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9. Third-order Nonlinearities: Four-wave mixing Third-harmonic generation Induced gratings Phase conjugation Nonlinear refractive index Self-focusing Self-phase modulation Continuum generation

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Third-harmonic generation 33 THG Medium We must now cube the input field: Third-harmonic generation is weaker than second-harmonic and sum-frequency generation, so the third harmonic is usually generated using SHG followed by SFG, rather than by direct THG.

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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 each input beam, the medium will generate interesting sum frequencies. THG medium Signal #1 Signal #2

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Nonlinear medium Signal #1 Signal #2 Third-order difference-frequency generation: Self-diffraction Consider some of the difference-frequency terms:

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Sample medium Signal The excite-probe geometry One field can contribute two factors, one E and the other E*. This will 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, E 1, however.

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Nonlinear medium Wave plate yielding 45˚ polarization Signal 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 that’s orthogonally polarized to the input beam E 1. If E 1 is horizontally polarized, the signal will be vertically polarized: This arrangement is also automatically phase-matched. It’s also referred to as polarization spectroscopy due to its many uses in both ultrafast and frequency-domain spectroscopy. The input beam is the signal beam direction is rejected by polarizer!

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x The irradiance of two crossed beams is sinusoidal, inducing a sinusoidal absorption or refractive index in the medium––a diffraction grating! An induced grating results from the cross term in the irradiance: Many nonlinear-optical effects can be considered as induced gratings. xxxx Time-independent fringes

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Diffraction off an induced grating A third beam will then diffract into a different direction. This results in a beam that’s the product of E 1, E 2 *, and E 3 : This is just a generic four-wave-mixing effect. x

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Induced gratings Phase-matching condition: Assume: The diffracted beam has the same frequency and k-vector magnitude as the probe beam, but its direction will be different. Nonlinear medium Diffracted signal ex1 ex2 sig pr pr ex z pr sig but

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Phase-matching induced gratings Phase-matching condition: z-component: x-component: ex pr ex z pr sig x The minus sign is just the excite-probe effect. The “Bragg Condition”

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Induced gratings with different frequencies Nonlinear medium Diffracted signal Phase-matching condition: 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. See my thesis!

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Induced gratings with plane waves and more complex beams (of the same frequency) A plane wave and a very distorted wave A plane wave and a slightly distorted wave All such induced gratings will diffract a plane wave, reproducing the distorted wave: Two plane waves A plane wave and a slightly distorted wave E 2 and E 3 are plane waves.

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One of the write beams has a complex spatial pattern that is the image. Different incidence angles correspond to different fringe spacings, so different views of the object are stored as different fringe spacings. A third beam (a plane wave) diffracts off the grating, acquiring the image information. In addition, different fringe spacings yield different diffraction angles– –hence 3D! The light phase stores the angular info. Holography is an induced-grating process.

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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. A normal mirror leaves the sign of the phase unchanged A phase-conjugate mirror reverses the sign of the phase The second traversal through the medium cancels out the phase distortion caused by the first pass! Phase conjugation

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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-reversed beam!

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Degenerate Four-Wave Mixing Consider only processes with three input frequencies and an output frequency that are identical. Degenerate four-wave mixing gives rise to an amazing variety of interesting effects. Some are desirable. Some are not. Some are desirable some of the time and not the rest of the time. Because the k-vectors can have different directions, we’ll distinguish between them (as well as the fields):

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Single-Field Degenerate Four-Wave Mixing Single-field degenerate four-wave mixing gives rise to “self” effects. These include: Self-phase modulation Self-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:

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Degenerate 4WM means a nonlinear refractive index. So the refractive 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/ ): since

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Nonlinear refractive index (cont’d) Usually we define a “nonlinear refractive index”, n 2 : The refractive index in the presence of linear and nonlinear polarizations: Assume that the nonlinear term << n 0 : Now, the usual refractive index (which we’ll call n 0 ) is: So: since:

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The nonlinear refractive index magnitude and response time A variety of effects give rise to a nonlinear refractive index. Those that yield a large n 2 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 refractive indices!

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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: Near beam center: The nonlinear refractive index,, causes beams to self-focus. The phase delay vs. radial co-ordinate will be:

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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 beam 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 undergoes small-scale self-focusing. Each tiny bump in the beam undergoes its own separate self-focusing, yielding a tightly focused spot inside the beam, called a “filament.” Intensity Position

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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 pulse travels, the more modulation occurs. A flat phase vs. time yields the narrowest spectrum. If we assume the pulse starts 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 a “Continuum.” That is: Pulse intensity vs. time

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The instantaneous frequency vs. time in SPM A 10-fs, 800-nm pulse that’s experienced self-phase modulation with a peak magnitude of 1 radian.

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Self-phase-modulated pulse in the frequency domain The same 10-fs, 800-nm pulse that’s experienced self-phase modulation with a peak magnitude of 1 radian. It’s easy to achieve many radians for phase delay, however.

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A highly self-phase-modulated pulse A 10-fs, 800-nm pulse that’s 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.

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Continua created by propagating 500-fs 625nm pulses through 30 cm of single-mode fiber. The Supercontinuum Laser Source, Alfano, ed. Low Energy Medium Energy High Energy Broadest spectrum occurs for highest energy. Experimental Continuum spectrum in a fiber

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Continuum Generation Simulations Input Intensity vs. time (and hence output phase vs. time) Output spectrum: Instantaneously responding n 2 ; maximum SPM phase = 72π radians The Supercontinuum Laser 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 inflection points, which yield maximum and minimum frequencies.

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Continuum Generation Simulation Output phase vs. time (≠ input intensity vs. time, due to slow response) Output spectrum: Noninstantaneously responding n 2 ; maximum SPM phase = 72π radians The Supercontinuum Laser Source, Alfano, ed. Asymmetry in phase vs. time yields asymmetry in spectrum.

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Experimental Continuum Spectra Input wavelength 625-nm (70 fs and 2 ps) pulses in Xe gas L = 90 cm The Supercontinuum Laser Source, Alfano, ed. Data taken by Corkum, et al. p = 15 & 40 atm

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Ultraviolet continuum 4-mJ 160-fs 308-nm pulses in 40 atm of Ar; 60-cm long cell. Lens focal length = 50 cm. Good quality output mode. The Supercontinuum Laser Source, Alfano, ed.

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308 nm input pulse; weak focusing with a 1-m lens. The Supercontinuum Laser Source, Alfano, ed. UV Continuum in Air! Continuum is limited when GVD causes the pulse to spread, reducing the intensity.

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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 news: Pulse shapes are uncontrollable. Theory is struggling to keep up with experiments. In a bulk medium, spectral broadening is not really that great— you need a log scale to see it… In a bulk medium, continuum can be high-energy, but it’s a mess spatially. In a fiber, continuum is clean, but it’s low-energy. In hollow fibers, things get somewhat better. Main problem: dispersion spreads the pulse, limiting the spectral broadening.

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Microstructure optical fiber

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Microstructure optical fibers modify dispersion.

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The spectrum extends from ~400 to ~1500 nm and is relatively flat (when averaged over time). This continuum was created using 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. The continuum from microstructure optical fiber is ultrabroadband.

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Continuum is quite beautiful!

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Other third-order nonlinear-optical effects Raman scattering Two-photon absorption

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Abstract Although the sine-Gordon equation was originally obtained for the description of four wave-mixing in transmission geometry, it describes self-diffraction.

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