版權(quán)說明:本文檔由用戶提供并上傳,收益歸屬內(nèi)容提供方,若內(nèi)容存在侵權(quán),請進(jìn)行舉報(bào)或認(rèn)領(lǐng)
文檔簡介
9.MeasuringUltrashortLaserPulsesII:FROGTheMusicalScoreandtheSpectrogramFrequency-ResolvedOpticalGating(FROG)1Dvs.2DPhaseRetrievalFROGasa2DPhase-retrievalProblemSecond-harmonic-generation(SHG)FROG(andothergeometries)MeasuringtheshortesteventevercreatedSingle-shotFROG,XFROG,andTREEFROGPerhapsit’stimetoaskhowresearchersinotherfieldsdealwiththeirwaveforms…Consider,forexample,acousticwaveforms.Autocorrelationandrelatedtechniquesyieldlittleinformationaboutthepulse.It’saplotoffrequencyvs.time,withinformationontopabouttheintensity.Themusicalscorelivesinthe“time-frequencydomain.”Mostpeoplethinkofacousticwavesintermsofamusicalscore.IfE(t)isthewaveformofinterest,itsspectrogramis:whereg(t-t)isavariable-delaygatefunctionandtisthedelay.Withoutg(t-t),SpE(w,)wouldsimplybethespectrum.SpectrogramSSpectrogramFrequencyFrequencyTimeDelaySpectrogramsforLinearlyChirpedPulsesPropertiesoftheSpectrogramThespectrogramresolvesthedilemma!Itdoesn’tneedtheshorterevent!AlgorithmsexisttoretrieveE(t)fromitsspectrogram.Thespectrogramessentiallyuniquelydeterminesthewaveformintensity,I(t),andphase,(t).
Thereareafewambiguities,buttheyare“trivial.”Thegateneednotbe—andshouldnotbe—significantlyshorterthanE(t). Supposeweuseadelta-functiongatepulse:=TheIntensity.Nophaseinformation!“Polarization-Gate”GeometrySpectro-meterFrequency-ResolvedOpticalGating(FROG)Trebino,etal.,Rev.Sci.Instr.,68,3277(1997).KaneandTrebino,Opt.Lett.,18,823(1993).FROGFrequencyFrequencyTimeDelayFROGTracesforLinearlyChirpedPulsesFrequencyFrequencyIntensityTimeDelayFROGTracesforMoreComplexPulsesFrequencyDelay
Unfortunately,spectrograminversionalgorithmsrequirethatweknowthegatefunction.SubstitutingforEsig(t,)intheexpressionfortheFROGtrace:yields:Esig(t,)E(t)|E(t–)|2where:g(t–)
|E(t–)|2TheFROGtraceisaspectrogramofE(t).IfEsig(t,),isthe1DFouriertransformwithrespecttodelaytofsomenewsignalfield,Esig(t,W),then:SowemustinvertthisintegralequationandsolveforEsig(t,W).Thisintegral-inversionproblemisthe2Dphase-retrievalproblem,forwhichthesolutionexistsandisunique.Andsimplealgorithmsexistforfindingit.andTheinputpulse,E(t),iseasilyobtainedfromEsig(t,W):E(t)
Esig(t,)Instead,considerFROGasatwo-
dimensionalphase-retrievalproblem.Stark,ImageRecovery,AcademicPress,1987.1DPhaseRetrieval:SupposewemeasureS(w)anddesireE(t),where:GivenS(kx,ky),thereisessentiallyonesolutionforE(x,y)!!!Itturnsoutthatit’spossibletoretrievethe2Dspectralphase!.GivenS(w),thereareinfinitelymanysolutions
forE(t).Welackthespectralphase.2DPhaseRetrieval:SupposewemeasureS(kx,ky)anddesireE(x,y):TheseresultsarerelatedtotheFundamentalTheoremofAlgebra.WeassumethatE(t)andE(x,y)areoffiniteextent.1Dvs.2DPhaseRetrievalStark,ImageRecovery,AcademicPress,1987.TheFundamentalTheoremofAlgebrastatesthatallpolynomialscanbefactored:fN-1zN-1+fN-2zN-2+…+f1z+f0=fN-1(z–z1)
