3.1. Characterization of the parametric instabilities involved

The two main instabilities of interest here are SRS and TPD instability. The resonant SRS process consists of the decomposition of a laser photon (${\it\omega}_{o}$) into a backscattered frequency-downshifted photon (${\it\omega}_{1}$) and a forward travelling electron plasma wave (${\it\omega}_{p}$) (EPW). It fulfils the following conditions for the frequency and wavevector:

(1)$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}}\displaystyle {\it\omega}_{o}\, & =\, & \displaystyle {\it\omega}_{1}+{\it\omega}_{p}\\ \displaystyle \mathbf{k}_{o}\, & =\, & \displaystyle \mathbf{k}_{1}+\mathbf{k}_{p}.\end{array} & & \displaystyle\end{eqnarray}$$

The frequency of the EPW follows from the dispersion relation as

(2)$$\begin{eqnarray}{\it\omega}_{p}\approx {\it\omega}_{\text{pe}}(1+3k_{epw}^{2}{\it\lambda}_{D}^{2})^{1/2},\end{eqnarray}$$

where ${\it\omega}_{\text{pe}}=(4{\it\pi}ne^{2}/m_{e})^{1/2}$ is the local electron plasma frequency and ${\it\lambda}_{D}=v_{\text{the}}/{\it\omega}_{\text{pe}}=(kT_{e}/4{\it\pi}ne^{2})^{1/2}$ is the Debye length, which in practical units is given as ${\it\lambda}_{D}=7.43\times 10^{2}T_{e}^{1/2}n^{-1/2}~\text{cm}$, with $T_{e}$ in eV and $n$ in $\text{cm}^{3}$; $v_{\text{the}}=(T_{e}/m_{e})^{1/2}$ is the electron thermal velocity. The electromagnetic $k$-vector of the backscattered light is given as $k_{1}=({\it\omega}_{1}/c)\sqrt{1-{\it\omega}_{\text{p e }}^{2}/{\it\omega}_{1}^{2}}$.

The TPD is the decomposition of the laser photon (${\it\omega}_{o}$) into two plasmons (${\it\omega}_{p1},{\it\omega}_{p2}$). The plasmon frequencies are determined by the dispersion relation for the electron plasma waves, Equation (2), and the matching conditions:

(3)$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}}\displaystyle {\it\omega}_{o}\, & =\, & \displaystyle {\it\omega}_{p1}+{\it\omega}_{p2}\\ \displaystyle \mathbf{k}_{o}\, & =\, & \displaystyle \mathbf{k}_{p1}+\mathbf{k}_{p2}.\end{array} & & \displaystyle\end{eqnarray}$$

Figure 4 shows the geometry of the wavevectors for TPD. The wavevectors of the two plasmons ($\mathbf{k}_{1},\mathbf{k}_{2}$) obey the relations

(4)$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}}\displaystyle \mathbf{k}_{1y}\, & =\, & \displaystyle -\mathbf{k}_{2y}\equiv \mathbf{k}_{y}\\ \displaystyle \mathbf{k}_{1,2x}\, & =\, & \displaystyle \frac{k_{o}}{2}\pm {\rm\Delta}\mathbf{k}\\ \displaystyle {\rm\Delta}\mathbf{k}^{2}\, & =\, & \displaystyle \frac{\mathbf{k}_{o}^{2}}{4}+\mathbf{k}_{y}^{2}.\end{array} & & \displaystyle\end{eqnarray}$$

Figure 4.

Geometry of the $k$-vectors involved in the TPD instability. The decay of a photon into two plasmons can be realized in two possible ways while preserving energy and momentum. This particular geometry applies in 2D and helps with the interpretation of the phase space diagrams. In reality, 3D, the number of possible $k$-vectors is infinite lying on an asymmetric cone around the laser $k$-vector.

