Filamentation instability in high temperature plasmas Open Access

Currently, there is growing interest to study plasmas at high temperature plasmas. Indeed, with the advancements in laser technology, powerful laser pulses are now evaluable, and their interaction with solid targets, as in inertial confinement fusion (ICF), produces high temperature plasmas up to $20 keV$⁠. At such high temperatures, electron rest energy $mec2$ becomes non-negligible compared to the electron thermal energy $Te$⁠, where $me$ is the electron rest mass, c is the speed of light, and $Te$ is the electron temperature in energy units. The relevant parameter that quantifies the relativistic effects is therefore $z=mec2Te$⁠. Thus, as temperature increases, the relativistic parameter z decreases, and relativistic effects become more significant.

In this high plasma temperature range, the relativistic effects must be taken into account, and in such plasmas several physical phenomena should be revisited.^1–6 Many works dedicated to this new relativistic regime have been reported in the literature. Using simulations, Langdon and Hinkel¹ studied the nonlinear evolution of backscattering Brillouin and Raman instabilities in high temperature plasmas $(3–20 keV)$⁠. In particular, the effects of the rescattering of these instabilities on the generation of hot electrons and scattering losses have been highlighted. In Ref. 2, the authors derived from the relativistic Vlasov equation the dispersion relations for electrostatic and electromagnetic (EM) waves. Using a different approach, the Vlasov equation was solved analytically in Ref. 3, and the dispersion relation of the electrostatic waves was also deduced. Bers et al. presented in Ref. 4 a new formulation for the damping rate of electron plasma waves in relativistic plasma temperatures $(5–15 keV)$⁠. This result was used to investigate forward and backward Raman backscattering instabilities in the context of inertial confinement fusion. Likewise, Li et al.⁵ revisited the laser hosing instability in relativistic plasmas taking into account both thermal and fluid relativistic effects. The model equations were used to study short-pulse and long-pulse laser hosing instabilities using a variational method approach. Similarly, Zhao et al.⁶ studied Raman and Brillouin parametric instabilities induced by high intensity laser pulses propagating into an underdense relativistic plasma.

In inertial confinement fusion, it is well known that in laser-created plasmas, the EM modes couple nonlinearly with plasma eigenmodes such as electron plasma wave or ion acoustic wave, to drive parametric instabilities. These instabilities depend mainly on the nature of the interacting modes and can be broadly classified into 3-mode couplings such as Brillouin instabilities (coupling of the EM wave with an ion acoustic wave) and Raman instabilities (coupling of the EM wave with an electron plasma wave), and to 4-mode couplings such as the filamentation instability (FI) and the modulational instability. The parametric instabilities play a detrimental role in the context of inertial confinement fusion, since they mainly reduce the heating rate by the laser beam of the target and prevent an efficient compression of the target. Numerous analytical and numerical works devoted to these instabilities have been reported in the literature in order to reduce these damaging effects.^7,8 The present work focuses on the FI which plays a central role in laser plasma coupling. The local increase in the intensity of the laser beam can alter transport properties and generate hot electrons. In addition, the interaction of the FI with other instabilities generated in the underdense plasma plays a detrimental role for achieving successful inertial confinement fusion.

The FI is triggered when a perturbation in the intensity of the laser beam propagating through plasma begins to grow in amplitude, forming filaments. In this work, we consider both sources of laser beam filamentation: the inverse bremsstrahlung absorption and the ponderomotive force. For both mechanisms, the ponderomotive and thermal forces push electrons from the center of the filament to the edges, creating a density depression that locally modifies the spatial refractive index profile. The refractive index, given by $nref=1−nencr$⁠, where $ne$ is the electron density and $ncr$ is the critical density, becomes higher in the filament's center than at the edges. As a result, the laser beam is refracted toward the center, reinforcing the intensity perturbation and further driving the FI.

The FI has been extensively studied in the literature,^9–12 and all these studies have focused on plasmas with non-relativistic temperature. The theoretical models used are based on hydrodynamic equations coupled with Maxwell's equations. It has been shown that in highly collisional plasmas close to thermodynamic equilibrium, the spatial growth rates of the FI are relatively low typically about $2 cm−1$ as reported in Ref. 9. Taking into account non-local effects in the fluid model, the authors have highlighted significant spatial growth rate of the FI. Similar work has been presented in Refs. 10 and 11 in weakly collisional plasmas. In addition, the interaction of the FI with other parametric instabilities was also considered. In particular, the interplay between FI and Brillouin instability was studied in Ref. 12.

