非线性光学英文版课件：Lecture 14- Self-Focusing of Light and Other Self-Action Effects

2011-8-29DepartmentofPhysicsSoutheastUniversityLecturefourteen:

Self-FocusingofLightandOtherSelf-ActionEffects

(Page329)Inthislecture,weexploreseveralprocessesofpracticalimportancethatoccurasaresultoftheintensity-dependentrefractiveindex.I)Self-focusing:anintensebeamoflightmodifiestheopticalpropertiesofamaterialmediuminsuchamannerthatthebeamiscausedtocometoafocuswithinthematerial.A.Self-ActionEffectsofLight

Thusthematerialactsasifitwereapositivelens,causingthebeamtocometoafocuswithinthematerial.(Self-focusingeffect)DiffractionSelf-trappingII)

Self-trapping:abeamoflightpropagateswithaconstantdiameterasaconsequenceofanexactbalancebetweenself-focusinganddiffractioneffects.Ananalysisshowsthatself-trappingcanoccuronlyifthepowercarriedbythebeamisexactlyequaltotheso-calledcriticalpowerforself-trapping:(1)

Example:CalculationofthecriticalpowerforCS2

Letusbeginouranalysisofself-actioneffectsbydevelopingasimplemodeloftheself-focusingprocess.Forthepresent,weignoretheeffectsofdiffraction;theseeffectsareintroducedlater.Theneglectofdiffractionisjustifiedifthebeamdiameterorintensity(orboth)issufficientlylarge.Collimatedbeam

B.ModelforSelf-FocusingofLight

Fermat’sprinciple:

(2)Fermat’sprinciplethentellsusthat

(3)

(4)

Forsmallerlaserpowers,theself-focusingdistancecanbeestimatedbynotingthatthebeamconvergenceangleisreducedbydiffractioneffectsandisgivenapproximatelyby

(5)

(6)Infact,oneexpectsself-trappingtooccurwhenthetendencyofabeamtospreadasaconsequenceofdiffractionispreciselybalancedbythetendencyofthebeamtocontractasaconsequenceofself-focusingeffects.Theconditionforself-trappingcanthusbeexpressedmathematicallyasastatementthatthediffractionanglebeequaltotheself-focusinganglethatis,(7)

fromwhichwehave

C.MathematicalDescriptionofSelf-ActionEffectsSelf-actioneffectscanbedescribedmorerigorouslybymeansofthenonlinearopticalwaveequation.Forthepresentweconsidersteady-stateconditionsonly,aswouldapplyforexcitationwithacontinuous-wavelaserbeam.Undertheseconditions,thenonlinearwaveequation

canbereducedtotheparaxialwaveequation(8)

Equation(8)isjustthewell-knownnonlinearSchrödingerequation(NLS).WeconsiderfirstthesolutionofEq.(8)forabeamthatisallowedtovaryinonetransversedimensiononly.(Thereexistsnoanalyticalsolutionsfortwodimensions).(9)Hencethisequation(applicableforplanarwaveguide)becomes

Asonecanverify(willbeleftasanexercise),Eq.(9)possessesananalyticalsolutionoftheform(10)(11)wherethewidthofthefielddistributionisgivenbyandtherateofnonlinearphaseacquisitionisgivenby(12)

ThesolutiongivenbyEq.(10)issometimesreferredtoasaspatialsoliton,becauseitdescribesafieldthatcanpropagateforlongdistanceswithaninvariantt