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PrefacetotheFourthEdition.....................................................xv
PrefacetotheThirdEdition......................................................xvii
PrefacetotheSecondEdition.....................................................xix
PrefacetotheFirstEdition.......................................................xxi
Chapter1:TheNonlinearOpticalSusceptibility..................................1
1.1IntroductiontoNonlinearOptics ..............................................1
1.2DescriptionsofNonlinearOpticalProcesses .................................4
1.2.1Second-HarmonicGeneration... .........................................4
1.2.2Sum-andDifference-FrequencyGeneration............................6
1.2.3Sum-FrequencyGeneration..............................................7
1.2.4Difference-FrequencyGeneration........................................8
1.2.5OpticalParametricOscillation.. .........................................9
1.2.6Third-OrderNonlinearOpticalProcesses...............................10
1.2.7Third-HarmonicGeneration..............................................10
1.2.8Intensity-DependentRefractiveIndex...................................11
1.2.9Third-OrderInteractions(GeneralCase)................................11
1.2.10ParametricversusNonparametricProcesses.............................13
1.2.11SaturableAbsorption.....................................................14
1.2.12Two-PhotonAbsorption..................................................15
1.2.13StimulatedRamanScattering.............................................16
1.3FormalDefinitionoftheNonlinearSusceptibility.. ..........................16
1.4NonlinearSusceptibilityofaClassicalAnharmonicOscillator.............20
1.4.1NoncentrosymmetricMedia..............................................21
1.4.2Miller’sRule. .............................................................26
1.4.3CentrosymmetricMedia..................................................27
1.5PropertiesoftheNonlinearSusceptibility ....................................32
1.5.1RealityoftheFields .......................................................33
1.5.2IntrinsicPermutationSymmetry.........................................34
1.5.3SymmetriesforLosslessMedia..........................................34
1.5.4FieldEnergyDensityforaNonlinearMedium..........................35
1.5.5Kleinman’sSymmetry....................................................37
1.5.6ContractedNotation.......................................................38
1.5.7EffectiveValueof d(deff ) ................................................40
1.5.8SpatialSymmetryoftheNonlinearMedium............................41
1.5.9InfluenceofSpatialSymmetryontheLinearOpticalPropertiesofa MaterialMedium..........................................................41
1.5.10InfluenceofInversionSymmetryontheSecond-OrderNonlinear Response ..................................................................42
1.5.11InfluenceofSpatialSymmetryontheSecond-OrderSusceptibility...44
1.5.12NumberofIndependentElementsof χ (2) ijk (ω3 ,ω2 ,ω1 ) ..................45
1.5.13DistinctionbetweenNoncentrosymmetricandCubicCrystalClasses.45
1.5.14DistinctionbetweenNoncentrosymmetricandPolarCrystalClasses..50
1.5.15InfluenceofSpatialSymmetryontheThird-OrderNonlinearResponse50 1.6Time-DomainDescriptionofOpticalNonlinearities...
1.7Kramers–KronigRelationsinLinearandNonlinearOptics.................56
1.7.1Kramers–KronigRelationsinLinearOptics............................56
1.7.2Kramers–KronigRelationsinNonlinearOptics.........................59
2.1TheWaveEquationforNonlinearOpticalMedia............................65
2.2TheCoupled-WaveEquationsforSum-FrequencyGeneration.
2.4Quasi-Phase-Matching(QPM)
2.7.1ApplicationsofSecond-HarmonicGeneration..........................98
2.8Difference-FrequencyGenerationandParametricAmplification..
2.9OpticalParametricOscillators.
2.9.1InfluenceofCavityModeStructureonOPOTuning...................105
2.10NonlinearOpticalInteractionswithFocusedGaussianBeams..............109
2.10.1ParaxialWaveEquation...................................................109
2.10.2GaussianBeams...........................................................110
2.10.3HarmonicGenerationUsingFocusedGaussianBeams.................112
2.11NonlinearOpticsatanInterface...............................................116
2.12AdvancedPhaseMatchingMethods ..........................................121
Chapter3:Quantum-MechanicalTheoryoftheNonlinearOpticalSusceptibility..137
3.1Introduction ....................................................................137
3.2SchrödingerEquationCalculationoftheNonlinearOpticalSusceptibility.138
3.2.1EnergyEigenstates........................................................139
3.2.2PerturbationSolutiontoSchrödinger’sEquation........................140
3.2.3LinearSusceptibility ......................................................142
3.2.4Second-OrderSusceptibility. .............................................144
3.2.5Third-OrderSusceptibility. ...............................................146
3.2.6Third-HarmonicGenerationinAlkaliMetalVapors....................148
3.3DensityMatrixFormulationofQuantumMechanics..
