Ion heating in Laser interacting with magnetized plasma

TL;DR

This paper investigates how laser interactions with strongly magnetized plasma lead to ion heating, revealing resonant absorption mechanisms related to plasma modes and the influence of laser polarization.

Contribution

It provides a detailed analysis of ion heating via resonant mechanisms in magnetized plasma using PIC simulations, highlighting the roles of different plasma modes and laser polarization.

Findings

Resonant absorption occurs near plasma mode frequencies.

Ion heating is most efficient close to resonance frequencies.

Laser polarization significantly affects absorption efficiency.

Abstract

The ion heating mechanism in the context of laser interacting with plasma immersed in a strong magnetic field is studied. The magnetic field is chosen to be strong for laser electromagnetic field propagation inside the plasma to be governed by the magnetized dispersion relation. Both X and RL mode configurations have been studied in detail using Particle - In - Cell (PIC) simulations. It is shown that the energy absorption process is governed by a resonant mechanism wherein the laser frequency matches with an underlying mode in the plasma. For X and RL mode configurations, these correspond to lower hybrid and ion cyclotron resonance, respectively. The absorption, however, is found to be most efficient at frequencies close to but not exactly matching with the resonance frequency. An understanding of the same has been provided. The role of laser polarization has been studied in detail.

Figures

Tables

Table 1. Table 1: Values of simulation parameters in normalized and standard units

Parameters	Normalized value	Values in SI unit
Plasma Parameters
$n_0$	$1.0$	$1.34\times {10}^{20}c{m}^{-3}$
${\omega }_{pe}$	$1.0$	$0.67\times {10}^{15}rad/s$
${\omega }_{pi}$ ($M/m=25$)	$0.2$	$0.13\times {10}^{15}rad/s$
Laser Parameters
${\omega }_l$	$0.30$	$0.2\times {10}^{15}rad/s$
${\lambda }_l$	$21$	$9.42\mu m$
Intensity	$a_0=0.5$	$3.5\times {10}^{19}W/m^2$

Table 2. Table 2: List of simulation cases, incident and transmitted laser polarization, and external magnetic field parameters

Cases	Incident Laser	Magnetic Field Parameters			EM in Plasma
	${\alpha }_i$	$B_0$	${\omega }_{ce}$	${\omega }_{ci}$	${\alpha }_t$
Case A	Linear ${\alpha }_i=0$	$B_0=2.5$	$2.5$	$0.1$	R-wave ${\alpha }_t=+1$
Case B	Linear $\alpha =0$	$B_0=10$	$10$	$0.4$	R and L-waves ${\alpha }_t=\pm 1$
Case C	RCP $\alpha =+1$	$B_0=2.5$	$2.5$	$0.1$	R-wave ${\alpha }_t=+1$
Case D	LCP $\alpha =-1$	$B_0=10$	$10$	$0.4$	L-wave ${\alpha }_t=-1$

Full text

1 Introduction

In recent years there has been a lot of progress (both fundamental and technological) in the area of laser plasma interaction studies [1]. The development of low frequency pulsed $CO_{2}$ lasers [2, 3, 4] and the possibility of generating strong magnetic fields (e.g. of the order of Kilo Tesla has already been achieved [5] and there are proposals to generate Mega Tesla [6] fields) are two developments which have opened up possibilities of carrying out experiments in a new direction involving laser interacting with magnetized plasma. This regime is of importance for fundamental explorations and also has rich implications for frontier applications such as direct heating of ions and neutron production in table top devices[7, 8, 9, 10, 11, 12, 13, 14, 15]. In this paper we explore the prospect of parametric scattering process associated with a laser pulse interacting and propagating inside a magnetized plasma. The applied external magnetic field has been chosen to be along the laser propagation direction. This particular geometry supports the propagation of Right (R) and Left (L) circularly polarised electromagnetic waves inside the plasma for the specific range of pass band frequencies in their dispersion curves.