To our knowledge, this instability has not been studied quantitatively in hot plasmas with temperatures typically exceeding 5 keV. At these high temperatures, relativistic effects can no longer be neglected, and the FI must be revisited in the context of inertial confinement fusion (ICF). This issue is the subject of the present work.

As mentioned above, the Raman instability (see Refs. 1, 4, and 6) has been studied in very hot plasmas. The present work therefore extends the study of the parametric instabilities in such plasmas.

We used new relativistic and collisionless hydrodynamic model equations that incorporate purely kinetic effects. This model serves as the counterpart to a kinetic model based on the Vlasov equation. The dissipation of the hydrodynamic modes in plasmas is included through the thermal conductivity and the viscosity coefficient, which account for the well-known Landau damping.^13,14 Additionally, we emphasize that new transport coefficients, namely, the convective heat flux and temperature anisotropy coefficients, are taken into account in this collisionless hydrodynamic model. These coefficients become negligible in the collisional approximation, and therefore they are not accounted for in the standard hydrodynamic equations.

Using these hydrodynamic model equations and Maxwell's equations, we investigate the FI in a new regime that is not only relativistic but also collisionless. Most previous models, whether relativistic or non-relativistic, are based on the continuity equation and the momentum and energy balance equations in the collisional regime. Additional assumptions on the pressure gradient term (choice of the adiabatic parameter, isothermal or adiabatic approximation, etc.) are used to make the system of equations self-consistent. These assumptions are valid only in the collisional regime. The present collisionless model which describes perturbed plasmas with respect to equilibrium is derived without any complementary assumptions.

This work is organized as follows. In Sec. II, we present the hydrodynamic model which includes both collisionless and relativistic effects. These equations are split into low-frequency equations and high-frequency equations including the wave equation of EM modes. Then, in Sec. III, we calculate the thermal and ponderomotive drives. Section IV is devoted to the dispersion relation of the FI and to the numerical analysis of the growth rate. Finally, we give in Sec. V a discussion and a summary of this work.

The FI of an EM wave arises from nonlinear coupling between the EM wave and the plasma. The main assumptions used in this hydrodynamic model are the consideration of the collisionless transport and relativistic temperature effects on plasma electrons. Ions, because of their strong inertia, are considered as non-relativistic.

The model used to describe the plasma is based on ion and electron hydrodynamic equations, while the EM wave as usual is described by the Maxwell's equations. It is assumed that the EM wave consists of a high-frequency (HF) electric field of constant amplitude of zero order, and a small perturbed electric field of first order. In complex notations, they are written, respectively, as $E→HF0=E0 exp−iω0t+ik→0·r→e→x$ and $δE→HF=δE+ exp−iω+t+ ik→+·r→e→x+δE− exp−iω−t+ik→−·r→e→x$⁠. The parameters defining these fields are the amplitudes $E0$⁠, $δE+$⁠, and $δE−$⁠, the frequencies $ω0$⁠, $ω+$⁠, and $ω−$⁠, and the wave vectors $k→0$⁠, $k→+$⁠, and $k→−$⁠. The wave vector directions are specified as follows: $k→0=k0e→z$⁠, $k→∓=k∓ze→z+ k∓ye→y$⁠. The unperturbed plasma is assumed to be homogeneous, defined by the electron density $ne0$⁠, electron temperature $Te0$⁠, ion density $ni0$⁠, and ion temperature $Ti0$⁠. In the presence of the EM wave, a coupling occurs between the EM wave and the plasma's eigenmodes. At lowest order, the perturbed electron and ion hydrodynamic quantities are $δne,δTe,δV→e$ and $δni,δTi,δV→i$⁠, corresponding, respectively, to the density, temperature, and fluid velocity for each species. All hydrodynamic quantities are assumed to be described by Fourier modes of the form, $δT,δn,δV→∼ exp −iωt+ik→·r→$⁠, where $ω$ is the frequency and $k→$ is the wave vector of the hydrodynamic modes. The modes considered in this work are resonant modes which ensure efficient wave–plasma coupling. These modes satisfy the resonance conditions $ω∓=ω ∓ ω0$ and $k→∓=k→ ∓ k→0$⁠.

The system of hydrodynamic equations describing the perturbed state of the plasma is divided into two groups: low-frequency equations and high-frequency equations. These two timescales are defined by $ω−1$ for the low-frequency hydrodynamic modes and by $ω0−1$ for the high-frequency modes, with $ω0≫ω$⁠. We now present each set of equations.

• The relativistic and collisionless electron hydrodynamic equations that describe the perturbed state are the continuity equations and the momentum and energy balance equations,¹⁵ 

  $−iωδne+ik→·δV→e ne0=0,$

  (1)

  $−iωδV→eG ne0me+ik→ne0δTe+Te0δne+ik→· δΠ¯¯e=qene0δE→+δF→pe+iωc2δq→e,$

  (2)

  $−iω32n0eHδTe + iωTe0δne+ik→·δq→e=δj→HF·E→HF,$

  (3)

  where $me$ is the electron mass, $qe$ is the electron charge, $δΠ¯¯e$ is the stress tensor, $δF→pe$ is the ponderomotive force, $δq→e$ is the heat flux, and $δj→HF·E→HF$ is the inverse bremsstrahlung absorption rate averaged on the time scale $T=2πω0$⁠, where $j→HF$ is the high-frequency current density. Moreover, we have used the parameters $G=K3z0K2z0$⁠, $Kn(z0)$ being the $nth$-kind modified Bessel function and $H=23−1+z021−G2+5z0G$ where here $z0=mec2Te0$⁠. The nonlinear ponderomotive and inverse bremsstrahlung absorption terms are coupling terms between the EM wave and the plasma, and they are the driven terms for the FI.

  For the sake of clarity, we present the explicit expressions of the heat flux and the stress tensor in  Appendix A. The expressions of these transport quantities for electrons are

  $δqxe=−KTene0cikkδTe−αVene0Te0δVe,$

  (4)

  $δΠxxe=−αTene0δTe−ηene0mecikkδVe.$

  (5)

  The transport coefficients are the thermal conductivity $KTe$⁠, the viscosity coefficient $ηe$⁠, which both represent the dissipation terms, and $αVe$ and $αTe$⁠, respectively, the convective and temperature anisotropy coefficients. We found that the Onsager symmetry is fulfilled, i.e., $αTe=αVe$⁠. For electrons, the explicit expressions of these four coefficients are given by Eqs. (A3)–(A6). We represent in Fig. 1 these transport coefficients as functions of the relativistic parameter $z0$⁠, where we can see that these coefficients increase with increasing relativistic effects (or decreasing $z0$⁠).

• Similarly, the non-relativistic and collisionless ion hydrodynamic equations read

  $−iωδni+ik→·δV→i ni0=0,$

  (6)

  $−iωδV→ini0mi+ik→ni0δTi+Ti0δni+ik→· δΠ¯¯i=qini0δE→,$

  (7)

  $−iω32ni0δTi+Ti0δni+52ni0Ti0ik→·δV→i+ik→·δq→i=0,$

  (8)
  where $mi$ and $qi$ are, respectively, the ion masse and the ion charge. Due to the ion high inertia, the relativistic effects $mic2Ti0≫1$⁠, the ion ponderomotive force, and the absorption of EM energy by ions are negligible. The non-relativistic transport coefficients are calculated in Ref. 14, and they read

  $δqxi=−KTini0vtiikkδTi−αVini0Ti0δVi,$

  (9)

  $δΠxxi=−αTini0δTi−ηini0mivtiikkδVi,$

  (10)

  where $KTi=952π$⁠, $ηi=252π$⁠, and $αTi=αVi=25$⁠.

• Poisson's equation relates the electrostatic field to the plasma density perturbation

  $ik→·δE→=qeδne+qiδniε0,$

  (11)

  where $ε0$ is the vacuum permittivity.

• The high-frequency hydrodynamic equations only concern electrons (low inertia), and they can be written as

  $G∂V→eHF0∂t=qemeE→HF0−νeiGV→eHF0,$

  (12)

  $G∂δV→eHF+∂t=qeme δE→HF+−νeiGδV→eHF+,$

  (13)

  $G∂δV→eHF−∂t=qeme δE→HF−−νeiGδV→eHF−,$

  (14)

  where $νei$ is the electron–ion collision frequency.¹⁵ 

• The last set of equation is the wave equation of EM modes. To determine this equation, we use the Maxwell–Faraday equation which connects the high-frequency electric $δE→HF∓$ to the high-frequency magnetic fields $δB→HF∓$⁠,

  $∇→×δE→HF∓=−∂δB→HF∓∂t,$

  (15)
  and the Maxwell–Ampère equation

  $∇→×δB→HF∓=μ0δj→HF+1c2∂δE→HF∓∂t,$

  (16)
  where $μ0=1/ε0c2$ is the magnetic permeability of the vacuum, and

  $δj→HF=qe ne0δV→eHF++δV→eHF−+qe δneV→eHF0$

  (17)

  is the high-frequency current density.

