Stimulated Brillouin Backscattering for Laser-Plasma Interaction in the Strong Coupling Regime

Stimulated Brillouin Backscattering For Laser-Plasma Interaction In The Strong Coupling Regime

S. Weber1,2,3,?, C. Riconda1,2, V.T. Tikhonchuk1

1Centre Lasers Intenses et Applications, UMR 5107 CNRS-Université Bordeaux 1-CEA, Université Bordeaux 1, 33405 Talence, France 2Laboratoire pour l’Utilisation Lasers Intenses/Physique Atomique des Plasmas Denses, UMR 7605 CNRS-CEA-Ecole Polytechnique-Université Paris VI, Université Paris VI, 75252 Paris, France 3 Centre de Physique Théorique, UMR 7644 CNRS-Ecole Polytechnique, Ecole Polytechnique, 91128 Palaiseau, France ? email: [email protected]

The strong coupling (sc) regime of stimulated Brillouin backscattering (SBS) is characterized by relatively high laser intensities and low electron temperatures. In this regime of laser plasma interaction (LPI) the pump wave determines the properties of the electrostatic wave. The present contribution intends to present several aspects of this regime. Up to now sc-SBS has received little attention due to the fact that research on LPI was dominated by standard inertial confinement fusion (ICF) relevant parameters. However, sc-SBS has several interesting applications and might also open up new approaches to ICF. One application is plasma-based optical parametric amplification (POPA) which allows for the creation of short and intense laser pulses. POPA has several advantages with respect to other approaches, such as Raman-based amplification, and overcomes damage threshold limitations of standard OPA using optical materials. Recent experiments are approaching the sc-regime even for short laser wavelengths. Simula- tions operating above the quarter-critical density have shown that new modes (e.g. KEENs, electron- acoustic modes, solitons etc.) can be excited which couple to the usual plasma modes and provide new decay channels for SBS.

1. Introduction Stimulated Brillouin scattering plays an important role in laser-plasma interaction in general and ICF in particular [1]. It has been studied in a regime of relatively low laser intensities for a long time. At low intensities SBS is characterized by an eigenmode of the plasma. However, interesting new eﬀects arise if one operates in the so-called strong coupling regime (see following section). This sc-SBS regime is very rich in physics phenomena, some of which might have application in laser-plasma interaction in general and some also in ICF-schemes. In the time of CO2-lasers this regime was already attained [2] due to the much longer laser wavelength of λo 10 µm. The determining parameter for 2 ≈ most eﬀects is Iλo and as present-day laser systems operate with wavelengths in general one order of magnitude smaller, λo 1 µm, much higher laser intensities are required, which, however, have become available.≈

2. The Strong Coupling Regime The sc-SBS regime [3] is characterized by low plasma temperatures, in general a few hundred electronvolts, and high laser intensities, albeit non-relativistic. In this regime the plasma response is no longer an eigenmode but a quasimode characterized by

1 + i√3 ω = (k2v2 ω2 /ω )1/3. (1) sc 2 o osc pi o One enters this regime provided 2 vosc 2 > 4kocsωo/ωpi. (2) ve

Here ko and ωo are wavevector and frequency of the laser light. In the above two formulae the functional dependencies of the relevant parameters, ωpi √ni, vosc √Io and v √T , allow to access this regime by adjusting density n ,∼electron temp∼erature T e ∼ e e e and laser intensity Io. Under these conditions the Brillouin instability has a much faster growth rate and a strongly modifies frequency of the excited ion-acoustic wave. In all the subsequent results presented the plasma density was chosen above the quarter- critical density in order to exclude Raman backscattering and therefore have the possibility to study “pure” Brillouin effects. In a recent series of papers [4, 5, 6] the strong coupling SBS-regime has been explored in some detail using 1D and 2D full-PIC simulations. The simulations performed indicate a series of new effects which were unknown up to now in the context of SBS and nonrelativistic intensities:

1. The SBS-reflectivity in the sc-regime exhibits a bursty, quasi-regular structure which is the precursor state for the subsequent phenomena occurring. The final state is a quasi-stationary, low-level saturated reflectivity.

2. The appearance of solitons in the wake of a relativistically strong and ultrashort laser pulse propagating in a plasma is a well-known phenomena. The present work has shown that solitons can also be created with the help of SBS for incident laser intensities which are not relativistic.

