耦合模理论

耦合模理论及其在微波和光纤技术中的应用（研究生课程用）钱景仁中国科学技术大学二零零五年I目录绪言(Preface)……………………………………………………………………1第一章耦合模的一般理论§1.1耦合模方程…………………………………………………………………6§1.2强耦合与弱耦合……………………………………………………………11§1.3周期性耦合…………………………………………………………………18§1.4耦合模与简正模……………………………………………………………29§1.5缓变参数情况下本地简正模广义理论……………………………………33§1.6理想模、本地简正模和超本地简正模……………………………………37§1.7耦合器应用举例……………………………………………………………42§1.8临界界面附近和稳相点附近的耦合模方程………………………………46第二章闭合波导中的耦合模问题§2.1介质填充波导………………………………………………………………51§2.2缓变表面阻抗和阻抗微扰…………………………………………………59§2.3弯曲波导……………………………………………………………………64第三章光纤中的耦合模问题§3.1光纤中的简正模式…………………………………………………………68§3.2耦合模理论的推广…………………………………………………………80§3.3非理想光纤的耦合模方程…………………………………………………81§3.4用闭合波导理论来研究开波导……………………………………………86第四章螺旋光纤及弯曲光纤§4.1螺旋光纤的耦合模分析……………………………………………………89§4.2单模传输条件下的螺旋光纤………………………………………………93§4.3弯曲光纤……………………………………………………………………98第五章耦合功率方程§5.1多模波导和多模光纤的传输特性…………………………………………104§5.2多模波导中的耦合功率方程………………………………………………105§5.3多模光纤传输中的耦合功率方程…………………………………………107中文参考文献……………………………………………………………………109英文参考文献……………………………………………………………………11089PrefaceWhatisthecoupled-modetheory?Isitacommontheoryinphysics?Wavesandvibrationphenomenaarepopularinphysicsasweknowsuchasmechanicalvibrations,acousticwaves,lightwaves,microwavesandradiowaves.Furthermore,connectionorcouplingamongsystemsisalsoageneralruleinuniverse.Everythingpresupposestheexistenceofsomeotherthing.Cause-effectrelationsandaction-reactionrelationsaregenerallyexistedamongsystemsintheuniverse.Itisobviousthattherearen’tanyidealwaveswhichexistindependentlyanddonotchangetheiramplitudesanddirections.Arealwaveorvibrationisalwaysconnectedwithasourceorotherwaves.Now,itisnecessarytodescribehowthesewavesorvibrations(oscillations)coupletoeachother,andhowtheiramplitudeschangewiththetimeorthedistance.Toillustratetheprincipleofthecouplingbetweenwavesorvibrations(oscillations),let’stakependulumsasanexample.Fig.aApendulumcanvibrate,thatistosayitswingsfromsidetoside.Wecangiveitapushandthenitwillvibrateatafixedspeedoratacertainfrequency.IftwopendulumswithsamefrequencyarehungonastringandoneofthemissetswingingasshowninFig.a,itwillswinglessandlessuntilitstopsaltogether,whiletheotherpendulumwillswinghigherandhigheruntilitreachesamaximum.Thentheprocesswillbereverseduntilthefirstpendulumreachesamaximumandthesecondcomestorestoncemore.Thiscyclerepeatsitselfagainandagain.Itwouldrepeatinfinitelyiftherewerenolossesinthesystem.stopstring89Thisisatypicalexperimentperformedinmostearlyphysicscourses.IhaddoneitwhenIwasinmiddleschool.tAA1A21periodA1A2At1Fig.bFrequenciesarethesame.Fig.cFrequenciesaredifferent.Ifthesetwopendulumshavedifferentfrequencies,thentransferofenergybetweenthemwillnotbecomplete,andthefirstpendulumwillnotstopintheprocess.WecanplotagraphtoexpresstheprocessasshowninFig.bandFig.c.Theabscissarepresentsthetime,andtheordinateArepresentstheamplitudeofeachpendulum.Iftheinitialconditionsatt=0areasfollows:()()1201,00AA==,WecanseethevariationsoftheamplitudesofthetwocoupledpendulumsinFig.bandFig.c,respectively,whentheirfrequenciesarethesameanddifferent.Thetimespacingbetweentwoadjacentmaxima(orminima)istheperiodoftheprocess,whichisdeterminedbythecouplingbetweenthetwopendulums.Thestrongerthecouplingis,theshortertheperiodis.Thecouplingbetweenthetwopendulumsiscausedbythefactthatthependulumsareconnectedtoasamestring,andanyvibrationofoneofthependulumswillhaveaneffectontheotherthroughthestring.Ithasbeenrecognizedthatcoupledtransmissionlines,coupledelectricalcircuits,coupledopticalfibersandcoupledwaveguidesareanalogoustocoupledpendulums.Thevariationsoftheamplitudesofwavesarethesameasshowninthefigures,butnowtheabscissarepresentsdistanceinstantoftime.Sometimesthecouplingisnotbetweenthesamekindofwavesoroscillations,forexample,inatravelingwavetube,aspace-chargewaveandanelectromagneticwave89coupletoeachother.Inacrystal,anelectricalvibrationwillcauseamechanical(oracoustic)vibrationandviceversa.Thereshouldbesomegeneralrulesorthereisageneralizedtheorytodescribethesecouplingproblems.Itisthesocalledcoupled-modetheory.Here,modemeansoneofthemodelsofwaveforms.Inthetheory,allthecoupled-modeorcoupled-vibrationproblemsareformulatedbyasetofcoupled-modeequations,whicharesimultaneousdifferentialequationsoffirstorderwithvariableorconstantcoefficients.Incaseoftwomodes,theycanbewrittenasfollows:()()()()()()11122221jjjjdAzAzcAzdzdAzAzcAzdzββ=−+=−+Whereiβandcarefunctionsofzingeneralcase.Whennmodesorwavesshouldbeconsideredinacouplingproblem,ndifferentialequationswillbeusedinsteadoftwo.Acommonmethodinelectromagnetictheoryisthemodalapproachinwhichthenormalmodesofthesystem(thosefieldswhichpropagateunchangedexceptinphase)arefound.Thisinvolvessolvingthewaveequationadaptedtotheparticulargeometryofthesystem,andmatchingsolutionsattheboundariestogivethenormalmodesoreigensolutions.Anyfieldofthesystemcanthenbeexpandedintermsofthenormalmodes,withtheexpansioncoefficientsdeterminedbycertainboundaryconditionse.g.initialconditions.Thismodal-expansionoreigenvectormethodisphysicallyintuitiveandstraightforwardinprinciple,butmodalsolutionsofthewaveequationcanonlybefoundforalimitednumberofidealsystemsofrelativelysimplege