Optimized laser ion acceleration at the relativistic critical density surface

Before looking at the details of the simulation results, we briefly review the one-dimensional RTF-RPA process. For small values of a₀ the laser pushes forward the front surface via HB-RPA. For larger a₀ the laser can penetrate into the plasma by turning the plasma relativistically transparent at the front. The velocity of the point of reflection of the laser can be modelled by:

$$\begin{equation} \beta_\mathrm{f} = \begin{cases} \beta_\mathrm{HB} & \textrm{if } v_\mathrm{HB} \gt v_\mathrm{S} \\ \beta_\mathrm{S} & \textrm{if } v_\mathrm{S} \gt v_\mathrm{HB} \end{cases}, \end{equation} \tag{ 1 }$$

where $\beta_\mathrm{HB} = \frac{\Sigma}{1+\Sigma}$ is the HB velocity with $\Sigma = \sqrt{\frac{Pa^2}{2~m_\mathrm{i} n_\mathrm{i}}}$ [26], and

$$\begin{equation} \beta_\mathrm{S} = K_+ - K_- -1, \end{equation} \tag{ 2 }$$

is the velocity of the relativistic critical density front with

$$\begin{equation} K_\pm(a,n_\mathrm{e}) = \left[\frac{a(x_\mathrm{f},t)}{b(n_\mathrm{e})} \left(\sqrt{1+\frac{a(x_\mathrm{f},t)}{27 b(n_\mathrm{e})}}\pm 1\right)\right]^{1/3}, \end{equation} \tag{ 3 }$$

and $b = \pi^2 P n_\mathrm{e}/16$ [38]. The lower subscript f refers to quantities of the RTF where the laser is reflected. The coefficient P refers to the polarization, see below. The plasma is assumed to be homogeneous with electron density $n_\mathrm{e} = Z n_\mathrm{i}$. Here, we introduced the empiric factor P to approximate the acceleration of the front velocity for both linear laser polarization (P = 1) and circular polarized light (P = 2). For circular polarized light equations (2) and (3) were found to be in good agreement with numerical simulations [38]. Here, we find that also for linear polarization we obtain a good agreement of the model with our simulations when introducing the factor P, cp. figure 2 [44]. For $a\gg b$ the above equation can be approximated to the simple form [38]:

$$\begin{equation} \beta_\mathrm{f}\approx \frac{1}{\sqrt{1+P\pi^2n_0/a}}. \end{equation} \tag{ 4 }$$

For the case shown in figure 2 with $a_0 = 50$ and $n_\mathrm{e} = 8$ the predicted velocity of the RTF from equation (2) is $\beta_\mathrm{f} = 0.6$, which compares well to a maximum value of $\beta_\mathrm{f}^\mathrm{PIC} = 0.57$ in the simulation.

Upon laser reflection a charge separation field is set up at the RTF which can reflect downstream ions with peak field strength [38]

$$\begin{equation} \hat E_x(t) = \sqrt{2P\frac{1-\beta_\mathrm{f}(t)}{1+\beta_\mathrm{f}(t)}}a(x_\mathrm{f}(t),t). \end{equation} \tag{ 5 }$$

While the laser pulse intensity ramps up, the velocity at which the laser turns the plasma transparent increases, i.e. the RTF accelerates and can catch up with the ions. The charge separation field at the front then reflects and further accelerates them repeatedly.

We now combine the above in order to obtain an analytic closed description of the RTF-RPA process in 1D. The acceleration of the ions $dp/dt$ during the laser pulse irradiation can be calculated from the usual HB velocity as long as $\beta_\mathrm{HB}\lt\beta_\mathrm{S}$, and later for larger laser strengths $a(x,t)$ by the equation of motion (EOM) with:

$$\begin{equation} \frac{dp}{dt} = Z E_x(x_i,t). \end{equation} \tag{ 6 }$$

We approximate E_x to be spatially constant around the front position, $E_x(t)\cong \hat E_{x}(x_\mathrm{f}(t),t)$ if $\left|x-x_\mathrm{f}\right| \lt L/2$, and $E_x = 0$ otherwise. We find empirically from our simulations shown below that the characteristic length L over which the accelerating field extends can be well approximated by a quarter plasma wavelength in the upstream region,

$$\begin{equation} L\approx \lambda_\mathrm{p}/4\approx \lambda_\mathrm{L}/\left(4\sqrt{n_0/\gamma}\right), \end{equation} \tag{ 7 }$$

where $\gamma\approx \sqrt{1+a_0^2/2}$ is the average Lorentz factor of the electrons in the upstream region.