3. The creation of solitons, whether transient or stationary, are accompagnied by strong local density depletions (cavities).

4. At the same time energy is transferred to the plasma: the electrons are heated and the ions are accelerated.

5. The result is a very irregular plasma with strong density ﬂuctuations. Such a plasma induces a loss of spatial and temporal coherence.

6. Finally, the effect of bursty reflectivity was exploited in a controlled way in order to amplify short laser pulses in a very efficient way.

The interest generated by these numerical experiments has induced the experimental community to perform a series of experiments on the sc-SBS (section 6).

3. Bursty Reflectivity, Saturation, Formation Of Solitons, Heating And Cavi- tation In the PIC-simulations performed [4, 5] the intensity was of the order of 1016 Wµm2/cm2, the plasma density was set to 0.3 nc and a realistic mass ratio of mi/me = 1836 was used. The most striking feature of the sc-SBS regime is the behaviour of the backscattered intensity: the reflectivity exhibits a spiky evolution in time, composed of short pulse with intensities of several tens of the incident laser intensity (see Fig. 1). It was shown [6] that above a certain threshold these high-intensiy pulses inside the plasma invoke a new 3-wave coupling decay process where a transverse electromagnetic wave is decomposed into a stationary transverse mode with zero group velocity and an electrostatic mode. Figure 1: Bursty reflectivity in the strong coupling regime: at a given location inside the plasma slab (red) and in vacuum in front of the plasma slab (blue).

The stationary transverse mode develops into a transverse soliton [7] associated with a strong plasma depletion, a cavity. The soliton- and cavity-formation process go along with a strong heating of the electrons and acceleration of the ions. It was found hat in one dimension the so created solitons are stationary structures which survive for tens of picoseconds. The main difference between 1D and 2D is the dynamics of the soliton. Whereas solitons are stable in 1D they are only transient in 2D [6]. However, in both cases the plasma heating, the saturation of the reflectivity and the cavitation process are present. The limited life-time of the solitons implies that the heating of the plasma is equally limited in time and that in due course laser transmission fully recovers. Figure 2 displays the transversally averaged reflectivity and transmission of the laser in the two- dimensional case. The initially spatially coherent laser beams breaks up and temporal coherence is lost due to IAW-perturbations. Figure 3 gives a corresponding 3D view of the reflectivity and its clear saturation everywhere in the plasma at late times. The saturation of the reflectivity has also important consequences for standard ICF- applications as in single speckles the intensity might well reach the values used in the numerical simulations and therefore help to contribute to the ongoing discrepancy concerning the reflectivity in experiments, simulation and analytical analysis.

4. The Issue of Laser Beam Smoothing And Coherence Loss Figure 4 shows the how the plasma evolves from a cavity-dominated phase to a very irregular state with the remnants of the cavities still visible. An electromagnetic plane wave passing through such a plasma is submitted to a strong loss of spatial and temporal coherence. This is due to the plasma density varying strongly across the computational volume on a characteristic length scale of a few times the laser wavelength and the plasma per- turbation which have their origin in the IAW-activity generated by the Brillouin process 2.5

1.5 o I/I 1

0.5

0 0 2000 4000 6000 8000 10000 12000 14000 t [ω−1] o

Figure 2: Reﬂectivity (blue) and transmission (red) averaged over the transverse direction of the laser. The appearnce of the “white” areas means that in the transverse direction the elctro- magnetic wave no longer oscillates in phase due to the strong plasma density variations.

Figure 3: Bird’s eye view of the spatio-temporal evolution of the backscattered intensity. Figure 4: The appearance of cavities and the final strongly perturbed state of the plasma and their effect on a plane electromagnetic wave traversing it. evolve on a typical time scale of a few picoseconds. In summary, such a plasma induces a smoothing of the laser beam. Basically this is just another example of self-induced smoothing of a laser beam as was shown already to exist for much lower intensities below the self-focusing threshold [8]. The same happens here just in a more violent way. The standard ICF-schemes propose a limit to the intensity of the order of 1015 Wµm2/cm2. This has its origin in the fact that the reflectivity was expected to be too high and correspondingly the energy losses unacceptable. However, these analysis were done with- out taking proper account of kinetic effects. Only the advent of multi-dimensional and large-scale full-PIC simulations allowed to correct erroneous conceptions concerning the reflectivity at higher intensities. The plasma-induced smoothing of a laser beam at high intensity together with the low-level saturation of the reflectivity would also open up new ways for ICF. One would no longer be restricted to low intensities which has advantages for the development of hydrodynamic instabilities in the implosion phase of the pellet.

