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1、Nonlinear Optics,Why 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 photons,Nonline
2、ar 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 s
3、ee 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?,R
4、ecall 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 effe
5、cts occur?(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.,
6、Nonlinear 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
7、as distortion.,Sharp edges require higher frequencies in the Fourier series.,Nonlinear effects in atoms,So an electrons motion will also depart from a sine wave.,Another way to look at nonlinear optics is that the potential of the electron is an atom is not a simple quadratic potential.,Nonlinear op
8、tics and anharmonic oscillators,For weak fields,motion is harmonic,and linear optics prevails.For strong fields(i.e.,lasers),anharmonic motion occurs,and higherharmonics occur,both in the motion and the light emission.,A nucleus(in a molecule)also does not have a simple quadratic potential.So its vi
9、brational motion is also nonlinear:,Molecules Excited by Laser Light,Laser light causes molecules to vibrate in unison.We say that laser light polarizes the medium.,Accelerating charges emit light.Polarized matter emits light at the frequency at which it is oscillating.,If the motion isnt sinusoidal
10、,the medium will also emit the additional frequencies!,Maxwells Equations in a Medium,The polarization,P,contains the effect of the medium:,Sine waves of all frequencies are solutions to the wave equation;its the polarization that tells which frequencies will occur.The polarization is the driving te
11、rm for the solution to this equation.,These equations reduce to the(scalar)wave equation:,Inhomogeneous Wave Equation,Solving the wave equation in the presence of linear induced polarization,For low irradiances,the polarization is proportional to the incident field:,In this simple(and most common)ca
12、se,the wave equation becomes:,This equation has the solution:,The induced polarization only changes the refractive index.Dull.If only the polarization contained other frequencies,where w=c k and c=c0/n and n=(1+c)1/2,Using the fact that:,Simplifying:,Maxwells Equations in a Nonlinear Medium,Nonlinea
13、r 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=2nd harmonic!Harmonic generation is one of many exotic effects that can arise!,Sum-and difference-frequency generat
14、ion,Suppose 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 gen,2nd-harmonic gen,Sum-freq gen,Diff-freq gen,dc rectification,So:,Complicated nonlinear-optical effects can occur.,The more photons(i.e.,the higher the or
15、der)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 stronger.,Nonlinear-optical processesare often referred to as:N-wave-mixing processeswhere N is
16、the number ofphotons involved(including the emitted one).This is a six-wave-mixing process.,Emitted-lightfrequency,wsig,Induced polarization for nonlinear optical effects,Arrows pointing upward correspond to absorbed photons and contribute a factor of their field,Ei;arrows pointing downward correspo
17、nd to emitted photons and contribute a factor of the complex conjugate of their field:,Solving the wave equation in nonlinear optics,Recall the inhomogeneous wave equation:,Because its second-order in both space and time,and P is a nonlinear function of E,we cant easily solve this equation.Indeed,no
18、nlinear differential equations are really hard.Well have to make approximations,Take into account the linear polarization by replacing c0 with c.,Separation-of-frequencies approximation,The total E-field will contain several nearly discrete frequencies,w1,w2,etc.So well write separate wave equations
19、 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 w2.,etc.,This will be a reasonable approximation even for relatively broadband ultrash
20、ort pulses,The non-depletion assumption,Well 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 result,we need only consider the wave equation for the field and polarization osc
21、illating 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(wsig t ksig z)Well assume that the new pulse envelope wont change too rapidly.Th
22、is is the Slowly Varying Envelope Approximation(SVEA).If d is the length scale for variation of the envelope,SVEA says:d lsig,The Slowly Varying Envelope Approximation,Comparing Esig and its derivatives:,Well do the same in time:If t is the time scale for variation of the envelope,SVEA says:t Tsigwh
23、ere 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 the time scale for variation of the envelope,SVEA says:t Tsigwhere Tsig is one
24、 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 and t.Also,neglect the 1st derivative of the polarization envelope(its small co
25、mpared to the wsig2Psig term).We must keep Esigs first derivatives,as well see in the next slide,x,Esig(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 cancel the complex exponentials,leaving:,The Slowly Varying Envelope Approximatio
26、n,Substituting the remaining derivatives into the inhomogeneous wave equation for the signal field at w0:,Slowly VaryingEnvelope Approximation,Dividing by 2iksig:,Including dispersion in the SVEA,We can include dispersion by Fourier-transforming,expanding ksig(w)to first order in w,and transforming
27、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 short(i.e.,very broadband)pulses.,Transforming to a moving co-ordinate s
28、ystem,Define a moving co-ordinate system:zv=z tv=t z/vg,Transforming 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!,Integrating the SVEA,Usually,Psig=Psig(z,t),and even this simple equation can
29、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 distance.,when Psig is constant,We choose wsig to be the sum of the input
30、 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)s,The k-vector mag of light at this frequency is:,Phase-matching,That kpol may not be the same as
31、 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 the SVEA in this case over the length of the medium(L)yields:
32、,The SVEA becomes:,Phase-matching(continued),So:,L,Isig,Dk large,Dk small,Sinusoidal dependence of SHG intensity on length,Large Dk,Small Dk,Notice how the intensity is created as the beam passes through the crystal,but,if Dk isnt zero,newly created light is out of phase with previously created ligh
33、t,causing cancellation.,The ubiquitous sinc2(DkL/2),Phase mismatch almost always yields a sinc2(Dk L/2)dependence.,Recall that:,Multiplying and dividing by L/2:,Isig,Dk,More of the ubiquitous sinc2(DkL/2),To maximize the irradiance,we must try to set Dk=0.This is phase-matching.,Dk,Esig,The field strength:,The irradiance(intensity):,Phase-matching=Conservation laws for photons in nonlinear optics,Adding the frequencies:is the same as energy conservation if we multiply both sides by:,So phase-matching is equivalent to conservation of energy and momentum!,Adding the ks conserves momentum:,
