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1、Chapter3PropagationandfocusingofopticalfieldsInthischapterwewillusetheangularspectrumrepresentationoutlinedinSection??todiscussfielddistributionsinstronglyfocusedlaserbeams.Thesameformalismisappliedtounderstandhowthefieldsinagivenreferenceplanearemappedtothefarfield.Thetheoryisrelev
2、antfortheunderstandingofconfocalandmultiphotonmicroscopy,singlemoleculeexperiments,andtheunderstandingofresolutionlimits.Italsodefinestheframeworkfordifferenttopicstobediscussedinlaterchapters.3.1FieldPropagatorsInSection??wehaveestablishedthatinahomogeneousspace,thespatialspectru
3、mEˆofanopticalfieldEinaplanez=const.(imageplane)isuniquelydefinedbythespatialspectruminadifferentplanez=0(objectplane)accordingtothelinearrelationshipEˆ(kx,ky;z)=Hˆ(kx,ky;z)Eˆ(kx,ky;0).(3.1)whereHˆistheso-calledpropagatorinreciprocalspaceHˆ(k,k;z)=e±ikzz,(3.2)xyalsoreferredtoastheo
4、pticaltransferfunction(OTF)infreespace.Rememberthatthelongitudinalwavenumberisafunctionofthetransversewavenumber,i.e.k=[k2−(k2+k2)]1/2,wherek=nk=nω/c=n2π/λ.The±signindicateszxyothatthefieldcanpropagateinpositiveand/ornegativezdirection.Eq.(3.1)canbeinterpretedintermsoflinearrespo
5、nsetheory:Eˆ(kx,ky;0)istheinput,Hˆisafilter12CHAPTER3.PROPAGATIONANDFOCUSINGOFOPTICALFIELDSfunction,andEˆ(kx,ky;z)istheoutput.Thefilterfunctiondescribesthepropaga-tionofanarbitraryspatialspectrumthroughspace.Hˆcanalsoberegardedastheresponsefunctionbecauseitdescribesthefieldatzdueto
6、apointsourceatz=0.↔Inthissense,itisdirectlyrelatedtotheGreen’sfunctionG.ThefilterHˆisanoscillatingfunctionfor(k2+k2)k2.Thus,iftheimageplaneissufficientlyxyseparatedfromtheobjectplane,thecontributionofthedecayingparts(evanescentwav
7、es)iszeroandtheintegrationcanbereducedtothecirculararea(k2+k2)k2oftheoriginalfieldarefilteredoutxyduringpropagationandtheinformationonhighspatialvariationsgetslost.Henc
8、e,thereisalwaysalossofinformationonthewayofprop
