**4. Results**

We follow the evolution of the system with a close look to the evolution in two regions of the domain, a first one at about a quarter of the length in the domain, and a second one closer to the center of the domain. We will point out important differences in the initial evolution of the spectra between these two regions depending on the level of the round-off errors which now act as a perturbation in the noiseless Vlasov code. So the initial evolution of the SRFS and SRBS is not uniform through the domain, and consequently this affect the initial evolution of the KEEN waves which, as we shall show, develops from the beginning of the Raman scattering.

### **4.1. Evolution of the system in the first quarter of the length of the plasma domain**

For the parameters used in these simulations, the SRFS plasma mode with *keFλDe* =0.0666 is very weakly damped [32]. No seed or initial perturbation is added to stimulate the heavily damped SRBS mode with *keBλDe* =0.3377. We present in Figure(1a-2a) and Figure(4,top left) a contour plot of the electron distribution function at a position *x* between *x* ∈(280,300), at about a quarter of the length of the domain, at a time *t*= 351, 468 and 761 respectively. We see in these figures a modulation with wavelength *λeF* =2*π* / *keF* =5.925, which is the weakly damped forward scattered mode. Figure (2a) and Figure (4,top left) shows small vortices appearing around *pxe* =0.183. The phase velocity of the SRBS plasma wave *υeB* =*ωeB* / *keB* =0.218, corre‐ sponding to a momentum *peB* =*υeBγeB* =0.2233 (where in this case the relativistic factor *γeB* =1 / 1−*υeB* <sup>2</sup> ), is different from the position of the observed small vorticities appearing around *pxe* =0.183 in Fig.(2a) and Figure(4,top left). To identify the spatial modes present in Figures(1a-2a), we present in Figure (1c-2c) the spatial Fourier transform of the longitudinal electric field at the time *t*=351 and 468 respectively, in the domain *x* ∈(250,410). We identify the dominant mode with *keF* =1.06 of the SRFS plasma wave. Since we have a linear polariza‐ tion, we have in the longitudinal perturbation a mode present with 2*k*<sup>0</sup> =6.674 (at twice the wavenumber of the pump, appearing at 6.676 in our results in Figure (1c)). This is due to the fact that if we have a linearly polarized wave: *E* <sup>→</sup> =(0,*Ey*,0), we can write in a linear analysis with *Ey* =*E*0cos(*ψ*), *ψ*=(*kx* −*ωt*), and Faraday's law is:

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear… http://dx.doi.org/10.5772/57476 261

$$\frac{\partial \vec{B}}{\partial t} = (0, 0, -\frac{\partial E\_y}{\partial \mathbf{x}}) \tag{19}$$

Then *B* <sup>→</sup> =(0,0,*Bz*) with *Bz* <sup>=</sup> *<sup>B</sup>*0cos(*ψ*), and *B*<sup>0</sup> <sup>=</sup>*E*0*<sup>k</sup>* / *<sup>ω</sup>*. From *<sup>E</sup>* → <sup>⊥</sup> = −∂*a* → <sup>⊥</sup> / ∂*t* and *p* → <sup>⊥</sup> =*a* → <sup>⊥</sup>, we get *p* <sup>→</sup> =(0,*py*,0), with *py* <sup>=</sup> <sup>−</sup> *<sup>p</sup>*0sin(*ψ*), and *p*<sup>0</sup> <sup>=</sup>*E*<sup>0</sup> / *<sup>ω</sup>*. The longitudinal Lorentz force is *pyBz* <sup>=</sup> <sup>−</sup> <sup>1</sup> <sup>2</sup> *<sup>k</sup> <sup>p</sup>*<sup>0</sup> 2 sin(2*ψ*). This drives a longitudinal response at the 2nd harmonic of the laser wave.

0

= -= - =

2 (from which we get *ωsF* =2.480), and 1 <sup>+</sup> *ksB*

5.398 3.3343 2 ;.0637 3.3343 1.0645 2.2698

The results in Eqs.(17-18) obey the dispersion relation for the electromagnetic wave:

We follow the evolution of the system with a close look to the evolution in two regions of the domain, a first one at about a quarter of the length in the domain, and a second one closer to the center of the domain. We will point out important differences in the initial evolution of the spectra between these two regions depending on the level of the round-off errors which now act as a perturbation in the noiseless Vlasov code. So the initial evolution of the SRFS and SRBS is not uniform through the domain, and consequently this affect the initial evolution of the KEEN waves which, as we shall show, develops from the beginning of the Raman scattering.

**4.1. Evolution of the system in the first quarter of the length of the plasma domain**

For the parameters used in these simulations, the SRFS plasma mode with *keFλDe* =0.0666 is very weakly damped [32]. No seed or initial perturbation is added to stimulate the heavily damped SRBS mode with *keBλDe* =0.3377. We present in Figure(1a-2a) and Figure(4,top left) a contour plot of the electron distribution function at a position *x* between *x* ∈(280,300), at about a quarter of the length of the domain, at a time *t*= 351, 468 and 761 respectively. We see in these figures a modulation with wavelength *λeF* =2*π* / *keF* =5.925, which is the weakly damped forward scattered mode. Figure (2a) and Figure (4,top left) shows small vortices appearing around *pxe* =0.183. The phase velocity of the SRBS plasma wave *υeB* =*ωeB* / *keB* =0.218, corre‐ sponding to a momentum *peB* =*υeBγeB* =0.2233 (where in this case the relativistic factor

around *pxe* =0.183 in Fig.(2a) and Figure(4,top left). To identify the spatial modes present in Figures(1a-2a), we present in Figure (1c-2c) the spatial Fourier transform of the longitudinal electric field at the time *t*=351 and 468 respectively, in the domain *x* ∈(250,410). We identify the dominant mode with *keF* =1.06 of the SRFS plasma wave. Since we have a linear polariza‐ tion, we have in the longitudinal perturbation a mode present with 2*k*<sup>0</sup> =6.674 (at twice the wavenumber of the pump, appearing at 6.676 in our results in Figure (1c)). This is due to the

<sup>2</sup> ), is different from the position of the observed small vorticities appearing

<sup>→</sup> =(0,*Ey*,0), we can write in a linear analysis with

=- = - = (18)

<sup>2</sup> =5.2588=*ωsB*

2 (from which we get

0

2.293). These results are very close to what is calculated in Eq.(17).

*sB eB sF eF kkk k kk*

1 + *ksF*

**4. Results**

*γeB* =1 / 1−*υeB*

fact that if we have a linearly polarized wave: *E*

*Ey* =*E*0cos(*ψ*), *ψ*=(*kx* −*ωt*), and Faraday's law is:

<sup>2</sup> =6.152=*ωsF*

260 Computational and Numerical Simulations

We note in Figs.(1c) a mode with a wavenumber 5.616. These results are confirmed in Figure (3), where we present the spatial Fourier modes at the time *t*=527 in the same domain *x* ∈(250,410), and where the mode with a wavenumber 5.616 appears with its growing harmonics at 11.232 and 16.81. This mode at *kKEEN* =5.616 is different from the value of *keB* =5.398 for the SRBS plasma wave, and will be further discussed and identified as a KEEN wave, responsible for the small vortices we see in Figure (2a) and Figure (4,top left). We show on a logarithmic scale in Figure (1b) the distribution function around *pxe* =0.183, spatially averaged over a length *λKEEN* =2*π* / *kKEEN* =1.118 (which is the width of the small vortices we see in Figure (2a) and Figure (4,top left)) around *x*=280. At this stage at time *t*=351, it shows on a logarithmic scale the straight line of a Maxwellian. However, in Figure (2b) at time *t*=468, and in Figure (4,bottom right) at time *t*=761, the distribution function shows a slightly distorted (but not fully flattened) distribution function, which lower the damping rate and which can facilitate the excitation of the mode around *pxe* =0.183. The entire distribution function, spatially averaged over a length *λKEEN* =1.118 around *x*=280, is shown at time *t*=761 in Figure (4,bottom left). Figure (4,top right) shows a plot of the longitudinal electric field in *x* ∈(280,300), showing a wavelength *λeF* =2*π* / *keF* =5.925 with a small modulation with *λKEEN* =1.118.

