超快光学-第05章-色散

Angulardispersionandgroup-velocitydispersionPhaseandgroupvelocitiesGroup-delaydispersionNegativegroup-delaydispersionPulsecompressionChirpedmirrorsDispersionandUltrashortPulsesDispersioninOpticsThedependenceoftherefractiveindexonwavelengthhastwoeffectsonapulse,oneinspaceandtheotherintime.GroupdelaydispersionorChirpd2n/dl2Angulardispersiondn/dlBothoftheseeffectsplaymajorrolesinultrafastoptics.Dispersionalsodispersesapulseintime:Dispersiondispersesapulseinspace(angle):vgr(blue)vgr(red)qout(blue)qout(red)Whentwofunctionsofdifferentfrequencyinterfere,theresultisbeats.0021exp()(Re{exp(}))totEiEttitEtakingE0tobereal.21aveaveAddingwavesoftwodifferentfrequenciesyieldstheproductofarapidlyvaryingcosine(ave)andaslowlyvaryingcosine().0000()Re{exp()exp()}Re{exp()[exp()exp()]}Re{2exp()cos()}aveaveaveatvotetEittEittEitititEittE0()2cos()cos()avetottEttE112222aveandLet:Whentwowavesofdifferentfrequencyinterfere,theresultisbeats.Indiv-idualwavesSumEnvel-opeIrrad-iancetime2p/InphaseOutofphaseInphaseOutofphaseInphaseWhentwowavesofdifferentfrequencyinterfere,theyalsoproducebeats.0222011122211010(,)Re{}2222(,)Re{exp()exexp()pex()(ptottaveaveavoteaveaveEikztkkztkztEkkikzEizkzttEiztkzkkkEELetandSimiliarly,andSo:000)}Re{exp()exp[()]}Re{2cos()}exp()exp()cos()2cos()aveaveaveaveaveaveaveikztikzttEikztikztEkztEkzttkzttakingE0tobereal.Traveling-WaveBeatsIndiv-idualwavesSumEnvel-opeInten-sityInphaseOutofphaseInphaseOutofphaseInphaseztimeInphaseOutofphaseSeeingBeatsIt’susuallyverydifficulttoseeopticalbeatsbecausetheyoccuronatimescalethat’stoofasttodetect.Thisiswhywesaythatbeamsofdifferentcolorsdon’tinterfere,andweonlyseetheaverageintensity.However,asumofmanyfrequencieswillyieldatrainofwell-separatedpulses:Indiv-idualwavesSumIrrad-iance2p/maxPulseseparation:2p/min0cos(,)2Puls()e)(totaveavekzztEzttkEGroupVelocityvgd/dkLight-wavebeats(continued):Etot(z,t)=2E0cos(kavez–avet)Pulse(kz–t)Thisisarapidlyoscillatingwave:[cos(kavez–avet)]withaslowlyvaryingamplitude:[2E0Pulse(kz–t)]Thephasevelocitycomesfromtherapidlyvaryingpart:v=ave/kaveWhatabouttheothervelocity—thevelocityofthepulseamplitude?Definethegroupvelocity:vg/kTakingthecontinuouslimit,wedefinethegroupvelocityas:carrierwaveamplitudeGroupvelocityisnotequaltophasevelocityifthemediumisdispersive(i.e.,nvaries).01021122vgckckknknkEvaluatethegroupvelocityforthetwo-frequencycase:00121212120,vv,vv/ggcckknnnnkknnncnIfphasevelocityIfwherek1andk2arethek-vectormagnitudesinvacuum.PhaseandGroupVelocitiesUnrealisticUnrealisticInvacuumMostcommoncaseUnrealisticPossiblevgd/dkNow,isthesameinoroutofthemedium,butk=k0n,wherek0isthek-vectorinvacuum,andniswhatdependsonthemedium.Soit'seasiertothinkofastheindependentvariable:Usingk=n()/c0,calculate:dk/d=(n+dn/d)/c0vgc0/(ndn/d)=(c0/n)/(1+/ndn/d)Finally:Sothegroupvelocityequalsthephasevelocitywhendn/d=0,suchasinvacuum.Butnusuallyincreaseswith,sodn/d0,and:vgvCalculatingthegroupvelocity1v/gdkdvv/1gdnndThegroupvelocityislessthanthephasevelocityinnon-absorbingregions.vg=(c0/n)/(1+dn/d)=v/(1+dn/d)Exceptinregionsofanomalousdispersion(neararesonanceandwhichareabsorbing),dn/dispositive.Sovgvformostfrequencies!00200000022000000222/(2/)2v/12v/1ggddndnddddcccdcccdnnndccnlllppllpplppUsethechainrule:Now,,so:Recallingthat:wehave:200002dnndclllpor:Calculatinggroupvelocityvs.wavelengthWemoreoftenthinkoftherefractiveindexintermsofwavelength,solet'swritethegroupvelocityintermsofthevacuumwavelengthl0.000v/1gcdnnndllRecallthattheeffectofalinearpassiveopticaldevice(i.e.,windows,filters,etc.)onapulseistomultiplythefrequency-domainfieldbyatransferfunction:?Eout()H()?Ein()whereH()isthetransferfunctionofthedevice/medium:()()exp[()]HHHBiSincewealsowriteE()=√S()exp[-i()],thespectralphaseoftheoutputlightwillbe:out()H()in()Wesimplyaddspectralphases.SpectralPhaseandOpticalDevicesNotethatweCANNOTaddthetemporalphases!out(t)H(t)in(t)H()?Ein()?Eout()Opticaldeviceormediumexp[()/2]Lforamedium~TheGroup-VelocityDispersion(GVD)Thephaseduetoamediumis:H()=n()k0L=k()LToaccountfordispersion,expandthephase(k-vector)inaTaylorseries:21000002()()()()...kLkLkLkListhegroupvelocitydispersion.000()v()k001()v()gk1()vgdkd1()vgdkdThefirstfewtermsareallrelatedtoimportantquantities.Thethirdoneisnew:thevariationingroupvelocitywithfrequency:TheeffectofgroupvelocitydispersionGVDmeansthatthegroupvelocitywillbedifferentfordifferentwavelengthsinthepulse.vgr(blue)vgr(red)Becauseultrashortpulseshavesuchlargebandwidths,GVDisabiggerissuethanforcwlight.CalculationoftheGVD(intermsofwavelength)Recallthat:20002ddcllp2000002dddddddcdlllpl000v/gdncndll200000011v2gdddnndcdcdllpll20020002ddnncddllpll22002200