专业英语-波动光学 2

第四章波动光学WaveOpticsWaveOptics4.1ElectromagneticWaves&ComplexExpressionLightisatransverse,electromagneticwavecharacterizedbytime-varyingelectricandmagneticfields.Thefieldspropagatehandinhand;itisusuallysufficienttoconsidereitheroneandignoretheother.Itisconventionaltoretaintheelectricfield,largelybecauseitsinteractionwithmatterisinmostcasesfarstrongerthanthatofthemagneticfield.Itisnotgenerallynecessarytoretainthevectornatureofthefieldunlesspolarizationeffectsarespecificallyknowntobeimportant.Thus,wewillgenerallybeabletodescribelightwiththescalarequationincomplexform,whereAistheamplitude,kisthewavevector,ωistheangularfrequencyandΦisthephaseoflightfield,respectively.4.2DiffractionHuygen'sPrinciple:eachpoint(orinfinitesimalarea)onapropagatingwavefrontitselfradiatesasmallsphericalwavelet.Thewaveletspropagateashort(really,infinitesimal)distance,andtheirresultantgivesrisetoa"new"wavefront.Thenewwavefrontrepresentsmerelythepositionoftheoriginalwavefrontafterithaspropagatedashortdistance.惠更斯原则：传输波前上每一点（或说无限小一个区域）会辐射出一个子球面波。这些子球面波传输一小段距离（实际上是无限小一段距离）后会合成一个"新的"波前。这个新的波前其实代表的是原波前传播一小段距离后的位置。Morespecifically,Huygen'sconstructionisshownabove.Thewavefrontinthiscaseisapartofaplanewavewhichhasjustbeenallowedtopassthroughanaperture.Thesphericalwaveletscombinetoproduceawavefrontlyingalongtheircommontangent.Thenewwavefrontisnearlyplaneandnearlyidenticalwiththeoriginalwavefront.Attheedges,however,itdevelopssomecurvatureowingtotheradiationoftheendpointsawayfromtheaxis.Succeedingwavefrontstakeonmoreandmorecurvature,asshown,andeventuallythewavefrontbecomesspherical.Wethenspeakofadivergingwavediffractedbyanaperture.上图是惠更斯原理的一个具体（例子）：一束平面波入射到一个小孔上，未达小孔时（子波的）波前是平面波的一部分，即（各点的）子球面波相叠加后产生了一个位于其共有切线上的（总）波前。而在（孔）边缘处（总波前）出现了一点弯曲，其由远离光轴的（子波）辐射引起。而如图所示，之后总波前会越来越弯曲，最终变成球面波。我们称这种波经过小孔而发散的现象为衍射。SuperpositionofWaves（波的叠加）Considertwowaves,derivedfromthesamesource,butcharacterizedbyphasedifferenceΦ.TheymaybewrittenasForconvenience,weallowthemtohavethesameamplitude.Ifthewavesaresuperposed,theresultantelectricfieldisItiswellknownthattheintensityofelectromagneticwaveinvacuumistheproductionoftheelectricfieldamplitudeanditscomplexconjugation.WemayimmediatelywriteTheintensityofthesuperposedbeamsvarybetween0andtwicethesumoftheintensitiesoftheindividualbeams.TheexactvalueatanypointinspaceortimedependsontherelativephaseΦ.Inparticular,whenΦ=2mπwhenΦ=(2m+1)πwheremisanyintegerorzero.Becauseenergymustbeconserved,werealizedthatwecannotachieveconstructiveinterferencewithoutfindingdestructiveinterferenceelsewhere.Interferenceoftwouniformwavesmaythereforebringaboutquitecomplicateddistributionsofenergy.引起：Bringabout;Giveriseto;Cause;Arouse;Stirup;Leadto4.2.1Single-SlitDiffractionThesingle-slitdiffractionisshowninonedimensioninFig.1.WeappealtoHuygen'sconstructionandassumethateachelementdsoftheslitradiatesasphericalwavelet.TheobservingscreenislocatedadistanceLawayfromtheaperture,andweseektheintensityofthelightdiffractedatangleθtotheaxis.ThecenteroftheapertureislocatedadistancerfromtheobservationpointP.Theopticalpathdifference(OPD)betweenthepathsfromθandfromtheelementds(ats)iss·sinθ,inFraunhoferapproximation.TheelectricfieldatParisingfromtheelementisWeobtaintherinthedenominatorbyrealizingthattheelementisessentiallyapointsource.Wecannotdropitfromthephasetermk(r+s·sinθ)becauseverysmallchangesofs·sinθcausepronouncedchangesofthephaseofthewaverelativetothatofanotherwave.ThetotalfieldatPisthesumofthefieldsduetoindividualelements.Ifthedimensionoftheslitisbanditscenter,s=0,thisisjusttheintegralwhereconstanttermshavebeenremovedfromtheintegral.Theintegrandisoftheformexp(a·s),sotheintegraliseasilyevaluatedIfwemultiplybothnumeratoranddenominatorbyb,anddefineβ=kbsinθ/2,wemaywriteMore-properanalysis,basedonelectromagnetictheoryandatwo-dimensionalformulationwouldincludeanadditionalfactorofiλintheexpressionforE(θ),buttheimportantpartisthevariablesinβ/β.Iftheviewingscreenisthefocalplaneofalens,thenthefirstminimumislocatedadistancefromthecenterofthepattern,whichextendsinthedirectionperpendiculartotheedgesoftheaperture.Over80%ofthediffractedlightfallswithin2λf'/bofthecenterofthepattern,andthefirstsecondarymaximumisabout5%asintenseastheprincipalmaximum.Similaranalysiscanbecarriedoutwithacircularapertureintwo-dimensions.Theresultissimilar,exceptthatthepatternisadisk,knownastheAirydisk,withradiusdefinedbythefirstzeroaswhereDisthediameteroftheaperture.ItisthefinitesizeoftheAirydiskthatlimitsthetheoreticalresolvingpowerofanyopticalsystem.4.2Interference（干涉）Supposeamonochromaticplanewave(acollimatedbeam,orabeamwithplanewavefronts)isincidentontheopaquescreen.Twoinfinitesimalslitsadistancedaparthavebeencutintothescreen.Eachslitbehaveasapointsource,radiatinginalldirections.WesetupanobservingscreenagreatdistanceLawayfromtheslits.Lightfrombothslitsfallsonthisscreen.TheelectricfieldatapointPisthesumofthefieldsoriginatingfromeachslit,i.e.,whereAistheamplitudeofthewavesattheviewingscreenandr1,r2aretherespectivedistanceoftheslitsfromP.Becausethefactorexp(iωt)iscommontoalltermsandwillvanishfromtheintensity,weshallhereafterdropit.IfLissufficientlylarge,r1andr2areeffectivelyparallelanddifferonlybyopticalpathdifference(OPD)d·sinθ.ThusweobtainForsmallangles,sinθ=x/L,andtheinterferencepatternhasasquareofcosvariationwithx.Maximaandminimaoccurwhentheargumentofthecosineisanintegralmultipleofπandπ/2,respectively,orwhereOPD=mλ(constructiveinterference)andOPD=(m+1/2)λ(destructiveinterference).Infact,theinterferencefringesdonotextendinfinitelyfarfromtheaxis.Thisissoforatleasttworeasons:(a)thelightisnotpurelymonochromatic,and(b)theslitsarenotinfinitesimalinwidth.Thefirstrelatestothecoherenceofthelight.Thiseffectbringsaboutslightlydifferentfringepatternfromtherest,andatlargeanglesθthepatternsdonotcoincideexactly.Thisresultsinthewashingoutandeventualdisappearanceofthefringes.Thesecondeffecthastodowithdiffraction.Thisassumptionisvalidonlyforzeroslitwidth.Afiniteslitradiatesprimarilyintoaconewhoseaxisisthedirectionoftheincidentlight.Forthisreason,Theintensityofthepatternfallsnearlyto0forlargeθ.实际上干涉条纹不会无限延展到离轴很远处，至少有两点原因（限制了这种延展）：（a）光不能做到真正的单色（b）缝（也）不是无限窄。第一个（原因）牵涉到光的相干性，其效应是（一种频率的）条纹与其他（频率）的（条纹在空间上）略有不同，（于是）在大的θ角下（不同频率的）条纹就不会精确重合，导致图样的褪色直至消失。第二个效应与衍射有关，以上的假设只在缝宽为零的情况下成立。宽不为零的缝的衍射主要分布在与入射方向同轴的一个立体角中，因此在大的θ角下图样强度下降到接近为零。Ifwegeneralizefromtwoslitstomany,wefindthattheOPDbetweenrayscomingfromadjacentslitsisdsinθ.Thus,theOPDbetweenthefirstandthejthslitis(j-1)dsinθ.Thetotalelectricfieldatapointonthedistantobservationscreenisasumofmanyterms,i.e.,whereNisthenumberofslitsandφ=kdsinθasbefore.Thus,theintensityoftheinterferencepatternisAtypicalinterferencepatternisillustratedinfigureshownintheright.Thesharppeaksareknownasprincipalmaximaandappearonlywhenmλ=dsinθ.Thisisknownasthegratingequation,andmisknownastheordernumber.Conference（相干）Untilnow,wehavealmostalwaysassumedlighttobecompletelycoherent,inthesensethatanyinterferenceexperimentresultedinhigh-qualityinterferencefringes.Ingeneral,thisisnotthecase,exceptwithcertainlasersources;thelightfrommostsourcesissaidtobeincoherentorpartiallycoherent.Whenconditionsaresuchthatthelightisincoherent,itisnotpossibletodetectinterferenceeffects.Adiscussionofwaveopticsisincompletewithoutconsideringtheconditionsthatmustexistforaninterferenceexperimenttobeperformedsuccessfully.到目前为止，我们总是假设光是完全相干的，意即（认为）所涉及的干涉实验都能获得高质量的条纹。（但）通常来说情况并非如此，除去特定的激光光源，大部分光源的光是非相干或部分相干的。在非相干光的条件下是不可能探测到干涉效应的，（因此）完备的关于波动光学（问题）的分析，必须要考虑到确保干涉实验能顺利实现的那些（额外）条件。Lightsourcesaretodayputintooneoftwocategories,lasersourcesandthermalsources.Atypicalthermalsourceisagasdischargelamp.Insuchalamp,lightisemittedbyexcitedatomsthatare,ingeneral,unrelatedtoeachother.Eachatomemitsrelativelyshortburstsorwavepackets.Ifanatomisexcitedseveraltimes,itcanemitseveralconsecutivewavepackets.Thesepacketsaregenerallyfarapart(comparedwiththeirduration)andareemittedrandomlyintime.Thepacketsemittedbyasingleatomthereforebearnoconstantphaserelationwitheachother.当前，光源被分为两大类：激光光源和热光源。气体放电灯就是一种典型的热光源，它的光是由互不相关联的激发态原子发射而出的。每个原子发射的是相对较短的脉冲或说是波包串。如果一个原子被激发起数次，它发射的将是几个连续的波包串，每串波包间通常距离很远（与他们自己的宽度相比），而且发出的时间也是随机的。所以（即使）是同一个原子发出的波包之间也没有固定的相位关系。Supposewetrytoperforminterferencebydivisionofamplitudewithpacketsemittedbyasingleatom.Thewavereflectedfromthesecondsurfaceisdelayedwithrespecttothefirstbecauseofthefinitespeedoflight.Ifthedelayisgreaterthanthedurationofthewavepacket,thetworeflectedpacketswillnotreachthedetectorsimultaneously.Therewillthereforebenointerferencepattern,andwewouldcomputetheintensityattheviewingscreenbyaddingtheintensities(notamplitudes)ofthereflectedwaves.（现在，）假设我们正通过分同一个原子发出的波包的振幅来实现干涉。因为光速有限，从第二个表面反射回来的波会比从第一个（表面反射）的波晚到。如果延迟长于波包宽度，两束反射波就不能同时到达探测器，也就不会有干涉图样。我们（这时）计算到达观测屏上的光强就应该将两束反射光的光强（而非振幅）相加。Thisstatementwouldbetrueevenifthewavepacketswereemittedsorapidlythatseveralpacketsenteredtheapparatusatthesametime.Becausethepacketsareemittedatrandom,theybearnodefinitephaserelation.WewouldsometimesdetectamaximumandsometimesaminimumforanygivenOPD.Overthelongterm,wewouldobserveconstantintensityandwouldregardthelightasincoherent.Similarly,thewavesfromoneatombearnodefiniterelationwiththewavesfromanyotheratom.Bypreciselythesamereasoning,weconcludethatthelightemittedbyoneatomisincoherentwiththatemittedbyanyotheratom.Superpositionofthewavesfromdifferentatomsisthereforedescribedbyaddingintensities,notamplitudes.就算（原子）以很快的速度发出波包，以至于一些波包能同时到达探测器，以上结论仍然是成立的。这是因为波包是随机发出的，没有固定的相位关系。给定一个OPD时，有时会探测到极大值，有时又会是极小值，（于是）在一个较长的时间段内，我们观测到的将会是一个固定的光强，只能认为光是非相干的。与此相似，一个原子发出的波与其他原子发出的波之间也没有固定的相位关系，所以结论是不同原子发出的光之间也是非相干的，原因与刚才完全相同。（所以）不同原子发出的波之间的叠加也只能由光强相加描述，不是振幅。