Figure (5) presents the spatial Fourier spectrum at time *t*=761 in *x* ∈(250,410). We note in Figure (5a) for the longitudinal wave, the mode with a wavenumber 5.616 has now developed important harmonics at 11.232 and 16.81. We also identify the dominant mode with *keF* =1.06 of the SRFS plasma wave. Since we have a linear polarization, we have in the longitudinal perturbation a mode present with 2*k*<sup>0</sup> =6.676 (at twice the wavenumber of the pump).

At this stage, we look in Figure (5b) to the wavenumber spectrum of the forward electromag‐ netic wave *E* <sup>+</sup> in the domain *x* ∈(250,410). We identify the dominant pump wave, appearing at *k*<sup>0</sup> =3.338 (3.334 in our theoretical results). We can also identify the contribution of the SRFS plasma wave at *ksF* =2.277 (2.2698 in our theoretical results in Eq.(18)). We have also a small peak at *kAS* =4.398, which corresponds to the anti-Stokes coupling *kAS* =*k*<sup>0</sup> + *ke*−*AS* = 3.334 + 1.064 = 4.398 in our theoretical results (*ke*−*AS* =1.064 is the plasma wavenumber for the anti-Stokes coupling, already present in the wide dominant peak *keF* =1.06 in Figure (5a)). The frequency *ωAS* =4.51 of the anti-Stokes wave is calculated from the relation *ωAS* <sup>2</sup> =1 <sup>+</sup> *kAS* <sup>2</sup> , in close agreement with the value calculated from the relation *ωAS* <sup>=</sup>

*ω*<sup>0</sup> + *ωe*−*AS* =3.481 + 1.0066=4.487, where the frequency *ωe*−*AS* =1.0066 associated with the anti-Stokes plasma wave is essentially the same as the already excited SRFS plasma wave at *ωeF* =1.0066. These frequencies will be verified when studying the spectrum in Figure (6,7).

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear…

and the backward propagating wave *E* <sup>−</sup>

and *E* <sup>−</sup>

in Figure (5c) shows the

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263

<sup>2</sup> =0.183, which is where

at the same position *x*=300,

are

, due to the

**Figure 3.** Spatial Fourier spectrum at the time *t*=527 in *x* ∈(250,410)

strictly decoupled. In a plasma, there is a very weak coupling between *E* <sup>+</sup>

same peaks at 3.338, 2.2778, 4.398 as in Figure (5b), but at a much lower level (the peak at 4.398 is barely visible, and the peak at 3.338 corresponding to the pump is almost two orders of magnitude smaller in Figure (5c) compared to Figure (5b)), with the exception of the peak at 2.277 which is reaching almost the same level in Figure (5c) as in Figure(5b). This peak of the backward wave at 2.277 in Fig.(5c) couples with the forward direction pump in Figure (5b) at 3.338 to give *k*<sup>0</sup> = −*ksF* + *kKEEN* , or a plasma wavenumber *kKEEN* =3.338 + 2.277=5.615 (which is the peak we identified before appearing at 5.616 in Figures (1c,2c,5a)). We have identified this mode as belonging to the KEEN wave (Afeyan *et al.*, 2004 and Afeyan *et al*., 2013a-f) when discussing above the spectrum. The frequency *ωKEEN* of this mode verifies *ω*<sup>0</sup> =*ωsF* + *ωKEEN* , or *ωKEEN* = 3.481 – 2.474= 1.007, and the phase velocity *υKEEN* =*ωKEEN* / *kKEEN* = 0.18, which

the small vortices appearing in Figs.(2-4) are located. These vortices are similar to what is

We present in Figure (6a) the frequency spectrum of the forward propagating wave *E* <sup>+</sup> recorded at the position *x*=300 between *t1*=664 and *t2*=824. We identify the pump frequency *ω*<sup>0</sup> =3.495 (*ω*<sup>0</sup> =3.481 in our theoretical results), and two small peaks for the forward scattered mode *ωsF* =2.474 (see Eq.(17)), and the anti-Stokes mode *ωAS* =4.487=*ω*<sup>0</sup> + *ωe*−*AS* =3.481 + 1.0066.

during the same time. It shows the peaks with the forward wave at 3.495 (at much lower level

nonlinearity of the medium. So the wavenumbers spectrum of *E* <sup>−</sup>

corresponds to a momentum *pKEEN* =*υKEEN γKEEN* =*υKEEN* / 1−*υKEEN*

Figure (6b) shows the frequency spectrum of the backward wave *E* <sup>−</sup>

In free space, the forward propagating wave *E* <sup>+</sup>

presented for instance in references [18-20].

**Figure 1.** a) Contour plot of the electron distribution function in *x* ∈(280,300)at *t*=351; b) Distribution function at *t*=351, spatially averaged around *x*=280 over a length of λ*KEEN* =1.118 ; 1c) Spatial Fourier spectrum at time *t*= 351 in *x* ∈(250,410).

**Figure 2.** a) Contour plot of the electron distribution function in *x* ∈(280,300) at *t*=468; b) Distribution function at *t*=468, spatially averaged around *x*=280 over a length of λ*KEEN* =1.118

*ω*<sup>0</sup> + *ωe*−*AS* =3.481 + 1.0066=4.487, where the frequency *ωe*−*AS* =1.0066 associated with the anti-Stokes plasma wave is essentially the same as the already excited SRFS plasma wave at *ωeF* =1.0066. These frequencies will be verified when studying the spectrum in Figure (6,7).

**Figure 3.** Spatial Fourier spectrum at the time *t*=527 in *x* ∈(250,410)

(a)

(b) (c)

(a) (b)

**Figure 2.** a) Contour plot of the electron distribution function in *x* ∈(280,300) at *t*=468; b) Distribution function at

*t*=468, spatially averaged around *x*=280 over a length of λ*KEEN* =1.118

**Figure 1.** a) Contour plot of the electron distribution function in *x* ∈(280,300)at *t*=351; b) Distribution function at *t*=351, spatially averaged around *x*=280 over a length of λ*KEEN* =1.118 ; 1c) Spatial Fourier spectrum at time *t*= 351 in

*x* ∈(250,410).

262 Computational and Numerical Simulations

In free space, the forward propagating wave *E* <sup>+</sup> and the backward propagating wave *E* <sup>−</sup> are strictly decoupled. In a plasma, there is a very weak coupling between *E* <sup>+</sup> and *E* <sup>−</sup> , due to the nonlinearity of the medium. So the wavenumbers spectrum of *E* <sup>−</sup> in Figure (5c) shows the same peaks at 3.338, 2.2778, 4.398 as in Figure (5b), but at a much lower level (the peak at 4.398 is barely visible, and the peak at 3.338 corresponding to the pump is almost two orders of magnitude smaller in Figure (5c) compared to Figure (5b)), with the exception of the peak at 2.277 which is reaching almost the same level in Figure (5c) as in Figure(5b). This peak of the backward wave at 2.277 in Fig.(5c) couples with the forward direction pump in Figure (5b) at 3.338 to give *k*<sup>0</sup> = −*ksF* + *kKEEN* , or a plasma wavenumber *kKEEN* =3.338 + 2.277=5.615 (which is the peak we identified before appearing at 5.616 in Figures (1c,2c,5a)). We have identified this mode as belonging to the KEEN wave (Afeyan *et al.*, 2004 and Afeyan *et al*., 2013a-f) when discussing above the spectrum. The frequency *ωKEEN* of this mode verifies *ω*<sup>0</sup> =*ωsF* + *ωKEEN* , or *ωKEEN* = 3.481 – 2.474= 1.007, and the phase velocity *υKEEN* =*ωKEEN* / *kKEEN* = 0.18, which corresponds to a momentum *pKEEN* =*υKEEN γKEEN* =*υKEEN* / 1−*υKEEN* <sup>2</sup> =0.183, which is where the small vortices appearing in Figs.(2-4) are located. These vortices are similar to what is presented for instance in references [18-20].

We present in Figure (6a) the frequency spectrum of the forward propagating wave *E* <sup>+</sup> recorded at the position *x*=300 between *t1*=664 and *t2*=824. We identify the pump frequency *ω*<sup>0</sup> =3.495 (*ω*<sup>0</sup> =3.481 in our theoretical results), and two small peaks for the forward scattered mode *ωsF* =2.474 (see Eq.(17)), and the anti-Stokes mode *ωAS* =4.487=*ω*<sup>0</sup> + *ωe*−*AS* =3.481 + 1.0066.

Figure (6b) shows the frequency spectrum of the backward wave *E* <sup>−</sup> at the same position *x*=300, during the same time. It shows the peaks with the forward wave at 3.495 (at much lower level than in Figure (6a)) and at 2.474 (at essentially the same level as the SRFS mode in Figure (6a)), and a small peak for the weakly growing heavily damped SRBS wave at *ωsB* =2.28 (2.30 in our theoretical results). So the coupling *ω*<sup>0</sup> =*ωsF* + *ωKEEN* , *k*<sup>0</sup> = −*ksF* + *kKEEN* has resulted in an KEEN wave at (*ωKEEN* =1.007,*kKEEN* =5.616) which creates the small vortices we see in Figure (2a) and Figure (4a) around *pKEEN* =0.183, and has resulted in a stimulation of the backward light wave appearing at (*ω* =2.474,*k* =2.277) in Figure (6b) and in Figure (5c) respectively. Since the kinetic enhancement of the backward scattering is one of the main point in the investigation of Raman scattering, since it can remove a substantial amount of energy from the pump laser propagating through the plasma, we have here an example of a stimulated backward wave at the wavenumber and frequency of the SRFS wave, which is excited before the SRBS wave through the coupling with the KEEN wave at (*ωKEEN* ,*kKEEN* ). The wave at (*ωKEEN* ,*kKEEN* ) is trapping a population of electrons at the phase velocity of the wave as in Figure (2a) and in Figure (4a,left). The slope of the distribution function is not flattened at the phase velocity of the KEEN wave, but it is reduced as shown in Figure (4b,right), which allows the wave to be less damped, and to exist and to trap a population of electrons which supports the KEEN wave, and allows it to propagate, as long as the stimulation of the pump is present through the coupling with a backward wave.

**Figure 5.** Spatial Fourier spectrum at the time *t*=761in *x* ∈(250,410) for: a) the longitudinal plasma wave; b) the for‐

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265

.

; b) backward propagating wave *E* <sup>−</sup>

; c) the backward electromagnetic wave *E* <sup>−</sup>

ward electromagnetic wave *E* <sup>+</sup>

**Figure 6.** Frequency spectrum: a) forward propagating wave *E* <sup>+</sup>

**Figure 4.** Top left: Contour plot of the electron distribution function in *x* ∈(280,300) at *t*=761. Top right: Plot of the longitudinal electric field in *x* ∈(280,300). Bottom: Distribution function at *t*=761, spatially averaged around *x*=280 over a length of λ*KEEN* =1.118

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than in Figure (6a)) and at 2.474 (at essentially the same level as the SRFS mode in Figure (6a)), and a small peak for the weakly growing heavily damped SRBS wave at *ωsB* =2.28 (2.30 in our theoretical results). So the coupling *ω*<sup>0</sup> =*ωsF* + *ωKEEN* , *k*<sup>0</sup> = −*ksF* + *kKEEN* has resulted in an KEEN wave at (*ωKEEN* =1.007,*kKEEN* =5.616) which creates the small vortices we see in Figure (2a) and Figure (4a) around *pKEEN* =0.183, and has resulted in a stimulation of the backward light wave appearing at (*ω* =2.474,*k* =2.277) in Figure (6b) and in Figure (5c) respectively. Since the kinetic enhancement of the backward scattering is one of the main point in the investigation of Raman scattering, since it can remove a substantial amount of energy from the pump laser propagating through the plasma, we have here an example of a stimulated backward wave at the wavenumber and frequency of the SRFS wave, which is excited before the SRBS wave through the coupling with the KEEN wave at (*ωKEEN* ,*kKEEN* ). The wave at (*ωKEEN* ,*kKEEN* ) is trapping a population of electrons at the phase velocity of the wave as in Figure (2a) and in Figure (4a,left). The slope of the distribution function is not flattened at the phase velocity of the KEEN wave, but it is reduced as shown in Figure (4b,right), which allows the wave to be less damped, and to exist and to trap a population of electrons which supports the KEEN wave, and allows it to propagate, as long as the stimulation of the pump is present through the

**Figure 4.** Top left: Contour plot of the electron distribution function in *x* ∈(280,300) at *t*=761. Top right: Plot of the longitudinal electric field in *x* ∈(280,300). Bottom: Distribution function at *t*=761, spatially averaged around *x*=280

coupling with a backward wave.

264 Computational and Numerical Simulations

over a length of λ*KEEN* =1.118

**Figure 5.** Spatial Fourier spectrum at the time *t*=761in *x* ∈(250,410) for: a) the longitudinal plasma wave; b) the for‐ ward electromagnetic wave *E* <sup>+</sup> ; c) the backward electromagnetic wave *E* <sup>−</sup> .

**Figure 6.** Frequency spectrum: a) forward propagating wave *E* <sup>+</sup> ; b) backward propagating wave *E* <sup>−</sup>

In Figure (9a) we present in the wavenumber spectrum of the longitudinal electric field in the domain *x* ∈(250,410) at *t*=879 (to be compared with Figure (5a)). We note that the SRBS plasma peak at *keB* =5.398 has grown, and eclipsing the KEEN wave peak at *kKEEN* =5.61. We also see the persistence of the harmonic of the mode *kKEEN* =5.61 at 11.2. Figure (9c) shows the wave‐

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear…

see the now growing backscattered wave at *ksB* =2.08 (2.06 in our theoretical resuls), and the modes at 2.238 and 3.338. The signature of the now growing backscattered wave is seen in the

forward pump at *k*<sup>0</sup> =3.338, the forward scattered wave at *ksF* =2.2778, and the anti-Stokes mode at *kAS* =4.398. Figure (10a) shows a contour plot of the distribution function in the domain *x* ∈(320,339) at *t*=1289, in the final stage close to saturation. Figures (10b) and (10c) shows the spatially averaged distribution function over one wavelength of the SRBS plasma wave, *λeB* =2*π* / *keB* =1.16 , which is essentially the width of a vortex in Figure (10a)), at the left edge of the domain in Figure (10b), and in the middle of the domain of Figure (10c) respectively. We see a bump, with a minimum at the phase velocity of the SRBS plasma wave. With *keB* =5.398, *ωeB* =1.178, this corresponds to a phase velocity *υeB* =*ωeB* / *keB* =0.218, and to a momentum *peB* =*υeBγeB* =0.2233, which corresponds to the position of the local minimum we see in Figure (10b) and Figure (10c), and which corresponds also to about the position of the

There is also a local maximum on the bumpy plateau. We look in Figure (11a) to the wave‐ number spectrum of the longitudinal electric field at *t*=1289, close to saturation, in the domain *x* ∈(250,410). We still observe the modes previously discussed, the SRFS plasma wave at *keF* =1.06, and the SRBS plasma wave at *keB* =5.38 is now dominant, the harmonic of the pump wave at 6.676 is still present, but the modes at 10.799 and 16.18 are harmonics of the now dominant SBRS plasma wave at *keB* =5.38 (compare with Figure (5a)). We also see sidebands developing, which is common when positive slopes of the distribution function are formed [33]. Detailed analysis of the eigen-frequencies of the distribution functions with a shape similar to what is presented in Figure (10b) and Figure (10c) has been discussed in details in [22], where BAM and EAW have been identified. The growth of these modes will lead to the fusion of the vortices but they missed KEEN wave contributions and SKEENS processes which we have seen in the SRS physics we present here. Detailed studies at this stage is beyond the scope of the present work, since an accurate study in this case requires a fine grid in velocity space for the proper treatment of the trapped particles effects and of the merging of the vortices. We note however a peak in Figure (11a) at *k* =4.32. The associated frequency from Eq.(16) is *ω* =1.11 (we note the broad frequency spectrum in Figure (12a)), which corresponds to a phase velocity *υφ* =*ω* / *k* =0.25, and to a momentum *p<sup>φ</sup>* =*υφγ* =0.265, which corresponds to the local maximum in the middle of the bumpy plateau. We also have in Fig.(11a) a peak at *k* =3.33, which corresponds from Eq.(16) to a frequency *ω* =1.065, and to a phase velocity *υφ* =*ω* / *k* =0.32 and to a momemtum *p<sup>φ</sup>* =0.339. This corresponds to the second local minimum we see on the

, in the same domain *x* ∈(250,410), at *t*=879. We

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267

in Figure (9b) at *ksB* =2.05, together with the

number spectrum of the backward wave *E* <sup>−</sup>

wavenumber spectrum of the forward wav*e E+*

center of the big vortices in Figure (10a).

bumpy plateau in Figure (10).

**Figure 7.** Frequency spectrum for the longitudinal electric field.

Figure (7,left) shows the frequency spectrum for the longitudinal electric field recorded at the position *x*=300, during the same time between *t1*=664 and *t2*=824 as in Figure (6), and Figure (7,right) shows the frequency spectrum at the same position, but between *t1*=664 and *t2*=984. The comparison emphasizes the growth of the modes present during this phase. We can identify the mode for the SRFS plasma wave *ωeF* =1.0014 (1.006 in our theoretical results, see Eq.(17), which is also the peak at the KEEN wave *ωKEEN* ), and its harmonic at 2.0028. A smaller peak at 1.178 is for the SRBS plasma wave, heavily damped and which is still slowly growing, appears in Figure (7,right). We also note the mode at the harmonic of the pump 2 *ω*0 = 6.95 (see discussion around Eq.(19)), and its sub-harmonic at the pump frequency *ω*<sup>0</sup> =3.475. In Figure (7), we are still in the phase where the SRFS *k*<sup>0</sup> =*ksF* + *keF* is coupling the pump with the forward scattered wave, and the KEEN wave *k*<sup>0</sup> = −*ksF* + *kKEEN* is coupling the pump with the backward wave at −*ksF* , and are dominating, as shown in Figure (4). We can also write 2*k*<sup>0</sup> =*keF* + *kKEEN* , but 2*k*0= 6.676 is also present in Figure (5a) as the harmonic of the pump as previously explained, further forcing the SKEENS oscilla‐ tion at *kKEEN* =5.615 and the SRFS at *keF* =1.06.

We have so far shown in this early stage, that the excitation of the KEEN wave and the forward scattering are dominating. We present in Figure (8) the contour plots of the distribution function for *x* ∈(300,339), for *t*=761, 937 and 966. We see at *t*=761 in Figure (8a) the same pattern as in Figure (4, top left). However, in Figures (8b,c) we see a mode coming from the right, and propagating to the left while growing, which is the SRBS plasma wave with *keB* =5.398 and *ωeB* =1.178. We will discuss later where and when this mode is generated. Indeed in Figure (7,right), where the frequency spectrum is obtained by extending the time domain from *t1*=664 to *t2*=984, we see the mode at *ωeB* =1.178 appearing.

In Figure (9a) we present in the wavenumber spectrum of the longitudinal electric field in the domain *x* ∈(250,410) at *t*=879 (to be compared with Figure (5a)). We note that the SRBS plasma peak at *keB* =5.398 has grown, and eclipsing the KEEN wave peak at *kKEEN* =5.61. We also see the persistence of the harmonic of the mode *kKEEN* =5.61 at 11.2. Figure (9c) shows the wave‐ number spectrum of the backward wave *E* <sup>−</sup> , in the same domain *x* ∈(250,410), at *t*=879. We see the now growing backscattered wave at *ksB* =2.08 (2.06 in our theoretical resuls), and the modes at 2.238 and 3.338. The signature of the now growing backscattered wave is seen in the wavenumber spectrum of the forward wav*e E+* in Figure (9b) at *ksB* =2.05, together with the forward pump at *k*<sup>0</sup> =3.338, the forward scattered wave at *ksF* =2.2778, and the anti-Stokes mode at *kAS* =4.398. Figure (10a) shows a contour plot of the distribution function in the domain *x* ∈(320,339) at *t*=1289, in the final stage close to saturation. Figures (10b) and (10c) shows the spatially averaged distribution function over one wavelength of the SRBS plasma wave, *λeB* =2*π* / *keB* =1.16 , which is essentially the width of a vortex in Figure (10a)), at the left edge of the domain in Figure (10b), and in the middle of the domain of Figure (10c) respectively. We see a bump, with a minimum at the phase velocity of the SRBS plasma wave. With *keB* =5.398, *ωeB* =1.178, this corresponds to a phase velocity *υeB* =*ωeB* / *keB* =0.218, and to a momentum *peB* =*υeBγeB* =0.2233, which corresponds to the position of the local minimum we see in Figure (10b) and Figure (10c), and which corresponds also to about the position of the center of the big vortices in Figure (10a).

**Figure 7.** Frequency spectrum for the longitudinal electric field.

266 Computational and Numerical Simulations

tion at *kKEEN* =5.615 and the SRFS at *keF* =1.06.

to *t2*=984, we see the mode at *ωeB* =1.178 appearing.

Figure (7,left) shows the frequency spectrum for the longitudinal electric field recorded at the position *x*=300, during the same time between *t1*=664 and *t2*=824 as in Figure (6), and Figure (7,right) shows the frequency spectrum at the same position, but between *t1*=664 and *t2*=984. The comparison emphasizes the growth of the modes present during this phase. We can identify the mode for the SRFS plasma wave *ωeF* =1.0014 (1.006 in our theoretical results, see Eq.(17), which is also the peak at the KEEN wave *ωKEEN* ), and its harmonic at 2.0028. A smaller peak at 1.178 is for the SRBS plasma wave, heavily damped and which is still slowly growing, appears in Figure (7,right). We also note the mode at the harmonic of the pump 2 *ω*0 = 6.95 (see discussion around Eq.(19)), and its sub-harmonic at the pump frequency *ω*<sup>0</sup> =3.475. In Figure (7), we are still in the phase where the SRFS *k*<sup>0</sup> =*ksF* + *keF* is coupling the pump with the forward scattered wave, and the KEEN wave *k*<sup>0</sup> = −*ksF* + *kKEEN* is coupling the pump with the backward wave at −*ksF* , and are dominating, as shown in Figure (4). We can also write 2*k*<sup>0</sup> =*keF* + *kKEEN* , but 2*k*0= 6.676 is also present in Figure (5a) as the harmonic of the pump as previously explained, further forcing the SKEENS oscilla‐

We have so far shown in this early stage, that the excitation of the KEEN wave and the forward scattering are dominating. We present in Figure (8) the contour plots of the distribution function for *x* ∈(300,339), for *t*=761, 937 and 966. We see at *t*=761 in Figure (8a) the same pattern as in Figure (4, top left). However, in Figures (8b,c) we see a mode coming from the right, and propagating to the left while growing, which is the SRBS plasma wave with *keB* =5.398 and *ωeB* =1.178. We will discuss later where and when this mode is generated. Indeed in Figure (7,right), where the frequency spectrum is obtained by extending the time domain from *t1*=664

