2011-8-29DepartmentofPhysicsSoutheastUniversityLecturenineteen:

StimulatedBrillouinScattering

(page429)A.StimulatedScatteringProcessesWeseeinlasttwolecturesthatlightscatteringcanoccuronlyastheresultoffluctuationsintheopticalpropertiesofamaterialsystem.Alight-scatteringprocessissaidtobespontaneousifthefluctuations(typicallyinthedielectricconstant)areexcitedbythermalorbyquantum-mechanicalzero-pointeffects.

Incontrast,alight-scatteringprocessissaidtobestimulatedifthefluctuationsareinducedbythepresenceofthelightfield.Stimulatedlightscatteringistypicallyverymuchmoreefficientthanspontaneouslightscattering.Forexample,approximatelyonepartin105(十萬分之一)ofthepowercontainedinabeamofvisiblelightwouldbescatteredoutofthebeambyspontaneousscatteringinpassingthrough1cmofliquidwater.Inthislecture,weshallseethatwhentheintensityoftheincidentlightissufficientlylarge,essentially100%ofabeamoflightcanbescatteredina1-cmpathastheresultofstimulatedscatteringprocesses.Wewillstudystimulatedlightscatteringresultingfrominduceddensityvariationsofamaterialsystem.ThemostimportantexampleofsuchaprocessisstimulatedBrillouinscattering(SBS),whichisillustratedschematicallyinFig.9.1.1.Stokesfrequencyincidentlightacousticwave

Theresponseofthematerialsystemtothisinterferenceterm(發生

電致伸縮,electrostriction)canactasasourcethattendstoincreasetheamplitudeofthesoundwave.ThusthebeatingofthelaserwavewiththesoundwavetendstoreinforcetheStokeswave,whereasthebeatingofthelaserwaveandStokeswavestendstoreinforcethesoundwave.Underpropercircumstances,thepositivefeedbackdescribedbythesetwointeractionsleadstoexponentialgrowthoftheamplitudeoftheStokeswave.DescriptionofanSBSprocessTherearetwoconceptuallydifferentconfigurationsinwhichSBScanbestudied.OneistheSBSgeneratorshowninpart(a)ofFig.9.1.2.TheotheristheSBSamplifiershowninpart(b).OnlythelaserbeamisappliedexternallyBoththelaserandStokesfieldsareappliedexternallyInFigs.9.1.1and9.1.2,wehaveassumedthatthelaserandStokeswavesarecounterpropagating(反向傳播的).Infact,theSBSprocessleadstoamplificationofaStokeswavepropagatinginanydirectionexceptforthepropagationdirectionofthelaserwave.However,SBSisusuallyobservedonlyinthebackwarddirection,becausethespatialoverlapofthelaserandStokesbeamsislargestundertheseconditions.B.Electrostriction(電致伸縮效應)Electrostrictionisthetendencyofmaterialstobecomecompressedinthepresenceofanelectricfield,thatis,thetendencyofmaterialstobecomemoredenseinregionsofhighopticalintensity。Electrostrictionisofinterestbothasamechanismleadingtoathird-ordernonlinearopticalresponseandasacouplingmechanismthatleadstostimulatedBrillouinscattering.

MacroscopicdescriptionMicroscopicdescription

OriginofElectrostrictionC.ElectrostrictionduetoaStaticField

(1)Consequently,thefieldenergydensitychangesbytheamount(2)

Theworkdoneperunitvolumeisgivenby

(3)

(4)

(5)

D.ElectrostrictionduetoanOpticalWaveForanopticalfrequencyfield,Eq.(5)shouldbereplacedby(6)whereasbeforetheangularbracketsdenoteatimeaverageovermanyopticalperiods.

Letusconsiderthemodificationoftheopticalpropertiesofamaterialsystemthatoccursasaresultofelectrostriction.

(7)

(8)

(9)

Thecomplexamplitudeofthenonlinearpolarizationthatresultsfromthischangeinthesusceptibilitycanbegivenby

(10)E.StimulatedBrillouinScattering(SBS)Asdiscussedbefore,spontaneousBrillouinscatteringassumedthattheappliedfieldsaresufficientlyweakthattheydonotaltertheacousticpropertiesofthematerialsystem(thermallyexcited).Foranincidentlaserfieldofsufficientintensity,thereoccursastimulatedBrillouinscatteringprocess(seebelow)inwhichtheacousticpropertiesaremodifiedbythelaserbeam..StimulatedBrillouinscattering

(12)

SBSGeneratorSituation

(13)Henceforacolinearpropagation,weobtain

(14)

(15)ByuseofEqs.(12)and(15),theBrillouinfrequencybcomes

(16)

(16)

(17)

(18)

SBSAmplifierSituationLetusnextseehowtotreatthenonlinearcouplingamongthethreeinteractingwaves,especiallyforanSBSamplifier.WerepresenttheopticalfieldwithintheBrillouinmediumas(19)where(20)(21)Similarly,wedescribetheacousticfieldintermsofthematerialdensitydistribution

(22)

ByusingEq.(19)andperformingthetimeaverageoperation,thissourcetermisgivenby(24)Weassumethatthematerialdensityobeystheacousticwaveequation(23)

SoundWaveEquationanditsSolution

IfwenowintroduceEqs.(22)and(24)intotheacousticwaveequation(23)andassumethattheacousticamplitudevariesslowlyinspaceandtime,weobtaintheresult(25)

(26)OpticalWaveEquationanditsCouplingThespatialevolutionoftheopticalfieldsisdescribedbythewaveequation(27)Thetotalnonlinearpolarizationisgivenby(28)

(29)

Thesecontributionsaregivenbywhere(30)(31)WeintroduceEqs.(20),(21),(26),and(30)intothewaveequation(27),andobtainthecoupledwaveequations(withanSVAA)(32)(33)WeseefromEqs.(32)and(33)thatSBSisapuregainprocess,thatis,thattheSBSprocessisautomaticallyphase-matched.Forthisreason,itispossibletointroducecoupledequationsfortheintensitiesofthetwointeractingopticalwaves.

(34)(35)

(36)Propertiesof