The parametric processes have been studied extensively for the past several decades. A detailed discussion of the same in the context of plasmas and its relevance to nuclear fusion experiments can be found in a recent review article [16]. The parametric instability occurs when a high-amplitude pump electromagnetic wave couples with an electrostatic disturbance in the plasma and generates a scattered Electromagnetic radiation. The interaction of the pump and the scattered radiation accentuates the density disturbance and enhances the electrostatic perturbations, thereby creating a feedback loop for the instability to occur. Some early work in this area are as follows. Drake et al. [17], derived a compact dispersion relation for the wave interactions in the context of an unmagnetized plasma. They also discussed the limiting form of the dispersion relation leading to the various instabilities like Brillouin, Compton, and Raman scattering processes in plasma. In Raman scattering the excited electrostatic mode is an electron plasma wave, whereas for the Brillouin scattering process it is essentially an ion acoustic wave. The analytical expression for the growth rate and the threshold condition for the Raman and Brillouin scattering instability has been obtained by Liu et al. [18]. The occurrence of these Raman and Brillouin instabilities were directly shown for pulsed electromagnetic solitonic structures propagating in plasmas by Saxena et al. [19] and Sundar et al. [20] respectively with the help of fluid simulations. The application of Brillouin scattering process in the context of laser fusion has been highlighted in many works [21, 22, 23, 24]. For magnetized plasma theoretical studies have been carried out for the parametric instability by many authors. For instance, Stenflo [25, 26] and Shukla [27] provided a theoretical prediction of ionospheric heating experiments. Jaiman and Tripathi [28] analyzed the generation of Brillouin and Compton scattering in magnetized plasma for oblique propagation of electromagnetic pump wave with respect to the external magnetic field direction.

We provide here an evidence of parametric process occurring for an electromagnetic pulse propagating inside a magnetized plasma using particle-in-cell (PIC) simulations [29, 30]. The laser is chosen to be incident on an overdense plasma and it propagates parallel to the applied external magnetic field. This particular geometry supports the propagation of Right (R) and/or Left (L) circularly polarised electromagnetic (EM) waves even for an overdense plasma, provided the EM wave frequency lies in the appropriate pass band of their dispersion relation.

The manuscript has been organized as follows. Section 2 provides simulation details. In section 3, we present the details of observations made by simulating different cases of polarization of the incident EM pulse. These observations have been analyzed in detail in Section4 which provide evidence of the occurrence of parametric Brillouin back scattering process. In a recent work from our group [7] we had observed the generation of electrostatic fluctuations for a sharp profile of the laser pulse in the same geometry by the ponderomotive pressure of the laser pulse. Here, in contrast for a smoother laser profile, we observe electrostatic fluctuations getting generated by the Brillouin backscattering process. We point out in section5 that both these processes in fact take place. The ponderomotive pressure driven process is the first to occur and gets saturated if the profile is relatively smooth. However the fluctuations generated from this process mimic the effective temperature of electron fluid to drive the brillouin scattering process which occurs at a later stage. We summarize and conclude in section 6.

2 Simulation details

We have employed the OSIRIS-4.0 framework [31, 32, 33] for carrying out 1-D particle-in-cell (PIC) simulations to study the interaction of a laser with magnetized plasma. The schematic of the simulation geometry (not to scale) has been shown in Fig.1. The external magnetic field is directed along the laser propagation direction $\hat{x}$. A 1-D simulation box with dimension $L_{x}=4000c/\omega_{pe}$ has been chosen. Here $c$ is the velocity of light and $\omega_{pe}$ represents the plasma frequency. The number of particles per cell are taken to be $8$. The plasma boundary starts from $x=400c/\omega_{pe}$. There is vacuum region between $x=0$ to $400c/\omega_{pe}$ with a sharp plasma vacuum interface at $x=400c/\omega_{pe}$. The spatial resolution is taken as $100$ cells per electron skin depth and it corresponds to the grid size $\Delta x=0.01c/\omega_{pe}$. The laser is incident on the plasma target from left side. We consider a short-pulse laser of frequency $\omega_{l}=0.3\omega_{pe}$. The laser profile is Gaussian having rise and fall time of $200\omega_{pe}^{-1}$ ($400\omega_{pe}^{-1}$ for some simulations) with the peak intensity of $I=3.5\times 10^{19}Wm^{-2}$ (corresponding to a relativistic factor $a_{0}=0.5$). Boundary conditions are taken as absorbing in the longitudinal direction. The PIC code OSIRIS uses normalised values of the various parameters. We have followed the dynamics of both electrons and ions. To reduce the computational time, we carried out the simulations for a reduced mass of ions, which is chosen to be $25$ ( the value of $40$, $50$ are also chosen for some simulations) times heavier than electrons. Thus $m_{i}=25m_{e}$ ($40m_{e}$, $50m_{e}$ for some simulations), where $m_{i}$ and $m_{e}$ represent the rest mass of the ion and electron species respectively. We have provided the laser and plasma simulation parameters both in normalized units alongside their typical possible values in standard units in Table-1. As per the convention the frequencies have been normalised by electron plasma frequency ($\omega_{pe}$), length by electron skin depth ($c/\omega_{pe}$), and the electric and magnetic fields by $m_{e}c\omega_{pe}e^{-1}$.

We have considered different configurations of incident laser polarization by appropriately choosing the parameter $\alpha_{i}$ (it is [math] for linear, and $\pm 1$ for right (RCP), left (LCP) circular polarizations respectively). Here the incident and transmitted laser electric field is denoted as $\vec{E}_{i}=\tilde{E}_{i}\left(\hat{y}+i\alpha_{i}\hat{z}\right)e^{-i\omega t}$ and $\vec{E}_{t}=\tilde{E}_{t}\left(\hat{y}+i\alpha_{t}\hat{z}\right)e^{-i\omega t}$ respectively. Where $\alpha_{i}$ and $\alpha_{t}$ respectively denote the polarization of incident and transmitted EM pulse. The applied external magnetic field $B_{0}$ in normalized units has been varied from $2.5$ to $10$ for our simulation studies as tabulated in Table-2.

The four different cases of study carried out by us correspond to four different choices of the incident laser EM field parameters. For case (A) and case (B), a linearly polarized laser ($\alpha_{i}=0$) propagating along the $X$-direction is considered. The two choices of magnetic field ensures that for the given laser frequency and the choice of $B_{0}=2.5$ in case(A) only the $R$ mode lies in the pass band of the dispersion curve, while for higher value of $B_{0}=10$ in case(B) both $R$ and $L$ EM modes are in the pass band. The polarization of laser is taken RCP ($\alpha_{i}=+1$) and LCP ($\alpha_{i}=-1$) in case (C) and (D) respectively. The transmitted wave also has the same polarization as the incident wave in these two cases as listed in Table -2. A fifth and sixth possibility of choosing LCP for incident EM wave with an external magnetic field of $B_{0}=2.5$ and RCP for $B_{0}=10$ respectively, does not permit any EM wave propagation inside the plasma, and hence have not been reported. In Table-2, $\omega_{cs}=\frac{q_{s}B_{0}}{m_{s}}$ is the gyro- frequency of the species $s$ of the plasma (where $s=e,i$ correspond to electron and ion particles) in the external magnetic field $B_{0}$.

3 Observations

In a recent publication Goswami et al. [7] have employed similar geometry to study the interaction of a laser pulse with a magnetized plasma. The study had interestingly shown excitation of electrostatic perturbations at the electron plasma frequency. Since the background plasma is overdense, such a perturbation cannot be understood through a parametric excitation process. Instead it was shown that the difference in the ponderomotive pressure experienced by electrons and ions led to the charge separation leading to plasma oscillations. Here, on the other hand we demonstrate that with a choice of a smoother laser pulse profile (consequently weakening the ponderomotive pressure term) one can excite parametric instability. Since the plasma is overdense only the Brillouin scattering process is observed.

We show in Fig.2 the dispersion curves for both $R$ and $L$ waves for the two values of the magnetic field $2.5$ and $10$ respectively. The incident laser frequency of $\omega_{l}=0.3\omega_{pe}$ lies in the pass band of only the $R$ wave for the case of $B_{0}=2.5$. For the other case of $B_{0}=10$ the laser frequency lies in the pass band of both $R$ and $L$ waves. The two waves, however, have distinct phase and group speeds. The various subplots (A, B, C, D) of Fig.3 correspond to the four different cases (A, B, C, D) respectively of study listed in Table.2. The $y$ component of the electric field $E_{y}$ has been shown at three different times in this figure as a function of $x$. While in subplot(A) and (B) the incident pulse has linear polarization for (C) and (D) the incident pulse has been chosen to have Right and left circular polarization. The pulse in red color shows $E_{y}$ for the incident EM pulse at $t=0$. It is placed in the vacuum region initially. The plasma starts from $x=400c/\omega_{pe}$. The blue and green pulses in the figure depict the $E_{y}$ component of electric field at $t=1200\omega_{pe}^{-1}$ and $t=3800\omega_{pe}^{-1}$ respectively. For $B_{0}=2.5$ both linear and right hand circular polarised incident laser, shown in (A) and (C) subplots propagate as $R$ waves inside the plasma. This is so because the laser frequency lies in the stop band of $L$ wave. The left circularly polarised incident radiation in this case gets totally reflected for this particular case. For case (B) the incident linearly polarised EM pulse generates both $L$ and $R$ waves, as the frequency lies inside the pass bands of both the modes. The $L$ and $R$ waves get spatially separated as they propagate inside the plasma due to their different group speeds. The $R$ wave being faster it moves ahead. For case (D), the incident laser pulse is chosen to have left circular polarization. This pulse propagates as $L$ wave inside the plasma. Thus Case (A) and (C) show the propagation of pure $R$ waves inside the plasma and case (D) shows pure $L$ wave propagation. Case(B) on the other hand depicts the propagation of both $L$ and $R$ wave. Thus for these cases one can study separately the propagation of each of the $R$ and $L$ waves without getting hindered by the other wave in any fashion.

From Fig.3 it is clear that as the EM pulse propagates inside the plasma, electrostatic field disturbances get generated behind the pulse. These disturbances have a higher amplitude when the incident laser pulse has circular polarisation. This may be so as the power level of the propagating $L$ and $R$ waves for the linear case get low compared to the case of circular polarization. Associated with these electromagnetic disturbances one can observe electrostatic perturbation which are shown as the plot of $E_{x}$ in Fig.4 at three different times. The corresponding electron and ion charge density perturbations are shown in Fig.12 for $t=3800$. The zoomed plot alongside shows that the electron (in solid blue line) and ion (in red dash line) density perturbations have similar form. In the next section we try to characterize these perturbations and provide the evidence that they represent the Brillouin scattering phenomena.

4 Characterization of the observed excitation as a Brillouin process

We evaluate the spatial Fourier transform (FFT) spectra for the electromagnetic $E_{y}$ and electrostatic $E_{x}$ fluctuations. The spectral wavenumber peak for one particular case (D) of the $L$ wave propagation has been shown in Fig.6. The FFT of $E_{y}$ field using only the region of pump pulse region shows a peak at $k_{0}=0.41$ (which satisfies the Dispersion curve for the $L$ wave. However, when FFT is taken of the $E_{y}$ in a region just behind the main pulse the peak of the spectra occurs slightly shifted at $k_{1}=0.46$. This appears to be the scattered EM radiation. The fourier spectra of the electrostatic field $E_{x}$ on the other hand peaks at $k_{2}=0.87$. It is interesting to note that the wave-vector matching condition for the parametric process is satisfied as $\vec{k_{2}}=\vec{k_{0}}+\vec{k_{1}}$. The electrostatic perturbation being at shorter scales than the pump and scattered EM radiation shows that it is a back scattering process that is taking place. Furthermore, as shown in Fig.12, the electron and ion density perturbations seem to occur in phase indicating that it is a Brillouin process. Furthermore, we have also repeated these simulations for the case of static ions (corresponding to $m_{i}=\infty$). For such a case no scattering is observed.

The simulations were carried out for the case of a plasma which was cold. The question then arises is what plays the role of effective temperature for exciting an ion acoustic perturbation in the Brillouin process. We feel that the velocity acquired by the electrons as a result of ponderomotive pressure from the EM pulse may contribute for the same. In fact similar observations were noted in the work by Sundar et. al. [20] where they observed a Brillouin scattering process for a flat top solitonic structure in a fluid simulation. We will discuss more about the role the ponderomotive pressure term plays by choosing different laser profiles in section 5.

In Fig.7 we have shown the evolution of electrostatic field energy for Case(C) of $R$ wave propagation for two different ion to electron mass ratios (e.g. $25m_{e}$ and $50m_{e}$). The other parameters for the laser pulse and the plasma medium are the same. It is interesting to observe that the initial slope for the two cases are identical. Subsequently a second phase with a faster growth rate occurs in both the cases. The onset of second phase occurs at an earlier time for ion mass $m_{i}=25m_{e}$ compared to that of $m_{i}=50m_{e}$. Also, the growth rate as discerned from the slope in this phase for the lower ion mass is higher. This can be understood by realising that the initial small rise of the electrostatic energy arises as a result of ponderomotive forcing from the laser pulse, during which the electrons acquire a certain kinetic energy which acts as an effective temperature for facilitating the Brillouin scattering process. The second phase corresponds to the Brillouin scattering instability during which the two growth rate differs. After the second phase the nonlinear effects seem to set in which slows down the growth rate. In Fig.8 we have plotted the evolution of electrostatic energy for the case of $L$ wave propagation of Case (D). In this case also similar behaviour is observed.

We now discuss briefly the nonlinear phase of the instability. We choose a specific time of $t=3000$ for the Case D at which the evolution of electrostatic energy is in its third phase, showing a slow down in the growth. In Fig.9 we show the spatial fourier transform of the electrostatic field at various spatial blocks with respect to the pump pulse as illustrated in the figure. It can be observed that when the spatial FFT is taken just behind the pulse it shows a clear peak ($k_{2}=0.87$). At longer distances behind the pulse other peak in the spectra start appearing (e.g. $k=0.97$). Thereafter it becomes broad with multiple scales indicating the presence of nonlinear effects as expected. We have tracked the power of two different wave numbers by carrying out a Fourier transform in several blocks of localised spatial region and observe that while the power in $k=0.87$ satisfying the Brillouin condition maximizes in the spatial regime overlapping with the pump pulse, the power in the second mode picks up thereafter.

5 Role of smooth and sharp laser profiles

In the work by Goswami et al. [7], published earlier for the same geometry a very sharp laser profile was chosen. There we observed the ponderomotive force driven electrostatic flluctutions. For the present studies we have chosen a smoother laser profile. In the section we describe a study where we carry out simulations with varying laser profile. The observations corresponding to four different laser profiles have been shown in Fig.10 subplot(v). The sharpest profile has been denoted by the violet line with triangular dots. It can be observed from the subplot (iii) that there is an initial growth of electrostatic energy which is highest for this particular case. A zoomed picture has been shown in subplot(iv) of this region which also confirms the same. This is the time regiem where the initial ponderomotive pressure driven electrostatic fluctations occur. The thermalization of the effective electron kinetic energy acquired by this process leads to the effective temperature for the Brillouin process to occur later. The onset of the Brillouin process occurs earlier for the cases of the fourth profile where this happens fastest. Since the total energy associated with the pulse for these profiles may slightly differ. We have also compared the evolution of the electrostatic energy normalised by the total input energy from the laser pulse. It can be observed that the normalised electrostatic energy in the case of profile 4 is higher compared to all other profiles after the scattering process takes place.

6 Conclusion

With the help of OSIRIS-4.0, we have carried out a comprehensive PIC simulations studies to understand the interaction of a laser pulse propagating in an overdense magnetize plasma for which the external magnetic field is directed along the laser propagation direction. It has been shown that the laser energy can get converted to the electrostatic energy by two distinct processes. The ponderomotive force driven process occurs first and is obviously found to be dependent on the sharpness of laser profile. Such an excitation process was also shown in an earlier publication by us [7] where purposely a very sharp laser pulse profile was chosen. Here we provide evidence of the Brillouin scattering process taking place. We also observe that there is a synergy between the ponderomotive process leading to charge density fluctuations and the Brillouin back scattering process. The Brillouin process occurs subsequently and the timing of its onset is related to electrostatic energy gained during the first process.

7 Acknowledgment

The authors would like to acknowledge the OSIRIS Consortium, consisting of UCLA ans IST (Lisbon, Portugal) for providing access to the OSIRIS4.0 framework which is the work supported by NSF ACI-1339893. This research work has been supported by the Core Research Grant No. CRG/2018/000624 of Department of Scient and Technology (DST), Government of India. We also acknowledge support from J C Bose Fellowship Grant of A D (JCB-000055/2017) from the Science and Engineering Research Board (SFERB), Government of India. The authors thank IIT Delhi HPC facility for computational resources. Laxman thanks the Council for Scientific and Industrial Research (Grant no. -09/086(1442)/2020-EMR-I) for funding the research.

8 References