To proceed further, we need to calculate the driving terms for the FI in the relativistic approximation, specifically the absorption rate and the ponderomotive force.

$Zi$ is the ion charge number, and $lnΛ$ is the Coulomb logarithm. We note that in the non-relativistic limit, $z0≫1$⁠, $K3z0≫1∼1z0exp −z0$⁠, and collision frequency scales as $νei∼1Te03/2$⁠.

The ponderomotive force considered in this work is transverse, i.e., $δF→pe=δFpee→y$⁠, meaning we are not concerned here with the longitudinal component, $δF→pe=δFpee→z$⁠.

The hydrodynamic equations (1)–(3) and (6)–(8) and the wave equation (21) together with the expressions of the absorption rate (23) and of the ponderomotive force (32) are the self-consistent basic equations of this work. In the following, it will be used to study the FI in relativistic plasma temperature.

The coefficients $Ci=0–8k,ne0,Te0$ are expressed as functions of the plasma and EM wave parameters.

• Steady-state filamentation instability.

We should note that Eq. (37) can be useful for further applications, particularly in plasmas where the parameter $ZiTeTi$ has low values close to 1. We have checked numerically that the two expressions (37) and (38) give very close numerical results. In both equations, the negative term $∼−k2$ stands for the diffraction (a stabilizing term), while the two positive terms represent the source terms of the instability. They correspond, respectively, to the thermal and ponderomotive drive terms. They depend, respectively, on the relevant parameter $νav02$ and $v02$⁠, which stand, respectively, for the inverse bremsstrahlung absorption and the ponderomotive force.

It is important to note that in the thermal drive, dissipation is accounted for only through the thermal conductivity $KTe$⁠; the viscosity, which represents the second dissipative term, does not appear in Eq. (38). Moreover, additional transport coefficients, such as the electron convective coefficient $αVe$ as well as all ion transport coefficients, do not contribute to the spatial growth rate of the stationary FI.

We have numerically calculated in Figs. 2 and 3, the spatial growth rate as a function of the wavenumber $k$ for, respectively, the ponderomotive and thermal sources. The electron temperature and the corresponding laser intensity are

• $Te0=1 keV$ and $I=4.2×1013 W/cm2$ as in Ref. 9;

• $Te0=3 keV$ and $I=5×1014 W/cm2$ as in Ref. 12;

• $Te0=20 keV$ and $I=1016 W/cm2$ as in Ref. 1.

A relationship between temperature and laser intensity would allow expressing the growth rate only as a function of temperature $Teo$⁠, i.e., the relativistic parameter $zo$⁠. This would be particularly useful for interpreting the role of relativistic effects on the FI. However, to our knowledge, this relationship $I0=fI0$ requires developing a complete analytical hydrodynamic model that describes the laser–plasma interaction, which is beyond the scope of this work. We just note that a hydrodynamic model was derived by Fabbro et al.¹⁶ for non-relativistic temperature plasmas, leading to the scaling law $I0∼Te03/2$⁠. In addition, we assume also as in Ref. 10 that $Ti0≈Te04$⁠.

Figures 2 and 3 show that the growth rate exhibits a maximum for an optimal wavenumber $kopt$⁠. For the thermal and ponderomotive FI, the spatial growth rate scales, respectively, as $Γth∼Sthk−k4$ and $Γp∼Spk2−k4$⁠. For both sources, the diffraction term $∼k4$ becomes dominant over the source terms in the high wavenumber range, whereas in the small wavenumber range, the two source terms dominate the diffraction term.

The spatial growth rate in Fig. 2 driven by the ponderomotive force is independent of the relativistic transport coefficients $KTe$⁠, $ηe$⁠, and $αTe$⁠. These modes are therefore not subject to damping due to thermal conduction or viscosity. However, the inertia caused by the relativistic effects, which is contained into the parameter $G2$⁠, will reduce the growth rate. This occurs as if the effective mass increases, thereby decreasing the oscillation velocity of electrons in the high-frequency EM field. We note that the parameter $G$ is the counterpart of the Lorentz factor $1−v2/c2−1/2$ for fluid relativistic effects. Figure 4 illustrates its variation with respect to the relativistic parameter $z0$⁠. In the non-relativistic limit, we find $G=1$⁠, and within the temperature range considered in this work, $G$ increases slowly as $z0$ decreases. Finally, the relativistic quiver velocity $v0/G$ increases with increasing laser intensity, and it is reduced by relativistic inertial effects.

In Fig. 3, where only the thermal drive is considered, the spatial growth rate decreases as the temperature increases. This behavior can be attributed to the increase in the dissipative transport coefficients $KTe$ and $αTe$ (see Fig. 1), the rise in the inertia parameter $G$⁠, and the decrease in the absorption rate $νa$ [Eq. (24)].

To better understand the behavior of the growth rates with respect to temperature, we derive their scaling laws. Using Eq. (38) and the scaling law of Ref. 15, $I0∼v02∼Te3/2$⁠, we can determine the maximum growth rate of the ponderomotive and thermal source terms with respect to temperature, i.e., with respect to relativistic effects. Although this scaling law is valid for non-relativistic plasmas, it is assumed that it remains valid for moderately relativistic plasmas since the electron mass inertia factor G does not vary significantly in the temperature range used in this work. The scaling law of the growth rates are $Γth∼Sthk−k4$ and $Γp∼Spk2−k4$⁠, where $Sth=12Gvte2c2 νav02ω021−αTevteKTe$ and $Sp=12vte2c2 ωpe2v02G2$⁠. We deduce readily $kopt,th=Sth41/3$ and $kopt,p=Sp21/2$ which give, respectively, the maximum growth rates $Γth∼Sth2/3$ and $Γp∼Sp$⁠. A simple analysis of these growth rates shows that $Γth∼1Te3/2$ and $Γp∼Te1/2$⁠, where we have used $νa∼1/T0e3/2$⁠. Thus, the behavior of the spatial growth rate with respect to the temperature is opposite for the thermal and ponderomotive drives. As a conclusion, from Figs. 2 and 3, the ponderomotive FI dominates the thermal FI and even more significantly when the plasma temperature increases.

Moreover, in the ultrarelativistic limit, $z0→0$⁠, $G→∞$⁠, $αT→1$⁠, and $vteKTe→π6c$⁠; thus, the thermal and ponderomotive sources should decrease drastically, and we expect that in strong relativistic regime the FI should be stabilized.

We give in Fig. 5 the maximum spatial growth rate and the corresponding optimum wavenumber which maximizes this growth rate, as a function of the position defined by the electron density in the underdense plasma. We present the results for the temperatures $Te0=1 keV$ and $Te0=20 keV$ and using only the dominant ponderomotive source. The plasma parameters are given in Fig. 2.

The results in Fig. 5 illustrate the dependence of the maximum growth rate and the corresponding optimum wavenumber on electron density in homogeneous plasmas. This analysis allows for a qualitative study of what occurs in an inhomogeneous plasma. If we assume that the plasma is inhomogeneous with a characteristic spatial scale length $L$⁠, the local approximation $L≫1$ is well verified, where $k$ is the FI wavenumber. Typically, $L$ is of the order of a hundred micrometers, and $k$ is greater than $105 m−1$⁠. Therefore, the dispersion relation of FI, derived for a homogeneous plasma, remains valid in inhomogeneous plasmas where the local approximation is verified. Furthermore, as seen in Fig. 5, the growth rate increases significantly with electron density. For instance, for $Te0=20 keV$⁠, the growth rate is approximately $Γ ≈ 3.8×105 m−1$ at the layer $nenc=0.1$ and about $Γ ≈ 9.4×105 m−1$ at the layer $nenc=0.6$⁠. This is due to the decrease in the wavenumber $k0$ of the EM wave via the dispersion relation of EM waves in plasmas. Additionally, we observe that the optimum wavenumber increases with electron density. We have also checked that these wavenumbers are consistent with the collisionless approximation $kλei>1$ used in the fluid model, since for $nenc=0.6$ we found $λei≈1.5×10−4 m$ and about $k≈105 m−1$⁠.

• Non-stationary filamentation instability.

In this section, we focus on the non-stationary modes of the FI. To do these, we must numerically solve Eq. (B1) in  Appendix B. This equation is an eighth-order polynomial with real coefficients for the even powers of ω and purely imaginary coefficients for the odd powers. By changing the variable ω by $Ω=iω$⁠, the coefficients of the polynomial are all real. We used standard numerical methods to calculate the roots of this polynomial,¹⁷ assuming a real wavenumber $k$ and a complex frequency $ω=ωr+iωi$⁠, where $ωr$ is the frequency and $ωi$⁠, if positive, would be the temporal growth rate. The results obtained for the complex frequency $ω$ are represented in Figs. 6 and 7 for, respectively, the frequency $ωrk$ and the growth rate $ωik$ for two plasma temperatures $Te0=3 keV$ and $Te0=20 keV$⁠. These results account for both the thermal and ponderomotive sources of instability.