The threshold for the TPD instability in an inhomogeneous plasma profile^{[Reference Liu and Rosenbluth43, Reference Simon, Short, Williams and Dewandre44]} is temperature dependent and is given as

(5)$$\begin{eqnarray}I_{16}^{\text{TPD}}\approx 0.5T_{e}L_{n}^{-1}{\it\lambda}_{o}^{-1}.\end{eqnarray}$$

Here, $T_{e}$, is in units of kilo electron volts and $I_{16}$ is the laser intensity in units of $10^{16}~\text{W}~\text{cm}^{-2}$; $L_{n}$ and ${\it\lambda}_{o}$ are given in units of microns. The TPD instability is excited in the vicinity of the quarter critical density and develops as an absolute instability as the slow plasma waves do not escape the resonance. In contrast, the threshold for SRS excitation near $n_{c}/4$ is not a function of $T_{e}$ and is given by^{[Reference Kruer17, Reference Menyuk, El-Siragy and Manheimer45]}

(6)$$\begin{eqnarray}I_{16}^{\text{SRS}}\approx 12L_{n}^{-4/3}{\it\lambda}_{o}^{-2/3}.\end{eqnarray}$$

Depending on the electron temperature, the thresholds for the two instabilities can be quite similar and develop in competition. A high-frequency hybrid instability (HFHI) can develop^{[Reference Afeyan and Williams46, Reference Afeyan and Williams47]}, leading to a co-existence of electrostatic and electromagnetic modes. The growth rate for Raman is given as^{[Reference Forslund, Kindel and Lindman48]} ${\it\gamma}_{\text{SRS}}={\it\gamma}_{o}\text{min}(1,{\it\gamma}_{o}/{\it\gamma}_{\text{tot}})$, with ${\it\gamma}_{\text{tot}}={\it\gamma}_{L}+{\it\gamma}_{\text{coll}}$. Here, ${\it\gamma}_{L}$ and ${\it\gamma}_{\text{tot}}$ are the damping rates due to Landau damping and collisions, respectively. In the case when Raman is above threshold, i.e., ${\it\gamma}_{\text{SRS}}^{2}>{\it\gamma}_{L}{\it\gamma}_{\text{coll}}$, the growth rate reduces to ${\it\gamma}_{\text{SRS}}={\it\gamma}_{o}$. Even for the lowest temperature used in the simulations, $500~\text{eV}$, the collisional damping rate is two orders of magnitude smaller than the Landau damping rate for EPWs, which, for backward SRS, is given as

(7)$$\begin{eqnarray}{\it\gamma}_{L}/{\it\omega}_{o}=0.14\frac{(n_{e}/n_{c})^{1/2}}{(k_{epw}{\it\lambda}_{D})^{3}}\exp (-1/(2k_{epw}^{2}{\it\lambda}_{D}^{2})).\end{eqnarray}$$

Here, $k_{epw}=k_{o}(\sqrt{1-n_{e}/n_{c}}+\sqrt{1-2\sqrt{n_{e}/n_{c}}})$ is the electron plasma wavevector and one has the following expression for $k_{o}^{2}{\it\lambda}_{D}^{2}$ in practical units:

(8)$$\begin{eqnarray}k_{o}^{2}{\it\lambda}_{D}^{2}=1.9\times 10^{-3}T_{e}/(n_{e}/n_{c}),\end{eqnarray}$$

with $T_{e}$ in keV. It should be noted that $k_{o}$ is the value of the electromagnetic wavevector in vacuum.

Evaluation of these expressions at a density of $0.04\,n_{c}$ and for $T_{e}=500~\text{eV}$ results in a Landau damping rate of ${\sim}1\times 10^{-2}{\it\omega}_{o}^{-1}$, i.e., much larger than the collision rate quoted in Section 2. The largest damping rate will correspond to the largest temperature, $10~\text{keV}$, and is of the order of ${\sim}1\times 10^{-3}{\it\omega}_{o}^{-1}$ at the quarter critical density $n_{e}/n_{c}=0.25$. However, even in the most damped case, since the collision frequency is so low as discussed in Section 2, the growth rate of Raman remains significantly above threshold and is therefore given by^{[Reference Forslund, Kindel and Lindman48]}

(9)$$\begin{eqnarray}{\it\gamma}_{\text{SRS}}/{\it\omega}_{o}={\it\gamma}_{o}/{\it\omega}_{o}=\frac{(k_{epw}/k_{o})(v_{osc}/c)}{2(\sqrt{n_{c}/n_{e}}-1)^{1/2}},\end{eqnarray}$$

where $v_{osc}=eE_{o}/{\it\omega}_{o}m_{e}$ is the quiver velocity of the electrons in the laser electric field. For an intensity of $I{\it\lambda}_{o}^{2}=1.2\times 10^{15}~\text{W}~{\rm\mu}\text{m}^{2}~\text{cm}^{-2}$ and a density of $n_{e}/n_{c}=0.25$ one obtains a growth rate for SRS of ${\it\gamma}_{\text{SRS}}/{\it\omega}_{o}\approx 1.5\times 10^{-2}$.

At very low electron plasma temperature, a few hundred eV, the Langmuir decay instability (LDI) can play an important role as it saturates TPD and SRS activity. Although of limited importance in the case of SI which operates in multi-keV conditions, it could be relevant for present-day LPI experiments for SI, as these take place at a much lower temperature. Therefore it is presented in some more detail.

The LDI was predicted in the 1960s^{[Reference Oraevskii and Sagdeev49, Reference Ichikawa50]} and in LPI experiments it was verified more recently by direct observation^{[Reference Michel, Depierreux, Stenz, Tassin and Labaune36, Reference Depierreux, Fuchs, Labaune, Michard, Baldis, Pesme, Hüller and Laval51–Reference Kline, Montgomery, Bezzerides, Cobble, DuBois, Johnson, Rose, Yin and Vu53]}, although its experimental existence had already been conjectured before^{[Reference Obiki, Itatani and Otani54]}, and it has been observed indirectly due to the ion-acoustic wave (IAW) damping on SRS^{[Reference Drake and Batha55–Reference Montgomery, Afeyan, Cobble, Fernandez, Wilke, Glenzer, Kirkwood, MacGowan, Moody, Lindman, Munro, Wilde, Rose, Dubois, Bezzerides and Vu59]}.

The LDI induces a decay of the pump plasma wave $(k_{epw},{\it\omega}_{p})$ into an IAW characterized by $(2k_{epw}-{\it\delta}k,{\it\omega}_{cs})$ (travelling in the direction of the original EPW) and an anti-Stokes daughter EPW (travelling in the opposite direction) having a frequency close to the original frequency (downshifted by ${\it\omega}_{cs}$) and a wavevector given by $k_{epw1}=k_{epw}+{\it\delta}k$, where the correction ${\it\delta}k$ has the form

(10)$$\begin{eqnarray}{\it\delta}k={\textstyle \frac{2}{3}}(Zm_{e}/m_{i})^{1/2}/{\it\lambda}_{D}.\end{eqnarray}$$

Here, ${\it\lambda}_{D}=(kT_{e}/4{\it\pi}ne^{2})^{1/2}$ is the Debye length, which in practical units is given as ${\it\lambda}_{D}=7.43\times 10^{2}T_{e}^{1/2}n^{-1/2}~\text{cm}$, with $T_{e}$ in eV and $n$ in $\text{cm}^{3}$. A necessary condition for LDI to take place is therefore that^{[Reference Oraevskii and Sagdeev49, Reference Bonnaud, Pesme and Pellat60]}

(11)$$\begin{eqnarray}k_{epw}{\it\lambda}_{D}>\frac{2}{3}\left(\frac{Zm_{e}}{m_{i}}\right)^{1/2}\approx 0.03.\end{eqnarray}$$

For all the cases considered in Table 1 this is clearly the case. However, the higher the temperature, the more the EPWs are damped. The LDI is in general most active at low temperature, provided Equation (11) is fulfilled.

The maximum growth rate is given by

(12)$$\begin{eqnarray}{\it\gamma}_{\text{LDI}}=\frac{1}{2}\frac{eE_{p}}{m_{e}{\it\omega}_{p}}\frac{1}{v_{\text{the}}}\sqrt{{\it\omega}_{p1}{\it\omega}_{cs}}.\end{eqnarray}$$

Here, $E_{p}$ is the electrostatic field associated with the EPWs, ${\it\omega}_{p},{\it\omega}_{p1}$ represent the two plasmons, and

(13)$$\begin{eqnarray}{\it\omega}_{cs}^{2}=k_{cs}^{2}c_{s}^{2}(1/(1+k_{cs}^{2}{\it\lambda}_{D}^{2})+3T_{i}/ZT_{e})\end{eqnarray}$$

is the IAW frequency. In order for LDI to develop, the growth rate has to be above a threshold given by the linear damping of electron plasma and ion-plasma waves: ${\it\gamma}_{\text{LDI}}>\sqrt{{\it\gamma}_{epw}{\it\gamma}_{iaw}}$. In the regime where electron kinetic effects are neglegible, LDI, LDI cascade and Langmuir wave collapse tend to dominate the nonlinear evolution of the EPW, participating in the saturation of the SRS instability. If the EPW density fluctuations are below the wavebreaking limit, neglecting the kinetic effect is equivalent to considering $k_{epw}^{2}{\it\lambda}_{D}^{2}\ll 1$. In such regimes it is thus expected to observe LDI, and its influence on SRS^{[Reference Karttunen61–Reference Russell, DuBois and Rose66]}.

For the intensity considered in this study SBS takes place in the weak-coupling regime even for the lowest temperature used, i.e., Equation (13) is the correct dispersion relation for the IAWs with $k_{cs}\approx 2k_{po}$, with $k_{po}=k_{o}\sqrt{1-n_{e}/n_{c}}$. The growth rate for SBS in this regime is given as ${\it\gamma}_{SBS}=k_{po}v_{osc}{\it\omega}_{pi}/\sqrt{2{\it\omega}_{cs}{\it\omega}_{o}}$, which in practical units is

(14)$$\begin{eqnarray}{\it\gamma}_{SBS}/{\it\omega}_{o}=3.1\times 10^{-2}\sqrt{I_{16}{\it\lambda}_{o}^{2}}\left(\frac{n_{e}}{n_{c}}\right)^{1/2}\left(\frac{Z}{A}\right)^{1/4}\frac{1}{T_{e}^{1/4}}.\end{eqnarray}$$

Here, ${\it\omega}_{pi}=\sqrt{4{\it\pi}n_{i}Z^{2}e^{2}/m_{i}}$ is the ion plasma frequency, $Z$ is the atomic charge and $A$ is the atomic mass number.

In the following the simulation results are analysed with respect to the behaviour of the reflectivity of the incident laser beam, the induced activity of the parametric instabilities, and the phase space and Poynting vector. These issues are of course strongly interdependent.

3.2. Overall scenario

Two fundamental issues have to be addressed as far as the simulations are concerned.

• ∙ The relative importance of TPD and SRS. One would expect that the colder the plasma the stronger the TPD.

• ∙ Which mechanism is saturating the TPD and SRS activity at the quarter critical density $n_{c}/4$? Again this will be temperature dependent.

With respect to the first point it is indeed found that the higher the temperature the more pronounced SRS becomes, although it is a negligible energy loss mechanism as far as reflectivity is concerned. With respect to the second point one can observe a clear transition from LDI-induced saturation of Raman at $500~\text{eV}$ to saturation due to cavitation and density fluctuations for temperatures of $2~\text{keV}$ and above. For the low-temperature case c8 there is LDI and TPD activity but no SRS.

SBS is present for any temperature but decreases in importance the higher the temperature. The general conclusion of this set of simulations is that the plasma temperature plays a crucial role as far as the LPI scenario is concerned. Already a factor of two in the electron plasma temperature can significantly affect the relative importance of the parametric instabilities and their effect on the plasma dynamics and laser absorption. Realistic simulations and future experiments for SI require LPI to take place at the right temperature.

The overall energy balance is affected by the amount of the energy of the incident laser beam reflected due to SRS and SBS, Section 3.3, as well as by absorption into collective plasma modes and hot electrons, Sections 3.5 and 3.6. The time scales of the simulations allow study of the saturation of these instabilities related to electron modes, Section 3.4. It has been shown^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Yan, Li and Ren67]} that the development and subsequent saturation of the electron-related instabilities takes place as a recurrent scenario.