There is also a local maximum on the bumpy plateau. We look in Figure (11a) to the wave‐ number spectrum of the longitudinal electric field at *t*=1289, close to saturation, in the domain *x* ∈(250,410). We still observe the modes previously discussed, the SRFS plasma wave at *keF* =1.06, and the SRBS plasma wave at *keB* =5.38 is now dominant, the harmonic of the pump wave at 6.676 is still present, but the modes at 10.799 and 16.18 are harmonics of the now dominant SBRS plasma wave at *keB* =5.38 (compare with Figure (5a)). We also see sidebands developing, which is common when positive slopes of the distribution function are formed [33]. Detailed analysis of the eigen-frequencies of the distribution functions with a shape similar to what is presented in Figure (10b) and Figure (10c) has been discussed in details in [22], where BAM and EAW have been identified. The growth of these modes will lead to the fusion of the vortices but they missed KEEN wave contributions and SKEENS processes which we have seen in the SRS physics we present here. Detailed studies at this stage is beyond the scope of the present work, since an accurate study in this case requires a fine grid in velocity space for the proper treatment of the trapped particles effects and of the merging of the vortices. We note however a peak in Figure (11a) at *k* =4.32. The associated frequency from Eq.(16) is *ω* =1.11 (we note the broad frequency spectrum in Figure (12a)), which corresponds to a phase velocity *υφ* =*ω* / *k* =0.25, and to a momentum *p<sup>φ</sup>* =*υφγ* =0.265, which corresponds to the local maximum in the middle of the bumpy plateau. We also have in Fig.(11a) a peak at *k* =3.33, which corresponds from Eq.(16) to a frequency *ω* =1.065, and to a phase velocity *υφ* =*ω* / *k* =0.32 and to a momemtum *p<sup>φ</sup>* =0.339. This corresponds to the second local minimum we see on the bumpy plateau in Figure (10).

In Figure (11c) we present the wavenumber spectrum for the backward wave *E* <sup>−</sup>

; c) the backward electromagnetic wave *E* <sup>−</sup>

Figure (11b) we present the spectrum of the forward wave *E* <sup>+</sup>

ear stage which is beyond the scope of this work.

ward electromagnetic wave *E* <sup>+</sup>

in the domain *x* ∈(250,410), which shows the now dominant backward mode at *ksB* =2.042, and a trace of the pump wave at *k*<sup>0</sup> =3.338, and of the anti-Stokes mode at 4.398. And in

**Figure 9.** Spatial Fourier spectrum at the time *t*=878 in *x* ∈(250,410) for: a) the longitudinal plasma wave; b) the for‐

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.

We see the peak of the pump wave at *k*<sup>0</sup> =3.338. We also see a peak at 2.042 which corresponds to the peak in Figure (11c). We have the peak of the forward scattered mode at *ksF* =2.277, and the peak for the anti-Stokes *kAS* =4.398. There is a small peak appearing at 8.718, which grew after the growth of the mode at *ksB* =2.042 in Figure (11c). We can write 8.718= 10.8-2.042 (*ksB* =2.042) or 8.718=6.676+2.042, which also happens to be at the harmon‐ ic of the anti-Stokes at 4.398. These results belong to the further evolution of the nonlin‐

at *t*= 1289,

in the domain *x* ∈(250,410).

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269

**Figure 8.** Contour plots of the distribution function for *x* ∈(300,339), at: a) *t*=761; b) t=937; c) t= 966.

**Figure 9.** Spatial Fourier spectrum at the time *t*=878 in *x* ∈(250,410) for: a) the longitudinal plasma wave; b) the for‐ ward electromagnetic wave *E* <sup>+</sup> ; c) the backward electromagnetic wave *E* <sup>−</sup> .

In Figure (11c) we present the wavenumber spectrum for the backward wave *E* <sup>−</sup> at *t*= 1289, in the domain *x* ∈(250,410), which shows the now dominant backward mode at *ksB* =2.042, and a trace of the pump wave at *k*<sup>0</sup> =3.338, and of the anti-Stokes mode at 4.398. And in Figure (11b) we present the spectrum of the forward wave *E* <sup>+</sup> in the domain *x* ∈(250,410). We see the peak of the pump wave at *k*<sup>0</sup> =3.338. We also see a peak at 2.042 which corresponds to the peak in Figure (11c). We have the peak of the forward scattered mode at *ksF* =2.277, and the peak for the anti-Stokes *kAS* =4.398. There is a small peak appearing at 8.718, which grew after the growth of the mode at *ksB* =2.042 in Figure (11c). We can write 8.718= 10.8-2.042 (*ksB* =2.042) or 8.718=6.676+2.042, which also happens to be at the harmon‐ ic of the anti-Stokes at 4.398. These results belong to the further evolution of the nonlin‐ ear stage which is beyond the scope of this work.

**Figure 8.** Contour plots of the distribution function for *x* ∈(300,339), at: a) *t*=761; b) t=937; c) t= 966.

268 Computational and Numerical Simulations

We present in Figure (12) the frequency spectra between *t1*=1132 and *t2*=1292, at the position *x*=300. The dominant peak in Figure (12a) is now for the SRBS plasma wave at *ωeB* =1.178. However, this peak is broad and includes peaks at 1.006 of the SRFS plasma wave *ωeF* and the KEEN wave *ωKEEN* . The other peaks are at 2.356, 3.534 (these peaks are very close to second and third harmonic of the now dominant SRBS plasma wave at *ωeB* =1.178). We have also a peak at 4.55, and the peak at the harmonic of the pump at 2*ω*0= 6.951. In Figure (12c) we see the now dominant backward mode at *ωsB* =2.277, and the trace of the pump at 3.495. The trace of the anti-Stokes mode is negligible. In Figure (12b) we identify the pump wave in the forward direction at *ω*0=3.495, and the trace of the backward wave at 2.277. The anti-Stokes peak at *ωAS* =4.487 appears negligible. The broad spectrum at 2.277 in Figure (12b,c) would also include

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**Figure 12.** Frequency spectrum recorded at *x*=300, between *t1*=1132 and *t2*=1292 : a) longitudinal plasma wave; b)

We present in Figure (13) a plot of the longitudinal electric field at *t*=761 (at the time we present the spectrum in Figure (5), after about 39000 time steps). The maximum of the electric field in Figure (13) is between *x*=400 or *x*=500. We look in Figure (14) to the phase-space contour plot

; c) backward propagating wave *E* <sup>−</sup>

**4.2. Evolution of the system around the center of the domain**

the SRFS peak at 2.474.

forward propagating wave *E* <sup>+</sup>

**Figure 10.** At *t*=1289: a) contour plot of the distribution function in the domain *x* ∈(320,339); b) averaged distribu‐ tion function at *x*=320; c) averaged distribution function at *x*=329.

**Figure 11.** Spatial Fourier spectrum at the time *t*=1289 in *x* ∈(250,410) for: a) the longitudinal plasma wave; b) the forward electromagnetic wave *E* <sup>+</sup> ; c) the backward electromagnetic wave *E* <sup>−</sup> .

We present in Figure (12) the frequency spectra between *t1*=1132 and *t2*=1292, at the position *x*=300. The dominant peak in Figure (12a) is now for the SRBS plasma wave at *ωeB* =1.178. However, this peak is broad and includes peaks at 1.006 of the SRFS plasma wave *ωeF* and the KEEN wave *ωKEEN* . The other peaks are at 2.356, 3.534 (these peaks are very close to second and third harmonic of the now dominant SRBS plasma wave at *ωeB* =1.178). We have also a peak at 4.55, and the peak at the harmonic of the pump at 2*ω*0= 6.951. In Figure (12c) we see the now dominant backward mode at *ωsB* =2.277, and the trace of the pump at 3.495. The trace of the anti-Stokes mode is negligible. In Figure (12b) we identify the pump wave in the forward direction at *ω*0=3.495, and the trace of the backward wave at 2.277. The anti-Stokes peak at *ωAS* =4.487 appears negligible. The broad spectrum at 2.277 in Figure (12b,c) would also include the SRFS peak at 2.474.

**Figure 12.** Frequency spectrum recorded at *x*=300, between *t1*=1132 and *t2*=1292 : a) longitudinal plasma wave; b) forward propagating wave *E* <sup>+</sup> ; c) backward propagating wave *E* <sup>−</sup>

### **4.2. Evolution of the system around the center of the domain**

**Figure 11.** Spatial Fourier spectrum at the time *t*=1289 in *x* ∈(250,410) for: a) the longitudinal plasma wave; b) the

**Figure 10.** At *t*=1289: a) contour plot of the distribution function in the domain *x* ∈(320,339); b) averaged distribu‐

tion function at *x*=320; c) averaged distribution function at *x*=329.

270 Computational and Numerical Simulations

.

; c) the backward electromagnetic wave *E* <sup>−</sup>

forward electromagnetic wave *E* <sup>+</sup>

We present in Figure (13) a plot of the longitudinal electric field at *t*=761 (at the time we present the spectrum in Figure (5), after about 39000 time steps). The maximum of the electric field in Figure (13) is between *x*=400 or *x*=500. We look in Figure (14) to the phase-space contour plot

of the electron distribution function in *x* ∈(500,519). We see the small vortices of the KEEN wave similar to what has been presented and discussed in Figure (4) at the same time. However, Figure (14) shows the presence of the growing SRBS plasma wave absent from Figure (4). The reason is that at that time, the round-off errors in the region around *x*=400 or *x*=500 have reached a sufficiently high level to act as a perturbation, which stimulates the growth of the SRBS plasma wave, at the same time as the SKEENS we have identified and studied in section 4.1.

backward propagating plasma wave that we see appearing in the domain *x* ∈(300,339) in

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**Figure 14.** Phase-space contour plot of the electron distribution function in *x* ∈(500,519), a*tt*=761.

harmonics at 11.27 and 16.77. The wavenumber spectrum of the backward wave *E* <sup>−</sup>

So the pattern of excitation of the modes to the right of the domain, for instance in *x* ∈(460,619) in Figure (15) is now different from what has been observed to the left, in *x* ∈(380,520) in Figure (15) for instance or in Figure (4, top left). The wavenumber spectrum for the longitudinal electric field in the domain *x* ∈(400,560) a*t t*=820, during the period of the growth, is given in Figure (16). It shows similar peaks as in Figure (5a), with the exception that now the SBRS plasma wave at *keB* =5.38 dominates with respect to the KEEN wave peak at 5.61, with its

peak of the backward scattered wave with *ksB* =2.042, (*k*<sup>0</sup> = −*ksB* + *keB*) much higher than the peak of the wave at 2.277 (compare with Figure (5c)). We also have peaks at 3.338 and 4.398, excited by the coupling with the modes with the same wavenumbers for the forward mode pump with *k*<sup>0</sup> =3.338, and the small peak at 4.398 for the anti-Stokes wave (similar to what has been discussed for Figure(5)). The frequency spectra shows the frequencies of the growing modes, close to what we observe in Figures (6,7). A more complete description of the spectra

Figure (17) shows the longitudinal electric field profile at *t*=1289. We finally present in Figures (18,19) the wavenumber and frequency spectra at *t*=1289, at the end of the simulation, in the domain *x* ∈(400,560). Figure (18) should be compared to Figure (11). We still observe the modes previously discussed, the SRFS plasma wave at *keF* =1.06, the now dominant SRBS plasma wave at *keB* =5.39, the harmonic of the pump wave at 2*ω*0 =6.676, and the harmonics of the now dominant SRBS plasma mode *keB* at 10.799 and 16.18. We also see sidebands devel‐ oping, which is common when positive slopes of the distribution function are formed [33].

In Figure (18b,c) we present the wavenumber spectrum of the longitudinal electric field, the

, toward the end of the simulation at *t*= 1289, in

and the backward wave *E* <sup>−</sup>

, shows a

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273

Figures (8b,c).

is given in the end.

forward wave *E* <sup>+</sup>

**Figure 13.** Plot of the longitudinal electric field at *t*=761.

In Figure (15) we present in the domain *x* ∈(380,619), at *t*=820, the phase-space contour plots of the electron distribution function. We note the rapid growth of the SRBS plasma wave, in addition to the SKEENS. We observe in Figures (15a,b) the same pattern as in Figures.(8a,b). In Figure (15a), we see the dominant modes excited are the SRFS plasma wave with *keF* =1.0645 (wavelength *λsF* =5.9), and the KEEN wave associated with the small vortices with *kKEEN* =5.616 (with a wavelength *λeA* =1.118). These small vortices have a phase velocity *υKEEN* = *ωKEEN* / *kKEEN* =0.18, which corresponds to a momentum *pKEEN* =*υKEEN* / 1−*υKEEN* <sup>2</sup> =0.183, as previously discussed for Figure (4,top left). Again these vortices are similar to what is pre‐ sented in [18] for ponderomotively driven KEEN waves (see also [19-20]). Figure (15b) shows the SRBS plasma wave with *keB* =5.39 propagating from right to left. If we follow the different frames in Figure (15), we see that the heavily damped SRBS plasma wave with *keB* =5.39 seems to have been excited around *x* =500, at a time and position where it is stimulated by the now substantial level reached by the round-off errors. From this position, the excited SRBS plasma wave will propagate to the left and to the right, as in Figure (15), excited by the round-off errors perturbation, which now are growing to the left and to the right around *x* =500. It is this backward propagating plasma wave that we see appearing in the domain *x* ∈(300,339) in Figures (8b,c).

of the electron distribution function in *x* ∈(500,519). We see the small vortices of the KEEN wave similar to what has been presented and discussed in Figure (4) at the same time. However, Figure (14) shows the presence of the growing SRBS plasma wave absent from Figure (4). The reason is that at that time, the round-off errors in the region around *x*=400 or *x*=500 have reached a sufficiently high level to act as a perturbation, which stimulates the growth of the SRBS plasma wave, at the same time as the SKEENS we have identified and studied in

In Figure (15) we present in the domain *x* ∈(380,619), at *t*=820, the phase-space contour plots of the electron distribution function. We note the rapid growth of the SRBS plasma wave, in addition to the SKEENS. We observe in Figures (15a,b) the same pattern as in Figures.(8a,b). In Figure (15a), we see the dominant modes excited are the SRFS plasma wave with *keF* =1.0645 (wavelength *λsF* =5.9), and the KEEN wave associated with the small vortices with *kKEEN* =5.616 (with a wavelength *λeA* =1.118). These small vortices have a phase velocity *υKEEN* =

previously discussed for Figure (4,top left). Again these vortices are similar to what is pre‐ sented in [18] for ponderomotively driven KEEN waves (see also [19-20]). Figure (15b) shows the SRBS plasma wave with *keB* =5.39 propagating from right to left. If we follow the different frames in Figure (15), we see that the heavily damped SRBS plasma wave with *keB* =5.39 seems to have been excited around *x* =500, at a time and position where it is stimulated by the now substantial level reached by the round-off errors. From this position, the excited SRBS plasma wave will propagate to the left and to the right, as in Figure (15), excited by the round-off errors perturbation, which now are growing to the left and to the right around *x* =500. It is this

<sup>2</sup> =0.183, as

*ωKEEN* / *kKEEN* =0.18, which corresponds to a momentum *pKEEN* =*υKEEN* / 1−*υKEEN*

section 4.1.

272 Computational and Numerical Simulations

**Figure 13.** Plot of the longitudinal electric field at *t*=761.

**Figure 14.** Phase-space contour plot of the electron distribution function in *x* ∈(500,519), a*tt*=761.

So the pattern of excitation of the modes to the right of the domain, for instance in *x* ∈(460,619) in Figure (15) is now different from what has been observed to the left, in *x* ∈(380,520) in Figure (15) for instance or in Figure (4, top left). The wavenumber spectrum for the longitudinal electric field in the domain *x* ∈(400,560) a*t t*=820, during the period of the growth, is given in Figure (16). It shows similar peaks as in Figure (5a), with the exception that now the SBRS plasma wave at *keB* =5.38 dominates with respect to the KEEN wave peak at 5.61, with its harmonics at 11.27 and 16.77. The wavenumber spectrum of the backward wave *E* <sup>−</sup> , shows a peak of the backward scattered wave with *ksB* =2.042, (*k*<sup>0</sup> = −*ksB* + *keB*) much higher than the peak of the wave at 2.277 (compare with Figure (5c)). We also have peaks at 3.338 and 4.398, excited by the coupling with the modes with the same wavenumbers for the forward mode pump with *k*<sup>0</sup> =3.338, and the small peak at 4.398 for the anti-Stokes wave (similar to what has been discussed for Figure(5)). The frequency spectra shows the frequencies of the growing modes, close to what we observe in Figures (6,7). A more complete description of the spectra is given in the end.

Figure (17) shows the longitudinal electric field profile at *t*=1289. We finally present in Figures (18,19) the wavenumber and frequency spectra at *t*=1289, at the end of the simulation, in the domain *x* ∈(400,560). Figure (18) should be compared to Figure (11). We still observe the modes previously discussed, the SRFS plasma wave at *keF* =1.06, the now dominant SRBS plasma wave at *keB* =5.39, the harmonic of the pump wave at 2*ω*0 =6.676, and the harmonics of the now dominant SRBS plasma mode *keB* at 10.799 and 16.18. We also see sidebands devel‐ oping, which is common when positive slopes of the distribution function are formed [33].

In Figure (18b,c) we present the wavenumber spectrum of the longitudinal electric field, the forward wave *E* <sup>+</sup> and the backward wave *E* <sup>−</sup> , toward the end of the simulation at *t*= 1289, in

**Figure 16.** The wavenumber spectrum in the domain *x* ∈(400,560) a*t t*=820, for the longitudinal electric field.

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**Figure 17.** Longitudinal electric field profile at *t*=1289.

**Figure 15.** Phase-space contour plot of the electron distribution function in *x* ∈(380,619) at *t*=820.

the domain *x* ∈(400,560). We see in Figure (18c) for the wavenumber spectrum for the backward wave *E* <sup>−</sup> the dominant backward mode at *ksB* =2.042, and the trace of the pump wave at *k*<sup>0</sup> =3.338. We see a small peak of the forward scattered mode at 2.277 which corresponds to the peak in Figure (18b). We also see in Figure (18b) the peak of the forward scattered mode at *ksF* =2.277, the peak for the anti-Stokes *kAS* =4.3985, and the peak of the pump at *k*<sup>0</sup> =3.338. The small peak at 2.042 corresponds to the dominant backward mode at *ksB* =2.042 in Figure (18c). There is a small peak at 8.718 which grew after the growth of the mode at *ksB* =2.042 in Figure (18c). This mode can be the result of a forced oscillation. We can write 8.7= 10.76-2.042 (*ksB* =2.042), which is a coupling with the harmonic of the SRBS plasma wave at 2*keB* =10.78, or 8.718=6.676+2.042, which involves a coupling with 2*k*<sup>0</sup> =6.676 (the plasma wave excited at the harmonic of the pump). Note also that the mode at 8.718 is close to the harmonic of the anti-Stokes excited at 4.398.

**Figure 16.** The wavenumber spectrum in the domain *x* ∈(400,560) a*t t*=820, for the longitudinal electric field.

**Figure 17.** Longitudinal electric field profile at *t*=1289.

the domain *x* ∈(400,560). We see in Figure (18c) for the wavenumber spectrum for the

**Figure 15.** Phase-space contour plot of the electron distribution function in *x* ∈(380,619) at *t*=820.

at *k*<sup>0</sup> =3.338. We see a small peak of the forward scattered mode at 2.277 which corresponds to the peak in Figure (18b). We also see in Figure (18b) the peak of the forward scattered mode at *ksF* =2.277, the peak for the anti-Stokes *kAS* =4.3985, and the peak of the pump at *k*<sup>0</sup> =3.338. The small peak at 2.042 corresponds to the dominant backward mode at *ksB* =2.042 in Figure (18c). There is a small peak at 8.718 which grew after the growth of the mode at *ksB* =2.042 in Figure (18c). This mode can be the result of a forced oscillation. We can write 8.7= 10.76-2.042 (*ksB* =2.042), which is a coupling with the harmonic of the SRBS plasma wave at 2*keB* =10.78, or 8.718=6.676+2.042, which involves a coupling with 2*k*<sup>0</sup> =6.676 (the plasma wave excited at the harmonic of the pump). Note also that the mode at 8.718 is close to the harmonic of the anti-

the dominant backward mode at *ksB* =2.042, and the trace of the pump wave

backward wave *E* <sup>−</sup>

274 Computational and Numerical Simulations

Stokes excited at 4.398.

To identify the frequency spectra at the end of the simulation, we present in Figure (19) the frequency spectra between *t1*=1113 and *t2*=1273, at the position *x*=450. We identify in Fig. (19a) the local peak at *ωeF* =1.0014 of the SRFS plasma wave (1.006 in our theoretical value). The dominant peak is for the SRBS plasma wave at *ωeB* =1.178. The other peaks are at 0.1767, 2.18, and 2.356, 3.534, 4.732 (these last three very close to second, third harmonic and fourth harmonic of *ωeB* =1.178). We have also a peak at the harmonic of the pump 2*ω*0= 6.951. In Fig. (19c) we see the now dominant backward scattered mode at *ωsB* =2.277, and the trace of the pump at 3.495. The trace of the anti-Stokes is negligible. In Figure (19b) we see the pump wave, dominant in the forward direction, at *ω*<sup>0</sup> =3.495, and the trace of the backward wave at 2.277. We see also the forward scattered mode at 2.474. The anti-Stokes peak at *ωAS* =4.487 appears negligible. In our normalized units (velocity normalized to the velocity of light) *υ<sup>T</sup>* / *c* =0.06256 at *Te*=2keV. With *ω* =0.176, we get *k* =2.169, very close to the peak at 2.14 in Figure (18a), which is close to the harmonic of 1.06 in Figure (18a).

**Figure 19.** Frequency spectra recorded at the position *x*=450, between *t1*=1113 and *t2*=1273 for: a) the longitudinal

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear…

In laser fusion, the coupling and propagation of the laser beams in the plasma surrounding the pellet can be the scene of nonlinear processes such as parametric instabilities, which must be well understood and controlled to keep them at low levels, since they are detrimental to laser fusion because they can lead to losses of energy and illumination uniformity. Recent publications [22,23,34-40] have identified the need for a deeper understanding of laser-plasma interactions, and the importance of a kinetic treatment of the plasma, particularly in the regimes currently being approached by the new generation of lasers, and for the treatment of modes such as KEEN waves [18-20], even newer horizons are opened up. The old picture of EPWs and their evolution is now replaced by a much richer scenario of multiple harmonic waves transiently trapping, untrapping and retrapping paricle distributions that maintain the wave on average but without the need for flat distribution functions as in the canonical BGK

; c) the backward propagating wave *E* <sup>−</sup>

.

http://dx.doi.org/10.5772/57476

277

electric field; b) the forward propagating wave *E* <sup>+</sup>

**5. Conclusion**

mode setting of lore.

**Figure 18.** The wavenumber spectra in the domain *x* ∈(400,560) a*t t*=1289, for: a) the longitudinal electric field; b) the forward propagating wave *E* <sup>+</sup> ; c) the backward propagating wave *E* <sup>−</sup> .

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear… http://dx.doi.org/10.5772/57476 277

**Figure 19.** Frequency spectra recorded at the position *x*=450, between *t1*=1113 and *t2*=1273 for: a) the longitudinal electric field; b) the forward propagating wave *E* <sup>+</sup> ; c) the backward propagating wave *E* <sup>−</sup> .

