超快光学-第07章-非线性光学.ppt

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1、Nonlinear OpticsWhy do nonlinear-optical effects occur?Maxwells equations in a mediumNonlinear-optical mediaSecond-harmonic generationSum-and difference-frequency generationHigher-order nonlinear opticsThe Slowly Varying Envelope ApproximationPhase-matching and Conservation laws for photonsNonlinear

2、 optics isnt something you see everyday.Sending infrared light into a crystal yielded this display of green light(second-harmonic generation,SHG):Nonlinear optics allows us to change the color of a light beam,to change its shape in space and time,and tocreate ultrashort laser pulses.Why dont we see

3、nonlinear opticaleffects in our daily life?1.Intensities of daily life are too weak.2.Normal light sources are incoherent.3.The occasional crystal we see has the wrong symmetry(for SHG).4.Phase-matching is required,and it doesnt usually happen on its own.Why do nonlinear-optical effects occur?Recall

4、 that,in normal linear optics,a light wave acts on a molecule,which vibrates and then emits its own light wave,which interfereswith the original light wave.We can also imagine thisprocess in terms of the molecular energy levels,using arrows for thephoton energies:Why do nonlinear-optical effects occ

5、ur?(continued)Now,suppose the irradiance is high enough that many molecules are excited to the higher-energy state.This state can then act as the lower level for additional excitation.This yields vibrations at all frequencies corresponding to all energy differences between populated states.Nonlinear

6、 optics is analogous to nonlinear electronics,which we can observe easily.Sending a high-volume sine-wave(pure frequency)signal into a cheap amplifier or cheap speakers yields a truncated output signal,more of a square wave than a sine.This square wave has higher frequencies.We hear this as distorti

7、on.Sharp edges require higher frequencies in the Fourier series.Nonlinear effects in atomsSo an electrons motion will also depart from a sine wave.The potential gets very flat out at infinity,so the electrons motion can easily go nonlinear!Potential due to nucleusNucleusEnergyPositionAnother way to

8、look at nonlinear optics is that the potential of the electron is an atom is not a simple quadratic potential.Nonlinear optics and anharmonic oscillatorsFor weak fields,motion is harmonic,and linear optics prevails.For strong fields(i.e.,lasers),anharmonic motion occurs,and higherharmonics occur,bot

9、h in the motion and the light emission.A nucleus(in a molecule)also does not have a simple quadratic potential.So its vibrational motion is also nonlinear:Molecules Excited by Laser LightLaser light causes molecules to vibrate in unison.We say that laser light polarizes the medium.Accelerating charg

10、es emit light.Polarized matter emits light at the frequency at which it is oscillating.If the motion isnt sinusoidal,the medium will also emit the additional frequencies!Maxwells Equations in a MediumThe polarization,P ,contains the effect of the medium:Sine waves of all frequencies are solutions to

11、 the wave equation;its the polarization that tells which frequencies will occur.The polarization is the driving term for the solution to this equation.These equations reduce to the(scalar)wave equation:Inhomogeneous Wave EquationSolving the wave equation in the presence of linear induced polarizatio

12、nFor low irradiances,the polarization is proportional to the incident field:In this simple(and most common)case,the wave equation becomes:This equation has the solution:The induced polarization only changes the refractive index.Dull.If only the polarization contained other frequencieswhere w=c k and

13、 c=c0/n and n=(1+c)1/2Using the fact that:Simplifying:Maxwells Equations in a Nonlinear MediumNonlinear optics is what happens when the polarization is the resultof higher-order(nonlinear!)terms in the field:What are the effects of such nonlinear terms?Consider the second-order term:2w w=2nd harmoni

14、c!Harmonic generation is one of many exotic effects that can arise!Sum-and difference-frequency generationSuppose there are two different-color beams present:Note also that,when wi is negative inside the exp,the E in front has a*.2nd-harmonic gen2nd-harmonic genSum-freq genDiff-freq gendc rectificat

15、ionSo:Complicated nonlinear-optical effects can occur.The more photons(i.e.,the higher the order)the weaker the effect,however.Very-high-order effects can be seen,but they require very high irradiance.Also,if the photon energies coincide with the mediums energy levels as above,the effect will be str

16、onger.Nonlinear-optical processesare often referred to as:N-wave-mixing processeswhere N is the number ofphotons involved(including the emitted one).This is a six-wave-mixing process.Emitted-lightfrequencyw wsigInduced polarization for nonlinear optical effectsArrows pointing upward correspond to ab

17、sorbed photons and contribute a factor of their field,Ei;arrows pointing downward correspond to emitted photons and contribute a factor of the complex conjugate of their field:sigSolving the wave equation in nonlinear opticsRecall the inhomogeneous wave equation:Because its second-order in both spac

18、e and time,and P is a nonlinear function of E,we cant easily solve this equation.Indeed,nonlinear differential equations are really hard.Well have to make approximationsTake into account the linear polarization by replacing c0 with c.Separation-of-frequencies approximationThe total E-field will cont

