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1、Third-order nonlinearities: Four-wave mixing,Third-harmonic generationInduced gratingsPhase conjugationNonlinear refractive indexSelf-focusingSelf-phase modulationContinuum generation,Third-harmonic generation,We must now cube the input field:,Third-harmonic generation is weaker than second-harmonic
2、 and sum-frequency generation, so the third harmonic is usually generated using SHG followed by SFG, rather than by direct THG.,Noncollinear third-harmonic generation,We can also allow two different input beams, whose frequencies can be different.So in addition to generating the third harmonic of ea
3、ch input beam, the medium will generate interesting sum frequencies.,Third-order difference-frequency generation: Self-diffraction,Consider some of the difference-frequency terms:,Samplemedium,Signal,w,w,w,The excite-probe geometry,One field can contribute two factors, one E and the other E*. This w
4、ill involve both adding and subtracting the frequency and its k-vector.,This effect is automatically phase-matched!,The excite-probe beam geometry has many applications, especially to ultrafast spectroscopy. The signal beam can be difficult to separate from the input beam, E1, however.,Nonlinearmedi
5、um,Wave plateyielding 45polarization,Signal,w,w,w,Polarization gating,Here field #2 contributes two factors, one E and the other E*. But one is vertically polarized, while the other is horizontally polarized. This yields a signal beam thats orthogonally polarized to the input beam E1.,If E1 is horiz
6、ontally polarized, the signal will be vertically polarized:,This arrangement is also automatically phase-matched. Its also referred to as polarization spectroscopy due to its many uses in both ultrafast and frequency-domain spectroscopy.,The input beam in the signal beam direction is rejected by pol
7、arizer!,The irradiance of two crossed beams is sinusoidal, inducing a sinusoidal absorption or refractive index in the mediuma diffraction grating!,An induced grating results from the cross term in the irradiance:,Many nonlinear-optical effects can beconsidered as induced gratings.,Time-independent
8、fringes,Diffraction off an induced grating,A third beam will then diffract into a different direction.This results in a beam thats the product of E1, E2*, and E3:,This is just a generic four-wave-mixing effect.,Induced gratings,Phase-matching condition:,Assume:,The diffracted beam has the same frequ
9、ency and k-vector magnitude as the probe beam, but its direction will be different.,but,Phase-matching induced gratings,Phase-matching:,z-component:,x-component:,The minus sign is just the excite-probe effect.,The “Bragg Condition”,Induced gratingswith different frequencies,Phase-matching condition:
10、,This effect is called non-degenerate four-wave mixing. In this case, the intensity fringes sweep through the medium: a moving grating.,The set of possible beam geometries is complex. I wrote a thesis on this!,Acousto-optics involves diffracting light off a grating induced by an acoustic wave.,Acous
11、to-optics works because acoustic waves have about the same wavelengths as (visible) light waves. Such diffraction can be quite strong: 70%. Acousto-optics is the basis of useful devices.,An acoustic wave induces sinusoidal density, and hence sinusoidal refractive-index, variations in a medium.,Induc
12、ed gratings with plane waves and more complex beams (of the same frequency),All such induced gratings will diffract a plane wave, reproducingthe distorted wave:,E2 and E3 are plane waves.,Holography is an induced-grating process.,One of the write beams has a complex spatial patternthe image. Differe
13、nt incidence angles correspond to different fringe spacings. Different object views are stored as different fringe spacings.A third beam (a plane wave) diffracts off the grating, acquiring the image infor-mation. Different fringe spacings yield different diffraction angleshence 3D!,The light phase s
14、tores the angular info.,Phase conjugation,When a nonlinear-optical effect produces a light wave proportional to E*, the process is called a phase-conjugation process. Phase conjugators can cancel out aberrations.,The second traversal through the medium cancels out the phase distortion caused by the
15、first pass!,Distorting medium,A phase-conjugate mirror reverses the sign of the phase,A normal mirror leaves the sign of the phase unchanged,Phase conjugation = Time reversal,A light wave is given by:,If we can phase-conjugate the spatial part, we have:,Thus phase conjugation produces a time-reverse
16、d beam!,Degenerate four-wave mixing,Consider only processes with three input frequenciesand an output frequency that are identical.Identical frequencies = degenerate.,Degenerate four-wave mixing gives rise to an amazing variety of interesting effects. Some are desirable. Some are not.Some are desira
17、ble some of the time and not the rest of the time.,Because the k-vectors can have different directions, well distinguish between them (as well as the fields):,Single-field degenerate four-wave mixing,Single-field degenerate four-wave mixing gives rise to “self” effects. These include:Self-phase modu
18、lationSelf-focusing (whole-beam and small-scale)Both of these effects participate in the generation of ultrashort pulses!,If just one beam is involved, all the k-vectors will be the same,as will the fields:,So the polarization becomes:,Degenerate 4WM means a nonlinearrefractive index.,So the refract
19、ive index is:,Recall the inhomogeneous wave equation:,and the polarization envelope (the linear and nonlinear terms):,Substituting the polarization into the wave equation (assuming slow variation in the envelope of E compared to 1/w):,since,Nonlinear refractive index (contd),Usually we define a nonl
20、inear refractive index, n2:,The refractive index in the presence of linear and nonlinear polarizations:,Assume that the nonlinear term n0:,Now, the usual refractive index (which well call n0) is:,So:,So:,since:,The nonlinear refractive index magnitude and response time,A variety of effects give rise
21、 to a nonlinear refractive index.Those that yield a large n2 typically have a slow response.,Thermal effects yield a huge nonlinear refractive index through thermal expansion due to energy deposition, but they are very very slow. As a result, most media, including even Chinese tea, have nonlinear re
22、fractive indices!,vibrational,Whole-beam self-focusing,This is precisely the behavior of a lens! But one whose focal power scales with the intensity.,If the beam has a spatial Gaussian intensity profile, then any nonlinear medium will have a spatial refractive index profile that is also Gaussian:,Ne
23、ar beam center:,The nonlinear refractive index, , causes beams to self-focus.,The phase delay vs. radial co-ordinate will be:,Small-scale self-focusing,Such filaments grow exponentially with distance. And they grow from quantum noise in the beam, which is always there.As a result, an intense ultrash
24、ort pulse cannot propagate through any medium without degenerating into a mass of tiny highly intense filaments,which, even worse, badly damage the medium.,If the beam has variations in intensity across its profile, it undergoessmall-scale self-focusing.,Each tiny bump in the beam undergoes its own
25、separate self-focusing, yielding a tightly focused spot inside the beam, called a “filament.”,Small-scale self-focusing vs. distance: Simulation,A somewhat noisy beam becomes a very noisy one.,Real examples of beam filamentation,All peak powers are in the 15 to 35GW/cm2 range. All beams began life s
26、mooth!,The self-phase-modulated pulse develops a phase vs. time proportional to the input pulse intensity vs. time.,Self-phase modulation & continuum generation,The further the pulsetravels, the moremodulation occurs.,A flat phase vs. time yields the narrowest spectrum. If we assume thepulse starts
27、with a flat phase, then SPM broadens the spectrum.This is not a small effect! A total phase variation of hundreds can occur! A broad spectrum generated in this manner is called Continuum.,That is:,Pulse intensity vs. time,The instantaneous frequency vs. time in SPM,A 10-fs, 800-nm pulse thats experi
28、enced self-phase modulation with a peak magnitude of 1 radian.,Self-phase-modulated pulse in the frequency domain,The same 10-fs, 800-nm pulse thats experienced self-phase modulation with a peak magnitude of 1 radian.,Its easy to achieve many radians for phase delay, however.,A highly self-phase-mod
29、ulated pulse,A 10-fs, 800-nm pulse thats experienced self-phase modulation with a peak magnitude of 10 radians,Note that the spectrum has broadened significantly. When SPM is very strong, it broadens the spectrum a lot. We call this effect continuum generation.,Continua created by propagating 500-fs
30、 625nm pulses through 30 cm of single-mode fiber.,The Supercontinuum Laser Source, Alfano, ed.,Broadest spectrum occurs for highest energy.,LowEnergyMediumEnergyHighEnergy,Experimental continuum spectrum in a fiber,Continuum generation simulations,Input Intensity vs. time (and hence output phase vs.
31、 time),Output spectrum:,Instantaneously responding n2; maximum SPM phase = 72p radians,The Super-continuumLaser Source, Alfano, ed.,Original spectrum is negligible in width compared to the output spectrum.,Oscillations occur in spectrum because all frequencies occur twice and interfere, except for i
32、nflection points, which yield maximum and minimum frequencies.,Dw,Continuum generation simulation,Output phase vs. time ( input intensity vs. time,due to slow response),Output spectrum:,Noninstantaneously responding n2; maximum SPM phase = 72p radians,The SupercontinuumLaser Source, Alfano, ed.,Asym
33、metry in phase vs. timeyields asymmetry in spectrum.,Experimental continuum spectra,625-nm (70 fs and 2 ps) pulses in Xe gas,L = 90 cm,The SupercontinuumLaser Source, Alfano, ed.,Data taken by Corkum, et al.,p = 15 & 40 atm,Ultraviolet continuum,4-mJ 160-fs 308-nm pulses in 40 atm of Ar; 60-cm long
34、cell.,Lens focal length= 50 cm.,Good quality output mode.,The SupercontinuumLaser Source, Alfano, ed.,308 nm input pulse; weak focusing with a 1m-focal-length lens.,The Super-continuumLaser Source, Alfano, ed.,UV Continuum in Air!,Continuum is limited when GVD causes the pulse to spread, reducing th
35、e intensity.,Continuum in air,Beam profile of a high-power beam (1000Pcr) after 15m. Note the multiple filamentation.,Use a negatively chirped pulse to pre-compensate for the dispersion of air. Then “temporal focusing” occurs. The self-focusing can compensate diffraction: light bullets! Usually this
36、 only happens for pieces of the beam: filamentation, and its messy.,Conical emission from a fs beam in air, near the critical power Pcr.,Continuum Generation:Good news and bad news,Good news:It broadens the spectrum, offering a useful ultrafast white-light source and possible pulse shortening.Bad ne
37、ws:Pulse shapes are uncontrollable.Theory is struggling to keep up with experiments.In a bulk medium, continuum can be high-energy, but its a mess spatially.In a fiber, continuum is clean, but its low-energy.In hollow fibers, things get somewhat better.Main problem: dispersion spreads the pulse, lim
38、iting the spectral broadening.,Microstructure optical fiber,Microstructure optical fibers modify dispersion.,The continuum from microstructure optical fiber is ultrabroadband.,The spectrum extends from 400 to 1500 nm and is relatively flat (when averaged over time).,This continuum was created using
39、unamplified Ti:Sapphire pulses.J.K. Ranka, R.S. Windeler, and A.J. Stentz, Opt. Lett. Vol. 25, pp. 25-27, 2000,Cross section of the microstructure fiber.,Continuum is quite beautiful!,Other third-order nonlinear-optical effects,Raman scattering,Two-photon absorption,Higher-Order Nonlinear Optics: The Death Star,Eight-wave mixing,