The EOM can be easily integrated numerically to obtain the ion velocity and energy. For example, we can derive the optimal values for the pulse duration, density or laser amplitude with respect to the final ion energy, see the following sections.

In the optimum case with the maximum ion energy for a given laser, the EOM can easily be integrated analytically. The maximum possible ion energy is dictated by the maximum velocity of the RTF $\beta_\mathrm{f,max}$, which is a function of a₀ and $n_\mathrm{e}$. The local optimum pulse durations and plasma densities are given when the accelerated ions are reflected the last time exactly when the laser maximum reaches the RTF. Then, they whiteness the full electrostatic potential around the RTF at its maximum strength.

For these optimum cases we start integrating the EOM after the ions have obtained the maximum RTF velocity with respective energy $m_i\gamma_0$, $\gamma_0 = \left(1-\beta_\mathrm{f,max}^2\right)^{-1/2}$. We make a variable transformation:

$$\begin{equation} \frac{1}{dt} = \left(1-\beta_\mathrm{f}\right)\frac{1}{d\phi}, \end{equation} \tag{ 8 }$$

and then integrate from $\phi_0 = 0$ to $\phi_\mathrm{max} = L$. We assume $E_{x}(t) = E_0$ is constant to first order around the laser maximum during the short post-acceleration phase. After some algebra we obtain:

$$\begin{equation} \gamma_\mathrm{max} = \sqrt{\frac{A^2}{\left(1-\beta_\mathrm{f,max}\right)^2}+1}, \end{equation} \tag{ 9 }$$

with $A = \gamma_0~\beta_\mathrm{f,max} \left( 1 - \beta_\mathrm{f,max} \right) + Z E_0~L / M_i$. The optimum maximum energies of protons are plotted in figure 3 as a function of laser strength a₀ and plasma density n₀.

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Generally, the final ion energy is sensitive to the exact position in the longitudinal field behind the RTF at the time of the laser peak arrival. In these cases, the EOM needs to be integrated numerically piecewise for each reflection in order to obtain the position of the fastest ions in the charge separation sheath.

The pulse duration at which RTF-RPA works is a function of laser amplitude a₀ and plasma density n₀. For laser pulses too short, the ions cannot keep up during the ramp-up, for pulses too long the ions energy will depend on the exact timing of the RTF dynamics and ion acceleration dynamics. To find the optimum pulse duration for specific values of a₀ and plasma density n₀, we selected five exemplary values for a₀ and varied the plasma density and laser pulse duration. For each set, we integrate the EOM and obtain the final hydrogen kinetic energy, shown in the right columns of figures 4(a)–(e). Above, in figure 2, we have shown a simulation for $a_0 = 50$ and $n_0 = 8n_\mathrm{c}$. Intuitively, at constant a₀ one would expect the ion energy to increase for smaller plasma densities, since the laser front can then propagate faster and the co-moving ions reach a higher final energy. What we find from the numerical solutions of the EOM is in fact such an increase, but only for the envelope along the line of the respective optimum pulse durations. Only at certain values of the plasma density the highest ion energies expected from equation (9) are obtained. These local maxima resemble those optimum conditions when the ions are reflected exactly at the time when the laser maximum reaches the RTF and for which equation (9) was derived. This can be seen in panels (f)–(h). In panel (f) the ion are only reflected once, but the density is smaller than optimum so that the ions are reflected before the laser maximum is reached. Panels (g) and (h) show the respective optimum cases for two and three reflections. Hence, each leaf-shaped area in the $n_\mathrm{e}-t_\mathrm{FWHM}$-space corresponds to a specific number of reflections. When the reflection occurs exactly at the laser maximum arrival, the local optima in the respective leaf w.r.t. ion maximum energy is observed.

To confirm these results we performed a series of 1D PIC simulations with varying initial plasma densities and pulse durations at different peak laser strengths. The results are plotted in figure 4 for the different laser strengths. We extracted the proton energies at $140~\textrm{fs}$ after laser peak arrival on target, when the laser pulse has passed. For a fixed pulse duration, in our parameter range the maximum proton energies due to RTF-RPA are generally obtained at the respective smallest plasma density at which RTF-RPA is still active (onset of RTF-RPA and white dash lines). Only at the highest simulated laser strength we observe similarly high ion energies by ion wave breaking [41, 42]) at smaller density, which is beyond the scope of this paper. The optimum combination of pulse duration and plasma density does not change much with laser strength. It roughly follows the empiric conjecture:

$$\begin{equation} t^\mathrm{opt}_\mathrm{FWHM} \approx m_i \left(P n_0\right)^{-3/2}, \end{equation} \tag{ 10 }$$

and is in very good agreement with the semi-analytic model, despite its simplistic assumptions. The exact position of the local optimum along the line given by this equation is quite sensitive to the exact timing of when the ions are reflected the last time, which translates into narrow regions of high proton energy.

