A new mechanism of direct coupling of laser energy to ions

There are many well-known schemes by which the laser energy can get absorbed in the plasmas [1, 2]. For instance, resonance absorption, Brunel, $\to {J}{\times}\to {B}$ heating schemes etc [3–9]. All these schemes rely on the dynamical response of the lighter electron species in the presence of the oscillating field of a high frequency laser. Consequently, the dominant transfer of energy from laser occurs in electron species. However, there are many applications (e.g. fast ignition of fusion targets [10–12], cancer radiotherapy [13], proton radiography [14, 15], production of warm dense matter [16] etc) where it is desirable to have energetic ions. This has led to efforts seeking efficient mechanisms of ion heating and/or acceleration [17–21]. In many previous studies, a two step approach has been envisaged for this particular objective. The laser energy is first absorbed by electrons and its transfer to ions depends on the electron–ion classical collisional and/or anomalous processes. Since Rutherford's collision cross section decreases with increasing electron energy, this process is inefficient at high energies [22]. Many ideas of anomalous transfer of energy from electron to ion have also been proposed [23–30]. There have been studies where indirect ways of coupling laser energy into ions via ponderomotive force [31, 32] and formation of collisionless shock [33] have been shown. In this work, we propose a novel scheme by which direct transfer of laser energy to ions is possible.

The new mechanism relies on the application of a strong external magnetic field normal to the laser propagation and also the polarization direction. Strength of the magnetic field is chosen such that it restricts the motion of electrons which are tied to the magnetic field lines but the heavier ion species remain un-magnetized and are relatively free to move in response to the laser electric field. This happens when the electron gyroradius is smaller than the laser wavelength whereas ion gyroradius is larger compared to it, as depicted in the schematic of figure 1. The frequency condition ω_ce > ω_l > ω_ci needs to be satisfied. This requirement demands that the applied external magnetic field strength B₀ should be greater than m_ecω_l/e.

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The $\to {E}{\times}\to {B}$ drift velocity in the presence of an externally applied magnetic field and an oscillating electric field is given by

$$\begin{equation}{\to {V}}_{\to {E}{\times}\to {B}}\left(t\right)=\frac{{\omega }_{\mathrm{c}s}^{2}}{{\omega }_{\mathrm{c}s}^{2}-{\omega }_{\mathrm{l}}^{2}}\frac{\to {E}\left(t\right){\times}\to {B}}{{B}^{2}}\end{equation} \tag{ 1 }$$

as ω_ce > ω_l > ω_ci, the drift velocity of the two species differ [34]. Here, the suffix s = e, i represents the electron and ion species respectively. Thus, ω_ce and ω_ci stands for the electron and ion cyclotron frequency respectively. Furthermore, ω_l stands for the laser frequency. As mentioned earlier the magnetic fieldis chosen so that the condition ω_ce > ω_l > ω_ci is satisfied. This ensures that the electrons motion is predominantly governed by $\to {E}{\times}\to {B}$ drift given by equation (1). Ions, on the other hand, being un-magnetized are essentially accelerated by the electric field acting on them. Magnetic field has negligible role on them.

The difference in the drift velocities of electrons and ions gives rise to a current. Since, the laser electric field and the externally applied magnetic field, both are perpendicular to the propagation direction, the electrons drift produces current along the laser propagation direction. This current has a spatial variations at the laser wavelength due to its dependence on the electric field of the applied laser. This spatial variation leads to a finite divergence of the current which drives a charge density fluctuation given by the continuity equation:

$$\begin{equation}\frac{\partial \rho }{\partial t}+\to {\nabla }.\to {J}=0.\end{equation} \tag{ 2 }$$

This generates electrostatic plasma mode in the system, thereby, the energy of the laser gets transferred to plasma. This is the underlying main principle on which our new mechanism has been proposed. It should be noted when the applied magnetic field is zero, ${\to {V}}_{\to {E}{\times}\to {B}}\approx 0$ (as ω_cs = 0) for both the species, and no charge separation will be created in the plasma. In the other limit of a very large magnetic field, both ions and electrons are magnetized and move with identical drift velocity ${\to {V}}_{\to {E}{\times}\to {B}}$ following equation (1). Thus in this limit also one should expect no charge density fluctuation. Thus, in both the limits the proposed mechanism of laser energy absorption will be inoperative. We have carried out PIC (particle-in-cell) simulations through OSIRIS-4.0 to support this mechanism.

