2011-8-29DepartmentofPhysicsSoutheastUniversityLecturethirteen:

PropagationthroughIsotropicNonlinearMediaLetusnowconsiderthepropagationofabeamoflightthroughamaterialwhosenonlinearpolarizationisgivenby

(1)Asweshowlater,onlylinearlyorcircularlypolarizedlightistransmittedthroughsuchamediumwithitsstateofpolarizationunchanged.Whenellipticallypolarizedlightpropagatesthroughsuchamedium,theorientationofthepolarizationellipserotatesasafunctionofpropagationdistanceduetothenonlinearinteraction.

Theelectricfieldvectorofsuchabeamcanalwaysbedecomposedintoalinearcombinationofleft-andright-handcircularcomponentsas(2)(3)wherethecircular-polarizationunitvectorsaredefinedby

left-handcircularpolarizationright-handcircularpolarization

(4)

ThenbymeansofEqs.(2)and(4),wefindthat(5)

(6)Sothenonlinearpolarization,Eq.(1),canberewrittenas(7)

(8)(10)

(9)Forconvenience,wesummarytheseresultsas(12)wherewehaveintroducedtheeffectivenonlinearsusceptibilities(11)Ontheotherhand,wenotethatthefieldmustsatisfythewaveequation:

(13)where

(14)(15)Obviously,Eqs.(14)and(15)canbewrittenas

(16)(17)

(18)

(19)InsertionofEq.(11)intoEq.(19)yields(20)

Further,Eq.(20)canberewrittenas

(21)wherewehaveintroducedtheeffectiverelativepermittivity

(22)

(23)

(24)Hence,therefractiveindicesforleft-andright-handcircularpolarizedcomponentsofthebeamareslightlydifferent,thatis

(25)(26)Thedifferenceintheirrefractiveindicesisgivenby(27)

Sincetheleft-andright-handcircularcomponentspropagatewithdifferentphasevelocities,thepolarizationellipseofthelightwillrotateasthebeampropagatesthroughthenonlinearmedium.Inordertodeterminetheangleofrotation,weexpressthefieldamplitudeas

(28)

Theconnectionbetweentwocoordinatesystemsis

(29)thatis,(30)

Asmentionedabove,therearetwocasesinwhichthepolarizationellipsedoesnotrotate.

Inthiscasethenonlinearpolarizationiswhichleadstoanonlinearrefractiveindex(31)

Theothercaseinwhichthereisnorotationisthatoflinearlypolarizedlight.

Sincelinearlypolarizedlightisacombinationofequalamountsofleft-andright-handcircularcomponents,thatis

wefindfromEq.(24)that