超快光学-第06章-光的时间空间

Spatio-temporaldistortionsRaymatricesTheGaussianbeamComplexqanditspropagationRay-pulse“Kosten-bauder”matricesTheprismpulsecompressorGaussianbeaminspaceandtimeandthecomplexQmatrixSpatio-temporalcharacteristicsoflightandhowtomodelthemininininxtoutoutoutoutxt0010001ABECDFGHIOpticalsystem↔4x4Ray-pulsematrixSpatio-temporaldistortionsOrdinarily,weassumethatthepulse-fieldspatialandtemporalfactors(ortheirFourier-domainequivalents)separate:(,,,)(,,)()xyztExyztExyzEtwherethetildeandhatmeanFTswithrespecttotandx,y,zˆˆ(,,,)(,,)()xyzxyzxyztEkkkEkkkE(,,,)(,,)()xyztExyzExyzE垐(,,,)(,,)()xyzxyzxyztEkkktEkkkEtExample00exp()exp()ikzit00exp()()ikz00()()()exp()xyzkkkkit00()()()()xyzkkkkAngulardispersionisanexampleofaspatio-temporaldistortion.000垐(,,,)[,,,)](xxxyzyzdkkdEkkkEkkInthepresenceofangulardispersion,theoff-axisk-vectorcomponentkxdependson:wherekx0()isthemeankxvs.frequency.PrismInputpulseAngularlydispersedoutputpulsexz0xdkdkx0(red)kx0(yellow)etc.Spatialchirpisaspatio-temporaldistortioninwhichthecolorvariesspatiallyacrossthebeam.Propagationthroughaprismpairproducesabeamwithnoangulardispersion,butwithspatialdispersion,oftencalledspatialchirp.Prismpairsareinsidenearlyeveryultrafastlaser.PrismpairInputpulseSpatiallychirpedoutputpulseSpatiallychirpedoutputpulseInputpulseTiltedwindowSpatialchirpisdifficulttoavoid.Simplypropagatingthroughatiltedwindowcausesspatialchirp!Becauseultrashortpulsesaresobroadband,thisdistortionisverynoticeable—andoftenproblematic!Howtothinkaboutspatialchirpwherex0isthecenterofthebeamcomponentoffrequency.0dxdzxSupposewesendthepulsethroughasetofmonochromaticfiltersandfindthebeamcenterposition,x0,foreachfrequency,.x0(1)x0(2)x0(3)x0(4)x0(5)x0(6)x0(7)x0(8)x0(9)00(,,,)[,),,(]dxxdExyzEyzPulse-fronttiltisanothercommonspatio-temporaldistortion.Phasefrontsareperpendiculartothedirectionofpropagation.Becausethegroupvelocityisusuallylessthanphasevelocity,pulsefrontstiltwhenlighttraversesaprism.PrismAngularlydispersedpulsewithpulse-fronttiltUndistortedinputpulseAngulardispersioncausespulse-fronttilt.Angulardispersioncausespulse-fronttiltevenwhengroupvelocityisnotinvolved.Diffractiongratingsalsoyieldpulse-fronttilt.Gratingshaveabouttentimesthedispersionofprisms,andtheyyieldabouttentimesthetilt.Thepathissimplyshorterforraysthatimpingeonthenearsideofthegrating.Ofcourse,angulardispersionandspatialchirpoccur,too.DiffractiongratingAngularlydispersedpulsewithpulse-fronttiltUndistortedinputpulsePulse-FrontTiltfromaGratingTheextradistancetraveledbytheraythatimpingesonthebackedgeofthegratingisd,wheredisthelengthofthegrating.But,inthetimeittakesforthisraytotravelthisextradistance,thedistancetraveledbytheraythatimpingesonthefrontedgeisalsod.Foradiffractiongrating,it’simportanttouseagrazing(large)incidenceangle(forlargestPFT).Wealsorequireadiffractedangleofabout0º.Otherwisewecan’timagethegratingontoaplanelater.Sothemaximumusefulpulse-fronttiltangleachievableusingagratingisgivenby:jtan(j)=d/d,orj=~45º.Pulse-FrontTiltfromanEtalonThelowestray,whichpassesstraightthroughtheetalon,experiencesessentiallyzeroextradelay.Sothatwecanlaterimagetheetalontoaplane,theexitangle,andhenceincidenceangle,mustbesmall.Thedistancextraveledupwardontheetaloninoneroundtripis2tsin≈2t,wheretisthethicknessoftheetalon.So,ifdisthewidthoftheetalon,thenumberofroundtripsbeforethepulsebumpsintothetopedgeoftheetalonis:N=d/x=d/(2t),jSothetotalextradistancetraveledbythehighest(andmostdelayed)rayis:N×2t=[d/(2t)]×2t=d/.Pulse-FrontTiltfromanEtalon(cont’d)Thetangentofthetotalpulse-fronttiltanglejisthenthetotalextradistancetraveledbytheuppermostraydividedbythewidthoftheetalond:tan(j)=(d/)/d=1/jtan(j)=1/Wecantakeintoaccounttherefractiveindexnoftheetalonmaterialbynotingthat,insidetheetalon,→/n,andalsotheextradistance(actuallytime)traveledeffectivelyincreasesbyafactorofn(assumingthatthegroupvelocityisslower,aboutc/n),yielding:tan(j)=n2/Notethat,asthetangent→∞,thetiltanglejgoesto90º.Interestingly,theetalon’swidthandthicknesscancelout.Modelingpulse-fronttiltPulse-fronttiltinvolvescouplingbetweenthespaceandtimedomains:00(,,,)[,,,]()dttxxdExyztExyzxForagiventransversepositioninthebeam,x,thepulsemeantime,t0,variesinthepresenceofpulse-fronttilt.Pulse-fronttiltoccursafterpulsecompressorsthataren’talignedproperly.0dtdxAngulardispersionalwayscausespulse-fronttilt!00垐(,,,)[(),,,]xyzxyzEkkkEkkk0(,,,)(,,,)ExyztExyztxAngulardispersionmeansthattheoff-axisk-vectordependson:where=dkx0/dwhichisjustpulse-fronttilt!InverseFourier-transformingwithrespecttokx,ky,andkzyields:00()(,,,)(,,,)ixExyzExyzeInverseFourier-transformingwithrespectto(or0)yields:usingtheshifttheoremusingtheshifttheoremagainThecombinationofspatialandtemporalchirpalsocausespulse-fronttilt.DispersivemediumSpatiallychirpedinputpulsevg(red)vg(blue)Spatiallychirpedpulsewithpulse-fronttilt,butnoangulardispersionThetheoremwejustprovedassumednospatialchirp,however.Soitneglectsanothercont