As pointed out earlier, the experimental implementation of this mechanism requires an application of an external magnetic field of magnitude B₀ > m_eω_lc/e. For a typical laser of 1 μm wavelength this translates to a requirement of magnetic field of the order of 10⁴ Tesla. The CO₂ lasers which have a 10 μm wavelength, reduces the requirement by ten times. Lately, there has been a rapid technological progress in achievinghigh magnetic field in the laboratory [35, 36]. The value of 1.2 kilo Tesla [37] at Institute for Solid State Physics at the University of Tokyo, Japan has already been achieved. Thus, it is only a matter of time before the requirement for carrying out experiments in this domain would be possible.

We use OSIRIS-4.0 framework [38–40] for our particle-in-cell (PIC) studies. The schematic (not to scale) in figure 1 shows the field configuration used in the simulation. The $\hat{x}$ is the propagation direction of the laser. The laser electric field is along and the applied external magnetic field is along as shown in the figure. A 2D rectangular simulation box with dimensions L_x = 3000c/ω_pe and L_y = 100c/ω_pe has been chosen. The number of particles per cell are taken to be 4. The plasma boundary starts from x = 500c/ω_pe onward. There is vacuum between x = 0 to x = 500c/ω_pe. The spatial resolution is taken as 10 cells per electron skin depth corresponding to a grid size Δx = 0.1c/ω_pe and time step for calculations is taken tobe ${\Delta}t=0.02{\omega }_{\mathrm{p}\mathrm{e}}^{-1}$. There is a sharp plasma–vacuum interface and laser is incident from left side on the plasma target. We consider a p-polarized, plane short-pulse laser of wavelength λ_l = 9.42 μm corresponding to CO₂ laser. The choice of long-wavelength laser is simply for the sake of definiteness and is motivated by the fact that it reduces the requirement of the externally applied magnetic field. The mechanism, however, is independent of the laser wavelength. The normally incident laser is propagating along $\hat{x}$, centred at x = 250c/ω_pe and has a pulse length ranging from x = 0 to 500c/ω_pe. The peak intensity of laser is taken to be I = 3.5 × 10¹⁵ W cm⁻² with a Gaussian profile having rise and fall time of $205{\omega }_{\mathrm{p}\mathrm{e}}^{-1}$ each. The laser is taken to be infinite in transverse direction in most of the runs. This is chosen purposefully to avoid edge effects which might interfere with the effects arising from ${\to {V}}_{\to {E}{\times}\to {B}}$ which we wish to clearly identify. We have, however, also carried out a few simulation with laser pulse having finite transverse extent. The characteristics features do not differ significantly for the two runs. Boundary conditions for fields and particle are periodic in transverse direction and absorbing in longitudinal direction. The number density of the plasma is taken to be n₀ = 3 × 10²⁰ cm⁻³ for which the electron plasma frequency is equal to 10¹⁵ rad s⁻¹. Hence the plasma is overdense for the incident laser pulse.

We have followed the dynamics of both electrons and ion species. The simulations are, however, carried out at a reduced mass of ions which is taken to be 25 times heavier than electrons (i.e. m_i = 25m_e, where m_i and m_e denote the rest mass of the ion and electron species respectively). This mass ratio has been chosen to reduce the computational time. We use normalised units henceforth, unless otherwise stated explicitly. The time is normalised by the inverse of electron plasma frequency corresponding to the chosen plasma density of n₀. The laser frequency is chosen to be ω_l = 0.2ω_pe, which is smaller than the electron plasma frequency but matches with the ion plasma frequency ω_pi with mass ratio of m_i/m_e = 25 chosen here. This choice has been made to observe the possible resonant excitation of ion mode. We have, however, also carried out simulations with laser frequencies of ω_l = 0.15ω_pe and ω_l = 0.12ω_pe which differ from the ion plasma frequency. The length is normalised by electron skin depth c/ω_pe and the magnetic field by m_ecω_pe/e. The external magnetic field has been chosen in such a way that ω_ce > ω_l > ω_ci. To satisfy this condition the value of magnetic field is chosen to be B₀ = 2.5 m_ecω_pee⁻¹. For this field, the electrons gyrate at a frequency 2.5ω_pe. The simulation geometry is specifically chosen to avoid the possibility of any well-known absorption schemes to be operative. The laser is incident normal to the sharp plasma interface. This ensures the absence of resonance and vacuum heating schemes. Also, the role of $\to {J}{\times}\to {B}$ electron heating is made negligible by choosing the laser intensity to lie in the non-relativistic domain of a₀ = eE/m_eω_lc < 1 (a₀ = 0.5 corresponding to CO₂ laser). The simulation values of various laser and plasma parameters in both normalised and standard units have been provided in table 1.

We first provide a comparison between two cases, namely (A) for which the applied magnetic fieldB₀ = 0 and (B) for which B₀ = 2.5. The choice of B₀ = 2.5 ensures that the condition ω_ce > ω_l > ω_ci is satisfied. For these simulations there is no transverse variation of the laser pulse and hence the simulation configuration is symmetric in the y coordinate, we choose to depict the fields as a function of the coordinate x in the figures.

In figure 2, the three subplots in the first row depict the plot of the time-dependent magnetic field associated with the laser and the self-consistent response of the plasma medium as a function of x at three distinct time (i) t = 0 showing the initial laser pulse field when it does not touch the plasma surface, (ii) t = 350 when the laser has touched the plasma and (iii) t = 700 when the laser pulse has been reflected from the plasma surface and the plasma is left with the remnant self-consistent excitations. The dashed red line denotes the case (A) for which B₀ = 0 and the solid blue line corresponds to case (B) with B₀ = 2.5.It can be observed from the figure that in case (A) there is no penetration of the laser magnetic field in the plasma and almost a complete reflection of the pulse occurs. However, for case (B) when the external applied field B₀ is finite and equal to 2.5, a part of the laser field shows clear penetration in the plasma medium. In the same figure the second row of three subplots show the plasma current J_x as a function of x for the two cases. Third and last row depicts the electrostatic field E_x generated in the plasma medium. It is evident from the plots of J_x and E_x that while the plasma gets stirred significantly by the laser field in the presence of applied external magnetic field in case (B), for case (A) the plasma continues to remain quiescent even after interacting with the laser pulse. This is as expected by our proposed mechanism.

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We now compare the evolution of various energies for the two cases in figure 3. We indicate distinct time incidents by vertical lines in this figure. The first two solid black lines denote the times t₁ = 100and t₂ = 380, where t₂ − t₁ region indicates the time interval during which the laser interacts with the plasma. Thereafter, laser field gets reflected from the plasma surface. At around t₃ = 500 denoted by the black dashed line, the laser reflects from the plasma surface and at t₄ = 780 indicated by black dotted line the laser field leaves the simulation box. Evolution of the electromagnetic field energy has been shown by the blue dashed and solid lines for case (A) and case (B) respectively. The left side of the y-axis in the figure represents scale for this particular energy. The electromagnetic field energy remains constant associated with the laser pulse till t₁. At t₁ the laser pulse interacts with the plasma surface. There is a fall of electromagnetic energy from t₁ to t₂, the time for which the laser pulse interacts with the plasma medium.It is evident from figure that the electromagnetic energy decreases more in case (B) than in case (A) during this period. From t₂ to t₃ the laser pulse reflects from the plasma surface but remains within simulation box therefore, the electromagnetic energy remains constant. At t₃ the left edge of the reflected laser pulse touches the left boundary of the simulation box. Thereafter there is a continuous decay in the field energy till t₄ at which the pulse completely leaves the simulation box. Thus, while the first fall of the electromagnetic field energy between t₁ to t₂ is associated with the laser pulse interacting with the plasma medium, the second fall of the electromagnetic energy between t₃ to t₄ is due to the laser pulse leaving the simulation box. The laser energy transfer, therefore, essentially occurs between t₁ to t₂. Further, it should be noted that the energy transfer to plasma medium is more in case (B) with externally applied magnetic field. Clearly, this energy transfer to plasma would be in the form of field and kinetic energies. Another interesting thing to observe is that even after the laser pulse leaves the system, electromagnetic energy for case (B) remains finite in contrast to case (A) where it falls off to zero. This shows that the laserinteraction creates some electromagnetic field fluctuation in the plasma medium in case (B).

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The evolution of kinetic energy of electrons and ions are shown by green and red color line plots respectively in the same figure 3. The dashed and dotted lines are for case (A) and the solid line with dots correspond to case (B). For these kinetic energy plots, the right y-axis defines the scale. In absence of applied magnetic field in case (A) the gain in ion energy is much smaller compared to that of electrons. However, for case (B) in the presence of magnetic field the kinetic energy of ions is much higher than that of electrons. The total energy absorbed by plasma in this case is also higher.

One can infer that the laser interacts with plasma in case (B) in a more efficient fashion and excites certain disturbances in the plasma. Disturbances that are excited in case (B) do have an electrostatic characteristic as evident from the evolution of the x component of the electric field E_x shown in figure 4(a) at location x = 540c/ω_pe (which is deep in the bulk region of the plasma medium). For case (A) the amplitude of the electrostatic field E_x is negligible. The frequency associated with the regular oscillation in the electric field E_x is identified by carrying a Fourier transformation which is shown in figure 4(b). We take the Fourier transform of self generated electric field (E_x) in bulk plasma (at x = 540) after laser hasreflected back from the plasma boundary. The peak in Fourier transform for the two cases appear at very distinct frequencies. In case (A) the spectrum shows a peak at ω = 1 which is the electron plasma frequency. On the other hand, for case (B) the peak in the frequency is shifted at a lower side (corresponds to a value close to ∼0.2) which is indicative of an ion dominated mode, the ion plasma frequency being 0.2. The peak frequency is found to be dependent on externally applied magnetic field which suggests the presence of lower hybrid mode. We have also carried out simulations with a few other values of laser frequencies for example ω_l = 0.15, 0.12 in addition to 0.2. The FFT for all these cases have been compared in figure 5 and it can be observed that despite differences in the laser frequency the electrostatic mode that is excited has the same frequency.

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The oscillations in electrostatic field are a result of space charge fluctuations excited in the plasma.The number density and the velocity fluctuations associated with the two species have been shown in the first and second row of the figure for the two species of electron and ions by blue and red colour lines respectively as a function of $\hat{x}$ at three different times (figure 6). The third row of subplots shows the current density fluctuations. It is clear from the plot that except for the short distance near the plasma boundary at x = 500, the ion and electron density oscillations and their x component of the velocities are in tandem. These oscillations propagate deeper in the plasma with time. The laser interacting at the plasma boundary excites a space charge fluctuations at the boundary which then travels in the bulk region of the plasma.

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We now compare the theoretical value of the velocity as expected from the mechanism put forward by us with that obtained from simulation for both the species (figure 7). It can be observed from the figure that the electron velocity is pretty low compared to the ion velocity. The electrons are magnetized and the x component of its velocity follow the theoretical expression of equation (1). It can be observed (from the left subplot of figure 7) that there is a good agreement between the theoretical expression and the value obtained in the simulation. The heavier ion species is un-magnetized as per our choice of magnetic field value and hence its dominant velocity along x is ultimately provided by the direct acceleration due to the x component of electric field vector E_x which gets created self consistently by the space charge separation. The close match between theoretical and simulation are in confirmation with our proposed scheme.

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In figure 8 we show a plot of the velocity and density fluctuations of the two species as a function of time for x = 540 (a point in the bulk region of the plasma). The plots correspond to six different values of the magnetic field. From the plots it is clear that at low values of the magnetic field velocity fluctuations are small. At an intermediate value of the magnetic field B₀ = 2, 2.5, 3.0 the velocity fluctuations increase and the difference between electron and ion velocity is also high. The number density fluctuations are also high for these cases. However, at very high value of magnetic field of B = 10 the electron and ion velocities are almost similar and the number density fluctuation is small. It should be noted that at the intermediate value of B = 2, 2.5, 3.0 only the required condition of ω_ce > ω_l > ω_ci is satisfied. For B = 10, the ions too get magnetized. This is consistent with our proposed mechanism in which the number density fluctuation are supposed to be excited by the difference in the $\to {E}{\times}\to {B}$ velocity of the two species. The difference in drift velocity of the two species vanishes in both the limits of low and high magnetic fields. This is yet another confirmation of the proposed mechanism.

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These number density fluctuations associated with ion time scale of response peak at an intermediate value of B₀ (the applied magnetic field). The mode disappears both at the low magnetic field and also at the higher values of the magnetic field. To understand the role of magnetic field on laser absorption, we show in table 2 the total energy absorbed by the two species (electrons and ions) from the laser for various values of the applied magnetic field. From this table, it is clear that the energy absorption in ions is low both at the low value of magnetic field and also at the high value of the magnetic field. This has also been depicted in the plot of figure 9 showing the total energy absorption by electrons and ions at various values of the external magnetic field. We have also shown variation of x-component of total current, J_x with applied magnetic field and observe that ion absorption follows the profile of J_x as a function of the applied magnetic field. Here the values are to be taken as indicative of the qualitative trend. The quantitative values will differ if real mass ratio of the two species are chosen.

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The particle count of the two species as a function of their acquired kinetic energy has been shown in figure 10 for the two cases (A) and (B). From this figure one can infer clearly that energy absorption by ions in the presence of external magnetic field is higher.

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We have chosen to show in figure 11 some particle related diagnostics for a set of 10 randomly chosen electrons (represented by blue color) and 10 ions (shown by red color). In subplot (a) their locations have been shown at t = 0 and subplot (b) and (c) show their particle trajectories from initial time to t = 500and t = 998 respectively. Since there are several particles in the frame the details of the trajectory of each particle is not evident from these plots. We have, therefore, added another subplot (d) in the figure where we have chosen to represent the trajectory of one particular set of electron and ion in an expanded x and y scale. From this plot it can be observed that the particles have an oscillatory motion in space. In subplot (e) we plot the x location of two particles (electron and ion) as a function of time which clearly shows the oscillatory motion of ions and electrons. The amplitude of ion oscillation is higher than that of electrons. A small steady drift along the positive x direction for both of them can also be observed initially. This is because for a finite temporal electromagnetic wave pulse it is well known that a charge particle gets displaced along the laser propagation direction. The particles, however, oscillate with increasing amplitude.

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An estimate of average speed along x for the two species have been provided. In subplot (f) of figure 11 we plot the total distance d_x covered in x by the particles whose position and trajectories have been depicted in other subplots (a–c) of figure 11. Here ${d}_{x}={\sum }_{i}\vert \left({\Delta}{x}_{i}\right)\vert ={\sum }_{i}\sqrt{{{\Delta}{x}_{i}}^{2}}$, where i represents the ith time step and Δx_i is the displacement of the particle in ith time step. We have also shown the expanded plot of d_x in the inset of this subplot (f) for a clear view. From the slope of the plot of d_x with time we have estimated the average speed of ions and electrons which have been listed in the plot as v_si = m₁ = 7 × 10⁻³ and v_se = m₂ = 2.4 × 10⁻³. These values compare well with the average speed that can be estimated from figure 6, in terms of the maximum amplitude and assuming the variation to be sinusoidal. Thus, ${v}_{{\mathrm{a}\mathrm{v}}_{\mathrm{i}}}=\sqrt{{\left({v}_{\mathrm{m}\mathrm{i}}\right)}^{2}/2}=6.8{\times}1{0}^{-3}$ and ${v}_{{\mathrm{a}\mathrm{v}}_{\mathrm{e}}}=\sqrt{{\left({v}_{\mathrm{m}\mathrm{e}}\right)}^{2}/2}=2.0{\times}1{0}^{-3}$ which compare well with v_si and v_se respectively. Here v_mi and v_me are the observed amplitude of electron and ion velocities shown in figure 6.

We have also carried out simulations with a laser of finite transverse extent to see the importance if any of 2D effects associated with the edge of the laser pulse. The laser pulse is chosen to have Gaussian profile in both $\hat{x}$ and the transverse direction.

The tranverse FWHM of the laser spot size is 150c/ω_pe. Other laser parameters are kept same as before. A rectangular box with dimensions L_x = 2000c/ω_pe and L_y = 2500c/ω_pe is chosen.

The 2D color plots depicting the case of finite transverse laser beam and the laser with infinite extent has been shown in figure 12. It can be observed that the ion density and current structures are similar in both the cases. The peak frequency in FFT of E_x also points at the same frequency in both cases (figure 13). The fractional count of electrons and ions as a function of acquired energy has been shown in figure 14 for the finite and infinite transverse laser pulse. From this figure it is clear that the energy acquired by ions is higher than that of electrons for both cases. The % of laser absorption for this particular simulation corresponding to B_z = 2.5 has been mentioned in the table 2 in brackets alongside which is similar to that of infinite laser case.

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The results of the previous section clearly demonstrate that the laser energy gets preferentially coupled to ions in the presence of external magnetic field. The space charge fluctuations created by the difference in the $\to {E}{\times}\to {B}$ velocity of the two species in presence of oscillating electric and static external magnetic field appears responsible for the excitation of the electrostatic mode in the plasma medium.

While we have primarily focused here on the illustration of a fundamentally new concept, it should be realised that in recent years there has been rapid progress in the creation of high magnetic fields in the laboratory. Magnetic field of the order of 1.2 kilo Tesla has already been produced in laboratory [37]. One expects that in near future this limit will increase. We feel that it is only a matter of time when we will witness magnetic field of the order of 10 s of kilo Tesla at which the mechanism presented in this manuscript can be experimentally verified in laboratory with a long wavelength CO₂ lasers which have now already been made to operate in the pulsed high-intensity mode [41–44].

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. AD would like to acknowledge her J C Bose fellowship grant JCB/2017/000055 and the CRG/2018/000624 Grant of DST for the work. The simulations for the work described in this paper were performed on Uday, an IPR Linux cluster. The authors would also like to thank Dr R K Bera and Laxman Goswami for fruitful discussions.A special thanks to Mr Omstavan Samant for his intuitive ideas and inputs in preparing certain diagnostics. We would also like to thank anonymous referees whose comments helped us improve the manuscript.