非线性光学英文版课件：Lecture 18- Spontaneous Brillouin Scattering

2011-8-29DepartmentofPhysicsSoutheastUniversityLectureeighteen:

SpontaneousBrillouinScatteringBrillouinscattering:thescatteringoflightfromsoundwaves—thatis,frompropagatingpressure(andthusdensity)waves.Brillouinscatteringcanalsobeconsideredtobethescatteringoflightfromacousticphonons.

(2)

Theequationofmotionforapressurewaveiswellknownfromthefieldofacousticsandisgivenby(1)

A.WaveEquationofSound

(3)

(4)

(5)Animportantspecialcaseoftheuseofthisformulaisthatofanidealgas,forwhichtheequationofstate(状态方程)isgivenby(6)

(7)

(8)

(9)Wethusseethatthevelocityofsoundisoftheorderofthemeanthermalvelocityofthemoleculesofthegas.Thevelocityofsoundforsomecommonopticalmaterials透明树脂乙醇Asanillustrationofthenatureoftheacousticwaveequation(1),weconsiderthepropagationofthewave(10)throughanacousticmedium.

(11)Wecanrewritethisrelationas(12)

ThusfromEq.(12)weobtain(13)wherewehaveintroducedthephonondecayrate(14)

(15)wherewehaveintroducedthesoundabsorptioncoefficient(16)Next,wecalculatetherateatwhichlightisscatteredoutofabeamoflightbytheseacousticwaves.Weassumethattheincidentopticalfieldisdescribedby(17)andthatthescatteredfieldobeysthedrivenwaveequation(18)

(19)B.LightScatteredbyaSoundWave

(20)Sincethedielectricconstanttypicallydependsmuchmorestronglyondensitythanontemperature,thesecondterminEq.(20)canbesafelyomitted.Thuswehave

(21)

(22)Herethefirsttermdescribesadiabaticdensityfluctuations(thatis,acousticwaves)andleadstoBrillouinscattering.Thesecondtermdescribesisobaricdensityfluctuations(thatis,entropyortemperaturefluctuations)andleadstoRayleigh-centerscattering.

(23)Asaresult,accordingtoEqs.(19),(21)and(23),wehave(24)

(25)

C.SpontaneousBrillouinScatteringAsinEq.(10),wetakeatypicalcomponentofthethermallyexcitedpressuredisturbancewithintheinteractionregiontobegivenby(26)Therethepolarization(24)becomes

(27)BysubstitutingEq.(27)intoEq.

(18),wefindthatthescatteredfieldmustobeythewaveequation

(28)ThefirstterminthisexpressionleadstoStokesscattering;thesecondtoanti-Stokesscattering.Westudythesetwocontributionsinturn.D.StokesScattering(seefirstterm)Thepolarizationisseentohaveacomponentwithwavevector(29)andfrequency(30)

(31)

(32)

(33)InorderforEqs.(31),(32)and(33)tobesatisfiedsimultaneously,thesound-wavefrequencyandwavevectormusteachhaveaparticularvalueforanyscatteringdirection.

(34)

(35)

Themaximumfrequencyshiftisthusgivenby(36)Stokesscatteringcanbevisualizedasthescatteringoflightfromaretreatingacousticwave,seediagram(c).E.Anti-StokesScattering(see2ndterm)TheanalysishereisanalogoustothatforStokesscattering.Thepolarizationisseentohaveacomponentwithwavevectorandfrequency

(37)(38)

(39)

Anti-Stokesscatteringcanbevisualizedasscatteringfromanoncomingsoundwave,asshowninpart(c)ofFig.8.3.2.

(40)Hence,theacousticfrequencyisgivenby(41)

Note:wehavethusfarignoredattenuationoftheacousticwaveinouranalysis.

(42)Thustheratiooftheli