Direction and divergence control of laser-driven energetic proton beam using a disk-solenoid target

TL;DR

This paper proposes a laser-driven method to control the direction and divergence of energetic proton beams using a disk-solenoid target, enabling precise beam steering for applications in high-energy-density physics and medicine.

Contribution

It introduces a novel target design with a solenoid coil to achieve controlled proton beam direction and collimation through sheath field manipulation.

Findings

Proton beams can be directed parallel to the solenoid axis.

Overlap of sheath fields reduces proton spread angle.

Proton properties are tunable via solenoid parameters.

Abstract

A scheme for controlling the direction of energetic proton beam driven by intense laser pulse is proposed. Simulations show that a precisely directed and collimated proton bunch can be produced by a sub-picosecond laser pulse interacting with a target consisting of a thin solid-density disk foil with a solenoid coil attached to its back at the desired angle. It is found that two partially overlapping sheath fields are induced. As a result, the accelerated protons are directed parallel to the axis of the solenoid, and their spread angle is also reduced by the overlapping sheath fields. The proton properties can thus be controlled by manipulating the solenoid parameters. Such highly directional and collimated energetic protons are useful in the high-energy-density as well as medical sciences.

Figures

Equations

$$L_{\perp }(t)=\frac{v_{p_{{\perp }_0}}}{{\omega }_s}\sin \left\{{\omega }_s\left[t-(T_0+t_{\mathrm{acc}})\right]\right\},$$

$$v_{p_{\perp }}(t)=v_{p_{{\perp }_0}}\cos \left\{{\omega }_s\left[t-(T_0+t_{\mathrm{acc}})\right]\right\},$$

$$|v_{p_{\perp }}/v_{p_{\parallel }}|\le \tan (\varphi +\psi ).$$

$$L_{\perp }(t)=c_1\cos [{\omega }_h(t-T_1)]+c_2\sin [{\omega }_h(t-T_1)],$$

$$v_{p_{\perp }}(t)=-c_1{\omega }_h\sin [{\omega }_h(t-T_1)]+c_2{\omega }_h\cos [{\omega }_h(t-T_1)],$$

$$|v_{p_{\perp }}/v_{p_{\parallel }}|\le f(\eta )\tan (\varphi +\psi ),$$

Full text

Energetic laser-driven ion source with unique features, such as small device size and high brightness, is useful in radiography kug , warm-dense-matter generation pel , fast ignition of fusion core rot , isotope generation led , tumor therapy goi , brightness enhancement for conventional accelerators kru , etc. The target-normal sheath acceleration (TNSA) mechanism wil is widely investigated because of its undemanding laser and target parameter requirements mak ; cla ; bor ; fuc ; rob ; wag . In TNSA, an intense laser interacts with a target, generating hot relativistic electrons that penetrate through the latter and establish in the backside vacuum a TV/m sheath electric field, which accelerates the target-back ions to multi-MeV energies sna . However, the intrinsic target-normal direction and divergence of the TNSA protons limit their applications. Different methods have been proposed for collimation and manipulation of the TNSA protons, including the use of structured targets son ; heg ; sch ; kar1 ; schn ; sok ; bur ; pat ; cow ; che ; bar ; qia , electrostatic lens ton ; kar , laser prepulse lin , oblique laser incidence zho ; mor ; zei , etc., but simultaneously controlling the direction and the divergence angle of the TNSA protons remains difficult.

In this Letter, we propose to attach a solenoid to the back of a thin disk target to collimate and guide the TNSA protons. Three dimensional (3D) particle-in-cell (PIC) simulations show that a precisely directed proton bunch with small divergence angle can be obtained. The result is attributed to a uniquely structured sheath field created by the hot electrons from the foil, and is therefore quite different from the electromagnetic-pulse-induced field usually observed in the laser-target-interaction kar1 . The results agree quite well with that from an analytical model, which is also useful for tailoring the solenoid parameters in order to produce well directed and collimated high-energy proton bunches under given laser and target conditions.

The target configuration can be visualized in Fig. 1 for the electron and proton densities at $t=420$ fs. The helical plasma-wire solenoid is attached to the back of the disk foil at, for definitiveness, a $\psi=20^{\circ}$ angle. The radius, length, and number of turns of the solenoid are $r_{s}=3.5\mu$m, $h=10\mu$m, and $n=6$, respectively. The coil wire is of diameter $0.6\mu$m and total length $l=n\sqrt{(2\pi r_{s})^{2}+({h/n})^{2}}=132\mu$m. The radius and thickness of the foil are $6\mu$m and $1\mu$m, respectively. As proton source, a hydrogen dot of thickness $0.5\mu$m and diameter $1\mu$m is placed at the center of the rear foil surface. In the 3D PIC simulations with EPOCH arb , the foil and the solenoid are Cu2+ plasma, at densities $n_{i0}=20n_{c}$ and $n_{e0}=40n_{c}$, respectively. The density of the hydrogen dot is $1n_{c}$, where $n_{c}\sim 1.1\times 10^{21}{\rm cm}^{-3}/\lambda_{L}^{2}$ is the critical density, and $\lambda_{L}$ [$\mu$m] is the incident laser wavelength. To account for the laser prepulse, a $5\mu$m long preplasma of density $n_{e}=n_{e0}\exp(x/\delta)$, where $\delta=0.5\mu$m, is placed in front of the disk foil. A $y$ polarized Gaussian laser pulse of $\lambda_{L}=1.06\mu$m, intensity $2\times 10^{20}$ W/cm2, waist radius $3\mu$m, and duration $500$ fs is normally incident from the left boundary at $x=-8\mu$m. The laser has a flat-top temporal profile, with $3.5$ fs rising time. The simulation box is $23\times 16\times 16\mu$m3, with $1143\times 795\times 795$ grids. There are 7 macroparticles per cell for the solenoid target and 30 for the hydrogen dot. As shown in Fig. 1, at $t=420$ fs, a directed and well collimated proton bunch with cut-off energy $>20$ MeV is generated. The exit angle of protons with energy $>12$ MeV is around $26^{\circ}$ from the foil normal in the $(x,y)$ plane.

As the intense laser pulse impinges on the target, hot electrons are generated at the front surface and directly accelerated by the laser to high speeds wik ; she . They can easily penetrate through the foil and establish immediately behind it as well as around the surface of the solenoid wire an intense sheath electric field, roughly given by $E_{h}\sim T_{eh}/(er_{s})\sim 0.86\times 10^{12}$ V/m, where $T_{eh}$ is the temperature of hot electrons and $e$ is the elementary charge wil . This value agrees well with that from the simulation, as shown in Fig. 2(a). On the other hand, the electrons exiting the foil at where the solenoid is attached can propagate in the solenoid wire (a conductor) and create around its surface a local sheath field with magnitude $E_{t}\sim T_{et}/(eK)\approx 2.4\times 10^{12}$ V/m, where $T_{et}$ is the temperature of these secondary electrons and $K$ the spatial scale of their field. Thus in the overlapped region the sheath field on the solenoid surface can be as high as $E_{s}=E_{h}+E_{t}\approx 3.3\times 10^{12}$ V/m. The normal (to the solenoid surface) components of both $E_{h}$ and $E_{s}$ lead to a focusing force on the dot protons accelerated by the foil-back sheath field, as shown in Figs. 2(a) and (b), so that instead of propagating normal to the foil back-surface, the protons are directed and collimated by the solenoid, as can be seen in Fig. 2(b).