### **5. Conclusion**

To identify the frequency spectra at the end of the simulation, we present in Figure (19) the frequency spectra between *t1*=1113 and *t2*=1273, at the position *x*=450. We identify in Fig. (19a) the local peak at *ωeF* =1.0014 of the SRFS plasma wave (1.006 in our theoretical value). The dominant peak is for the SRBS plasma wave at *ωeB* =1.178. The other peaks are at 0.1767, 2.18, and 2.356, 3.534, 4.732 (these last three very close to second, third harmonic and fourth harmonic of *ωeB* =1.178). We have also a peak at the harmonic of the pump 2*ω*0= 6.951. In Fig. (19c) we see the now dominant backward scattered mode at *ωsB* =2.277, and the trace of the pump at 3.495. The trace of the anti-Stokes is negligible. In Figure (19b) we see the pump wave, dominant in the forward direction, at *ω*<sup>0</sup> =3.495, and the trace of the backward wave at 2.277. We see also the forward scattered mode at 2.474. The anti-Stokes peak at *ωAS* =4.487 appears negligible. In our normalized units (velocity normalized to the velocity of light) *υ<sup>T</sup>* / *c* =0.06256 at *Te*=2keV. With *ω* =0.176, we get *k* =2.169, very close to the peak at 2.14 in Figure (18a), which

**Figure 18.** The wavenumber spectra in the domain *x* ∈(400,560) a*t t*=1289, for: a) the longitudinal electric field; b)

.

; c) the backward propagating wave *E* <sup>−</sup>

is close to the harmonic of 1.06 in Figure (18a).

276 Computational and Numerical Simulations

the forward propagating wave *E* <sup>+</sup>

In laser fusion, the coupling and propagation of the laser beams in the plasma surrounding the pellet can be the scene of nonlinear processes such as parametric instabilities, which must be well understood and controlled to keep them at low levels, since they are detrimental to laser fusion because they can lead to losses of energy and illumination uniformity. Recent publications [22,23,34-40] have identified the need for a deeper understanding of laser-plasma interactions, and the importance of a kinetic treatment of the plasma, particularly in the regimes currently being approached by the new generation of lasers, and for the treatment of modes such as KEEN waves [18-20], even newer horizons are opened up. The old picture of EPWs and their evolution is now replaced by a much richer scenario of multiple harmonic waves transiently trapping, untrapping and retrapping paricle distributions that maintain the wave on average but without the need for flat distribution functions as in the canonical BGK mode setting of lore.

We showed for the first time in this study that a seemless transition occurs from Raman forward scatter, to the standing wave excited KEEN wave very near the backscattering plasma wave so that the distribution function is strongly modified by the KEEN wave before the EPW can be excited in SRBS. For the parameters we have investigated, the SRBS process is preceded by KEEN waves and then competes with SKEENS for supremacy and eventual merging. This rich physics was not observed when strong seeding of the backscattered wave prevented any detection of these intermediate processes.

Poisson system of equations. We expect to find interesting resonances, even with KEEN waves that have significantly lower phase velocities than the electron plasma waves. The work here showed the coevolution of SKEENS and SRBS for electrostatic waves whose phase velocities were so close that their vortical structures in phase-space directly overlap‐

Stimulated Raman Scattering with a Relativistic Vlasov-Maxwell Code: Cascades of Nonstationary Nonlinear…

http://dx.doi.org/10.5772/57476

279

The authors are grateful to the Centre de calcul scientifique de l'IREQ (CASIR) for computer time for the simulations presented in this work. BA would like to acknowledge the financial

assistance of DOE OFES HEDP program through a subcontract via UCSD.

1 Institut de recherche Hydro-Québec (IREQ), Varennes, Québec, Canada

[1] Atzeni S, Meyer-ter-Vehn J. The Physics of Inertial Fusion. Oxford; 2004.

geneous Plasmas I: Two Model Problems. Phys. Plasmas 1997; 4, 3788.

geneous Plasmas III: Two-Plasmon Decay. Phys. Plasmas 1997; 4, 3827.

[8] Sagdeev R Z, Galeev A A. Nonlinear Plasma Theory. W. A. Benjamin; 1969

[7] Kono M, Skoric M M. Nonlinear Physics of Plasmas. Springer; 2010

[3] Afeyan B, Williams E A. Variational Approach to Parametric Instabilities in Inhomo‐

[4] Afeyan B, Williams E A. Variational Approach to Parametric Instabilities in Inhomo‐ geneous Plasmas II: Stimulated Raman Scattering. Phys. Plasmas 1997; 4, 3803.

[5] Afeyan B, Williams E A. Variational Approach to Parametric Instabilities in Inhomo‐

[6] Afeyan B, Williams E.A. Variational Approach to Parametric Instabilities in Inhomo‐ geneous Plasmas IV: The mixed polarization high-frequency instability. Phys. Plas‐

[2] Kruer W L. The Physics of Laser Plasma Interactions. Westview; 2003.

and Bedros Afeyan2

2 Polymath Research Inc., Pleasanton, CA, USA

ped and were eventually mixed.

**Acknowledgements**

**Author details**

Magdi Shoucri1

**References**

mas 1997; 4, 3845.

The accurate representation and evolution of the particles distribution function provided by the Eulerian Vlasov code offers a powerful tool to study highly nonlinear nonstationary processes in high energy density plasmas. We have uncovered some distinctive features of KEEN waves participating in the Raman process, using a 1D Eulerian Vlasov-Maxwell code that relativistically evolves both ions and electrons. To avoid any interference from artificially distorted distribution functions or imposed linear wave seeding, we start the code from an initial Maxwellian distribution, and a very weak scattered light field standing wave pattern which is enough to trigger both SRFS and then SKEENS. The system evolves under the influence of a pump light wave which provides fluctuations from which SRBS eventu‐ ally develops. We identify in the early phase of the Raman interaction a reflected light that matches the backscattering of the pump laser off a KEEN wave whose fundamental harmonic has the same wavelength as the forward scattered light, and its appearance precedes the growth and saturation of SRBS. The evolution of the system is however modified with the results presented in section 4.2, close to the center of the simulation domain. In this region, the round-off errors have reached a level where they act as a perturbation, leading to the simultaneous appearance and growth of the SRBS process, in addition to the KEEN wave (see Figure (14)). So we have two distinct evolution scenarios of Raman scattering in the domain we study. To the right of the region *x* ∈(500,519) in Figure (14), we see a simultaneous growth of the SRBS plasma wave and the KEEN wave (see Figure (15)). And to the left of the region *x* ∈(500,519), the growth of the round-off errors acting as a perturbation leads to the appearance of SRBS plasma waves moving to the left in the backward direction. This is where the KEEN wave has already reached saturation, causing heating and relative flattening of the electron distribution function, which shows a structure with a trapped population of electrons. Note the harmonic structure associated with the SRBS mode *ωeB* =1.178, *keB* =5.38 in Figures (11a,12a). Recent publica‐ tions have pointed to the importance of 2D and 3D effects for a rigorous theory of SRS saturation [41]. This is beyond the scope of the present work. We have restricted our study to the initial phase of the evolution of Raman scattering, and we have shown that in this case scattering off a KEEN wave can produce a backward wave which contributes to the inflation of the Raman signal well before the SRBS starts growing on its own.

In future studies, we propose to investigate the physics of the interaction between SKEENS and SRBS, but eliminating the need for SRFS initiation. This can be achieved by driving the KEEN wave directly by the ponderomotive force generated by the beating of the pump and the appropriate seed electromagnetic wave. Driving KEEN waves directly and electromag‐ netically generalizes the work of Afeyan et al. [18-20] which has been based on the VlasovPoisson system of equations. We expect to find interesting resonances, even with KEEN waves that have significantly lower phase velocities than the electron plasma waves. The work here showed the coevolution of SKEENS and SRBS for electrostatic waves whose phase velocities were so close that their vortical structures in phase-space directly overlap‐ ped and were eventually mixed.
