37. 3rd order nonlinearities

Transcription

1 37. 3rd order nonlinearities Characterizing 3rd order effects The nonlinear refractive index Self-lensing Self-phase modulation Solitons When the whole idea of χ (n) fails Attosecond pulses!

2 χ () : New frequency components One hallmark of χ () nonlinear optics is the generation of new colors of light. We have seen this effect in detail in one specific case: second harmonic generation ( ) ( ) ( ) c ( ω ) P t E t = E os j t + E 0 0 We have also seen that χ () is zero for most materials, so these effects can only be seen for certain specific materials or situations. χ (3) effects are weaker, but they can occur in virtually any medium.

3 As with second order phenomena, we expect to find new frequency components at sum and difference frequencies: ( ) ( ) ( ) ( ) 3 (3) = ε0χ 1 3 P E t E t E t ( ) Third-order nonlinear effects ( jω )( )( ) 1t jω1 t jωt jωt j ω3 t j ω3 E e E e E e E e E e E e t = ε χ (3) * * * Consider the case where two of the components are equal: ω 1 = ω ( ω ω ) ( ω ω ) ( ω ) 3 = (3) (3) (3) P P P P two components at shifted frequencies one component at an unshifted frequency ω ω 1 ω 1 ω 3 ω 1 ω 3 χ (3) χ (3) If all three frequencies are equal (e.g., all are from a single laser beam): ( ) ( 3ω ) ( ω) 3 (3) (3) P = P + P again, a piece at the input frequency ω χ (3) The polarization field at an unshifted frequency is particularly interesting

4 Degenerate third-order nonlinear effects Considering only this unshifted polarization component (and assuming that χ () = 0), the total polarization is: TOT ( ) (1) ( ) (3) P ω = ε ( ) ( ) 0 χ E ω + 3χ E ω E ω We can define an effective susceptibility χ eff, such that P TOT = ε 0 χ eff E But (3) 3χ n n0 + E n = + 3 E (1) (3) χeff χ χ ω Then, the effective index of the medium: ( 1) ( ) n0 = 1+ χ is the usual refractive index. 0 ( ω ) neff = 1+ χ 3χ n = 1+ χ + 3χ E ω = 1+ χ 1+ E ω 1 + χ (3) ( ) ( ) (1) (3) (1) (1) ( ) = n 0 + a small intensity-dependent correction eff

5 Refractive index depends on intensity (3) 3χ n n0 + E n 0 ( ω) The refractive index of a χ (3) medium has an intensitydependent term. This is usually written: n = n 0 + n I where n n has units of inverse intensity, or m / Watt. It is usually very small. χ (3) Typical numbers for n : air m /W glass m /W If the incident radiation is very intense (i.e., approaching 1/n ), then the index of the medium changes. This can lead to self-induced effects. This is known as the optical Kerr effect.

6 How realistic is it to get to these intensities? For air: n = m /W So the interesting intensity range is: I = 1/n =.5 10 W/m. Suppose our light is focused to a spot size of 10 microns. Then: area = π R = m necessary power = I area = 10 1 W Suppose our pulse duration is τ P =100 femtoseconds. Then: necessary pulse energy = power τ P = 0. joules Thus: focusing a pulse with an energy of millijoules and a duration of 100 femtoseconds gives n I = 0.01 in air. The refractive index is changed by ~1% at the peak of the pulse. This is a big intensity, but it is achievable using high-energy short pulses. In solid and liquid materials, the threshold is lower, because n is bigger.

7 One result: self-lensing (or self-focusing) Suppose we focus a Gaussian beam into a χ (3) medium. The center of the beam is more intense than the wings intensity profile of Gaussian beam n = n + 0 ni optical thickness (i.e., travel time*c 0 ) Kerr medium x,y x,y A flat slab can act like a lens! I phase fronts I

8 Self-lensing can be used to generate femtosecond pulses If a pulse experiences additional focusing due to high intensity and the nonlinear refractive index, and we align the laser for this extra focusing, then a high-intensity beam will have better overlap with a gain medium. High-intensity pulse Mirror gain crystal Low-intensity pulse Additional focusing optics can arrange for perfect overlap of the high-intensity beam back in the gain crystal. But not the lowintensity beam! Thus, the non-linearity favors high intensities, which favors short pulse formation. It is now routine to generate pulses of less than 100 femtosecond duration, using this self-lensing mechanism.

9 Self-lensing and the formation of filaments Self-focusing can be a positive feedback effect, leading to an ever-increasing intensity at the center of the laser beam. Eventually, the intensity is high enough to ionize atoms. In air, ionizing atoms produces a plasma. This plasma then contributes to the refractive index, with a negative contribution: n = n 0 + n I n plasma If these two contributions offset each other, then a stable filament is formed. This filament can propagate for many meters!