One can combine equations (9) and (10) to obtain the maximum ion energy as a function of a₀ and $t_{FWHM}^{opt}$, cp. Figure 3 top horizontal axis. In the range between $a_0\gt n_0$ and $a_0\lesssim 100$ one obtains the approximate scaling of $E_{max}\propto a_0^{3/2}/n_0$ at constant t_FWHM . We can also evaluate the maximum ion energy as a function of laser pulse energy density (i.e. $a_0^2 t$), as indicated by the solid lines in figure 3. For a given fixed laser pulse energy density between 0.5 and $4~\textrm{J}\,\mu\textrm{m}^{-2}$ we find that the maximum possible ion energy is expected to be approximately independent of the laser intensity and pulse duration. In figure 5 the results of the numeric integration of the EOM are shown for constant laser pulse energy density matching that of the reference simulation in figure 2 (${\approx} 1.7~\textrm{J}\,\mu\textrm{m}^{-2}$) and one can readily see that in deed the optimum maximum ion energies at the local maxima do not change much with the pulse duration. For much larger laser pulse energies longer pulse durations would be favoured while for much smaller energies shorter pulse durations are more favourable.

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However, for very short pulse durations the synchronization between ions and the laser needs to be increasingly more precise, because the difference between too fast laser propagation (ions are left behind early in the laser upramp) and too slow laser propagation (ions are pushed too far ahead and the laser cannot catch up during the short pulse duration) becomes progressively smaller. This means, that in practice the parameter range where RTF-RPA is accessible gets progressively smaller and in the simulations RTF-RPA does effectively not occur for short pulse durations or low laser intensities. Instead, at the predicted optimum RTF-RPA condition we observe a direct transition between transparency and hole boring, with no RTF-RPA in between. The threshold pulse duration below which no RTF-RPA can be observed in the simulations increases with decreasing laser strength. While for $a_0 = 90$ we can observe RTF-RPA down to almost $t_\mathrm{FWHM}\cong 30~\textrm{fs}$, for $a_0 = 15$ we can only see it down to $60~\textrm{fs}$.

On the other hand, for longer pulse durations the density range at which we observe RTF-RPA increases due to the fact that the ions are pushed forward and reflected at the accelerating RTF multiple times. Slower laser intensity ramp-up in principle simply results in more reflections until the final energy is reached. Additionally, the maximum ion energy becomes less sensitive to the exact timing of the last reflection due to the temporally spread out maximum of the laser intensity and front velocity. Hence, it follows that the higher ion energies occurring at smaller plasma densities due to faster laser penetration require an increasingly longer pulse duration and hence higher laser pulse energy in order for the ions to keep up with the RTF. However, at the longest pulse durations we observe considerable density steepening by the laser pulse at the front, which slows down the acceleration of the RTF. This leads to a dephasing of the laser with the ions and inhibits the synchronization with the acceleration ions eventually. We performed an extensive simulation study testing negative density gradients to circumvent this problem, but due to a very unstable, unpredictive onset of the steepening the synchronization is very delicate between steepening and transparency—, practically, we could not achieve it in our simulations for long pulse durations.

In practice, this means that RTF-RPA is limited both for small as well as high laser pulse energies, which poses an ultimate limit to equation (9) and the maximum achievable ion energy. We can therefore expect the existence of an optimum pulse duration. In the simulations this occurs at the parameter combination at which the second or third ion reflection happens at the laser pulse maximum. In this sense, the case shown in figure 2 hence already resembles a close-to-optimum scenario for a Gaussian beam profile at the given pulse energy density.