3.3.1Example:Two-LevelAtom...............................................158
3.4PerturbationSolutionoftheDensityMatrixEquationofMotion..
3.5DensityMatrixCalculationoftheLinearSusceptibility .....................161
3.5.1LinearResponseTheory.. ................................................164
3.6DensityMatrixCalculationoftheSecond-OrderSusceptibility....
3.6.1 χ (2) intheLimitofNonresonantExcitation.............................178
3.7DensityMatrixCalculationoftheThird-OrderSusceptibility. ..............179
3.8ElectromagneticallyInducedTransparency...................................184
3.9Local-FieldEffectsintheNonlinearOptics..................................192
3.9.1Local-FieldEffectsinLinearOptics.....................................192
3.9.2Local-FieldEffectsinNonlinearOptics.................................194
Chapter4:TheIntensity-DependentRefractiveIndex.............................203
4.1DescriptionsoftheIntensity-DependentRefractiveIndex...................203
4.2TensorNatureoftheThird-OrderSusceptibility... ..........................209
4.2.1PropagationthroughIsotropicNonlinearMedia........................213
4.3NonresonantElectronicNonlinearities.... ....................................217
4.3.1Classical,AnharmonicOscillatorModelofElectronicNonlinearities.218
4.3.2Quantum-MechanicalModelofNonresonantElectronicNonlinearities218
4.3.3 χ (3) intheLow-FrequencyLimit........................................222
4.4NonlinearitiesDuetoMolecularOrientation .................................223
4.4.1TensorPropertiesof χ (3) fortheMolecularOrientationEffect.........229
4.5ThermalNonlinearOpticalEffects............................................231
4.5.1ThermalNonlinearitieswithContinuous-WaveLaserBeams..........233
4.5.2ThermalNonlinearitieswithPulsedLaserBeams.......................234
4.6SemiconductorNonlinearities .................................................235
4.6.1NonlinearitiesResultingfromBand-to-BandTransitions...............235
4.6.2NonlinearitiesInvolvingVirtualTransitions.............................241
4.7ConcludingRemarks...........................................................243
5.1NonlinearSusceptibilitiesCalculatedUsingTime-IndependentPerturbation Theory..........................................................................249
5.1.1HydrogenAtom...........................................................250
5.1.2GeneralExpressionfortheNonlinearSusceptibilityintheQuasi-Static Limit.......................................................................251
5.2SemiempiricalModelsoftheNonlinearOpticalSusceptibility .............255 ModelofBoling,Glass,andOwyoung ...............................................256
5.3NonlinearOpticalPropertiesofConjugatedPolymers.
5.4Bond-ChargeModelofNonlinearOpticalProperties...
5.5NonlinearOpticsofChiralMedia.............................................264
5.6NonlinearOpticsofLiquidCrystals..........................................266
6.1Introduction ....................................................................273
6.2DensityMatrixEquationsofMotionforaTwo-LevelAtom
6.2.1ClosedTwo-LevelAtom..................................................276
6.2.2OpenTwo-LevelAtom...................................................279
6.2.3Two-LevelAtomwithaNon-RadiativelyCoupledThirdLevel........279
6.3Steady-StateResponseofaTwo-LevelAtomtoaMonochromaticField...280
6.4OpticalBlochEquations.......................................................288
6.4.1HarmonicOscillatorFormoftheDensityMatrixEquations...........291
6.4.2Adiabatic-FollowingLimit...............................................293
6.5RabiOscillationsandDressedAtomicStates... .............................295
6.5.1RabiSolutionoftheSchrödingerEquation..............................296
6.5.2SolutionforanAtomInitiallyintheGroundState......................298
6.5.3DressedStates.............................................................302
6.5.4InclusionofRelaxationPhenomena.....................................305
6.6OpticalWaveMixinginTwo-LevelSystems.................................307
6.6.1SolutionoftheDensityMatrixEquationsforaTwo-LevelAtominthe PresenceofPumpandProbeFields......................................308
6.6.2NonlinearSusceptibilityandCoupled-AmplitudeEquations...........315
Chapter7:ProcessesResultingfromtheIntensity-DependentRefractiveIndex....321
7.1Self-FocusingofLightandOtherSelf-ActionEffects.
7.1.1Self-TrappingofLight....................................................324
7.1.2MathematicalDescriptionofSelf-ActionEffects.......................327
7.1.3LaserBeamBreakupintoManyFilaments..............................328
7.1.4Self-ActionEffectswithPulsedLaserBeams...........................333
7.2OpticalPhaseConjugation.....................................................334
7.2.1AberrationCorrectionbyPhaseConjugation............................336
7.2.2PhaseConjugationbyDegenerateFour-WaveMixing..................338
7.2.3PolarizationPropertiesofPhaseConjugation...........................345
7.3OpticalBistabilityandOpticalSwitching ....................................349
7.3.1AbsorptiveBistability.....
7.3.2RefractiveBistability
7.5PulsePropagationandTemporalSolitons....................................365
7.5.1Self-PhaseModulation....
7.5.2PulsePropagationEquation..............................................368 7.5.3TemporalOpticalSolitons................................................372
8.1.1FluctuationsastheOriginofLightScattering...........................382
8.1.2ScatteringCoefficient.....................................................384
8.1.3ScatteringCrossSection..................................................385
8.2MicroscopicTheoryofLightScattering......................................386
8.3ThermodynamicTheoryofScalarLightScattering....
8.3.1IdealGas...................................................................394
8.3.2SpectrumoftheScatteredLight..........................................395
8.3.3BrillouinScattering.......................................................395
8.3.4StokesScattering(FirstTerminEq.(8.3.36))...........................398
8.3.5Anti-StokesScattering(SecondTerminEq.(8.3.36))..................400
8.3.6RayleighCenterScattering...............................................402
8.4Acoustooptics..................................................................403
8.4.1BraggScatteringofLightbySoundWaves..............................403
8.4.2Raman–NathEffect.......................................................412
Chapter9:StimulatedBrillouinandStimulatedRayleighScattering..............419
9.1StimulatedScatteringProcesses.
9.3.1PumpDepletionEffectsinSBS..........................................431 9.3.2SBSGenerator............................................................433
9.3.3TransientandDynamicalFeaturesofSBS..............................436
9.4PhaseConjugationbyStimulatedBrillouinScattering.
9.5StimulatedBrillouinScatteringinGases.
9.6GeneralTheoryofStimulatedBrillouinandStimulatedRayleighScattering443 9.6.1Appendix:DefinitionoftheViscosityCoefficients.....................454
10.1TheSpontaneousRamanEffect
11.4IntroductiontothePhotorefractiveEffect. ....................................507
11.5PhotorefractiveEquationsofKukhtarevetal........... ......................508
11.6Two-BeamCouplinginPhotorefractiveMaterials...........................511
11.7Four-WaveMixinginPhotorefractiveMaterials.............................518
11.7.1ExternallySelf-PumpedPhase-ConjugateMirror.......................519
11.7.2InternallySelf-PumpedPhase-ConjugateMirror........................519
11.7.3DoublePhase-ConjugateMirror... ......................................520
11.7.4OtherApplicationsofPhotorefractiveNonlinearOptics................521
Chapter12:OpticallyInducedDamageandMultiphotonAbsorption.............523
12.1IntroductiontoOpticalDamage
12.2Avalanche-BreakdownModel.................................................524
12.3InfluenceofLaserPulseDuration...
12.4DirectPhotoionization.........................................................528
12.5MultiphotonAbsorptionandMultiphotonIonization........
12.5.1TheoryofSingle-andMultiphotonAbsorptionandFermi’sGolden Rule........................................................................530
12.5.2Linear(One-Photon)Absorption.........................................532
12.5.3Two-PhotonAbsorption..................................................535
13.3.1Self-Steepening...........................................................548
13.3.2Space–TimeCoupling....
13.3.3SupercontinuumGeneration..............................................551
13.4Intense-FieldNonlinearOptics................................................552
13.5MotionofaFreeElectroninaLaserField
13.7TunnelIonizationandtheKeldyshModel
13.8NonlinearOpticsofPlasmasandRelativisticNonlinearOptics.............560
13.9NonlinearQuantumElectrodynamics.
