NonlinearOpticsWhydononlinear-opticaleffectsoccur?Maxwell'sequationsinamediumNonlinear-opticalmediaSecond-harmonicgenerationSum-anddifference-frequencygenerationHigher-ordernonlinearopticsTheSlowlyVaryingEnvelopeApproximationPhase-matchingandConservationlawsforphotonsNonlinearOpticsWhydononlineNonlinearopticsisn’tsomethingyouseeeveryday.Sendinginfraredlightintoacrystalyieldedthisdisplayofgreenlight(second-harmonicgeneration,SHG):Nonlinearopticsallowsustochangethecolorofalightbeam,tochangeitsshapeinspaceandtime,andtocreateultrashortlaserpulses.Whydon'tweseenonlinearopticaleffectsinourdailylife?1.Intensitiesofdailylifearetooweak.2.Normallightsourcesareincoherent.3.Theoccasionalcrystalweseehasthewrongsymmetry(forSHG).4.Phase-matchingisrequired,anditdoesn'tusuallyhappenonitsown.Nonlinearopticsisn’tsomethiWhydononlinear-opticaleffectsoccur?Recallthat,innormallinearoptics,alightwaveactsonamolecule,whichvibratesandthenemitsitsownlightwave,whichinterfereswiththeoriginallightwave.Wecanalsoimaginethisprocessintermsofthemolecularenergylevels,usingarrowsforthephotonenergies:Whydononlinear-opticaleffecWhydononlinear-opticaleffectsoccur?(continued)Now,supposetheirradianceishighenoughthatmanymoleculesareexcitedtothehigher-energystate.Thisstatecanthenactasthelowerlevelforadditionalexcitation.Thisyieldsvibrationsatallfrequenciescorrespondingtoallenergydifferencesbetweenpopulatedstates.Whydononlinear-opticaleffecNonlinearopticsisanalogoustononlinearelectronics,whichwecanobserveeasily.Sendingahigh-volumesine-wave(purefrequency)signalintoacheapamplifierorcheapspeakersyieldsatruncatedoutputsignal,moreofasquarewavethanasine.Thissquarewavehashigherfrequencies.Wehearthisasdistortion.SharpedgesrequirehigherfrequenciesintheFourierseries.NonlinearopticsisanalogousNonlineareffectsinatomsSoanelectron’smotionwillalsodepartfromasinewave.Thepotentialgetsveryflatoutatinfinity,sotheelectron’smotioncaneasilygononlinear!PotentialduetonucleusNucleusEnergyPositionAnotherwaytolookatnonlinearopticsisthatthepotentialoftheelectronisanatomisnotasimplequadraticpotential.NonlineareffectsinatomsSoaNonlinearopticsandanharmonicoscillatorsForweakfields,motionisharmonic,andlinearopticsprevails.Forstrongfields(i.e.,lasers),anharmonicmotionoccurs,andhigherharmonicsoccur,bothinthemotionandthelightemission.Anucleus(inamolecule)alsodoesnothaveasimplequadraticpotential.Soitsvibrationalmotionisalsononlinear:NonlinearopticsandanharmoniMoleculesExcitedbyLaserLightLaserlightcausesmoleculestovibrateinunison.Wesaythatlaserlightpolarizesthemedium.Acceleratingchargesemitlight.Polarizedmatteremitslightatthefrequencyatwhichitisoscillating.Ifthemotionisn’tsinusoidal,themediumwillalsoemittheadditionalfrequencies!MoleculesExcitedbyLaserLig8Maxwell'sEquationsinaMediumThepolarization,P,containstheeffectofthemedium:Sinewavesofallfrequenciesaresolutionstothewaveequation;

it’sthepolarizationthattellswhichfrequencieswilloccur.Thepolarizationisthedrivingtermforthesolutiontothisequation.Theseequationsreducetothe(scalar)waveequation:InhomogeneousWaveEquationMaxwell'sEquationsinaMediuSolvingthewaveequationinthepresenceoflinearinducedpolarizationForlowirradiances,thepolarizationisproportionaltotheincidentfield:

