Optical Properties of Solids LM Herz Trinity Term 2012 ...

Interaction of Electromagnetic Radiation with MatterreflectedlightincidentlightAbsorptiontransmittedlightScattered light(phonons, impurities...)LuminescenceMEDIUMLM Herz - Optical Properties of SolidsTT2012 2

I Absorption and ReflectionMacroscopic ElectromagnetismMaxwell’sequations:(no net free charges)(non-magnetic)(ohmic conduction)Linear OpticsIn a linear, non-conducting medium:Solution:where:complex!Define complex refractive index:refractionabsorptionLM Herz - Optical Properties of SolidsTT2012 3

AbsorptionIntensity decay of wave:(Beer’s law)Define absorption coefficient:At normal incidence:ReflectionReflectivity R for a material’s surfacecontains information on its absorption!Relationship between components of ñ and ε randif(weak absorption):LM Herz - Optical Properties of SolidsTT2012 4

The classical dipole oscillator modelInside a material the electric field of an EM wave mayinteract with:• bound electrons (e.g. interband transitions)• ions (lattice interactions)• free electrons (plasma oscillations)Equation of motion for a bound electron in 1D:wherestationary solutions:Displacement of charge causes polarisation:(N: oscillator density)backgroundLM Herz - Optical Properties of SolidsTT2012 5

Real and imaginary part of ε rCan now calculateandOptical constants for a classical dipole oscillatorLM Herz - Optical Properties of SolidsTT2012 6

Local field correctionsIn a dense medium:• atoms experience “local field” composed of externalfield E and polarization from surrounding dipoles• treat interacting dipole as being at centre of spheresurrounded by a polarized dielectricClausius-Mossotti relationship:electricsusceptibilityper atomProblems with the classical oscillator model• no information on selection rulesneed quantum mechanics• interband transitions should depend on the density ofstates g(E)• One possible modification: “oscillator strength”(from QM)write:line shape of transition(from classical oscillator model)LM Herz - Optical Properties of SolidsTT2012 7

LM Herz - Optical Properties of SolidsTT2012 8

II Interband optical transitionsconduction bandfivalence bandTreat interband transitions through timedependentperturbation theory - Fermi’sgolden rule gives transition probability:joint densityof stateswith matrix element:wheredipole momentE-field of incident waveBlochwavefunctionsMatrix element for interband transitions:absorptiondeduce conditions for dipole allowed(direct) transitions.emissionConsider:(1) wavevector conservation(2) Parity selection rule(3) dependence on photon energyLM Herz - Optical Properties of SolidsTT2012 9

Conditions for direct interband transitions(1) wavevector conservationonly iforabsorptionemissionEtypically:“vertical”transitionsE gk(2) parity selection ruleodd parityonly if and have different parity!In a typical 4-valent system (e.g. group IV or III-V compound):pisolatedatomsp3-hybridp-antibondings-antibondingin crystalconduction bandsp-bondingE gvalence bands-bondingexpect to see strong absorption for these materialsLM Herz - Optical Properties of SolidsTT2012 10

(3) Dependence of transition probability on photon energyFinal state is an electron-hole pairwherereduced effective massIf M if is independent of the photon energy hω, the jointdensity of states contains the dependence of the transitionprobability on hω. For this case:Absorption coefficient(for direct transitions):Examples for direct semiconductorsa) InSb at 5K b) GaAs at 300Kdatadeviations frome.g. due tophonon absorption or non-parabolicity of the bands.LM Herz - Optical Properties of SolidsTT2012 11

Indirect interband transitions• Indirect gap: valence band maximum and conduction bandminimum lie at different wavevectors,• direct transitions across the indirect gap forbidden, butphonon-assisted transitions may be possible.(i) wavevector conservation:Ewith phonon wavevectorq phononE gk(ii) probability for indirect transitions:• perturbation causing indirect transitions is second orderoptical absorption much weaker than for directtransitions!• find absorption coefficientphonon absorptionor emissionLM Herz - Optical Properties of SolidsTT2012 12

Example for an indirect semiconductor: SiBand structure:mostlyindirect directtransitions transitionsE E1g E 2E 1E gE 2Absorption spectrum:LM Herz - Optical Properties of SolidsTT2012 13

LM Herz - Optical Properties of SolidsTT2012 14

III ExcitonsAbsorption coefficient of CuO 2 at 77K:Series of absorption peaksjust below the energy gapCoulomb interactionbetween electron and holegives rise to “excitonic”states (bound electron-holepairs)Wannier-Mott Excitonseh• weakly bound (free) excitons• binding energy ~ 10meV• common in inorganicsemiconductors (e.g. GaAs,CdS, CuO 2 ...)• particle moving in a mediumof effective dielectricconstant ε rFrenkel Excitonseh• strongly (tightly) bound excitons• binding energy ~ 0.1 – 1eV• typically found in insulators andmolecular crystals (e.g. rare gascrystals, alkali halides, aromaticmolecular crystals)• particle often localized on justone atomic/molecular siteLM Herz - Optical Properties of SolidsTT2012 15

Weakly bound (Wannier) ExcitonsSeparate exciton motion into centre-of-mass and relative motion:CM motion: exciton momentum:whereexciton mass:kinetic energy:Relative motion:Binding energy:where(Rydberg)reduced massExciton radius:wherea 0 = 0.529Å (Bohr radius)E-k diagram for the weakly bound exciton(a) uncorrelated electron-hole pair(one-electron picture)E(b) exciton (one-particle picture)Ewavevector conservation:;kwavevector conservation:...R Xn=2En=1gktransitions where light lineintercepts with“vertical transitions”forcontinuum onset at band edgeLM Herz - Optical Properties of SolidsTT2012 16

Exciton-PolaritonphotonEIIIexcitonk• Absorption occurs at point wherephoton dispersion intersects excitondispersion curve.• exciton-photon interaction leads tocoupled EM and polarization wave(polariton) travelling in the mediumaltered dispersion curve(2 branches)• But: if exciton damping (phononscattering...) is larger than excitonphotoninteraction we can treatphotons and excitons separately.Examples for weakly bound excitons: GaAs• sub-gap excitonic absorptionfeatures• exciton dissociation throughcollisions with LO phononsbecomes more likely athigher T → exciton lifetimeshortened and transition linebroadened• Coulomb interactionsincrease the absorption bothabove and below the gapLM Herz - Optical Properties of SolidsTT2012 17

Examples for weakly bound excitons: GaAsAt low temperature (here: 1.2K) and in ultra pure material, thesmall line width allows observation of higher excitonic transitions:n=1n=2n=3here:E gTightly bound (Frenkel) excitons• radius of weakly-bound excitons:model of bound e-h pair in dielectric mediumbreaks down when a n is of the order of interatomicdistances (Å)have tightly bound excitons for small ε r , large μ• tightly-bound electron-hole pair, typically located onsame unit (atom or molecule) of the crystal (but thewhole exciton may transfer through the crystal)• large binding energies (0.1 – 1eV) → excitons persist atroom temperature.LM Herz - Optical Properties of SolidsTT2012 18

Transition energies for tightly bound excitons• transition energies often correspond to those found in theisolated atom or molecule that the crystal is composed of• theoretical calculations may be based e.g. on tightbindingor quantum-chemical methods• often need to include effects of strong coupling betweenexcitons and the crystal lattice (polaronic contributions)Examples for tightly bound excitons: rare gas crystalsabsorption spectrum of crystalline Kr at 20K:E 1 =10.2eVE g =11.7eVE b =1.5eVNote: the loweststrong absorptionin isolated Kr is at9.99eVclose to lowestexcitonic transitionE 1 in crystalLM Herz - Optical Properties of SolidsTT2012 19

LM Herz - Optical Properties of SolidsTT2012 20

VI Low-dimensional systemsde Broglie wavelength for anelectron at room temperature:If we can make structured semiconductors on these lengthscales we may be able to observe quantum effects!Possible using e.g. molecular beam epitaxy (MBE) ormetal-organic chemical vapour deposition (MOCVD)E(k=0)growth direction zAl x Ga 1-x As (barrier)GaAs (well)Al x Ga 1-x As (barrier)GaAs substratedconfinedholesvalencebandE g(AlGaAs)E g(GaAs)E g(AlGaAs)conductionbandconfinedelectronsEffect of confinement on the DOSConfinement in a particular direction results in discrete energy states, butfree movement in other directions gives rise to continuum.→ Joint density of states g(hω) (for direct CB-VB transitions):3D (bulk) 2D (Q well) 1D (Q wire) 0D (Q dot)LM Herz - Optical Properties of SolidsTT2012 21

