The Nonlinear Optical Properties of Semiconductors

1 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFThe Nonlinear Optical Properties ofSemiconductorsDavid C. of Electronics and Electrical EngineeringThe Nonlinear Optical Properties of Semiconductors p. 1/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFOptical Susceptibility TensorP= 0[ (1)E+ (2)EE+ (3)EEE+..] (1)ij Linear refraction and absorption (2)ijk 1+ 2 Sum frequency generation 1 2 Difference frequency generation, rectification + 0 Electro-optic (Pockel s) effect (3)ijkl + + Third harmonic generation + Nonlinear refraction and absorption, 4WM + 0 + 0 Kerr electro-optic effect (DC)The Nonlinear Optical Properties of Semiconductors p. 2/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFContentsStructure of zincblende Semiconductors (1): Absorption & Refraction (2): Electro-optic effect (Pockels) & Frequencyconversion, SHG (3): Two-photon Absorption, Nonlinear Refraction& 4 wave mixingQuasi- (3): Carrier effects, Electro-optic effect(Kerr) & Thermal effectsThe Nonlinear Optical Properties of Semiconductors p.

2 3/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFWaveguide geometrySlab or rib waveguidesAssume weakly guidingAlso assume nonlinearity weak such thattransverse guided mode unchangedConventional orientation with[001]growth and[110]cleavage:xyzThe Nonlinear Optical Properties of Semiconductors p. 4/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFCrystal SymmetriesCommon compound Semiconductors in photonicshave a zinc-blende (cubic) structure43mNote different layer ordering for111and111 Introducing heterostructure, quantum well,breaks translational invariance in one directionThe Nonlinear Optical Properties of Semiconductors p. 5/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFBandstructure modelsBloch form for wavefunction with periodicuk(r) (r) =uk(r)eik rHamiltonian containsk pterm treat asperturbation2 parabolic bands: scalar modelKane model with singlet conduction band andtriplet valence band: vector but isotropicKane plus next highest conduction triplet:anisotropicThe Nonlinear Optical Properties of Semiconductors p.

3 6/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFAlGaAs bandstructureAlxGa1 xAs direct bandgap forx < Nonlinear Optical Properties of Semiconductors p. 7/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFk pmodels 15151ccv10 EEvccE The Nonlinear Optical Properties of Semiconductors p. 8/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFQuantum Theory of (n)Susceptibility expressions are derived using thequantum Liouville equation, with HamiltonianH=H0+ (e/m0)A (n)use sets of(n+ 1)-level systems and sumover(n+ 1)!time of (n)in Semiconductors requires:Electronic energies (valence and conductionbands)Momentum matrix elements between electronicstatesSummation overallstatesThe Nonlinear Optical Properties of Semiconductors p.

4 9/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFLinear absorption and refraction (1)ii( ) =e22m20~ 2 k,bands|ei pvc(k)|2 (1 vc(k) ( + i )+1 vc(k) + ( + i ))take limit 0: Im (1)&n Re (1)matrix element|pvc(k)|is approximately parabolic bands, absorption follows square-rootdensity-of-statesThe Nonlinear Optical Properties of Semiconductors p. 10/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OF (1)for 2-level -WL Nonlinear Optical Properties of Semiconductors p. 11/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFLinear refractionDispersion with parabolic bands gives Adachi formula ( ) =A{f(X) +12[EgEg+ ]3/2f(Xso)}+Bf(X) = Re(2 1 +X 1 X)/X2n= X=~ Eg= g Xso=~ Eg+ A(x) = + (x) = ( + + )eV = Nonlinear Optical Properties of Semiconductors p.

5 12/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFLinear refraction (300K) index vs wavelength (nm)x=0, , , Nonlinear Optical Properties of Semiconductors p. 13/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFSlowly varying envelope Maxwell s equations wave equationFourier transform:t Substitute, withk=n /cE( ) = E( ,z)eikzAssume envelope E( ,z)varies slowly incomparison to wavelength of lightsame as paraxial wave equation (BPM)d Edz= 2 E+i 2 02kPNLe ikzThe Nonlinear Optical Properties of Semiconductors p. 14/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFSecond-order nonlinearitiesNo second-order nonlinearity in media withinversion symmetryCrystal symmetry in cubic semiconductorsspecifies that the only non-zero tensor elementsare (2)xyz= (2)yzx= (2)zxyMore independent tensor elements inheterostructures and at surfacesThe Nonlinear Optical Properties of Semiconductors p.

6 15/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFPockels Electro-optic effectPockels Electro-optic effect is0 + andappears as a modification to the refractive indextransverse geometry, DC/RF field surface phase modulation for TE Optical polarisationconventionally use reducedrtensor notation sothat Optical path length change is nL=n3r41V L2dGaAs at = m,r41= pmV 1andn= Nonlinear Optical Properties of Semiconductors p. 16/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFFrequency conversion by (2) Optical frequency conversion (define 3= 1+ 2)sum frequency generation: 1+ 2 3second-harmonic generation: + 2 difference frequency generation: 3 1 2parametric amplification: 3 1 amplifies 1reduceddtensor notation,d= (2)( , ) epitaxial GaAs at = m,d14=94 pmV continual forward energy conversion, requirePhase-matching, phase velocities ofgenerating and generated waves must be Nonlinear Optical Properties of Semiconductors p.

7 17/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFSecond harmonic energy (eV)0100200300 (2)xyz( , ) (pmV 1)kmax= (eV)1/2kmax= (eV)1/2kmax= (eV)1 Irradiance k=0 DRDD k= / 0 The Nonlinear Optical Properties of Semiconductors p. 18/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFSecond harmonic generation 2 Simplest setup to characterise nonlinearinteractiontype-I configuration: TE fundamental TM SHtype-II configuration: TE and TM fundamental TE SHThe Nonlinear Optical Properties of Semiconductors p. 19/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFDifference Frequency Generation sp /2 ip /2 p i sChannel conversion for WDMP otentially integrate pump laser on chipThe Nonlinear Optical Properties of Semiconductors p.

8 20/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFParametric Amplification p s i sBroad-bandwidth amplifierOptical Parametric Oscillator for mid-IR usecavity and sbuilds up from noisePotentially integrate resonatorPotentially integrate pump laser on chipThe Nonlinear Optical Properties of Semiconductors p. 21/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OF (3)processesthird-harmonic generation (3)( , , )two-photon absorption = I ( ) =3 2 0n20c2Im (3)eff( , , ) Nonlinear refraction n=n2In2( ) =34 0n20cRe (3)eff( , , )DC Kerr effect (3)(0,0, )The Nonlinear Optical Properties of Semiconductors p. 22/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OF (3)symmetry considerations4 independent nonzero tensor elements for bulksemiconductors3 for single (3)xxxx( , , ), (3)xyxy( , , ), (3)xxyy( , , )Three measurements required to completelycharacterise each Nonlinear processBreaking symmetry, heterostructureintroduces many more independent nonzerotensor elementsThe Nonlinear Optical Properties of Semiconductors p.

9 23/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFTwo-photon Energy (eV)02x10 194x10 196x10 198x10 19Im (3) (m2V 2) Energy (eV)051015202530 Two-photon absorption (cm/GW)E||[111]E||[011]E||[001]k||[111]k ||[011]k||[001]CalculatedIm (3)tensor elements and spectra of for asE Nonlinear Optical Properties of Semiconductors p. 24/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFUltrafast Nonlinear RefractionThree measurements completely characterisenonlinear refraction in bulk (cubic) semiconductorStrengthnL2[001] n/I|[001]Anisotropy 2{nL2[001] nL2[011]}/nL2[001]Biref. param. {nL2[001] nC2(kk[100])}/nL2[001]n2scales asE 4gFor isotropic Kleinmann: = 0, = 1/3 GaAs at the half-gap (theory): = , = at the half-gap (exper.)

10 : = , = Nonlinear Optical Properties of Semiconductors p. 25/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFCoupled Propagation EquationsUsual configuration in semiconductor slabwaveguides has TEk[110] (cleavage planes) andTMk[001].i u z+ 2u x2+ u+h 1 2 uu + 1 2 vv iu+ 2 u v2= 0i v z+ 2v x2 v+h 1 2 uu +vv iv+ 2 u2v = 0uandvare the scaled electric field amplitudes forTE and TM. is proportional to the (structurally induced)birefringencenTM Nonlinear Optical Properties of Semiconductors p. 26/39 Research GroupOptoelectronicsGLASGOWUNIVERSITY OFCalculated (3)in GaAs-4x10-1904x10-19Re (3) (m2V-2) (eV)-4x10-1904x10-19Re (3) (m2V-2)xxxxxyxyxxyy(b)(a) /Eg-4x10-19-2x10-1902x10-194x10-19(m2V-2 )(c)(b)(a)Top: 14-band, bottom: 8-band (isotropic)The Nonlinear Optical Properties of Semiconductors p.