3.3. Reflectivity

Figure 5 summarizes the reflectivities for the four temperatures. The reflectivities are evaluated using $B_{z}$ integrated over the whole transverse direction. The use of $B_{z}$ instead of $E_{y}$ ensures that nothing is lost due to a possible opening angle of the backscattered light. There is basically no SRS activity for the cases c8 and h8. The substantial backscattering of the order of 10%–20% originates entirely from Brillouin. Increase of the temperatures strongly reduces SBS activity and increases SRS. However, the energy losses due to SRS even for the highest temperature case, h9, remain negligible. The bursty, spike-like nature of the SRS reflectivity is not related to inflationary SRS, as the corresponding frequency spectra relate the region of SRS activity to the vicinity of $n_{c}/4$, in accordance with standard absolute SRS excitation. In general, SRS is strongly Landau damped at temperatures of the order of a few keV. However, in the vicinity of the quarter critical density the wavevector of the backscattered light

(15)$$\begin{eqnarray}k_{1}\approx (1/c)\sqrt{{\it\omega}_{1}^{2}-{\it\omega}_{\text{p e }}^{2}}=({\it\omega}_{o}/c)\sqrt{1-2\sqrt{n/n_{c}}}\end{eqnarray}$$

goes to zero, such that the momentum balance gives $k_{epw}\approx k_{o}$. The resulting Landau damping rate, see Equations (7) and (8), is of the order of ${\it\gamma}_{L}/{\it\omega}_{o}\approx 10^{-3}$ (for the highest temperature case h9, $T_{e}=10~\text{keV}$). The threshold for SRS is independent of temperature, so one would expect that the SRS-induced part of the reflectivity will not change as the temperature is increased. However, locally TPD and SRS are competing for the pump and the threshold for TPD is strongly temperature dependent. As $T_{e}$ increases, TPD activity is reduced and SRS can take more energy out of the pump, thereby increasing the reflectivity.

Figure 5.

Reflectivities ($R=I/I_{o}$, i.e., reflected intensity over incident intensity at the centre of the speckle in the transverse direction) for the cases (a) c8, (b) h8, (c) h7 and (d) h9. The curves are ‘filled’ as the laser temporal period is resolved. The blue curve corresponds to SBS-like frequencies, summing the range $0.9{\it\omega}_{o}$–$1.1{\it\omega}_{o}$. The red curve corresponds to SRS-like frequencies, summing the range $0.0{\it\omega}_{o}$–$0.9{\it\omega}_{o}$. No frequencies are present in the interval $0.8{\it\omega}_{o}$–$0.9{\it\omega}_{0}$. Note: the time on the axis refers to the moment the reflected light crosses the boundary of the computational box; as the quarter critical density is located at $2200\,k_{o}^{-1}$, the light was actually refelected ${\sim}2200{\it\omega}_{o}^{-1}$ earlier.

Figure 6.

Poynting vector for the case h9 at $t{\it\omega}_{o}=13\,600$.

The total reflectivities, i.e., the SRS and SBS contributions combined, reduce strongly as the temperature is increased. For the cases c8 and h8 the average reflectivity is of the order of 10%–15%, whereas for the high-temperature cases, h7 and h9, the overall reflectivity is reduced to 1%–2% only.

SBS develops everywhere in the profile, up to $n_{c}/4$. At $n_{c}/4$ SBS is strongly inhibited due to the fact that the laser beam is randomized due to the cavitation process and LDI, and the associated strong density fluctuations (see, e.g., Figure 6 for the case h9).

By contrast, SRS develops predominantly at $n_{c}/4$. The simulated time scales are short for the SBS evolution. It is therefore unclear what the saturation mechanisms for SBS are on longer time scales. The bursty behaviour of the reflectivities as visible in Figure 5 has been observed before and was discussed in the literature^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Riconda, Weber, Tikhonchuk, Adam and Heron68]}. In the case of SRS it is related to hot electron production and cavity creation.

Figure 7.

Frequency spectra for the cases (a) c8, (b) h9 and (c) a zoom of (b). Note: (a) and (b) are on log scale whereas (c) is on linear scale.

Figure 7 shows the frequency spectra of the backscattered light, in agreement with the decomposition into SRS and SBS of the reflectivities. In the cold case, c8 (Figure 7(a)), the SRS signal is at $0.5{\it\omega}_{o}$ but is very weak in agreement with the reflectivity curves. In the hot case, h9 (Figure 7(b)), SRS is dominant. Clearly visible is a splitting of the SRS frequency around $0.5{\it\omega}_{o}$ with ${\rm\Delta}k\approx 0.013\,k_{o}$. This most likely originates from density perturbations induced by TPD and/or cavitation. The localization of the SRS signal around $0.5{\it\omega}_{o}$ is a signature of the fact that SRS originates from the vicinity of the quarter critical density. The downshifted backscattered light from SRS has the correct frequency to be trapped locally in density cavities^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Weber, Riconda, Klimo, Heron and Tikhonchuk20]}.

Figure 8.

Two-dimensional Fourier spectra of the electromagnetic field $B_{z}$ evaluated in the vicinity of $n_{c}/4$ for the cases c8 (a, c) and h9 (b, d) taken at times $t{\it\omega}_{o}=4000$ (a, b) and $t{\it\omega}_{o}=7000$ (c, d).

Figure 8 shows the Fourier transforms of the electromagnetic field in the plasma for the two extreme cases, c8 and h9. In Figure 8(a) one can observe a feature at $|k|=1.5\,k_{o}$ at large angle. This is a signature of TPD coupling with the laser and disappears at later time after saturation of TPD, as can be seen in Figure 8(b). The signal close to $0.86\,k_{o}$ stems from the incoming laser and the backscattered light due to SBS. The strong-signal opening angle increases in time due to side scattering and refraction from density perturbations. In the hot case, Figure 8(b, d), at later time two pronounced signals appear at $k_{\Vert }\approx 0.05\,k_{o}$ and $k_{\Vert }\approx 1.5\,k_{o}$. The first is the signature of SRS, as $k_{1}\ll 1$ close to $n_{c}/4$ (see the discussion above), the latter one is the coupling of electron plasma waves at ${\it\omega}\approx 0.5{\it\omega}_{o}$ with the laser. As expected, the signal is mostly in the parallel direction as SRS has a small opening angle.

3.4. Electron-related mode activity: SRS, TPD and LDI

As already mentioned, there is almost no SRS for the cases c8 and h8. Strong SRS is present for higher temperatures above $2~\text{keV}$. In general, it can be said that there is no SRS activity in the low-density part of the plasma corona. SRS is concentrated near the quarter critical density. The high-temperature case h9 has very strong absolute SRS activity but basically no TPD.

As can be seen from Table 2, the threshold for TPD is much lower then the threshold for SRS for the cases c8 and h8. As a consequence, TPD arises first and no or very little absolute SRS develops as TPD is initially depleting the pump, as can be seen in Figure 9. The figure clearly shows the absence of any laser beam in the density profile behind the quarter critical density, which however is a transient feature due to TPD saturation.

Table 2.

Temperature-dependent occurrence of LPI phenomena. The number of stars gives a rough ‘visual’ interpretation of the strength of the process occurring, with $\star \star \star \star$ strongest and $\star$ weakest. The numbers in the columns $I_{16}^{\text{TPD}}$ and $I_{16}^{\text{SRS}}$ are calculated from the corresponding Equations (5) and (6). The thresholds have to be compared with the laser intensity, which in units of $10^{16}~\text{W}~\text{cm}^{-2}$ is 1.2 for all cases. CAV $=$ cavitation.

As the SBS reflectivity data (see Figure 5(a)) show reflection of about 20%–30%, the remaining energy of order 70% at this time is absorbed in the plasma due to TPD.

At a later stage SRS still does not develop due to the strong, irregular small-scale density modulations induced by TPD. For the cold case c8, TPD is saturated by LDI which develops on the EPWs generated originally by TPD.

In the high-temperature cases (h7 and h9) the threshold for SRS is lower than for TPD and strong absolute SRS develops which induces cavitation and leads to saturation.

Figure 10(a) shows the Fourier transform of the ion density for the coldest case, c8, after TPD saturation. The large feature between $30^{\circ }$ and $40^{\circ }$ reproduces the IAWs which are the result of LDI decay of the plasmons excited by TPD and has $k$-vectors that are roughly twice the original $k$-vectors from the EPWs. The extent is due to the large spread of the $k$-vectors of the EPWs. The smaller feature with $k_{\Vert }$ close to zero results from the low-frequency beating of symmetrically excited plasmons, e.g., $\text{EPW}_{1}$ and $\text{EPW}_{1}^{\prime }$ in Figure 4, summing up to purely perpendicular fluctuations^{[Reference Riconda, Weber, Klimo, Heron and Tikhonchuk69]}. The very small feature at $k_{\Vert }\approx 1.8\,k_{o}$ and in an almost perfect forward direction is the IAW signal generated by SBS (also visible in (b) for case h8).

