1、Second-order nonlinear-optical effectsSymmetry issuesPhase-matching in SHGPhase-matching bandwidthGroup-velocity mismatchNonlinear-optical crystalsPractical numbers for SHGElectro-opticsDifference-frequency generation and optical parametric generationFirst demonstration of second-harmonic generation
2、P.A.Franken,et al,Physical Review Letters 7,p.118(1961)The second-harmonic beam was very weak because the process was not phase-matched.First demonstration of SHG:the dataThe actual published resultsInput beamThe second harmonicNote that the very weak spot due to the second harmonic is missing.It wa
3、s removed by an overzealous Physical Review Letters editor,who thought it was a speck of dirt and didnt ask the authors first.Symmetry in second-harmonic generationFor this to hold,c(2)must be zero for media with inversion symmetry.Most materials have inversion symmetry,so you just dont see SHGor an
4、y other even-order nonlinear-optical effectevery day.E(t)E 2(t)Esig(x,t)c(2)E 2(x,t)If we imagine inverting space:Esig(x,t)-Esig(x,t)E(x,t)-E(x,t)Now,if the medium is symmetrical,c(2)remains unchanged.So:-Esig(x,t)c(2)-E(x,t)2 =c(2)E(x,t)2 Esig(x,t)Phase-matching in second-harmonic generationHow doe
5、s phase-matching affect SHG?Its a major effect,another important reason you just dont see SHGor any other nonlinear-optical effectsevery day.Sinusoidal dependence of SHG intensity on lengthLarge DkSmall DkThe SHG intensity is sharply maximized if Dk=0.which will only be satisfied when:Unfortunately,
6、dispersion prevents this from ever happening!Phase-matching second-harmonic generationSo were creating light at wsig=2w.022()polkkncww00(2)()(2)sigsigsigknnccwwwwsigpolkk(2)()nnwww2wFrequencyRefractive indexAnd the k-vector of the polarization is:The phase-matching condition is:The k-vector of the s
7、econd-harmonic is:We can now satisfy the phase-matching condition.Use the extraordinary polarizationfor w and the ordinary for 2w.Phase-matching second-harmonic generation using birefringenceBirefringent materials have different refractive indices for different polarizations.Ordinary and extraordina
8、ry refractive indicescan be different by up to 0.1 for SHG crystals.(2)()oennwww2wFrequencyRefractive index neonne depends on the propagation angle,so we can tune for a given w.Some crystals have ne no,so the opposite polarizations work.Noncollinear phase-matching02cos2()cospolpolkkkkzkncww02(2)sigk
9、ncww(2)()cosnnwwcossinkkzkx-cossinkkzkx But:So the phase-matching condition becomes:zxDropping the“o”and“e”subscripts for generality.Phase-matching bandwidthPhase-matching only works exactly for one wavelength,say l0.Since ultrashort pulses have lots of bandwidth,achieving approximate phase-matching
10、 for all frequencies is a big issue.The range of wavelengths(or frequencies)that achieve approximate phase-matching is the phase-matching bandwidth.4()()(/2)knnllllD-0l02lWavelengthRefractive index ne no22()(/)sinc(/2)sigILLk LlDRecall that the intensity out of an SHG crystal of length L is:where:()
11、/2(nnll2llIsigDkCalculation of phase-matching bandwidth4()()(/2)knnllllD-00000041()1()()(/2)(/2)2knnnnllllllllllD-When the input wavelength changes by l,the second-harmonic wavelength changes by only l/2.The phase-mismatch is:Assuming the process is phase-matched at l0,lets see what the phase-mismat
12、ch will be at l=l0+l.xxBut the process is phase-matched at l0 00041()()(/2)2knn lllllD-to first order in lFirst:0000441411/llll lll-Now this term yields only second-order terms and so can be neglected.The sinc2 curve will decrease by a factor of 2 when Dk L/2=1.39.So solving for the wavelength range
13、 that yields|Dk|widlerDifference-Frequency Generation:Optical Parametric Generation,Amplification,Oscillationw1w3w2Optical Parametric Amplification(OPA)w1w1w3w2Optical Parametric Generation(OPG)Difference-frequency generation takes many useful forms.mirrormirrorOptical Parametric GenerationEquations
14、 are just about identical to those for SHG:2(2)*11232112(2)*22132222(2)33122331v21v21v2i k zgi k zgi k zgEiE E eztc kEiE E eztc kEiE E eztc kwcwcwcD D-D -where:ki=wave vector of ith wave Dk=k1+k2-k3 vgi=group velocity of ith waveThe solutions for E1 and E2 involve exponential gain!OPAs etc.are ideal
15、 uses of ultrashort pulses,whose intensities are high.Phase-matching applies.We can vary the crystal angle in the usual manner,or we can vary the crystal temperature(since n depends on T).Free code to perform OPO,OPA,and OPG calculationsPublic domain software maintained by Arlee Smith at Sandia Nati
16、onal Labs.Just web-search SNLO.You can use it to select the best nonlinear crystal for your particular application or perform detailed simulations of nonlinear mixing processes in crystals.Functions in SNLO:1.Crystal properties 2.Modeling of nonlinear crystals in various applications.3.Designing of
17、stable cavities,computing Gaussian focus parameters and displaying the help file.Optical Parametric GenerationResults using the nonlinear medium,periodically poled RbTiOAsO4Sibbett,et al.,Opt.Lett.,22,1397(1997).signal:idler:An ultrafast noncol-linear OPA(NOPA)Continuum generates an arbitrary-color
18、seed pulse.NOPA specsCrystals for far-IR generationWith unusual crystals,such as AgGaS2,AgGaSe2 or GaSe,one can obtain radiation to wavelengths as long as 20 m.These long wavelengths are useful for vibrational spectroscopy.Gavin D.Reid,University of Leeds,and Klaas Wynne,University of Strathclyde10
19、m1 mWavelengthElsaesser,et al.,Opt.Lett.,23,861(1998)Difference-frequency generation in GaSeAngle-tuned wavelengthAnother 2nd-order process:Electro-opticsApplying a voltage to a crystal changes its refractive indices and introduces birefringence.In a sense,this is sum-frequency generation with a bea
20、m of zero frequency(but not zero field!).A few kV can turn a crystal into a half-or quarter-wave plate.VIf V=0,the pulse polarization doesnt change.If V=V,the pulse polarization switches to its orthogonal state.Abruptly switching a Pockels cell allows us to switch a pulse into or out of a laser.“Pockels cell”(voltage may be transverse or longitudinal)Polarizer