TheNonlinearOpticalSusceptibility
1.1IntroductiontoNonlinearOptics
Nonlinearopticsisthestudyofphenomenathatoccurasaconsequenceofthemodificationof theopticalpropertiesofamaterialsystembythepresenceoflight.Typically,onlylaserlight issufficientlyintensetomodifytheopticalpropertiesofamaterialsysteminthismanner.The beginningofthefieldofnonlinearopticsisoftentakentobethediscoveryofsecond-harmonic generationbyFrankenetal.(1961),shortlyafterthedemonstrationofthefirstworkinglaser byMaimanin1960.∗ Nonlinearopticalphenomenaare“nonlinear”inthesensethattheyoccur whentheresponseofamaterialsystemtoanappliedopticalfielddependsinanonlinearmanner onthestrengthoftheappliedopticalfield.Forexample,second-harmonicgenerationoccurs asaresultofthepartoftheatomicresponsethatscalesquadraticallywiththestrengthofthe appliedopticalfield.Consequently,theintensityofthelightgeneratedatthesecond-harmonic frequencytendstoincreaseasthesquareoftheintensityoftheappliedlaserlight.
Inordertodescribemorepreciselywhatwemeanbyanopticalnonlinearity,letusconsider howthedipolemomentperunitvolume,orpolarization ˜ P(t),ofamaterialsystemdependson thestrength ˜ E(t) ofanappliedopticalfield.† Inthecaseofconventional(i.e.,linear)optics,the inducedpolarizationdependslinearlyontheelectricfieldstrengthinamannerthatcanoften bedescribedbytherelationship
∗ Itshouldbenoted,however,thatsomenonlineareffectswerediscoveredpriortotheadventofthelaser.The earliestexampleknowntotheauthoristheobservationofsaturationeffectsintheluminescenceofdyemolecules reportedbyG.N.Lewisetal.(1941).
† Throughoutthetext,weusethetilde(~)todenoteaquantitythatvariesrapidlyintime.Constantquantities, slowlyvaryingquantities,andFourieramplitudesarewrittenwithoutthetilde.See,forexample,Eq.(1.2.1). NonlinearOptics. https://doi.org/10.1016/B978-0-12-811002-7.00010-2 Copyright©2020ElsevierInc.Allrightsreserved.
wheretheconstantofproportionality χ (1) isknownasthelinearsusceptibilityand 0 isthe permittivityoffreespace.∗ Innonlinearoptics,theopticalresponsecanoftenbedescribedasa generalizationofEq.(1.1.1)byexpressingthepolarization
P(t) asapowerseriesinthefield strength ˜ E(t) as
Thequantities χ (2) and χ (3) areknownasthesecond-andthird-ordernonlinearopticalsusceptibilities,respectively.Forsimplicity,wehavetakenthefields ˜ P(t) and ˜ E(t) tobescalar quantitiesinwritingEqs.(1.1.1)and(1.1.2).InSection 1.3 weshowhowtotreatthevectornatureofthefields;insuchacase χ (1) becomesasecond-ranktensor, χ (2) becomesathird-rank tensor,andsoon.InwritingEqs.(1.1.1)and(1.1.2)intheformsshown,wehavealsoassumed thatthepolarizationattime t dependsonlyontheinstantaneousvalueoftheelectricfield strength.Theassumptionthatthemediumrespondsinstantaneouslyalsoimplies(throughthe Kramers–Kronigrelations† )thatthemediummustbelosslessanddispersionless.Weshallsee inSection 1.3 howtogeneralizetheseequationsforthecaseofamediumwithdispersionand loss.Ingeneral,thenonlinearsusceptibilitiesdependonthefrequenciesoftheappliedfields, butunderourpresentassumptionofinstantaneousresponsewetakethemtobeconstants.
Weshallreferto P (2) (t) = 0 χ (2) E 2 (t) asthesecond-ordernonlinearpolarizationandto P (3) (t) = 0 χ (3) E 3 (t) asthethird-ordernonlinearpolarization,andsoonforhigher-order terms.Weshallseelaterinthissectionthatphysicalprocessesthatoccurasaresultofthe second-orderpolarization ˜ P (2) aredistinctfromthosethatoccurasaresultofthethird-order polarization ˜ P (3) .Inaddition,weshallshowinSection 1.5 thatsecond-ordernonlinearopticalinteractionscanoccuronlyinnoncentrosymmetriccrystals—thatis,incrystalsthatdo notdisplayinversionsymmetry.Sinceliquids,gases,amorphoussolids(suchasglass),and evenmanycrystalsdisplayinversionsymmetry, χ (2) vanishesidenticallyforsuchmaterials, andconsequentlysuchmaterialscannotproducesecond-ordernonlinearopticalinteractions. Ontheotherhand,third-ordernonlinearopticalinteractions(i.e.,thosedescribedbya χ (3) susceptibility)canoccurforbothcentrosymmetricandnoncentrosymmetricmedia.