(z–z2)…(z–zN–1)TheFundamentalTheoremofAlgebrafailsforpolynomialsoftwovariables.Onlyasetofmeasurezerocanbefactored.fN-1,M-1yN-1zM-1+fN-1,M-2yN-1
zM-2+…+f0,0=?Whydoesthismatter?Theexistenceofthe1DFundamentalTheoremofAlgebraimpliesthat1Dphaseretrievalisimpossible.Thenon-existenceofthe2DFundamentalTheoremofAlgebraimpliesthat2Dphaseretrievalispossible.PhaseRetrievalandtheFundamentalTheoremofAlgebra1DPhaseRetrievalandtheFundamentalTheoremofAlgebraTheFouriertransform{F0,…,FN-1}ofadiscrete1Ddataset,{f0,…,fN-1},is:wherez=e–ikTheFundamentalTheoremofAlgebrastatesthatanypolynomial,fN-1zN-1+…+f0,canbefactoredtoyield:fN-1(z–z1)
(z–z2)…(z–zN–1)SothemagnitudeoftheFouriertransformofourdatacanbewritten:|Fk|=|fN-1(z–z1)
(z–z2)…(z–zN–1)|wherez=e–ikComplexconjugationofanyfactor(s)leavesthemagnitudeunchanged,butchangesthephase,yieldinganambiguity!So1Dphaseretrievalisimpossible!PhaseRetrieval&theFundThmofAlgebra2polynomial!2DPhaseRetrievalandtheFundamentalTheoremofAlgebraTheFouriertransform{F0,0,…,FN-1,N-1}ofadiscrete2Ddataset,{f0.0,…,fN-1,N-1},is:wherey=e–ikandz=e–iqButwecannotfactorpolynomialsoftwovariables.Sowecanonlycomplexconjugatetheentireexpression(yieldingatrivialambiguity).Onlyasetofpolynomialsofmeasurezerocanbefactored.So2Dphaseretrievalispossible!Andtheambiguitiesareverysparse.PhaseRetrieval&theFundThmofAlgebra2Polynomialof2variables!Esig(t,)E(t)|E(t–)|2Esig(t,)GeneralizedProjectionsEsig(t,)E(t)AlgorithmFindsuchthatandisascloseaspossibletoEsig(t,)DeLongandTrebino,Opt.Lett.,19,2152(1994)CodeisavailablecommerciallyfromFemtosoftTechnologies.WemustfindsuchthatandisascloseaspossibletoEsig(t,).ApplyingtheSignalFieldConstraintThewaytodothisistofindthefield,E(t),thatminimizes:OncewefindtheE(t)thatminimizesZ,wewritethenewsignalfieldas:Thisisthenewsignalfieldintheiteration.DeLongandTrebino,Opt.Lett.,19,2152(1994)KohlertracesSecond-harmonicgeneration(SHG)isthestrongestNLOeffect.Second-Harmonic-GenerationFROGKaneandTrebino,JQE,29,571(1993).DeLongandTrebino,JOSAB,11,2206(1994).FrequencyFrequencyTimeDelaySHGFROGtracesaresymmetricalwithrespecttodelay.FrequencyFrequencyIntensityTimeDelaySHGFROGtracesforcomplexpulsesFrequencyDelaySHGFROGMeasurementsofaFree-ElectronLaserSHGFROGworksverywell,eveninthemid-IRandfordifficultsources.Richman,etal.,Opt.Lett.,22,721(1997).Time(ps)Wavelength(nm)507651125148-4-2024IntensitySpectralIntensityPhase(rad)SpectralPhase(rad)043215043215ShortestpulselengthYearShortestpulsevs.yearPlotpreparedin1994(byErichIppen,MIT)reflectingthestateofaffairsatthattime.Themeasuredpulsespectrumhadtwohumps,andthemeasuredautocorrelationhadwings.10-fsspectraandautocorrsDatacourtesyofKapteynandMurnane,WSUDespitedifferentpredictionsforthepulseshape,boththeorieswereconsistentwiththedata.Twodifferenttheoriesemerged,andbothagreedwiththedata.FromHarveyet.al,Opt.Lett.,v.19,p.972(1994)FromChristovet.al,Opt.Lett.,v.19,p.1465(1994)FROGdistinguishesbetweenthetheories.Taft,etal.,J.SpecialTopicsinQuant.Electron.,3,575(1996).SHGFROGMeasurementsofa4.5-fsPulse!Baltuska,Pshenichnikov,andWeirsma,J.Quant.Electron.,35,459(1999).Pulseretrievalremainsequivalenttothe2Dphase-retrievalproblem.Manyinteractionshavebeenused,e.g.,polarizationrotationinafiber.GeneralizingFROGtoarbitrarynonlinear-opticalinteractionsFROGtracesofthesamepulsefordifferentgeometriesFROGgeometries:ProsandConsSecond-harmonicgenerationThird-harmonicgenerationTransient-gratingPolarization-gateSelf-diffractionmostsensitive;mostaccuratetightlyfocusedbeamsusefulforUV&transient-gratingexperimentssimple,intuitive,bestschemeforamplifiedpulsesusefulforUVGeneralizedProjectionswithanArbitraryMediumResponseDeLong,etal.,Opt.Lett.20,486(1995).Single-shotFROGSingle-ShotPolarization-GateFROGKaneandTrebino,Opt.Lett.,18,823(1993).Thiseliminatestheneedforabulkyexpensivespectrometeraswellastheneedtoalignthebeamthroughatinyentranceslit(whichwouldinvolvethreesensitivealignmentparameters)!Wecanusethefocusofthebeaminthenonlinearmediumastheentranceslitforahome-madeimagingspectrometer(inmulti-shotandsingle-shotFROGmeasurements).FROGallowsaverysimpleimagingspectrometer.PulsesfromfirsthalfofFROGCollimatingLensCameraorLinearDetectorArrayGratingImagingLensNonlinearMediumSignalpulseWhenaknownreferencepulseisavailable:
Cross-correlationFROG(XFROG)TheXFROGtrace(aspectrogram):UnknownpulseCameraE(t)Eg(t–t)SFGcrystalLensKnownpulseIfaknownpulseisavailable(itneednotbeshorter),thenitcanbeusedtofullymeasuretheunknownpulse.Inthiscase,weperformsum-frequencygeneration,andmeasurethespectrumvs.delay.Spectro-meterXFROGcompletelydetermin
溫馨提示
- 1. 本站所有資源如無特殊說明,都需要本地電腦安裝OFFICE2007和PDF閱讀器。圖紙軟件為CAD,CAXA,PROE,UG,SolidWorks等.壓縮文件請下載最新的WinRAR軟件解壓。
- 2. 本站的文檔不包含任何第三方提供的附件圖紙等,如果需要附件,請聯(lián)系上傳者。文件的所有權(quán)益歸上傳用戶所有。
- 3. 本站RAR壓縮包中若帶圖紙,網(wǎng)頁內(nèi)容里面會(huì)有圖紙預(yù)覽,若沒有圖紙預(yù)覽就沒有圖紙。
- 4. 未經(jīng)權(quán)益所有人同意不得將文件中的內(nèi)容挪作商業(yè)或盈利用途。
- 5. 人人文庫網(wǎng)僅提供信息存儲(chǔ)空間,僅對用戶上傳內(nèi)容的表現(xiàn)方式做保護(hù)處理,對用戶上傳分享的文檔內(nèi)容本身不做任何修改或編輯,并不能對任何下載內(nèi)容負(fù)責(zé)。
- 6. 下載文件中如有侵權(quán)或不適當(dāng)內(nèi)容,請與我們聯(lián)系,我們立即糾正。
- 7. 本站不保證下載資源的準(zhǔn)確性、安全性和完整性, 同時(shí)也不承擔(dān)用戶因使用這些下載資源對自己和他人造成任何形式的傷害或損失。
最新文檔
- 勞動(dòng)者安全生產(chǎn)協(xié)議書七篇
- 麻痹性腸梗阻病因介紹
- 非火器性顱腦開放傷病因介紹
- 陰虱病病因介紹
- 內(nèi)蒙古通遼市(2024年-2025年小學(xué)六年級語文)部編版隨堂測試(上學(xué)期)試卷及答案
- 中考?xì)v史復(fù)習(xí)方案專題突破知識專題六大國史
- (參考模板范文)申請報(bào)告
- 園藝植物種子生產(chǎn)教學(xué)課件
- 企業(yè)內(nèi)部凝聚力培訓(xùn)課件
- 保健按摩師中級練習(xí)題庫及答案
- 學(xué)校食堂輿情處置預(yù)案
- 2024年大學(xué)生信息素養(yǎng)大賽(省賽)考試題庫(含答案)
- 應(yīng)用語言學(xué)智慧樹知到答案2024年杭州師范大學(xué)
- Chinese Festivals (教學(xué)設(shè)計(jì))-2024-2025學(xué)年外研版(一起)英語五年級上冊
- 乙方和甲方對賭協(xié)議書范本
- 2024年人教版八年級數(shù)學(xué)(上冊)期末試卷及答案(各版本)
- 安全先進(jìn)個(gè)人事跡材料(7篇)
- 高職院校開設(shè)人工智能通識課程的必要性探究
- 2025高考數(shù)學(xué)一輪復(fù)習(xí)-41.1-橢圓的概念及基本性質(zhì)【課件】
- 《觀滄?!贰肚镌~》《天凈沙. 秋思》群文閱讀教學(xué)設(shè)計(jì) 2023-2024學(xué)年統(tǒng)編版語文七年級上冊
- 《烏魯木齊市國土空間總體規(guī)劃(2021-2035年)》
評論
0/150
提交評論