Sb95667

Fig. 3.5. Experiment of SRS excitation

In the Fig.3.5 is shown the scheme of one of the first experiments on SRS observation. Pumping laser radiation was focused into the SRS-active medium. The output radiation was collimated by the lens and was sent to the prism spectrograph. One could have seen on the screen the residual pumping radiation, the intense first Stokes component and weaker first anti-Stokes and second Stokes components.

3.3. Stimulated Brillouin scattering

The spontaneous scattering of light on the thermal acoustic waves was first predicted by L.Mandelstam in 1918, but was not published. It was rediscovered by Brilloin in 1922 and now is known as Brillouin scattering (in Russian – Mandel- stam-Brillouin scattering). Its stimulated analogue (SBS), where the acoustic wave is amplified by intense light action, was demonstrated in 1964.

The first experiment on SBS demonstration is shown in the Fig.3.6. The Q- switched giant pulse from ruby laser with the frequency ω2 was focused into the quartz crystal. The acoustical wave with the frequency ωs in the range of several GHz (the so called hypersonic wave) was excited in the crystal, scattering the light with the frequency ω2−ωs. The Fabrys-Perot interferometers made it possible to register both the primary and scattered wave, which have had a very small (in optical measure) spectral shift. Both acoustical and scattered waves were propagating in definite directions and were observed when the laser radiation power have exceeded the definite threshold level.

Spontaneous Brillouin scattering originates from the so called noise hypersonic waves. Hypersound has the frequency in the range 108-109 Hz and the wave-

32

length in the micrometer range. On this scale the matter is subjected to everexisting temperature and pressure fluctuations, which cause the said noise hypersound, scattering the incident light. If the incident light intensity is sufficiently high, it can amplify the hypersonic waves via the so called electrostriction mechanism, stimulated by interference of incident and spontaneously scattered light. Under definite geometry this amplification gets the exponential growth, transforming the spontaneous scattering process into stimulated one. Its nature and features have much in common with SRS effect.

Fig. 3.6. First experiment on SBS observation

Let us consider the elementary volume of matter, which is subjected by the electric field of the light wave dx dy dz. Let some point inside this volume shifts from its equilibrium position in u(x, t), producing the single dimension deformation ∂u/∂x. Let us introduce phenomenological constant γ, which characterizes the variation of dielectric permittivity due to deformation:

δε = −γ(∂u∂x). (3.19) In the presence of deformation the density of stored energy is changed in – 1/2 γ (∂u/∂x)E2. Variation of stored energy due to deformation produces the excess

pressure

Here η is the attenuation constant, accounting for the acoustical losses; Т is the volume elasticity module of the medium; ρ is the medium density; T = (1/ρ) (dρ/dp). Let us assume that both electric fields and the acoustic wave are the plane waves, propagating in the arbitrary directions:

Here r1, r2, rs are the distances along the propagation directions k1, k2, ks, where ri

Let us now replace in (3.21) x by rs, and we shall get:

From (3.23) results that ωs = ω2 − ω1, ks = k2 − k1, and the right side of (3.23) is transformed into

and the wave equation (3.23) now looks like

where vs is the velocity of acoustic waves propagation ( vs2 = T / r ).

34

Let us calculate as an example the parameters of SBS for quartz. Here

Т = 5 · 1010 N/m2 is the typical value of elasticity module for the solid state bod-

phire. So the threshold power can be evaluated as Iпор ≈ 1011 W/cm2.

3.4. Nonlinear effects, accompanying the stimulated scattering

We have said already that SRS and SBS effects arouse only if the radiation

intensity exceeds some threshold. However, often the measured value of threshold is lower than the expected one. The difference between theory and experiment can be rather large; in some liquids these values differ in one hundred and more times. The reason the self-focusing effect. Due to this effect the beam diameter in the medium is gradually reducing and finally the beam is collapsing to “focal point”. In this point the power density of the laser radiation can be very high and can lead to material damage. In particular, this effect is very important in the high power pulsed solid-state lasers, because it can lead to active element damage. Let us discuss this effect in more details.