Quantum well with infinite potential barriersSchrödinger’s eqn inside the well:outside the well:wavefunction along z:with wavevectorconfinement energy:Bandstructure modifications from confinementbulk GaAs(Γ-valley)EelectronsJ=1/2GaAs QWEE 2,eE 1,ekJz =3/2heavyholes J z =1/2lightholesholesJ=3/2E 1,lhE 1,hhbandstructure for 2D motion isaltered (quantum confinementleads to valence band mixing)LM Herz - Optical Properties of SolidsTT2012 22

Optical transitions in a quantum wellas before, matrix element:wavefunctions now:valence/conduction bandBloch functionhole/electronwavefunction along z(i) changes slowly over a unit cell (compared to u c , u v )(ii) unless (k-conservation)dipole transition criteria (as before)electron-hole spatial overlap in wellSelection rules for optical transitions in a QW(i) wavevector conservation:(ii) parity selection rule: and must differ in parity(iii)asbeforeneed sufficient spatial overlap between electron and hole wavefunctionsalong the z-direction. For an infinite quantum well:unlessN.B.: expect some deviation in finite quantum wells!(iv) unless and have equal parityLM Herz - Optical Properties of SolidsTT2012 23

(v) energy conservation:hh1 – e1lh1 – e1hh2 – e2e2e1E ghh1hh2 lh1photonenergyand forbandgap ofwellmaterialconfinementenergy ofelectron/holekineticenergy inplane of QW(at the band edge):alloweddipoletransitionsExample: absorption of a GaAs/AlAs QWAbsorption of GaAs/AlAs MQW (d=76Å) at 4K:hh1-e1(X)lh1-e1(X)e1-hh3(X)hh2-e2(X)lh2-e2(X)hh3-e3(X)e3-hh1(X)hh2-e2 (continuum)lh1-e1 (continuum)lh2-e2 (continuum)hh1-e1 (continuum)below each onset ofabsorption: excitonicfeatures (X)above onset: flatabsorption, since 2Djoint density of statesindependent of hωdeviation from Δn=0,in particular at high ELM Herz - Optical Properties of SolidsTT2012 24

Influence of confinement on the excitonbulk GaAshhlhn=1T=300Kn=2GaAs MQWConfinement bringselectron and hole closertogether.enhanced excitonbinding energyincreased oscillatorstrengthLM Herz - Optical Properties of SolidsTT2012 25

LM Herz - Optical Properties of SolidsTT2012 26

V Optical response of a free electron gasClassic Lorentz dipole oscillator model (again):solutions are as before, but with(no retaining force!)dielectric constant:with plasma frequencyand background dielectric constantreal and imaginary part of the dielectric constant:AC conductivity of a free electron gasCan re-write equation of motion as:electron with momentum p is accelerated by field but loosesmomentum at rateobtain electron velocity:and usingAC conductivitywhere(DC conductivity)andoptical measurements of ε r equivalent to those of AC conductivity!LM Herz - Optical Properties of SolidsTT2012 27

Low-frequency regimeAt low frequency of the EM wave, or :and one may approximate:Skin depth (distance from surface at which incident power hasfallen to 1/e):For Cu at 300K: σ 0 =6.5×10 7 Ω -1 m -1δ = 8.8mm @ ν = 50Hzδ = 6.2μm @ ν = 100MHzHigh-frequency regimeIn a typical metal: N ≈ 10 28 –10 29 m -3 , σ 0 ≈ 10 7 Ω -1 m -1Drude model predicts: γ≈10 14 s -1At optical frequencies: (weak damping)and(i)(ii)largely imaginarywave mostly reflectedlargely realwave partly transmitted,weak absorption ( )LM Herz - Optical Properties of SolidsTT2012 28

Reflectivity in the high-frequency regimedopedsemiconductors:large backgrounddielectic constant( ) fromhigher-energyinterband transitionsmost metals:(if no strongoptical transitions athigher photonenergy)Example: Reflection from Alkali metalsMetalN(10 28 m -3 )ω p /2π(10 15 Hz)λ p(nm)λ UV(nm)Li4.701.95154205Na2.651.46205210K1.401.06282315Rb1.150.96312360Cs0.910.86350440measured at low T calculated from measured UVtransmission cut-off• high reflectivity up to UV wavelengths• good agreement between measurement and Drude-Lorentz modelLM Herz - Optical Properties of SolidsTT2012 29

Example: Reflection from transition metalsMetalN(10 28 m -3 )ω p /2π(10 15 Hz)λ p(nm)Cu8.472.61115Ag5.862.17138Au5.902.18138measured at low Tcalculated fromThese transition metals should be fully reflective up to deep UVBut we know: Gold appears yellow, Copper redReflection of light from Au, Cu and Al:D-L model forω p =9eV (Au)AuCuD-L model forω p =15.8eV (Al)AlDrude-Lorentz model doesnot account fully foroptical absorption oftransition metals(especially in the visible)need to considerbandstructure(damping has weakeffect at thesefrequencies)LM Herz - Optical Properties of SolidsTT2012 30

Example: Reflection from CopperElectronic configuration of Cu: [Ar] 3d 10 4s 1Transitions (in visible range of spectrum) between relativelydispersionless bands of tightly bound 3d electrons and half-filledband of 4s-electrons:E f3dbandsstrong interband absorption for → copper appears red !Example: Reflection from doped semiconductorsFree-carrier reflectivity of InSb:forwhereCan determine effective mass of majority carriersfrom free carrier absorptionLM Herz - Optical Properties of SolidsTT2012 31

Example: Free-carrier absorption in semiconductorsFor free carriers in the weak absorption regime ( ):predict:But experiments on n-type samples show:whereDeviations arise from:• intraband transitions involving phonon scattering• in p-type semiconductors: intervalence band absorption• absorption by donors bound to shallow donors or acceptorsExample: Impurity absorption in semiconductorsIn doped semiconductors the electron (hole) and the ionizedimpurity are attracted by Coulomb interaction hydrogenic systemAbsorption of Phospor-doped silicon at 4.2K:conduction bandn=2n=n=1valence bandObserve Lyman series for transitions from 1s level ofPhosphor to p levels, whose degeneracy is lifted as aresult of the anisotropic effective mass of the CB in SiLM Herz - Optical Properties of SolidsTT2012 32

PlasmonsAt the plasma edge ( ):What happens at this frequency?• Polarization induced by the EM wave:where• WavevectorP is equal and opposite to incident fieldAt the Plasma edge a uniform E-field in the materialshifts the collective electron w.r.t the ionic lattice!Plasma oscillations for ε r =0:applied EM waveresulting chargedistribution+ + + + +- - - - -induced macroscopicpolarization+ + + + ++ + + + +- - - - -+ + + + +- - - - -- - - - -• For a (transverse) wavevector , the resulting chargedistribution corresponds to a longitudinal oscillation ofthe electron gas with frequency ω p !• The quantum of such collective longitudinal plasmaoscillations is termed a plasmon.LM Herz - Optical Properties of SolidsTT2012 33

Example: Plasmons in n-type GaAsLight scattered from n-type GaAs at 300K:plasmonemissionplasmonabsorptionEnergy conservation:from data:Expectfrom:LM Herz - Optical Properties of SolidsTT2012 34

VI Optical studies of phononsDispersion relation for a diatomic linear chain:transverse optical (TO) phonon:0lightdispersionopticalbranches(1LO + 2TO)acousticbranches(1LA + 2TA)transverse acoustic (TA) phonon:EM radiation is a transverse wave with wavevectorand can thus interact directly only with TO modes in polarcrystals near the centre of the Brillouin zone.Harmonic oscillator model for the ionic crystal latticeDiatomic linear chain under the influence of an external electric field:Equations of motion:x -x +wherefrequency of TO mode near centre of Brillouinzone (with effective spring constant C)reduced massrelative displacement of positive and negative ionsLM Herz - Optical Properties of SolidsTT2012 35

Add damping term to account for finite phonon lifetime:Displacement of ions induces polarization P = NQxDielectric constant (as before):Rewrite this result in terms of the static ( ) and the high-frequency( ) limits of the dielectric constant:Lattice response in the low-damping limitLong phonon lifetimes:Consider Gauss’s law. In the absence of free charge:orwave must betransverse ( )longitudinal wavepossible ( )LM Herz - Optical Properties of SolidsTT2012 36

What happens at ε r =0 ?Again: Wavevector of EM wave in medium:+ + + + +- - - - -+ + + + +- - - - -+ + + + +- - - - -→ all ions of same chargeshift by the same amountthroughout the medium→ result can be seen as atransverse wave ( )with k ≈ 0 or as alongitudinal wave ( )in orthogonal direction.The Lyddane-Sachs-Teller relationshipAt ε r =0 the induced polarization corresponds to alongitudinal wave, i.e. ε r (ω LO )=0Lyddane-Sachs-TellerrelationshipAnd fromfollows:→ In polar crystals the LO phonon frequency is always higherthan the TO phonon frequency→ In non-polar crystals, and the LO and TO phononmodes are degenerate (at the Brillouin zone centre)LM Herz - Optical Properties of SolidsTT2012 37

Dielectric constant and Reflectivity for undamped lattice“Reststrahlen” band(R=1)Influence of dampingLattice Reflectivity:For finite phonon lifetime(γ≠0) at resonance:→ Reststrahlen band nolonger fully reflective→ general broadening offeaturesLM Herz - Optical Properties of SolidsTT2012 38

Measurements of IR reflectivityFourier transform infrared spectroscopy (FTIR):• measure interference patternI(d) as a function of mirrordisplacement d• I(d) gives Fourier transformof sample transmission T(ν)multiplied with systemresponse S(ν):detectorsampleIR sourceM2dM1Example: reflection spectra for zinc-blende-type latticesmeasured at 4.2Kmeasured at RTLM Herz - Optical Properties of SolidsTT2012 39

Phonon-PolaritonsExamine more closely the dispersion relations for phonons and theEM wave near the Brillouin zone centre:If no coupling occurred:• But: coupling between TOphonon and EM wave leads tomodified dispersion• resulting wave is mixedmode with characteristics ofTO polarization and EM wave• LO phonon dispersion remainsunchanged as it does not coupleto the EM wavePhonon-Polariton dispersion:upper polaritonbranch→ two branches for ω(k)Reststrahlen band(perfect reflection,no mode can propagate)lower polaritonbranchLM Herz - Optical Properties of SolidsTT2012 40

Inelastic Light Scattering• Scattering of light may be caused by fluctuations of thedielectric susceptibility χ of a medium• time-dependent variation of χ may be caused by elementaryexcitations, e.g. phonons or plasmons• scattering from optical phonons is called Raman scatteringand that from acoustic phonons Brillouin scattering• if u(r,t) is the displacement (of charge) associated with theexcitation, the susceptibility can be expressed in terms of aTaylor series:Polarization in the medium:letlight wave with frequency ωlattice wave with frequency ω qwhereunscattered polarization waveAnti-Stokes scatteringStokes scatteringLM Herz - Optical Properties of SolidsTT2012 41

Energy & momentum conservation for inelastic scatteringScattering process:energy conservation:ω in , k inω out , k outω q , qwave vector conservation:Anti-Stokes scattering requires absorption of a phonon andtherefore sufficiently high temperature. In general the ratio ofAnti-Stokes to Stokes scattering intensities is given by:Maximum momentum transfer in backscattering geometry, where:Inelastic light scattering probes phonons with small wave vectorRaman spectroscopy: Experimental detailsRequire detection of optical phonons within typical frequencyrange 1cm -1 < ω p < 3000cm -1→ need excitation source (laser) with sufficiently narrowbandwidth→ need detection system with high dispersion and abilityto suppress elastically scattered lightTypical set-up:S4detectorM5filterpolarizerLaseranalyzerG2M3G1S2,3M4M2doublegratingspectrometersampleS1M1LM Herz - Optical Properties of SolidsTT2012 42

Raman spectra for zinc-blende-type semiconductorsBrillouin scattering: Experimental detailsRequire detection of acoustic phonons near the centre of theBrillouin zone where ω q =v ac q → need to be able to measure shiftsof only a few cm -1 !Set-up based on a MultipassInterferometer:Brillouin spectrum for Si(100):LM Herz - Optical Properties of SolidsTT2012 43

Phonon lifetimesExperimental evidence for finite phonon lifetimes fromi. Reflectivity measurements: R

VII Optics of anisotropic mediaA medium is anisotropic if its macroscopic optical propertiesdepend on directionExamples:anisotropicisotropicgas, liquid,amorphoussolidpolycrystalline crystalline liquid crystalε r and χ in an anisotropic mediumPolarizability now depends on direction in which E-field is applied→ relative electric permittivity ε r and susceptibility χ now tensors:or→ D and E no longer necessarily point into the same direction!But can always find coordinate system for which off-diagonalelements vanish, in which case:In the directions of these principal crystal axes E and D are parallel.LM Herz - Optical Properties of SolidsTT2012 45

Propagation of plane waves in an isotropic mediumAmpere’s and Faraday’s law for plane waves:whereis the wavevector in free space.Choosing a coordinate system along the crystal’s principal axesyields:homogeneous matrix equation, requireSolving the matrix equation () yields:This provides a dispersion relationshipCan obtain the refractive index from the ratio of phasevelocities in vacuo and inside medium:LM Herz - Optical Properties of SolidsTT2012 46

Propagation of plane waves in uniaxial crystalsIn uniaxial crystals (optic axis along z): ,Two solutions:(i) Sphere:for ordinary ray(polarized ⊥ to k-z plane)(ii) ellipsoid of revolutionk 3k-directionn o k 0n(θ)kn o k 00θn o k 0n e k 0k2for extraordinary ray(polarized in k-z plane)Ordinary vs extraordinary rays in uniaxial crystalsk 3θn o k 0n o k 0k 3θSSHE,Dkk 2EDn o k 0n e k 0 k 2Hk(a) ordinary ray:E, D polarized ⊥ to planecontaining k and the optic axis;Refractive index:(b) extraordinary ray:E, D polarized in planecontaining k and the optic axisLM Herz - Optical Properties of SolidsTT2012 47

Comments on wave propagation in uniaxial crystals1) Faraday’s & Ampere’s law for plane waves in dielectrics:D is normal to both k and HH is normal to both k and EN.B.: this does not imply E ⊥ k!2) All fields are of the formwavefronts are ⊥ to k.3) The phase velocity v is in the direction of k with4) As usual, the group velocity isv g is normal to the k-surface!5) The pointing vector is normal to E and HCan show: for small Δk Snormal to k-surface6) From (5) and (1) follows that E is parallel to the k-surface.k 0e-rayϕαRefraction at the surface of a uniaxial crystalβkcrystalairopticaxisphase matching condition:N.B.: Snell’s law holds for the directions ofk in the media, but this is not necessarilythe direction of ray propagation!Example: Double refraction at normal incidence:LM Herz - Optical Properties of SolidsTT2012 48

Linear optics:VIII Non-linear OpticsPolarization depends linearly on the electric field:Electrons experience harmonic retaining potentialrefractive index n, absorption coefficient α, reflectivityR independent of incident EM wave’s intensityBut:If E-fields become comparable to those binding electrons in theatom, anharmonic (non-linear) effects become significant.For an H-atom:need EM wave intensityPossible with tightly focused laser beams!The non-linear susceptibility tensorFor a medium in whichwe may in general write:linearnon-linear partnow power-dependent!In an anisotropic medium, non-linear response will depend ondirections of E-fields wrt the crystal:Second-order non-linear polarization components:Third-order non-linear polarization components:LM Herz - Optical Properties of SolidsTT2012 49

Non-linear medium response to sinusoidal driving fieldlow appliedfield amplitude:high appliedfield amplitude:If and the applied field , then:second-order nonlinearity: rectification and frequency doublingthird-order nonlinearity: frequency triplingSecond-order nonlinearities (NL)Treatment of non-resonant 2 nd order NL within oscillator model:Assume anharmonic potential:Equation of motion:Use trial solution:Assume→Obtaindisplacementamplitudes:LM Herz - Optical Properties of SolidsTT2012 50

Calculating the induced polarization:linear susceptibility, as beforesecond-order non-linear susceptibilityCan re-write the second-order non-linear susceptibility as:→ materials with large linear susceptibilty also have alarge non-linear susceptibility→ in a centrosymmetric medium, U(x) = U(-x)and therefore C 3 = 0 and χ (2) =0second-order nonlinearities only occur inmedia that lack inversion symmetry!(This may also be shown directly from the definitionof P(2) - see question sheet.)LM Herz - Optical Properties of SolidsTT2012 51

The second-order non-linear coefficient tensor d ij27 components in χ (2) ijkBut some of these components must be the same(e.g. χ (2) xyz E y E z = χ (2) xzy E z E y , so χ (2) xyz = χ (2) xzy becauseordering of fields is arbitrary)Second-order response can be described by the simplertensor d ij , i.e.In many cases crystal symmetry requires that most of thecomponents of d ij vanish.2 nd order NL: Frequency (three-wave) mixingPresume two waves are travelling in the medium, withThe induced polarization is:sum-frequencygenerationdifference-frequencygenerationFeynman diagrams for second-order nonlinear frequency mixing:ω 1ω 2ω 1 + ω 2ω 1- ω 2ω 1 - ω 2LM Herz - Optical Properties of SolidsTT2012 52

2 nd order NL: Frequency doublingConsider the generation of second harmonics in more detail:Maxwell’sequations:Wave equation:Consider propagation of second-harmonic wave in z-direction:Let this wave be generated from two fundamental waves:Obtain specific wave equation:Assume that the variation of the complex field amplitude is small(slowly varying envelope approximation):Obtain DE for increase of the second harmonic along the directionof propagation:whereis the phase mismatch betweenfundamental and second harmonic waveLM Herz - Optical Properties of SolidsTT2012 53

2 nd order NL: Phase matching conditionsE(ω)E(ω)E(2ω)0 LFor efficient frequency conversion, weneed the fundamental wave and the higherharmonic to be in phase throughout thecrystal, i.e. whereThe second-harmonic field at length L for arbitraryis:For constant fundamental wave amplitudes (thin crystal) thesecond harmonic intensity is then given by:Second-harmonic intensity after propagation through crystal oflength L without phase matchingFirst intensity minimum at:But: dispersion in media meansthat in general:Example: Sapphire→ need too thin a crystal toachieve efficient 2 ndharmonics generationLM Herz - Optical Properties of SolidsTT2012 54

2 nd order NL: Phase matching in a uniaxial crystalIn general, in the birefringent medium, but since therefractive index now depends on the direction of propagation andwave polarization wrt the optic axis, for some geometries we mayhave→ phase matching!n o2ωn oωθn eωphase-matchingdirectionn e2ωHere, phase matching occurs for the fundamental travelling asextraordinary (polarization in plane) and the 2 nd harmonic asordinary (polarization ⊥ to plane) withThird-order nonlinearitiesThird-order effects become important in centrosymmetric (e.g.isotropic media) whereFor three waves with frequencies the third-ordernonlinear polarization isgenerates a wave with(a) four-wave mixingω2ω 1ω 3ω 1 + ω 2 + ω 3(c) Optical Kerr effectωω-ωω(b) frequency triplingω ωω-ωω Sω S3ωe.g.:(d) stimulated Raman scatteringLM Herz - Optical Properties of SolidsTT2012 55

3 rd order NL: The optical Kerr effectOptical Kerr effect:andno phase mismatchIn an isotropic medium (χ (2) =0):Refractive index:wherelight intensityRefractive index varies with light intensity!Example: Kerr lensingOptical Kerr effect:at high intensity ILaser beam with spatially varyingprofile (e.g. Gaussian beam)experiences in the medium a higherrefractive index at the centre of thebeam than the outside → mediumacts as a lens!n(x)Propagation of an intense gaussian beam through a Kerr medium:xLM Herz - Optical Properties of SolidsTT2012 56

3 rd order NL: Resonant nonlinearitieshνConsider medium with optical transition atresonance with incident wave of Intensity Ifind that absorption decreases as higherstate become more populated:Absorption coefficientforfor(weak absorption)Can view saturable absorption as a third-order optical nonlinearity!LM Herz - Optical Properties of SolidsTT2012 57