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1、15ECEN4616/5616Jan30,2013ScalarDiffractionPlaneWaveSpectrum:Inthepreviouslecture,weshowedthatageneralplanewavesolutiontotheElectro-Magneticwaveequationcanbewrittenas:Or,incomplexexponentialnotation:,wherethedotproductbetweenthepositionandwavevectorshasbeenexpanded,andtheReal()functionisassu
2、med.Thisplanewaveisuniquelyidentifiedbyitsintersectionwiththex,y,z=0planeatt=0,ifthewavelengthisknown.Thisintersectionisdefinedbytheequations:,or,wherethevariablesubsitutions:u=kx/2p,v=ky/2phavebeenmade,whereu,varespatialfrequenciesandtheexpressionfitsintothedefinitionoftheFourierTransformb
3、etweenafunctionandthespatialfrequenciesthatcomposeit,viathetransformpair:Note:Thespatialfrequencyrepresentationabove(boxedequation)describestheintersectionofaplanewavewiththex,y,z=0planeatt=0–itisnotaplanewave,norisitasolutionoftheEMWaveequation.Itis,infact,astatic2-Dfunctionthatchangessinu
4、soidallyinaparticulardirectionwithaparticularcyclelength.IntheDiscreteFourierTransform,eachspatialfrequencyhasanintegralnumberofcyclesoverthearray.Anexampleofaspatialfrequencyfunctionisshownintwoviewsbelow:15ECEN4616/5616Jan30,2013Theresultofthesemanipulationsisthatwecandecomposeaharmonic(s
5、inusoidalintime)fieldinanapertureintoasetofspatialfrequencies,eachofwhichuniquelyidentifiesaplanewave(giventheharmonicfrequency,orthewavelengthoftheplanewaves).Thiscollectionofplanewaveswecallthe“PlaneWaveSpectrum”(PWS)ofthefield.Thissetofplanewaves,intersectingatthex,y,z=0planewouldre-crea
6、tethefield.Wecan,ofcourse,usetheFourierTransformtodecomposeany2Dfunctionintospatialfrequencies–however,identifyingthespatialfrequencyspectrumasaPlaneWaveSpectrumonlymakessenseifthefieldiscreatedbyEMwaves(aswithanilluminatedaperture),orcanbethesourceofEMwaves(suchasanantenna).CalculatingPSFs
7、fromthePupilFunction(review):Everyopticaldesignprogramwillgiveapupilfunctionasanamplitudeandphasedistortionfromanidealsphericalwaveattheexitpupilofanopticalsystem.Inthissituation,thePointSpreadFunction(PSF)oftheopticalsystemcanbecalculatedsimplybyfinding