In order to go circumvent the limits set by hole boring and plasma density steepening on the feasible laser pulse durations, we need another approach to reach smaller plasma densities than just increasing the laser pulse duration. We propose an inhomogeneous target plasma density to match the velocity of the RTF to that of the accelerating ions. A higher density at the target front is employed to achieve this goal by slowing the laser penetration during the initial phase. Specifically, in the following we study a two-layer target consisting of a thin dense front compression layer and a low-density bulk rear. The goal is to set the bulk density to a much lower density than the optimum RTF-RPA density of a homogeneous density to facilitate a high terminal velocity of the RTF. The injection problem, i.e., acceleration rate matching of the ions, is managed by the dense front layer. Experimentally, such a front layer could, for example, be realized by dedicated laser prepulse compression or a tailored laser pulse rising edge contrast profile. Since this compression layer is of higher density than the remaining bulk, the laser needs some time to bore through. This gives the ions enough time for the initial acceleration and to later keep up with the RTF once the laser bores through the front layer. Then, the RTF accelerates further to the final velocity in the less dense rear and—for the same laser parameters—the comoving ions should reach much higher energies than in the case of the respective optimum density homogeneous plasma.

The effectiveness of this method is summarized in a parameter scan in figure 6, again for the same laser parameters as in figure 2 with $a_0 = 50$ and $40~\textrm{fs}$ FWHM pulse duration. For a series of 1D PIC simulations of a plasma consisting purely of electrons and protons, we plot the maximum proton energy for different combinations of the front layer thickness and rear layer density. For panels (a), (b), (c) we defined the front compression layer density $n_1 = 14$, 28, 56, respectively. For finite values of the front layer thickness we find that the optimum density n₀ gets reduced. The optimum front layer thickness is shown by the white dashed lines, as calculated by numerically solving the EOM including the two stages of RTF-RPA first in the dense and secondly in the less dense plasma. It is again in good agreement with the simulations. The larger the front layer thickness d and density n₁, the larger is the reduction of the optimum n₀ for a given a₀. At the same time, we can observe a substantial increase in ion maximum energy, as predicted. For fixed laser intensity and pulse duration the optimum two-layer plasma profile is found at $n_0\cong 3n_\mathrm{c}$, $n_1 = 14n_\mathrm{c}$, $d\cong 1.8~\mu\textrm{m}$. Then, the maximum ion energy jumps to approximately $800~\textrm{MeV}$, which is twice the energy of the optimum at homogeneous plasma density using the same laser parameters.

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The reason for this large increase in ion energy lies in the higher terminal laser group velocity in less dense plasmas as before, but in this case there is no need for compensation by a longer pulse duration to facilitate more ion reflections. Instead, direct synchronization is achieved by the optimization of the plasma density profile. The ions can be synchronized at lower bulk density at a given short pulse duration, circumventing the limitation by dephasing due to pulse steepening at long pulse duration. Figure 7 shows the dynamics for the optimized two-layer RTF-RPA case. It can be seen that the front velocity reaches a maximum value exceeding $\beta_\mathrm{f}^\mathrm{PIC} \cong 0.77c$ (cf $\beta_\mathrm{f} = 0.79$, equation (2)), which compares to only $\beta_\mathrm{f}^\mathrm{PIC}\cong 0.57c$ in the optimum homogeneous density case, see figure 2. Again, panels (b) and (c) show the same characteristic dynamics as before: the most energetic ions comove with the point of laser reflection, which is located inside the plasma where the relativistically corrected electron density surpasses the critical density and a charge separation is induced. Note that now that the laser penetrates the plasma much faster and hence the ion acceleration lasts much longer than before for the optimum case of a homogeneous plasma density in figure 2. While at 50 fs after the laser maximum arrival the ion energy is approximately equal to that of the optimum homogenous density case, in the following 100 fs the ions now continue to be accelerated up to 800 MeV.