Inthissimple(andmostcommon)case,thewaveequationbecomes:Thisequationhasthesolution:Theinducedpolarizationonlychangestherefractiveindex.Dull.Ifonlythepolarizationcontainedotherfrequencies…wherew=ckandc=c0

/n

and

n=(1+c)1/2Usingthefactthat:Simplifying:SolvingthewaveequationintMaxwell'sEquationsina

Nonlinear

MediumNonlinearopticsiswhathappenswhenthepolarizationistheresultofhigher-order(nonlinear!)termsinthefield:Whataretheeffectsofsuchnonlinearterms?Considerthesecond-orderterm:

2w=2ndharmonic!Harmonicgenerationisoneofmanyexoticeffectsthatcanarise!Maxwell'sEquationsinaNonliSum-anddifference-frequencygenerationSupposetherearetwodifferent-colorbeamspresent:Notealsothat,whenwiisnegativeinsidetheexp,theEinfronthasa*.2nd-harmonicgen2nd-harmonicgenSum-freqgenDiff-freqgendcrectificationSo:Sum-anddifference-frequencyComplicatednonlinear-opticaleffectscanoccur.Themorephotons(i.e.,thehighertheorder)theweakertheeffect,however.Very-high-ordereffectscanbeseen,buttheyrequireveryhighirradiance.Also,ifthephotonenergiescoincidewiththemedium’senergylevelsasabove,theeffectwillbestronger.Nonlinear-opticalprocessesareoftenreferredtoas:N-wave-mixingprocesseswhereNisthenumberofphotonsinvolved(includingtheemittedone).Thisisasix-wave-mixingprocess.Emitted-lightfrequencywsigComplicatednonlinear-opticalInducedpolarizationfornonlinearopticaleffectsArrowspointingupwardcorrespondtoabsorbedphotonsandcontributeafactoroftheirfield,Ei;arrowspointingdownwardcorrespondtoemittedphotonsandcontributeafactorofthecomplexconjugateoftheirfield:sigInducedpolarizationfornonliSolvingthewaveequationinnonlinearopticsRecalltheinhomogeneouswaveequation:Becauseit’ssecond-orderinbothspaceandtime,andPisanonlinearfunctionofE,wecan’teasilysolvethisequation.Indeed,nonlineardifferentialequationsarereallyhard.We’llhavetomakeapproximations…Takeintoaccountthelinearpolarizationbyreplacingc0withc.SolvingthewaveequationinnSeparation-of-frequenciesapproximationThetotalE-fieldwillcontainseveralnearlydiscretefrequencies,w1,w2,etc.Sowe’llwriteseparatewaveequationsforeachfrequency,consideringonlytheinducedpolarizationatthegivenfrequency:whereE1andP1aretheE-fieldandpolarizationatfrequencyw1.whereE2andP2aretheE-fieldandpolarizationatfrequencyw2.etc.ThiswillbeareasonableapproximationevenforrelativelybroadbandultrashortpulsesSeparation-of-frequenciesapprThenon-depletionassumptionWe’llalsoassumethatthenonlinear-opticaleffectisweak,sowecanassumethatthefieldsattheinputfrequencieswon’tchangemuch.Thisassumptioniscallednon-depletion.Asaresult,weneedonlyconsiderthewaveequationforthefieldandpolarizationoscillatingatthenewsignalfrequency,wsig.whereEsigandPsigaretheE-fieldandpolarizationatfrequencywsig.Thenon-depletionassumptionWeWe’llwritethepulseE-fieldasaproductofanenvelopeandcomplexexponential:

