### Rigid sphere model of field driven cluster

In the RSM, cluster is assumed as a pre-ionized spherical nano-plasma of radius *R* and fixed ionic charge density \(\rho _\mathrm {i}\). RSM has been widely used for LCI^{29,30,31,32,33,34,37,67} without \(\varvec{B}_{ext}\). In this work we first include \(\varvec{B}_{ext}\) in RSM to study its effects. Ions provide the potential \(\phi (r)\) with the space-charge field

$$\varvec{E}_{sc} (\varvec{r}) = \left\{ {\begin{array}{*{20}l} \omega _\mathrm {M}^{2} \varvec{r} \hfill & \text {if }r \le R \hfill \\ \omega _\mathrm {M}^{2} R^3 \varvec{r}/{r^3} \hfill & \text {if }r > R \hfill \\ \end{array} } \right.$$

(1)

in which electrons interact in addition to the applied laser field (\(\varvec{E}_l, \varvec{B}_l\)) and external \(\varvec{B}_{ext}\). Dynamics of an electron obeys

$$\begin{aligned} \frac{d\varvec{p}}{dt}&= q\left[ \left( \varvec{E}_l + \varvec{E}_{sc}(\varvec{r}) \right) + \varvec{v}\times \left( \varvec{B}_l + \varvec{B}_{ext} \right) \right] \end{aligned}$$

(2)

$$\begin{aligned} \frac{d\varvec{r}}{dt}&= \varvec{v} = \frac{\varvec{p}}{\gamma m_0} \end{aligned}$$

(3)

$$\begin{aligned} \frac{d({\gamma } m_{0} c^2) }{dt}&= q {\varvec{v}}.\left( \varvec{E}_l + \varvec{E}_{sc}(\varvec{r}) \right) \end{aligned}$$

(4)

where \(\gamma = 1/\sqrt{1-v^2/c^2}= \sqrt{1+p^2/m_{0}^2 c^2}\) is the relativistic \(\gamma\)-factor for the electron, \(m_{0}, q, \varvec{r}, \varvec{v}, \varvec{p}\) are its rest-mass, charge, position, velocity and linear momentum respectively with \(m_0 = 1\), \(q = e = -1\) in a.u. Equations (1)â€“(3) represent a field driven three-dimensional non-linear oscillator. The coulomb part of \(\varvec{E}_{sc} \propto \varvec{r}/{r^3}\) restricts its analytical solution, except in some simplified linear case of \(\varvec{E}_{sc} \propto \varvec{r}\) with continuous (plane-wave) laser field only. For example, see direct laser acceleration (DLA) of electrons from an under-dense, pre-formed plasma channel^{65,68,69,70,71,72} assisted by auxiliary fields, e.g., magnetic wigglers, static electric and magnetic fields with \(I_0 >10^{18}\,\text{ W/cm}^{2}\) and corresponding normalized vector potential \(a_0 = \sqrt{I_0}/\omega c > 1\). To obtain electrons of MeV energies (or higher), the regime of \(a_0 > 1\) is an obvious choice. Such pre-formed plasma channels are very long (typically tens of \(\lambda\)) and relativistically intense laser has to propagate several \(\lambda\) which then sets up electro-static fields in the channel with associated self-generated quasi-static magnetic fields. Electrons are injected into the channel or drawn from the plasma itself and guided by the channelâ€™s fields and the applied laser field. If the ambient magnetic field is in the direction transverse to the laser polarization, then energy of electrons can be increased and ECR may happen if such magnetic field satisfies the ECR condition. This work, however, reports other unexplored regime of DLA with \(I_0 < 10^{18}\,\text{ W/cm}^{2}\) using short-pulsed light and a constant \(\varvec{B}_{ext}\) for an over-dense cluster plasma electrons.

The field \(\varvec{E}_{sc}\) imparts oscillatory motion in \(\varvec{r}\), whereas \(\varvec{B}_{ext}\) imparts rotation in the plane perpendicular to \(\varvec{B}_{ext}\) (in \(\varvec{r}_{\perp }\)) to an electron. Combining these two motions, the position dependent squared effective-frequency \({\omega }_{\mathrm {eff}}^2[\varvec{r}

(5)

The term \((\gamma \varvec{E}_{sc}/r + \Omega _{c0}^2 \varvec{\hat{r}_{\perp }}) \cdot \varvec{\hat{r}_{\perp }}/\gamma ^2\) represents motion due to combined space-charge and \(\varvec{v}\times \varvec{B}_{ext}\) field in \(\varvec{r}_{\perp }\) plane and \((\gamma \varvec{E}_{sc} \cdot \varvec{\hat{r}_{||}})/\gamma ^2 r\) represents motion in \(\varvec{\hat{r}_{||}}\) along \(\varvec{B}_{ext}\). The unit vectors \(\varvec{\hat{r}}, \varvec{\hat{r}_{\perp }}\) indicate frequencies are valid only for motions in those directions. