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1、Second-order nonlinear-optical effects,Symmetry issuesPhase-matching in SHGPhase-matching bandwidthGroup-velocity mismatchNonlinear-optical crystalsPractical numbers for SHGElectro-opticsDifference-frequency generation and optical parametric generation,First demonstration of second-harmonic generati
2、on,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 data,The actual published results,Input beam,The second harmonic,Note that the very weak spot due to the second harmonic
3、is missing. It was 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 generation,For this to hold, c(2) must be zero for media with inversion symmetry. Most materials have inversion symmetry, so yo
4、u just dont see SHGor any 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),Ph
5、ase-matching in second-harmonic generation,How does 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 length,Large Dk,Small Dk,The SHG intensity is sharply maximized
6、if Dk = 0.,which will only be satisfied when:Unfortunately, dispersion prevents this from ever happening!,Phase-matching second-harmonic generation,So were creating light at wsig = 2w.,Frequency,Refractive index,And the k-vector of the polarization is:,The phase-matching condition is:,The k-vector o
7、f the second-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 birefringence,Birefringent materials have different refractive indices for different polarizations. Ordinary and
8、extraordinary refractive indicescan be different by up to 0.1 for SHG crystals.,ne 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-matching,But:,So the phase-matching condition becomes:,Dropping the “o” a
9、nd “e” subscripts for generality.,Phase-matching bandwidth,Phase-matching only works exactly for one wavelength, say l0. Since ultrashort pulses have lots of bandwidth, achieving approximate phase-matching for all frequencies is a big issue.The range of wavelengths (or frequencies) that achieve appr
10、oximate phase-matching is the phase-matching bandwidth.,Wavelength,Refractive index,Recall that the intensity out of an SHG crystal of length L is:,where:,Calculation of phase-matching bandwidth,When the input wavelength changes by , the second-harmonic wavelength changes by only /2.,The phase-misma
11、tch is:,Assuming the process is phase-matched at 0, lets see what the phase-mismatch will be at = 0 + .,x,x,But the process is phase-matched at 0,to first order in d,First:,Now this term yields only second-order terms and so can be neglected.,The sinc2 curve will decrease by a factor of 2 when k L/2
12、 = 1.39. So solving for the wavelength range that yields |k | 2.78/L yields the phase-matching half-bandwidth dl/2.,Calculation of phase-matching bandwidth (contd),FWHM,2.78/L,-2.78/L,sinc2(DkL/2),Isig,Dk,Phase-matching efficiency vs. wavelength for BBO,These curves also take into account the (L/l)2
13、 factor. While the curves are scaled in arbitrary units, the relative magnitudes can be compared among the three plots. (These curves dont, however, include the nonlinear susceptibility, (2).,Phase-matching efficiency vs. wavelength for the nonlinear-optical crystal, beta-barium borate (BBO), for di
14、fferent crystal thicknesses:,10 m,100 m,1000 m,Note the huge differences in phase-matching bandwidth and efficiency with crystal thickness.,Phase-matching efficiency vs. wavelength for KDP,The curves for the thin crystals dont fall to zero at long wavelengths because KDP simultaneously phase-matches
15、 for two wavelengths, that shown and a longer (IR) wavelength, whose phase-matching ranges begin to overlap when the crystal is thin.,Phase-matching efficiency vs. wavelength for the nonlinear-optical crystal, potassium dihydrogen phosphate (KDP), for different crystal thicknesses:,10 m,100 m,1000 m
16、,The huge differences in phase-matching bandwidth and efficiency with crystal thickness occur for all crystals.,A thin crystal generates SH for all frequencies of the input pulse in the forward and other directions.,ThinSHG crystal,The more the propagation direction differs from the precise phase-ma
17、tching angle, the less the efficiency. This fall-off is faster for thicker crystals.,Phase-matching bandwidth (in SHG),A thick crystal generates SH for only some frequencies in the input pulse in each direction.,Thick SHG crystal,Polar plots of SH output intensity vs. angle for a given frequency,Gro
18、up-Velocity Mismatch is another way of describing the phase-matching bandwidth.,In the crystal, the two wavelengths have different group velocities.Define the Group-Velocity Mismatch (GVM):,Crystal,As the pulse enters the crystal:,As the pulseleaves the crystal:,Second harmonic created just as pulse
19、 enters crystal (overlaps the input pulse).,Group-velocity mismatch,Calculating GVM:,So:,But we only care about GVM when n(l0/2) = n(0).,Group-velocity mismatch lengthens the SH pulse.,Assuming that a very short pulse enters the crystal, the length of the SH pulse, t, will be determined by the diffe
20、rence in light-travel times through the crystal:,We always try to satisfy:,L /LD,Group-velocity mismatch pulse lengthening,Second-harmonic pulse shape for different crystal lengths:,Its best to use a very thin crystal. Sub-100-micron crystals are common.,Inputpulseshape,LD is the crystal length that
21、 doubles the pulse length.,Group-velocity mismatch numbers,Group-velocity mismatch limits bandwidth.,Lets compute the second-harmonic bandwidth due to GVM.Take the SH pulse to have a Gaussian intensity, for which t = 0.44. Rewriting in terms of the wavelength, t = t d/d1 = 0.44 d/d1 = 0.44 2/c0where
22、 weve neglected the minus sign since were computing the bandwidth, which is inherently positive. So the bandwidth is:,Calculating the bandwidth by considering the GVM yields the same result as the phase-matching bandwidth!,Alternative method for phase-matching: periodic poling,Recall that the second
23、-harmonic phase alternates every coherence length when phase-matching is not achieved, which is always the case for the same polarizationswhose nonlinearity is much higher.Periodic poling solves this problem. But such complex crystals are hard to grow and have only recently become available.,SHG eff
24、iciency,The second-harmonic field, E2, is given by:,The irradiance, I2, will be:,Dividing by the input irradiance, I1, to obtain the SHG efficiency:,Substituting in typical numbers:,Take Dk = 0,d c(2), and includes crystal additional parameters.,Serious second-harmonic generation,Frequency-doublingK
25、DP crystals atLawrence LivermoreNational LaboratoryThese crystals convert as much as 80% of theinput light to its second harmonic. Then additional crystals produce the third harmonic with similar efficiency!These guys are serious!,w1,w1,w3,w2 = w3 - w1,Parametric Down-Conversion(Difference-frequency
26、 generation),Optical Parametric Oscillation (OPO),w3,w2,signal,idler,By convention:wsignal widler,Difference-Frequency Generation: Optical Parametric Generation, Amplification, Oscillation,w1,w3,w2,Optical Parametric Amplification (OPA),w1,w1,w3,w2,Optical Parametric Generation (OPG),Difference-freq
27、uency generation takes many useful forms.,mirror,mirror,Optical Parametric Generation,Equations are just about identical to those for SHG:,where: ki = wave vector of ith wave Dk = k1 + k2 - k3 vgi = group velocity of ith wave,The solutions for E1 and E2 involve exponential gain!,OPAs etc. are ideal
28、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 calculations,Public domain software maintained by Arlee Smith at San
29、dia National 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
30、. Designing of stable cavities, computing Gaussian focus parameters and displaying the help file.,Optical Parametric Generation,Results using the nonlinear medium,periodically poled RbTiOAsO4,Sibbett, et al., Opt. Lett., 22, 1397 (1997).,signal:,idler:,An ultrafast noncol-linear OPA (NOPA),Continuum
31、 generates an arbitrary-color seed pulse.,NOPA specs,Crystals for far-IR generation,With unusual crystals, such as AgGaS2, AgGaSe2 or GaSe, one can obtain radiation to wavelengths as long as 20mm.These long wavelengths are useful for vibrational spectroscopy.,Gavin D. Reid, University of Leeds, and
32、Klaas Wynne, University of Strathclyde,10 mm,1 mm,Wavelength,Elsaesser, et al., Opt. Lett., 23, 861 (1998),Difference-frequency generation in GaSe,Angle-tuned wavelength,Another 2nd-order process: Electro-optics,Applying a voltage to a crystal changes its refractive indices and introduces birefringe
33、nce. In a sense, this is sum-frequency generation with a beam of zero frequency (but not zero field!). A few kV can turn a crystal into a half- or quarter-wave plate.,V,If V = 0, the pulse polarization doesnt change.,If V = Vp , 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,