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What is left is a discussion how exactly the highest energy ions are injected and trapped in the fast moving charge separation field at RTF. The proton phase space is shown in figure 8(a) for several timesteps before and during the RTF-RPA process for the case of optimum ion acceleration with a front compression layer, see figure 7. For comparison, we also show the accelerating electric fields and electron densities in panels (b) and (c), respectively. In panel (c), we scaled the density for each timestep individually for clarity, the thin dashed lines show the respective critical density $n_\mathrm{c}$ for reference. The ion acceleration starts early, already at 40 fs before the maximum arrival the ion gain energy at the front surface, while the electron density $n_\mathrm{e}$ is still seen to be quite similar to $n_\mathrm{e}/\gamma$. In this phase the laser pushes the surface forward while not yet being intense enough to accelerate electrons to relativistic energies. This changes at around 20 fs before the laser pulse maximum arrival. Now the laser penetrates into the compression layer plasma by turning it relativistically transparent at the surface. Up until between the laser maximum arrival and less than 10 fs later the most energetic ions can keep up with the RTF. At the same time, the compression layer rear surface expands into the less dense bulk similar to TNSA, also accelerating ions there. At that point, the compression layer has been turned fully transparent and the laser enters the less dense bulk. We can now observe three distinct ion acceleration groups, as the laser front sweeps through the bulk plasma. The first group, labeled R in figure 8 is that of bulk ions initially at rest. Those ions remain far too slow as they witness the charge separation field moving by. They are accelerated to only ${\approx} 0.1c$ while the front now passes by with 0.7 to 0.8 c. The second group of ions is pre-accelerated by TNSA at the compression layer rear surface (T). With their initial energy from TNSA they can comove somewhat longer with the laser front to reach ${\approx} 0.2c$ before they are also outrun by the RTF. Only the ions that were pre-accelerated by RTF-RPA in the compression layer to ${\approx} 0.4 c$ already (S1) can be trapped in the RTF witnessed in panels (b) and (c) (S2). They undergo continued and repeated acceleration by RTF-RPA in the bulk and are thus accelerated to the final RTF velocity in the bulk and correspondingly high energies (beyond $800\mathrm{M}\mathrm{eV}$ in this specific case). The shaded area indicates the large gain in energy by RTF-RPA for the optimum injected ions from the compression layer compared to the stationary bulk ions.

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2.3.1. Lower laser intensity.

We now want to discuss an important benefit of the front compression layer specifically at smaller laser intensities, $a_0 \lesssim 20$, and short-pulse durations, $t_\mathrm{FWHM} \lesssim 40~\textrm{fs}$, where RTF-RPA is less effective or even non-existent in homogeneous plasmas, cf figure 4. As a reason for this problem, we discuss that the matching between the RTF edge velocity and the ion acceleration cannot be established in these cases. To illustrate this, we show two cases with homogeneous density n₀ close to each other (4 and 5 n_c ) in figure 9 tor $a_0 = 15$ and $t_\mathrm{FWHM} = 40~\textrm{fs}$. The problem can readily be seen in that for low density ((a), (d)), the laser propagates too fast, while for a slightly higher density ((c), (f)) the ions are reflected in hole boring like acceleration and the laser cannot keep up with them. Even as we tried densities closer to each other we could not find RTF-RPA in between, i.e., there appears to be a sudden jump from transparency regime to hole boring.

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Introducing a finite pre-compression layer changes the situation drastically, see figure 10. We can now see the same qualitative behaviour as for high laser intensity. The pre-compression layer shifts the optimum density of the bulk to lower values. At the same time, the maximum ion energy increases with increasing thickness of the compression layer, reaching a maximum value of approximately twice that of the homogeneous plasma. At this optimum RTF-RPA is well established, as indicated by the close vicinity of the most energetic ions and the point of the laser reflection witnessed in figures 9(b), (e) and 11. The front compression layer keeps the laser slow at the beginning and gets faster after boring through, enabling the RTF-RPA process. Consequently, we can expect that for a pre-compressed target RTF-RPA can be effective even for low laser intensities, specifically in plasmas near the critical density and compression to modest density $7 \leqslant n_1 \leqslant 14$.

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2.3.2. 3D simulations.

It is important to note that the RTF-RPA process, and also the enhanced two-phase acceleration, can also be observed in 3D. This is to be expected since the RTF-RPA process is essentially a 1D process, but for future experimental realizations the applicability in 3D has to be checked and analyzed. We verified the effectiveness of the front compression layer by conducting exemplary 3D simulations for homogeneous and front compressed plasmas using PIConGPU [45]. The laser peak intensity and duration were kept as before in 1D, now with a tight focus of $3.1~\mu\textrm{m}$ FWHM. Figures 12 and 13 show the RTF edge and ion front velocities along with the profiles of electric fields, ion energies and plasma densities for the two optimal cases of a homogeneous and two-layered target, respectively. The ion energies are somewhat smaller in both cases than those observed in 1D geometry before, but again, we observe a substantial increase in the compression front plasma compared to the homogeneous plasma by more than $100~\textrm{MeV}$.

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Thus, the qualitative behaviour of the system with varying parameters does not essentially differ, though some quantitative differences can be noted: the thickness of the compression layer that enhances the final energies is higher than for the 1D optimum, and also the density of the RTF-RPA optimum with a homogeneous target is higher than in the 1D case. The latter is due to an increased laser strength due to self-focusing in the target, and transversally pushing the density away from the laser axis.