Esig(z,t)=Esig(z,t)exp[i(wsigt–ksig

z)]We'llassumethatthenewpulseenvelopewon’tchangetoorapidly.ThisistheSlowlyVaryingEnvelopeApproximation(SVEA).Ifdisthelengthscaleforvariationoftheenvelope,SVEAsays:

d>>

lsig

TheSlowlyVaryingEnvelopeApproximationComparingEsiganditsderivatives:We’llwritethepulseE-fieldWe’lldothesameintime:Iftisthetimescaleforvariationoftheenvelope,SVEAsays:

t>>

TsigwhereTsig

isoneopticalperiod,2p/wsig.TheSlowlyVaryingEnvelopeApproximation(continued)ComparingEsiganditstimederivatives:We’lldothesameintime:ThAndwe’lldothesameforthepolarization:

Psig

(z,t)=Psig

(z,t)exp[i(wsigt–ksigz)]

Iftisthetimescaleforvariationoftheenvelope,SVEAsays:

t>>

TsigwhereTsig

isoneopticalperiod,2p/wsig.TheSlowlyVaryingEnvelopeApproximation(continued)ComparingPsiganditstimederivatives:Andwe’lldothesamefortheComputingthederivatives:SVEA(continued)Neglectall2ndderivativesofenvelopeswithrespecttozandt.Also,neglectthe1stderivativeofthepolarizationenvelope(it’ssmallcomparedtothewsig2Psig

term).WemustkeepEsig’sfirstderivatives,aswe’llseeinthenextslide…xxxxEsig(z,t)=Esig(z,t)exp[i(wsig

t–ksig

z)]Similarly,Computingthederivatives:SVEANow,becauseksig=wsig/

c,thelasttwobracketedtermscancel.Andwecancancelthecomplexexponentials,leaving:TheSlowlyVaryingEnvelopeApproximationSubstitutingtheremainingderivativesintotheinhomogeneouswaveequationforthesignalfieldatw0:SlowlyVaryingEnvelopeApproximationDividingby2iksig:xxNow,becauseksig=wsig/c,IncludingdispersionintheSVEAWecanincludedispersionbyFourier-transforming,expandingksig(w)tofirstorderinw,andtransformingback.Thisreplacescwithvg:WecanincludeGVDalso,byexpandingto2ndorder,yielding:Wecanunderstandmostnonlinear-opticaleffectsbestbyneglectingGVD,sowewill,butthisextratermcanbecomeimportantforveryveryshort(i.e.,verybroadband)pulses.IncludingdispersionintheSVTransformingtoamovingco-ordinatesystemDefineamovingco-ordinatesystem: zv=ztv=t–z/vgTransformingthederivatives:TheSVEAbecomes:Cancelingterms,theSVEAbecomes:We’lldropthesub-script(v)tosimplifyourequations.Thetimederiva-tivescancel!xxTransformingtoamovingco-orIntegratingtheSVEAUsually,Psig=Psig

(z,t),andeventhissimpleequationcanbedifficulttosolve(integrate).Fornow,we’lljustassumethatPsigisaconstant,andtheintegrationbecomestrivial:Andthefieldamplitudegrowslinearlywithdistance.Theirradiance(intensity)thengrowsquadraticallywithdistance.whenPsigisconstantIntegratingtheSVEAUsually,PWechoosewsigtobethesumoftheinputw’s: ButthesignalE-fieldandpolarizationk-vectorsaren’tnecessarilyequal.Butthek-vectorofthepolarizationis:Sokpolmaynotbethesameasksig!Andwemaynotbeabletocanceltheexp(-ikz)’s…Thek-vectormagoflightatthisfrequencyis:wsigWechoosewsigtobethesumoPhase-matchingThatkpolmaynotbethesameasksigistheall-importanteffectofphase-matching.Itmustbeconsideredinall