Figure 9.

Poynting vector for the case c8 at $t{\it\omega}_{o}=3600$. The ‘hole’ behind the density layer around $n_{c}/4$ is clearly visible.

Figure 10.

Fourier transform of the ion density corresponding to Figure 11. (a) Case c8 at $t{\it\omega}_{o}=5600$, (b) case h8 at $t{\it\omega}_{o}=7000$, (c) case h7 at $t{\it\omega}_{o}=8300$ and (d) case h9 at $t{\it\omega}_{o}=13\,600$. It should be noted that the axes for the various cases differ as the $k$-vectors become shorter as the temperature increases.

The EPWs generated by TPD and the secondary waves generated by LDI evolve into turbulence, as observed in other recent simulations for SI^{[Reference Vu, DuBois and Bezzerides18, Reference Vu, DuBois, Russell and Myatt70–Reference Vu, DuBois, Russell, Myatt and Zhang72]} (see the discussion of real-space figures in Section 3.5). As the temperature increases (Figure 10(b–d)), the LDI-induced signal disappears and the purely perpendicular signal from the EPW beating retreats to smaller wavevector (longer wavelength).

3.5. Plasma cavitation

The role of cavitation was clearly identified in previous work related to SI^{[Reference Riconda, Weber, Tikhonchuk and Heron19–Reference Klimo, Weber, Tikhonchuk and Limpouch21]}. It is a fundamental mechanism of LPI which acts as a converter, transferring energy from the laser into kinetic energy of the plasma, and was originally identified in SBS activity^{[Reference Weber, Riconda and Tikhonchuk73]}. Cavities present strong local plasma perturbations which can act as a dynamic random phase plate and induce coherence loss to the incident laser beam, i.e., they act as a means of plasma smoothing^{[Reference Riconda, Weber, Tikhonchuk, Adam and Heron68]}. They are also essential in saturating parametric instabilities at the quarter critical density. Finally, they are an important mechanism for laser energy absorption. Figure 11 shows the ion density cavities for the various simulation cases. As was shown before^{[Reference Riconda, Weber, Tikhonchuk and Heron19]}, the presence of the beating of the TPD waves facilitates the cavitation process. As the threshold for TPD is a linear function of $T_{e}$, see Equation (5), one would expect cavitation to occur later, the higher the temperature is. This is confirmed by the figure, which shows for each of the cases the time slice of maximum cavity activity. The cavities can be filled with either electrostatic^{[Reference Vu, DuBois, Myatt and Russell71, Reference Vu, DuBois, Russell, Myatt and Zhang72]} or electromagnetic fields; for high laser intensity it was shown that they have a soliton-like structure^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Weber, Riconda, Klimo, Heron and Tikhonchuk20]}.

Figure 11.

Ion density near the quarter critical density (located at $xk_{o}\approx 2300$. (a) Case c8 at $t{\it\omega}_{o}=5600$, (b) case h8 at $t{\it\omega}_{o}=7000$, (c) case h7 at $t{\it\omega}_{o}=8300$ and (d) case h9 at $t{\it\omega}_{o}=13\,600$. It should be noted that the colour scale used is not the same for each of the four sub-figures in order to enhance the visibility of the structures.

The high-temperature case h9, in particular, Figure 11(d), shows very large crater remnants from previous cavitations. These density perturbations strongly affect the incident laser light and act as a dynamic random phase plate, inducing fluctuating speckles for densities above $n_{c}/4$ (see Figure 6), thereby randomizing the laser beam that is transmitted across the perturbed density layer. In previous work it was established that SRS-induced cavitation and TPD might help the cavitating process by providing initial density modulation which serves as seeds for SRS-induced cavitation. In this particular case the characteristic size of the cavities is of the order of the laser wavelength, which is in contrast to electrostatic cavities having typical sizes of the order of some ${\it\lambda}_{D}$. The particular regimes where either one or the other is dominant are not yet fully clarified.

As discussed above (see Section 3.4), SRS activity increases with increasing temperature (also observed in Ref. [Reference Yan, Li and Ren67]) at the quarter critical density. At the same time this also increases cavitation activity and size, as visible in Figure 11(a–d). In addition, a higher temperature implies a higher laser intensity such that the ponderomotive force can balance the plasma pressure so that cavities can be maintained.

The creation of the cavities is therefore an intricate interdependence of laser intensity, plasma profile, plasma temperature and growth rate of absolute Raman at $n_{c}/4$. From previous and present simulations it follows that the optimum is around $5~\text{keV}$ for cavity creation. Figure 11(c) has more cavities than the $10~\text{keV}$ case shown in Figure 11(d). The reason for this is that the TPD activity for the case h9 is reduced due to the higher threshold for exciting the instability, which, as said, facilitates the creation of cavities. The typical lifetime of these cavities is of the order of picoseconds, which implies that they are a recursive phenomenon.

Figure 12.

The transverse electron phase space as a function of the laser propagation direction for (a) case c8 at $t{\it\omega}_{o}=5600$, (b) case h8 at $t{\it\omega}_{o}=7000$ , (c) case h7 at $t{\it\omega}_{o}=7400$ and (d) case h9 at $t{\it\omega}_{o}=5200$. The time slice for h9 is taken at an early time as the electrons start to recirculate quickly.

3.6. Laser absorption into hot electrons

SRS, TPD and cavitation are all sources of hot electron production. However, the hot electrons differ as far as temperature and propagation direction are concerned^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Weber, Riconda, Klimo, Heron and Tikhonchuk20]}.

As discussed above, TPD is the dominant process in the cold case c8. As shown in Figure 4, the decay takes place in two directions, which is reflected in the phase space of the electrons, Figure 12(a). Clearly visible are the emerging clouds of accelerated electrons in the forward and backward directions. The presence of two partially separated features in phase space can be related to the excitation of plasma waves centred around different $k$-vectors, which has also been observed recently in other simulation work^{[Reference Yan, Ren, Li, Maximov, Mori, Sheng and Tsung74]}. A corresponding signature is found in the $p_{y}$-space (not shown here). By contrast, the hot case h9, Figure 12(d), shows only a strong signature of SRS-accelerated electrons in the forward direction. No clear signal from TPD is visible. The intermediate cases, h7 and h8, show contributions from both processes, TPD and SRS. Most of the incident laser energy at this stage is absorbed in the vicinity of $n_{c}/4$ by either collective modes or hot electrons. For the case c8, Figure 9, depletion of the pump occurs after $2300{\it\omega}_{o}^{-1}$, i.e., no transmission at all. At early time the transmissivity is on average 50%, and it is 10% at later time (around ${\sim}7000{\it\omega}_{o}^{-1}$). For case h8 one has 35% transmission at early time, and 15% at later time (same times as for the previous case c8). The case h7 is shown in detail in Figure 2(b) of Ref. [Reference Riconda, Weber, Tikhonchuk and Heron19] and has roughly 50% transmission. Finally, for the case h9 one has 25%–30% transmission increasing towards the end of the simulation. The hot electrons also act as a damping mechanism for EPWs and participate in the saturation of TPD and SRS. The distribution functions and temperatures are similar to those obtained previously^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Weber, Riconda, Klimo, Heron and Tikhonchuk20]}.

As was pointed out previously^{[Reference Batani, Baton, Casner, Depierreux, Hohenberger, Klimo, Koenig, Labaune, Ribeyre, Rousseaux, Schurtz, Theobald and Tikhonchuk10]}, the temperature characterizing the hot tail of the electron distribution function is less than $100~\text{keV}$ and is not dependent on the laser intensity but is determined by the resonant interaction of the electrons with the plasma waves generated by TPD and SRS. The laser intensity affects the number of hot electrons produced. The relevant parameters affecting the hot electron production are the phase velocity and the initial bulk electron temperature. Present and previous simulation work^{[Reference Riconda, Weber, Tikhonchuk and Heron19, Reference Weber, Riconda, Klimo, Heron and Tikhonchuk20]} indicates that the hot electron temperature increases with the initial bulk temperature of the plasma. However, a more detailed analysis would be required to quantify this relationship.