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1、Stronglyfocusedlightbeamsinteractingwithsingleatomsinfreespace1StronglyfocusedGaussianbeamsWeuseasetofeigenmodesoffourcommutingoperatorscorrespondingtothefollowingfourphysicalquantites:energyangularmomentuminzdirectionmomentuminthezdirectionhelicityThemodesarethuscharacterizedbythefournu
2、mbersThecompleteorthogonalsetofmodesisdefinedsuchthatthefreeelectricfield(thesolutionofthesource-freeMaxwellequations)canbeexpandedinthissetasisarbitrarycomplexamplitudes.Thefunctionistransverse,thesummationovervisashort-handnotationfor正交條件Fortheremainderofthissection,wewillonlyconsiderm
3、onochromaticbeamspropagatinginthepositivezdirection.wetakeasamodenumberinsteadof,andwereducesetofmodenumbersbyTheactionofthelensismodeledherebyassumingthatthefielddistributionoftheincomingfieldismultipliedbyalocalphasefactorAidealparaboliclenswouldberepresentedbyaphasefactorIntheplaneoft
4、helensIfweapproximatetheincomingbeambyacircularlypolarized(lowest-order)GaussianbeamwithRayleighrangeWhentheparaxiallimitisvalidfortheoutgoingbeam,i.e.,when,andcorrespond.Butalsooutsidetheparaxiallimit,thefocusedlightbeamischaracterizedbythetwoparametersand.Thelargestcomponentoftheoutput
5、fieldisthecomponent,whichisgivenbyWhentheparaxialapproximationisvalidfortheoutgoingbeam,wemaytakeoutafactorextendtheintegrationlimittoinfinityusedefiningOutsidetheparaxiallimit,thefocalplaneisnolongeratz=z0butmovestowardsthelensbyseveralwavelengths,furthermore,unlikeintheparaxialapproxim
6、ation,theshapeofthefieldisnotjustdeterminedbythevalueofbutdependsonaswellTheplotsofthetransverse-modeprofilesshowthatbeyondacertainpointthewidthofthefieldnolongerdecreaseswithstrongerfocusing.Onecannotfocusdownalaserfieldtobelowacertainlimit,roughlyabouthalfawavelength,withtheidealensfac
7、tor,nomatterhowsmallbecomes.Moreover,noonenotestheasymmetryoftheoutgoingbeamaroundthefocalplane,incontrasttoaparaxialbeamthatissymmetryunderreflectionsinthefocalplane,thisfactshouldnotbesurprising.FIG1(a)fieldstrengthofstronglyfocusedGaussianbeamsonthezaxisasafunctionNote