nonlinear-opticalproblems.Ifthek’sdon’tmatch,theinducedpolarizationandthegeneratedelectricfieldwilldriftinandoutofphase.where:IntegratingtheSVEAinthiscaseoverthelengthofthemedium(L)yields:TheSVEAbecomes:Phase-matchingThatkpolmaynoPhase-matching(continued)So:LIsigDklargeDksmallPhase-matching(continued)So:LSinusoidaldependenceofSHGintensityonlengthLargeDkSmallDkNoticehowtheintensityiscreatedasthebeampassesthroughthecrystal,but,ifDkisn’tzero,newlycreatedlightisoutofphasewithpreviouslycreatedlight,causingcancellation.SinusoidaldependenceofSHGiTheubiquitoussinc2(DkL/2)Phasemismatchalmostalwaysyieldsasinc2(DkL/2)dependence.Recallthat:MultiplyinganddividingbyL/2:Theubiquitoussinc2(DkL/2)PhaIsigDkMoreoftheubiquitoussinc2(DkL/2)Tomaximizetheirradiance,wemusttrytosetDk=0.Thisisphase-matching.DkEsigThefieldstrength:Theirradiance(intensity):IsigDkMoreoftheubiquitoussPhase-matching=ConservationlawsforphotonsinnonlinearopticsAddingthefrequencies:

isthesameasenergyconservationifwemultiplybothsidesby?:wsigSophase-matchingisequivalenttoconservationofenergyandmomentum!Addingthek’sconservesmomentum:Phase-matching=ConservationNonlinearOpticsWhydononlinear-opticaleffectsoccur?Maxwell'sequationsinamediumNonlinear-opticalmediaSecond-harmonicgenerationSum-anddifference-frequencygenerationHigher-ordernonlinearopticsTheSlowlyVaryingEnvelopeApproximationPhase-matchingandConservationlawsforphotonsNonlinearOpticsWhydononlineNonlinearopticsisn’tsomethingyouseeeveryday.Sendinginfraredlightintoacrystalyieldedthisdisplayofgreenlight(second-harmonicgeneration,SHG):Nonlinearopticsallowsustochangethecolorofalightbeam,tochangeitsshapeinspaceandtime,andtocreateultrashortlaserpulses.Whydon'tweseenonlinearopticaleffectsinourdailylife?1.Intensitiesofdailylifearetooweak.2.Normallightsourcesareincoherent.3.Theoccasionalcrystalweseehasthewrongsymmetry(forSHG).4.Phase-matchingisrequired,anditdoesn'tusuallyhappenonitsown.Nonlinearopticsisn’tsomethiWhydononlinear-opticaleffectsoccur?Recallthat,innormallinearoptics,alightwaveactsonamolecule,whichvibratesandthenemitsitsownlightwave,whichinterfereswiththeoriginallightwave.Wecanalsoimaginethisprocessintermsofthemolecularenergylevels,usingarrowsforthephotonenergies:Whydononlinear-opticaleffecWhydononlinear-opticaleffectsoccur?(continued)Now,supposetheirradianceishighenoughthatmanymoleculesareexcitedtothehigher-energystate.Thisstatecanthenactasthelowerlevelforadditionalexcitation.Thisyieldsvibrationsatallfrequenciescorrespondingtoallenergydifferencesbetweenpopulatedstates.Whydononlinear-opticaleffecNonlinearopticsisanalogoustononlinearelectronics,whichwecanobserveeasily.Sendingahigh-volumesine-wave(purefrequency)signalintoacheapamplifierorcheapspeakersyieldsatruncatedoutputsignal,moreofasquarewavethanasine.Thissquarewavehashigherfrequencies.Wehearthisasdistortion.SharpedgesrequirehigherfrequenciesintheFourierseries.NonlinearopticsisanalogousNonlineareffectsinatomsSoanelectron’smotionwillalsodepartfromasinewave.Thepotentialgetsveryflatoutatinfinity,sotheelectron’smotioncaneasilygononlinear!PotentialduetonucleusNucleusEnergyPositionAnotherwaytolookatnonlinearopticsisthatthepotentialoftheelectronisanatomisnotasimplequadraticpotential.NonlineareffectsinatomsSoaNonlinearopticsandanharmonicoscillatorsForweakfields,motionisharmonic,andlinearopticsprevails.Forstrongfields(i.e.,lasers),anharmonicmotionoccurs,andhigherharmonicsoccur,bothinthemotionandthelightemission.Anucleus(inamolecule)alsodoesnothaveasimplequadraticpotential.Soitsvibrationalmotionisalsononlinear:NonlinearopticsandanharmoniMoleculesExcitedbyLaserLightLaserlightcausesmoleculestovibrateinunison.Wesaythatlaserlightpolarizesthemedium.Acceleratingchargesemitlight.Polarizedmatteremitslightatthefrequencyatwhichitisoscillating.Ifthemotionisn’tsinusoidal,themediumwillalsoemittheadditionalfrequencies!MoleculesExcitedbyLaserLig40Maxwell'sEquationsinaMediumThepolarization,P,containstheeffectofthemedium:Sinewavesofallfrequenciesaresolutionstothewaveequation;

