillcreatea .png leofthegeometryanddielectricconstantprole.Theoption -S3 isascalingoption,whichinthiscasescalestheimage by3.TheimagecreatedbythiscommandisFigure17 80

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Figure18:MEEPexample.ctlgeometryanddielectricconstantprole. Anexampleofacommandthatwillcreateavisualofthe E z eldintensityproleis unix:~/Documents/MeepFolder$h5topng-S3-Zcdkbluered-ayarg-A example-eps-000000.00.h5example-ez-000200.00.h5 The -Zcdkbluered optionwillcreateacolorscalethatmakesareasofnegativeeld intensityblue,areaswheretheeldintesityiszerowhite,andareaswithpositiveeldintesityvaluesasred.The -ayarg-A willoverlaythedielectricproleandgeometryinlight gray.Theresultingimageofthissimulationis Figure19:MEEPexample.ctl E z eldintensityprole. 81