10 Optical filaments - the Teramobile A stable filament in air acts as a conductive channel, which is essentially a lightning rod. This can be used as a mobile lightning protection system. Teramobile guided and unguided lightning a self-guided filament induced in air by a high-power, infrared (800 nm) laser pulse

11 Another effect: pulses can modify their own spectral phase As a light beam propagates a distance z in a medium, it acquires a phase: ω φ = ωt kz = ωt n z c Optical Kerr effect: the refractive index depends on intensity: ω inst ω φ = ωt ( n ) 0 + ni z c Instantaneous frequency is equal to the time derivative of the phase: dφ n di t ωz = = ω dt c dt ( ) If intensity depends on time, then the pulse frequency changes with time! self-phase modulation

12 Self-phase modulation The nonlinear phase gives rise to an instantaneous frequency which depends on time: d ω z di( t) δω ( t) = ϕ NL = n dt c dt where: 0 ω( t) = ω0 δω( t) If the light is a pulse, then the instantaneous frequency is first smaller than, and then larger than, the central frequency ω 0.

13 Self-phase modulation: ω depends on t ω = constant throughout } ω = higher } ω = lower This can be extremely dramatic if the excursions of ω(t) away from its original value are large.

14 Self-phase modulation: spectral broadening If I(t) changes very rapidly (e.g., femtosecond pulse), then its derivative is large - so that the excursions of the frequency δω could be larger than the initial bandwidth of the pulse! The spectrum of the light gets broadened! microstructured optical fiber original pulse bandwidth new frequencies! output spectrum broadened by a factor of 100! Optics Letters, vol. 5, p. 5 (000)

15 Supercontinuum generation red light in white light out

16 Self-phase modulation + anomalous dispersion new frequency components at ω > ω 0 generated by the later (falling) edge of the pulse What if this occurs in a regime of anomalous dispersion? new frequency components at ω < ω 0 generated by the early (rising) edge of the pulse dn < 0 dω lower ω has lower velocity higher ω has higher velocity The new (lower) frequencies which are generated on the leading edge travel a bit slower, so the pulse catches up to them. The new (higher) frequencies which are generated on the trailing edge travel a bit faster, so they catch up to the pulse. Result: the pulse shape is stable! A solitary wave, or soliton

17 Solitons Soliton: a localized traveling wave whose intensity profile is stablized by the interplay of (linear) anomalous dispersion and non-linearity, so that its shape doesn t change as the wave propagates. Discovered in 1834: John Scott Russell observed solitary waves of water propagating for long distances along the Union canal in Scotland. a well-defined heap of water which continued its course along the channel apparently without change of form or diminution of speed. a recreation of his observation, on the John Scott Russell Acqueduct, 1995

18 Solitons are a solution to the nonlinear Schroedinger equation where E z D E = jd jξ E E t 1 k ω L and ξ π nl λ proportional to group velocity dispersion, GVD proportional to Kerr nonlinearity The envelope of the solution is a pulse: (, ) = E ( ) 0 sech t E z t τ with duration: 1 τ ξ A0 = D solution only exists if ξ/d < 0 - anomalous dispersion pulse duration is independent of propagation distance!

19 Solitons in optics Solitons in fiber optics are the basis for many telecommunications transmission systems. Dispersion of standard optical fiber anomalous Solitons can exist in standard fibers for λ > 1310 nm

20 When solitons collide Because of the non-linear nature of the equation, superposition does not hold for solitons. But they can still collide with each other, and when they do, they act a bit like particles! A soliton collision: The same collision, decomposed: A cool and easy-to-understand discussion about solitons:

21 Sometimes, the χ (n) approach fails Consider propagation of an intense pulse in a gas. Symmetry: all even orders of χ vanish P t E t E t E t (1) (3) 3 (5) 5 ( ) = χ ( ) + χ ( ) + χ ( ) +... This treatment assumes χ (3) >> χ (5) >> χ (7) We would expect each successive high harmonic to be exponentially weaker than the preceding one. Therefore, we would never expect observe very high-order harmonics.

22 High harmonic generation neon Counter-example (the first report): Kapteyn and Murnane, Phys. Rev. Lett., 79, 967 (1997) Very high-order harmonics can be observed. helium We cannot use a perturbative approach, i.e., the χ (n) picture fails completely. We must resort to an atomic description of the dynamics: 1. ionization of an atom. acceleration of a free electron 3. impact with the parent ion