19、ain several nearly discrete frequencies,w1,w2,etc.So well write separate wave equations for each frequency,considering only the induced polarization at the given frequency:where E1 and P1 are the E-field and polarization at frequency w1.where E2 and P2 are the E-field and polarization at frequency w

20、2.etc.This will be a reasonable approximation even for relatively broadband ultrashort pulsesThe non-depletion assumptionWell also assume that the nonlinear-optical effect is weak,so we can assume that the fields at the input frequencies wont change much.This assumption is called non-depletion.As a

21、result,we need only consider the wave equation for the field and polarization oscillating at the new signal frequency,wsig.where Esig and Psig are the E-field and polarization at frequency wsig.Well write the pulse E-field as a product of an envelope and complex exponential:Esig(z,t)=Esig(z,t)expi(w

22、sig t ksig z)Well assume that the new pulse envelope wont change too rapidly.This is the Slowly Varying Envelope Approximation(SVEA).If d is the length scale for variation of the envelope,SVEA says:d l lsig The Slowly Varying Envelope Approximation Comparing Esig and its derivatives:Well do the same

23、 in time:If t is the time scale for variation of the envelope,SVEA says:t t Tsigwhere Tsig is one optical period,2p/wsig.The Slowly Varying Envelope Approximation(continued)Comparing Esig and its time derivatives:And well do the same for the polarization:Psig(z,t)=Psig(z,t)expi(wsig t ksig z)If t is

24、 the time scale for variation of the envelope,SVEA says:t t T Tsigwhere Tsig is one optical period,2p/wsig.The Slowly Varying Envelope Approximation(continued)Comparing Psig and its time derivatives:Computing the derivatives:SVEA(continued)Neglect all 2nd derivatives of envelopes with respect to z a

25、nd t.Also,neglect the 1st derivative of the polarization envelope(its small compared to the wsig2Psig term).We must keep Esigs first derivatives,as well see in the next slidexxxxEsig(z,t)=Esig(z,t)expi(wsig t ksig z)Similarly,Now,because ksig=wsig/c,the last two bracketed terms cancel.And we can can

26、cel the complex exponentials,leaving:The Slowly Varying Envelope ApproximationSubstituting the remaining derivatives into the inhomogeneous wave equation for the signal field at w0:Slowly VaryingEnvelope ApproximationDividing by 2iksig:x xIncluding dispersion in the SVEAWe can include dispersion by

27、Fourier-transforming,expanding ksig(w)to first order in w,and transforming back.This replaces c with vg:We can include GVD also,by expanding to 2nd order,yielding:We can understand most nonlinear-optical effects best by neglecting GVD,so we will,but this extra term can become important for very very

28、 short(i.e.,very broadband)pulses.Transforming to a moving co-ordinate systemDefine a moving co-ordinate system:zv =z tv =t z/vgTransforming the derivatives:The SVEA becomes:Canceling terms,the SVEA becomes:Well drop the sub-script(v)to simplify our equations.The time deriva-tives cancel!x xIntegrat

29、ing the SVEAUsually,Psig=Psig(z,t),and even this simple equation can be difficult to solve(integrate).For now,well just assume that Psig is a constant,and the integration becomes trivial:And the field amplitude grows linearly with distance.The irradiance(intensity)then grows quadratically with dista

30、nce.when Psig is constantWe choose wsig to be the sum of the input ws:But the signal E-field and polarization k-vectors arent necessarily equal.But the k-vector of the polarization is:So kpol may not be the same as ksig!And we may not be able to cancel the exp(-ikz)sThe k-vector mag of light at this

31、 frequency is:wsigPhase-matchingThat kpol may not be the same as ksig is the all-important effect of phase-matching.It must be considered in all nonlinear-optical problems.If the ks dont match,the induced polarization and the generated electric field will drift in and out of phase.where:Integrating

32、the SVEA in this case over the length of the medium(L)yields:The SVEA becomes:Phase-matching(continued)So:LIsigDk largeDk smallSinusoidal dependence of SHG intensity on lengthLarge DkSmall DkNotice how the intensity is created as the beam passes through the crystal,but,if Dk isnt zero,newly created

33、light is out of phase with previously created light,causing cancellation.The ubiquitous sinc2(D DkL/2)Phase mismatch almost always yields a sinc2(Dk L/2)dependence.Recall that:Multiplying and dividing by L/2:IsigDkMore of the ubiquitous sinc2(D DkL/2)To maximize the irradiance,we must try to set Dk=

34、0.This is phase-matching.DkEsigThe field strength:The irradiance(intensity):Phase-matching=Conservation laws for photons in nonlinear opticsAdding the frequencies:is the same as energy conservation if we multiply both sides by:wsigSo phase-matching is equivalent to conservation of energy and momentum!Adding the ks conserves momentum: