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## Influence of the plasma density shape and laser intensity profile on the phase evolution

In this section we investigate the role played by an inhomogeneous plasma density profile on the evolution of the amplification of the seed. This section is organized as following: in subsection 3.2.1 we show that the modification of the pump and seed phases as they are propagating throughout a plasma with a not constant density profile is negligible if there is no coupling. In subsections 3.2.3 and 3.2.4 we study the phase evolution when the amplification happens in presence of triangular density profile and we show how a decreasing triangular plasma density (in the direction of propagation of the seed) improves the amplification. In subsection 3.2.5 we associate the intrinsic frequency chirp for the amplification in an inhomogeneous density profile to a value of the chirp parameter .

### Phase evolution for the propagation of pump and seed lasers without coupling

In this section we explore the phase evolution of the pump and seed in the case they are freely propagating in a linear plasma to asses the importance of the phase velocity variation due to the density gradient. In a general way, the pump and seed electric fields are defined as: 50 3 – Intrinsic frequency chirp in the strong coupling regime for SBS Ep / cos(kpx !pt) Es / cos(ksx !st) (3.6) In the case of a constant plasma profile the phases of the pump and seed are respectively: ‘p = kpx !pt ‘s = ksx !st (3.7) At given point x and time t we have: ‘p ‘s = (kp + ks)x (!p !s)t.

#### The role of a plasma density shape on the phase evolution

The comparison is made between two constant density profile, one with a maximum of the plasma density of nmax=nc = 0:1 (Fig.(3.7(a)), in the following we refer to this case as const1) and one with nmax=nc = 0:05 (Fig.(3.7(b)), case const2 in the following), a triangular one with the maximum shifted towards left (Fig.(3.7(c)), case trl) and nmax=nc = 0:1 and a triangular one with the maximum shifted towards right (Fig.(3.7(d)), case trr) and nmax=nc = 0:1. The choice of nmax=nc = 0:05 for the case const2 is dictated by the fact that for a linear ramp with 0 nmax=nc 0:1 the average density is 0.05 and thus it seems reasonable to make a comparison with a constant density of this value. The laser are now crossing on the right boundary of the plasma, at x = 650 m, in order to let the seed to explore the entire length of the plasma. In the following we make the hypothesis that the space coordinate is centered at xcross = 650 m, in a way that the position in the plasma is defined as ^x = xxcross: the seed laser is travelling towards ^x < 0. In Fig.(3.8)(a) we show the amplified seed in function of space at t = 2:2 ps for all the case of interest: the final value of the seed electric field amplitude is strongly influenced by the shape of the plasma density profile. As expected the seed is better amplified in the case const1 compared to const2 one. The cases const1 (black line) and trr (green line) are comparable in terms of final electric field amplitude and a bigger difference can be see if we compare the trr case with the const2 one. This means that the direction of the plasma density ramp compensates for the fact that the coupling can be locally weaker. In Fig.(3.8)(b) we present the seed amplification at x = 550 in function of time we see that the seed is growing faster for the case const1. To justify all these results, in the following we explore the evolution of the seed phase in function of the plasma density shape; we will show that the density profile acts on the coupling in a similar way to a chirp.

**Intrinsic frequency chirp due to a inhomogeneous plasma density profile**

In the previous subsections we showed how the evolution of the phase of the seed is influenced by the shape of the density profile. In terms of final value of the amplitude of the electric field of the amplified seed at the end of the simulations, we showed that the cases const1 and trr are comparable, even if in the triangular case the seed is exploring a lower value of averaged density and, indeed, trr shows a better amplification than the case const2, a constant profile with n=nc = 0:05. The averaged density value of the entire plasma explored by the seed is nav=nc = 0:05. Motivated by these results and by the improvement in the coupling when the right chirp is chosen for the pump, in this section we make a comparison between the seed phase variation for the cases with an inhomogeneous density profile with a possible value of the chirp parameter to be imposed to the pump laser in a constant density in order to find the same shapes and values of final electric field amplitude. In previous sections we introduced the chirp phase as: (x; t) = (k0(x x0) !0(t t0))2.

**Influence of realistic plasma density profile and laser shape on the SBS amplification**

In this section we show the results of the SBS amplification in the case of realistic plasma and laser configurations. In a first series of simulations we consider a gaussian plasma density profile with nmax=nc = 0:1 and a plasma length of L = 600m. As before, the pump and seed lasers are kept constant and at t = 0 they are crossing on the right side plasma, at x = 650 m, in a way to let the seed pulse to interact with the whole plasma length (Fig. (3.10)). In Fig.(3.11) we show the evolution of the seed electric field amplitude in function of time and in coincidence of the maximum of the plasma density, at x = 350, for different values of positive (Fig.(3.11)(a)) and negative (Fig.(3.11)(b)) values of the chirp parameter . As in the simulations presented in the previous sections, the highest intensity amplification is reached for negative values of . The best amplification for = 2 107, comparable with the optimal value found in Sec. 3.1, even if a bit smaller. For positive values of the amplification is strongly quenched. This is true even if is negative but large (jj > 6:5 107).

Notice that compared to the results found in Sec. 3.1 the seed amplification is now more sensitive to the chirp parameter values, even if this is small: the amplification of the seed in a gaussian shaped plasma with a chirped pump is now the results of the sum of these two effects. Fig. (3.12) show the shapes of the pump and seed at the and of the amplification in function of space and for = 0 (solid lines) and = 2 107: with a negative chirped pump the seed is slightly better amplified.

**Relative delay between the pump and seed laser pulses**

As discussed in Sec.4.2, correctly choosing the relative delay between pump and seed limits the pump energy loss by SRS backscattering. This relative delay is defined as the time before the seed starts to enter the simulation box since the beginning of the simulation. The growth rate of the instability sc = =(!sc) and the downshift ! = Re(!sc) = sc= p (Eq.(1.51)) depend on the local values of intensity and plasma density. In Fig. 4.5 we show the result of Eq.(1.51) for a gaussian pump of intensity Ip = 1015W=cm2 and duration FWHMp = 4:2 ps, and for a 1 mm gaussian plasma density profile (FWHMplasma 392 m) with nmax=nc = 0:1. The red and black lines in Fig. 4.5 (a) represent the normalized pump intensity at two different times, when the maximum of the pump reaches the left boundary and when the pump is in the middle of the simulation box, respectively; the green line is the plasma density. In Fig. 4.5(b) we show the values of growth rates sc=!0 = =(!s)=!0 for the SBS instability, as a function of the position in the plasma and for two arrival times (delay) of the pump (red and black lines respectively). When the peak of the pump is on the left side of the simulation box, sc=!0 0:0025 (red dashed line in Fig. 4.5(b) ) and !=!0 0:0014; when the peak of the pump is in the middle of the plasma, sc=!0 0:0028 (black dashed line in Fig. 4.5(b) ) and !=!0 0:0017. An optimal delay corresponds to the possibility of reaching higher growth rates, larger frequency spread and downshift of the backscattered wave . According to this linear analysis to maximize the sc-SBS coupling effect pump and seed lasers should cross at the center of the plasma density (black line in Fig. 4.5). However, the linear analysis does not take into account spontaneous losses. In simulations we find an optimal situation when the seed crosses the peak of the pump in the first half of the plasma, in a situation that is intermediate between the two examples of Fig. 4.5, as it will be discussed. In particular, when seed and pump cross in the middle of the plasma, the effective local intensity of the pump has already been reduced by the spontaneous backscattering and results in less efficient coupling. This effect is seen as well in experiments [34][36] but the actual optimal delay is reduced with respect to simulations because the noise is artificially increased in the latter.

**Table of contents :**

**1 Introduction and basic notions of laser-plasma interactions **

1.1 Laser Plasma Interaction

1.2 The Propagation of Light in a Plasma

1.3 Plasma waves

1.3.1 Electron plasma waves

1.3.2 Ion acoustic waves

1.4 Stimulated Brillouin and Raman scattering

1.4.1 Stimulated Brillouin Backscattering

1.4.2 Stimulated Raman Backscattering

**2 Energy flow directionality and time scale characterization in the strong coupling regime for SBS **

2.1 Weak coupling regime

2.2 Strong coupling regime

2.3 Different phases of the amplification in the strong-coupling limit

2.3.1 Initial seed growth

2.3.2 Exponential growth – ’Linear’ phase

2.3.3 Pump depletion and growth saturation

2.4 Conclusions

**3 Intrinsic frequency chirp in the strong coupling regime for SBS **

3.1 Effect of the chirp on the coupling

3.2 Influence of the plasma density shape and laser intensity profile on the phase evolution

3.2.1 Phase evolution for the propagation of pump and seed lasers without coupling

3.2.2 The role of a plasma density shape on the phase evolution

3.2.3 Density profile with n0(^x = 0) == 0

3.2.4 Density profile with n0(^x = 0) = 0

3.2.5 Intrinsic frequency chirp due to a inhomogeneous plasma density profile

3.2.6 Influence of realistic plasma density profile and laser shape on the SBS amplification

3.3 Conclusions

**4 Parametric studies of sc-SBS optimal coupling via one-dimensionalPIC simulations **

4.1 Simulations set-up

4.2 Shape of the density profile

4.3 Relative delay between the pump and seed laser pulses

4.4 Initial seed duration

4.5 Energy transfer and final duration of the seed

4.6 Conclusions

**5 Recent experiments on strong coupling SBS amplification and comparison with one dimensional PIC simulations **

5.1 Experiments and simulations comparison

5.1.1 Comparison between spectra of the amplified seed from simulations and experiments

5.1.2 Influence of amplification on SRS spectra

5.2 Conclusions

**6 Presentation of the new particle in code SMILEI and results of two dimensional simulations of sc-SBS amplification **

6.1 The Particle-In-Cell (PIC) method for collisionless plasmas

6.1.1 The PIC loop

6.2 Two dimensional simulation of sc-SBS amplification of a short laser pulse

6.2.1 Simulations set-up

6.2.2 Amplification results

6.3 Conclusions

Conclusions

**Communications related to this thesis **

**Appendices **

**A Relation between the chirp parameter and the duration of a gaussian laser beam **

**B SMILEI’s performance and capabilities **

**Bibliograph**y