We have already mentioned that for dielectric permittivity is valid either

ε = ε 0 + γ 1E or ε = ε0 + γ2 E 2 , hence the refraction index n ~ ε will also de-

pend upon Е, and thus upon the light intensity I ~ E 2 . In general case one can

this value is positive, and thus in the beam center, where I is larger than in periphery region, the value of n is also larger. In other words the Gaussian beam with high I will produce in the medium the region, whose properties are similar to the positive lens, providing self-focusing of the laser beam. This effect is known as the large scale self-focusing. In some media n < 0. In this case the nonlinear lens is negative, and the beam self-defocusing occurs.

The large-scale self-focusing competes with diffraction broadening of the beam, so the beam either will collapse after collection in one point, or will create the narrow channel, where the diffraction and self-focusing will be mutually compensated. Such compensation can take place after the beam collapse.

38

The multi-mode beams are usually self-focused into several channels, known as filaments. Similar filamentation of the beam occurs in the case of the so called small-scale self-focusing. It is caused by the growth of perturbations of the beam wavefront in the field of high-power light wave. This effect is especially often observed in the active (amplifying) medium. In this case the primary smooth (plane or Gauss) wavefront of the amplified beam, propagating through the active medium, is gradually breaking into several segments, and each of these segments subjects the self-focusing and filamentation. The small-scale self-focusing of laser radiation in the active medium is one of the main negative effects, which endanger the high-power solid-state lasers. Its prevention requires special measures.

Self-focusing requires high intensity of radiation, and thus it is realized usually in the pulsed mode. In this case one has also to take into account the variation of the laser pulse parameters during its propagation through the nonlinear medium. Let us consider the pulse propagation through the Kerr medium where

n = n0 + n2 E (t ) 2 . It is rather obvious that the pulse propagation through the me-

dium and field intensity growth will be accompanied by the growth of optical power of the nonlinear lens. The forward front of the laser pulse will pass through the non-perturbed medium and will not be subjected to self-focusing. The following parts of the pulse will feel the stronger and stronger influence of this effect. So the transverse section of the pulse will change with its propagation through the medium (see Fig.3.9). The backward part of the pulse will be narrowed due to selffocusing, while its forward part will expand due to diffraction

of the pulse.

Spectral content of laser pulse in this case is also subjected to very large modification. It is caused by phase self-modulation effect, whose nature is as follows. The phase of the wave is equal to:

39

It is illustrated by the Fig.3.10. The upper curve corresponds to the giant (Q- switched) pulse, and the bottom – to its frequency. In the pulse beginning its frequency ω = ω0 + ω is less than ω0 , then it grows nearly linearly, and at the pulse end ω > ω 0 . It means that the pulse, propagating through the nonlinear medium, accumulates the nearly linear frequency modulation. If we will now send such a pulse into the so called disperse line, which has the abnormal dispersion of the refraction index (it means that n falls with ω), the backward part of the pulse will have the group velocity which will be higher than that of the forward part of the pulse. In other words the backward part will move faster and the pulse will be self-sharpening, and its total duration will be reduced (compressed). The maximal rate of compression of the pulse with duration τ0 down to τmin is determined by the formula

S = ττ0 ≈ ∂2k n2Il ,

min ∂ω2

Here l is the medium length, and I is the intensity in the pulse maximum. Abnormal dispersion can be realized in the metal vapor nearby the absorption line, in special devices, comprised by two diffraction gratings, or in special single-mode optical fiber. In the latter case the low value of nonlinearity (n2 = 3.2 · 10–13 cm2 kW–1 ) is compensated by the possibility to organize propagation of the radiation with width of 5-10 μm along hundreds of meters. The so called pulse compressor, comprised by combination of diffraction gratings and such single-mode fiber, makes it possible to obtain S ≈ 100. So one can compress the pulses, produced by Q-switched lasers or by lasers with mode synchronization, down to durations in ten range of ~ 10 −14 − 10 −15 с, i.e. of just few cycles of optical electromagnetic wave. Such femtosecond laser are now actively developed.

40