5. Amplification Of Short Laser Pulses The sc-SBS mechanism can also be used to generate high-intensity pulses [9]. The gen- eration of short and intense light pulses is of importance to many scientific and technical applications. The most advanced technique, chirped-pulse amplification (CPA) [10], is limited by the constraints on the optical strength of compressor gratings. A natural way to overcome the limitations of solid optical materials is the use of gases, liquids and plasmas as amplifying medium. Compression schemes based on the backward parametric amplification in liquids or gases offer much higher damage thresholds but the pulse length is limited by the phonon period which is of the order of 100 picoseconds. It has therefore been proposed to use plasmas as amplifying medium where the damage threshold is much higher [11]. The plasma-based amplification and compression scheme is due to a three-wave coupling between a short seed pulse and a longer pump pulse delivering the energy. The role of the third wave is taken by either the electron plasma wave (stimulated Raman scattering (SRS)) or the ion-acoustic wave (IAW) (stimulated Brillouin scattering (SBS)). Using SRS for light pulse amplification has been explored already in some detail and has been experimentally verified [12, 13]. Compression of a light pulse in a plasma in the weak coupling regime of SBS has already been suggested. Historically SBS has been disregarded as the compressed pulse duration is limited to half an ion-acoustic wave period. However, the SBS response time can be decreased by changing over into the strong coupling regime. Short-pulse amplification using SBS has several advantages with respect to SRS. SBS produces a very small frequency down-shift of the order of ωpi ωo and therefore very little photon energy is lost during amplification. Also the IAW spectrum in the sc-regime is rather broad and the amplification gain is not too sensitive to the frequency mismatch. The SRS and sc-SBS amplifiers operate in complementary conditions. Whereas the typical SRS-gain is small and long amplifier lengths are required, the sc-SBS amplification is very efficient over very short interaction lengths of the order of 100 µm. Also SBS shows a weak sensitivity to plasma inhomogeneities and it can be performed under conditions where kinetic effects do not play a role. In contrast, the SRS amplification is limited by particle trapping and wavebreaking.

One of the main interest of Raman-based amplification is the possibility to compress −1 pulses down to characteristic times of the order of ωpe . However, SRS is limited to low plasma densities of a few per cent of the critical density. The sc-Brillouin amplification −1 −1 can achieve easily pulse durations of the order of ωpi 43 ωpe but for a much higher density of the order of 30% of the critical density. Thus∼ the SBS compression is only a factor 8 away from the SRS for a hydrogen plasma. Although kinetic processes play an important role for the post-interaction plasma dynamics, it can be shown that they do not intervene during the amplification process itself provided the characteristic time-scale for wavebreaking twb is larger than the FWHM of −1 the amplifying seed pulse which has as minimal value the inverse growth rate γsc for Brillouin in the sc-regime. Equating the two time scales therefore gives one theoretical limit for the amplification process:

3/2 2 −1 vosc mi ωpi γsc twb . (3) ∼ ⇒ c max ∼ me ωo 1.5 This expression can be written in the form vosc/c (n/nc)Z (mi/mp) mp/me. For a ∼ q density of 0.3 n , Z = 1 and λ = 1 µm one gets I = 2.2 1020 W/cm2. These values c o max × are strongly relativistic and it is therefore not easy to get into the wavebreaking limit. One example of a sc-SBS amplification process is shown in Fig. 5. Pump and seed pulse 16 2 15 2 were set to intensities of Ip = 1 10 W/cm and Is = 1 10 W/cm , respectively for a laser wavelength of 1µm). In ×the figure the left-going part× (direction of the travelling seed-pulse) of the total Maxwell-field was extracted. An amplification factor of 50 was obtained on an amplification length of less than 100 µm. The amplification process can be approximately treated by coupled equations for the three waves involved: the two electromagnetic waves travelling in opposite directions and the ion-acoustic wave:

2 ωpe δnp,s (∂t vg∂x) Ep,s = i Es,p, − 2ω0 ne ∗ 2 2 2 δnp 2 ZEpEs (∂t + 4kocs) = ko . (4) ne − 4πncmi By rendering the equations dimensionless, transforming into the frame of reference of the seed and looking for self-similar solutions one obtains:

2 dAp d B 4 ? 2 = iBAs, = ApA , dξ − dξ2 −3√3 s 50 1550ω−1 1450ω−1 −1 40 1350ω 1250ω−1 ω−1 30 1150 so −1 /I ω s 1050 I

0 0 100 200 300 400 500 600 x [c/ω ] o

Figure 5: The extracted left-going seed pulse being ampliﬁed.

ξ dA 3 s + A = iB?A . (5) 2 dξ 4 s − p The self-similar equations (see Fig. 6) do reproduce some of the essential features of the amplification process, such as the growth of density perturbations behind the amplified seed pulse and the characteristic π-structure, i.e. the oscillatory behavior, in the tail of the pulse. However, this model can not account for the initial phase of the amplification process which is characterized by pulse-stretching and faster than self-similar growth. The sc-SBS-based scheme for amplification belongs the class of optical parametric amplification. It is proposed to divide OPA-mechanisms according to their conversion efficiency into energy-loss OPA, such as Raman amplification and crystal-based OPA, and energy- loss-free approaches such as Brillouin. An efficient set-up to create high-intensity, short pulses could envisage a combination of standard OPCPA together with subsequent amplification stages using Brillouin and maybe Raman in plasmas.

6. An Experiment On Strong Coupling SBS At The LULI 100TW Laser Facility The various phenomena which were shown in the simulations to exist once LPI takes place in the strong coupling regime will be verified in a forthcoming experimental campaign on the LULI laser facility. The experimental set-up is rather complex as four laser beams are required (see Fig. 7). A first laser is used to create a gas-jet plasma. Care has to be taken to obtain a density above the quarter-critical density in order to avoid spurious Raman- effects. The second and third laser beam take up the roles of pump and seed for the amplification process. A fourth laser beam is used in ordet to create an energetic proton beam which is used as diagnostic to probe the plasma state. The cavities and transient solitons are associated with extremely strong electrostatic fields which will deflect the proton beam and allow a direct imaging of the cavities. In addition specific diagnostics will allow to determine the heating of the plasma and the induced coherence loss. The distortion of the proton beam is registered by radiosensitive films or micro-channel plates which then allow a reconstruction of the field intensity and distribution as well as their 1

0.8 pump

0.6 n/n

δ density , s

, I 0.4 p I

0.2 seed 0 0 2 4 6 ξ 8 10 12

Figure 6: The self-similar solution of the 3-wave coupling equations.

seed pulse laser for plasma creation

FRC Al target laser for proton beam protons gas jet

amplified seed

pump laser

Figure 7: Experimental layout for the verification of the found phenomena. temporal evolution. A head-on collision of pump and seed is not possible in an experiment. The two pulses are supposed to cross each other under a small angle as shown in Fig. 8. This only slightly affects the growth rate for the parametric instability. The laser parameters can adjusted in order to reproduce well the parameters used in the simulations. An important point is the plasma noise level as parasitic Brillouin, i.e. originating from the plasma fluctuations and not declenched by the seed pulse, will grow from it. In PIC-simulations the noise level is given by the number of particles per computational cell and is many order of magnitudes higher than in real life. This implies that in an actual experiment the length over which pump and seed interact can be much longer and the amplification correspondingly larger. In the experiment this will be achieved by varying the waist of the pump laser beam and/or the angle under which the two laser beams cross each other. The principal feasibility in two dimensions has been shown already using hybrid as well as hydrodynamic simulations. Two-dimensional PIC-simulations of the amplification process using a non-zero angle are 5°−10° pump 10−20 micron 80 micron

Rayleigh length seed 150−200 micron

Figure 8: Realistic two-dimensional conﬁguration for the ampliﬁcation process. under way.

7. Conclusion And Outlook The purpose of this brief article is to stimulate the interest of the scientific community to take up again issues related to Brillouin backscattering in the strong coupling regime. Not only is this field interesting in its own right but it also has potentially important spin-off, such as the amplification of short laser pulses. Also the laser/plasma parameters will be easily attained by the future large ICF-projects such as HiPER. Simulations being necessarily limited in some respects, the forthcoming experiments will certainly elucidate many aspects of sc-SBS.

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