Weshallseeinlatersectionsofthisbookhowtocalculatethevaluesofthenonlinearsusceptibilitiesforvariousphysicalmechanismsthatleadtoopticalnonlinearities.Forthepresent, wemakeasimpleorder-of-magnitudeestimateofthesizeofthesequantitiesforthecommon caseinwhichthenonlinearityiselectronicinorigin(see,forinstance,Armstrongetal., 1962). Onemightexpectthatthelowest-ordercorrectionterm ˜ P (2) wouldbecomparabletothelinear
∗ Exceptwhereotherwisenoted,weusetheSI(MKS)systemofunitsthroughoutthisbook.Theappendixtothis bookpresentsaprescriptionforconvertingamongsystemsofunits.
† See,forexample,LandauandLifshitz(1960)Section62orthediscussioninSection 1.7 ofthisbookfora discussionoftheKramers–Kronigrelations.
response P (1) whentheamplitudeoftheappliedfield E isoftheorderofthecharacteristic atomicelectricfieldstrength Eat = e/(4π 0 a 2 0 ),where e isthechargeoftheelectronand a0 = 4π 0 2 /me 2 istheBohrradiusofthehydrogenatom(here isPlanck’sconstantdivided by2π ,and m isthemassoftheelectron).Numerically,wefindthat Eat = 5.14 × 1011 V/m. Wethusexpectthatunderconditionsofnonresonantexcitationthesecond-ordersusceptibility χ (2) willbeoftheorderof χ (1) /Eat .Forcondensedmatter χ (1) isoftheorderofunity,andwe henceexpectthat χ (2) willbeoftheorderof1/Eat ,orthat
Similarly,weexpect χ (3) tobeoftheorderof χ (1) /E 2 at ,whichforcondensedmatterisofthe orderof
Thesepredictionsareinfactquiteaccurate,asonecanseebycomparingthesevalueswith actualmeasuredvaluesof χ (2) (see,forinstance,Table 1.5.3)and χ (3) (see,forinstance,Table 4.3.1).
Forcertainpurposes,itisusefultoexpressthesecond-andthird-ordersusceptibilitiesin termsoffundamentalphysicalconstants.Asjustnoted,forcondensedmatter χ (1) isofthe orderofunity.Thisresultcanbejustifiedeitherasanempiricalfactorcanbejustifiedmore rigorouslybynotingthat χ (1) istheproductofatomicnumberdensityandatomicpolarizability. Thenumberdensity N ofcondensedmatterisoftheorderof (a0 ) 3 ,andthenonresonant polarizabilityisoftheorderof (a0 )3 .Wethusdeducethat χ (1) isoftheorderofunity.Using theexpressionfor E quotedabove,wesimilarlyfindthat χ (2) (4π 0 )3 4 /m2 e 5 and χ (3) (4π 0 )6 8 /m4 e 10 .SeeBoyd(1999)forfurtherdetails.
Themostusualprocedurefordescribingnonlinearopticalphenomenaisbasedonexpressingthepolarization P(t) intermsoftheappliedelectricfieldstrength E(t),aswehavedone inEq.(1.1.2).Thereasonwhythepolarizationplaysakeyroleinthedescriptionofnonlinear opticalphenomenaisthatatime-varyingpolarizationcanactasthesourceofnewcomponents oftheelectromagneticfield.Forexample,weshallseeinSection 2.1 thatthewaveequationin nonlinearopticalmediaoftenhastheform
(1.1.5)
where n istheusuallinearrefractiveindexand c isthespeedoflightinvacuum.Wecan interpretthisexpressionasaninhomogeneouswaveequationinwhichthepolarization ˜ P NL associatedwiththenonlinearresponseactsasasourcetermfortheelectricfield E .Since ∂ 2 ˜ P NL /∂t 2 isameasureoftheaccelerationofthechargesthatconstitutethemedium,this