It is also necessary to consider the effect of the self-generated magnetic fields. The electron current in the solenoid wire is much larger than the Alfvén limit alf . The return current induced on the wire surface gives rise to a strong longitudinal magnetic field in the solenoid kay ; xia . Figures 2(c) and (d) show that the peak magnetic field (on the solenoid axis) is $B_{\parallel}\sim 1\times 10^{4}$ T. The corresponding proton gyroradius is $r_{pg}\sim m_{p}v_{p_{\perp}}/(eB_{\parallel})\approx 15.7\mu$m, where $m_{p}$ and $v_{p_{\perp}}\sim 0.05c$ are the proton rest mass and transverse velocity component, respectively. Since $r_{pg}\gg r_{s}$, the magnetic field should have little effect on the protons, as can be seen in Fig. 2(e) for three typical proton trajectories. Thus, the direction and collimation of the proton bunch are mainly determined by the electric fields $E_{h_{\perp}}$ and $E_{s_{\perp}}$.

It is thus of interest to see how $E_{h_{\perp}}$ and $E_{s_{\perp}}$ affect the proton dynamics. Since the solenoid electrons propagate at near light speed, the distance between the field boundary (i.e., the rough boundary between $E_{h_{\perp}}$ and $E_{s_{\perp}}$) and the foil back-surface at $t>T_{0}$ is $L_{b_{\parallel}}(t)\sim ch(t-T_{0})/l$, where $T_{0}$ is the time when the electrons start to propagate in the solenoid wire. The protons are first accelerated to a velocity $\bm{v}_{p_{0}}$ by the sheath field at the foil rear from $T_{0}$ to $(T_{0}+t_{\mathrm{acc}})$, after which they are no longer accelerated. The parallel and transverse proton velocities are then $v_{p_{\parallel_{0}}}=v_{p_{0}}\cos(\phi+\psi)={\rm const.}$ and $v_{p_{\perp_{0}}}=v_{p_{0}}\sin(\phi+\psi)$, respectively, where $\phi$ is the divergence angle of the protons and $\psi$ is the angle between the solenoid and the foil. Since protons with $v_{p_{\parallel_{0}}}\leq{ch}/({l-ct_{\rm acc}})$ cannot cross the field boundary, the protons (hereafter referred to as proton 1) is thus mainly governed by $E_{s_{\perp}}$ (or $E_{h_{\perp}}$ for those with small $v_{p_{\perp}}$). On the other hand, protons (hereafter referred to as proton 2) with $v_{p_{\parallel_{0}}}>ch/(l-ct_{\rm acc})$ can cross the field boundary at $t=T_{1}=T_{0}+t_{\rm acc}/(1-ch/(lv_{p_{\parallel_{0}}}))$. Type 1 protons satisfy $d^{2}L_{\perp}/{dt^{2}}=eE_{s_{\perp}}/{m_{p}}$, where $L_{\perp}$ is the distance between the proton and the solenoid axis. Assuming $E_{s_{\perp}}$ depends linearly on $L_{\perp}$, say, $E_{s_{\perp}}=-E_{s_{\perp},\max}L_{\perp}/r_{s}$, we obtain for $t>T_{0}+t_{\rm acc}$

[TABLE]

and

[TABLE]

for proton 1, where $\omega_{s}=\sqrt{eE_{s_{\perp},\max}/(m_{p}r_{s})}$. Therefore, the exit angle for proton 1 is centered along the $\psi$ direction, with the divergence angle given by

[TABLE]

The dynamics of proton 2 for $t\leq T_{1}$ is similar to that of proton 1. However, at $t=T_{1}$, proton 2 can cross the field boundary. Thereafter their motion is governed by $d^{2}L_{\perp}/dt^{2}=eE_{h_{\perp}}/m_{p}$. Assuming again that $E_{h_{\perp}}$ depends linearly on $L_{\perp}$, i.e., $E_{h_{\perp}}=-E_{h_{\perp},\max}L_{\perp}/r_{s}$, one obtains for proton 2

[TABLE]

[TABLE]

where $c_{1}=v_{p_{\perp_{0}}}\sin(\omega_{s}\tau)/\omega_{s}$, $c_{2}=v_{p_{\perp_{0}}}\cos(\omega_{s}\tau)/\omega_{h}$, $\tau=cht_{\rm acc}/(lv_{p_{\parallel_{0}}}-ch)$, and $\omega_{h}=\sqrt{eE_{h_{\perp},\max}/(m_{p}r_{s})}$. Accordingly, the exit angle of the type 2 protons for $t>T_{1}$ is centered at $\psi$, with the divergence angle given by

[TABLE]

where $f(\eta)=\sqrt{\left(E_{h_{\perp},\max}/E_{s_{\perp},\max}\right)\sin^{2}\eta+\cos^{2}\eta}$ and $\eta=\omega_{s}\tau$. Since $f(\eta)\leq 1$, the divergence of the type 2 protons is reduced after they cross the field boundary.

Figure 3(a) shows the trajectories of the two proton types. Although $E_{\perp}$ also evolves with time, only its distribution at $t=259$ fs is displayed in the background as a reference. Note that at this moment proton 2 is located at the field boundary, agreeing well with the boundary crossing time from the theory and shown in Fig. 3(b). We see that, except immediately behind the foil disk, proton 1 experiences negative $E_{h_{\perp}}$ at all times. On the other hand, proton 2 is decelerated by $E_{s_{\perp}}$ when it moves away from the solenoid center axis and accelerated by $E_{h_{\perp}}$ when it moves toward it.

Figures 3(c) and (d) show the trajectories of proton 1 and proton 2 in the $E_{\perp}$ versus $L_{\perp}$ space. In (c) we can see that proton 1 experiences a larger $E_{\perp}$ field on the way back to the solenoid axis than that when it moves away from the latter, resulting in increase of its divergence angle. In contrast, in (d) we see that proton 2 experiences a smaller $E_{\perp}$ on the way back to the axis than that when it moves away from it, so that its divergence angle decreases with time. Accordingly, the divergence angle of the TNSA proton bunch can be minimized by tailoring the solenoid parameters such that a maximum number of protons can cross the boundary between the two sheath fields.

The parameter $f(\eta)$ depends strongly on the solenoid parameters $h$, $n$, and $r_{s}$. If two of them are fixed, one can find the value of the third one in order to obtain the highest proton energy density, as shown in Fig. 4. For example, $n=6$ should be the optimal number of turns if $h=10\mu$m and $r_{s}=3.5\mu$m. Indeed, Fig. 5(a) from the simulation shows that with $n=6$, the resulting proton energy density is higher than that with $n=4$ and $n=8$. Figure. 5(b) for the exit angle of the protons versus the solenoid angle $\psi$ under optimized conditions shows that the direction of the proton bunch is well controlled if $\psi\leq 20^{\circ}$. However, both the directional preciseness and proton energy density decrease with increase of $\psi$, which is also consistent with the relations (3) and (6) of the analytical model.

In summary, we have proposed an effective scheme for directional control and collimation of intense laser-driven protons using a disk-solenoid target. Our simulations show that two partially overlapping sheath fields are induced by the hot electrons and they result in an electric field distribution that collimates and focuses the energetic protons in the solenoid to its axis (instead of the target normal direction). In fact, the divergence angle of the protons decreases when they cross the boundary region of the two sheath fields. The simulation results are in good agreement with that of an analytical model of the proton dynamics, which is also useful for tailoring the solenoid parameters for obtaining the desired proton energy and divergence angle. Highly collimated and precisely directed proton bunches are desirable for radiography, tumor therapy, warm dense matter generation, etc.

This work is supported by the National Key Program for R&D Research and Development, Grant No. 2016YFA0401100; the National Natural Science Foundation of China (NNSFC), Grant Nos. 11575031, 11575298, and 11705120, the China Postdoctoral Science Foundation 2017M612708, the National ICF Committee of China. K. J. would like to thank K. D. Xiao, C. Z. Xiao, R. Li, T. Y. Long, Y. C. Yang, R. X. Bai, and M. J. Jiang for useful discussions and help.