it’sthepolarizationthattellswhichfrequencieswilloccur.Thepolarizationisthedrivingtermforthesolutiontothisequation.Theseequationsreducetothe(scalar)waveequation:InhomogeneousWaveEquationMaxwell'sEquationsinaMediuSolvingthewaveequationinthepresenceoflinearinducedpolarizationForlowirradiances,thepolarizationisproportionaltotheincidentfield:

Inthissimple(andmostcommon)case,thewaveequationbecomes:Thisequationhasthesolution:Theinducedpolarizationonlychangestherefractiveindex.Dull.Ifonlythepolarizationcontainedotherfrequencies…wherew=ckandc=c0

/n

and

n=(1+c)1/2Usingthefactthat:Simplifying:SolvingthewaveequationintMaxwell'sEquationsina

Nonlinear

MediumNonlinearopticsiswhathappenswhenthepolarizationistheresultofhigher-order(nonlinear!)termsinthefield:Whataretheeffectsofsuchnonlinearterms?Considerthesecond-orderterm:

2w=2ndharmonic!Harmonicgenerationisoneofmanyexoticeffectsthatcanarise!Maxwell'sEquationsinaNonliSum-anddifference-frequencygenerationSupposetherearetwodifferent-colorbeamspresent:Notealsothat,whenwiisnegativeinsidetheexp,theEinfronthasa*.2nd-harmonicgen2nd-harmonicgenSum-freqgenDiff-freqgendcrectificationSo:Sum-anddifference-frequencyComplicatednonlinear-opticaleffectscanoccur.Themorephotons(i.e.,thehighertheorder)theweakertheeffect,however.Very-high-ordereffectscanbeseen,buttheyrequireveryhighirradiance.Also,ifthephotonenergiescoincidewiththemedium’senergylevelsasabove,theeffectwillbestronger.Nonlinear-opticalprocessesareoftenreferredtoas:N-wave-mixingprocesseswhereNisthenumberofphotonsinvolved(includingtheemittedone).Thisisasix-wave-mixingprocess.Emitted-lightfrequencywsigComplicatednonlinear-opticalInducedpolarizationfornonlinearopticaleffectsArrowspointingupwardcorrespondtoabsorbedphotonsandcontributeafactoroftheirfield,Ei;arrowspointingdownwardcorrespondtoemittedphotonsandcontributeafactorofthecomplexconjugateoftheirfield:sigInducedpolarizationfornonliSolvingthewaveequationinnonlinearopticsRecalltheinhomogeneouswaveequation:Becauseit’ssecond-orderinbothspaceandtime,andPisanonlinearfunctionofE,wecan’teasilysolvethisequation.Indeed,nonlineardifferentialequationsarereallyhard.We’llhavetomakeapproximations…Takeintoaccountthelinearpolarizationbyreplacingc0withc.SolvingthewaveequationinnSeparation-of-frequenciesapproximationThetotalE-fieldwillcontainseveralnearlydiscretefrequencies,w1,w2,etc.Sowe’llwriteseparatewaveequationsforeachfrequency,consideringonlytheinducedpolarizationatthegivenfrequency:whereE1andP1aretheE-fieldandpolarizationatfrequencyw1.whereE2andP2aretheE-fieldandpolarizationatfrequencyw2.etc.ThiswillbeareasonableapproximationevenforrelativelybroadbandultrashortpulsesSeparation-of-frequenciesapprThenon-depletionassumptionWe’llalsoassumethatthenonlinear-opticaleffectisweak,sowecanassumethatthefieldsattheinputfrequencieswon’tchangemuch.Thisassumptioniscallednon-depletion.Asaresult,weneedonlyconsiderthewaveequationforthefieldandpolarizationoscillatingatthenewsignalfrequency,wsig.whereEsigandPsigaretheE-fieldandpolarizationatfrequencywsig.Thenon-depletionassumptionWeWe’llwritethepulseE-fieldasaproductofanenvelopeandcomplexexponential:

Esig(z,t)=Esig(z,t)exp[i(wsigt–ksig

z)]We'llassumethatthenewpulseenvelopewon’tchangetoorapidly.ThisistheSlowlyVaryingEnvelopeApproximation(SVEA).Ifdisthelengthscaleforvariationoftheenvelope,SVEAsays:

d>>

lsig

TheSlowlyVaryingEnvelopeApproximationComparingEsiganditsderivatives:We’llwritethepulseE-fieldWe’lldothesameintime:Iftisthetimescaleforvariationoftheenvelope,SVEAsays:

t>>

TsigwhereTsig

isoneopticalperiod,2p/wsig.TheSlowlyVaryingEnvelopeApproximation(continued)ComparingEsiganditstimederivatives:We’lldothesameintime:ThAndwe’lldothesameforthepolarization:

Psig

(z,t)=Psig

(z,t)exp[i(wsigt–ksigz)]

Iftisthetimescaleforvariationoftheenvelope,SVEAsays:

t>>

TsigwhereTsig

isoneopticalperiod,2p/wsig.TheSlowlyVaryingEnvelopeApproximation(continued)ComparingPsiganditstimederivatives:Andwe’lldothesamefortheComputingthederivatives:SVEA(continued)Neglectall2ndderivativesofenvelopeswithrespecttozandt.Also,neglectthe1stderivativeofthepolarizationenvelope(it’ssmallcomparedtothewsig2Psig

term).WemustkeepEsig’sfirstderivatives,aswe’llseeinthenextslide…xxxxEsig(z,t)=Esig(z,t)exp[i(wsig

t–ksig

z)]Similarly,Computingthederivatives:SVEANow,becauseksig=wsig/

c,thelasttwobracketedtermscancel.Andwecancancelthecomplexexponentials,leaving:TheSlowlyVaryingEnvelopeApproximationSubstitutingtheremainingderivativesintotheinhomogeneouswaveequationforthesignalfieldatw0:SlowlyVaryingEnvelopeApproximationDividingby2iksig:xxNow,becauseksig=wsig/c,IncludingdispersionintheSVEAWecanincludedispersionbyFourier-transforming,expandingksig(w)tofirstorderinw,andtransformingback.Thisreplacescwithvg:WecanincludeGVDalso,byexpandingto2ndorder,yielding:Wecanunderstandmostnonlinear-opticaleffectsbestbyneglectingGVD,sowewill,butthisextratermcanbecomeimportantforveryveryshort(i.e.,verybroadband)pulses.IncludingdispersionintheSVTransformingtoamovingco-ordinatesystemDefineamovingco-ordinatesystem: zv=ztv=t–z/vgTransformingthederivatives:TheSVEAbecomes: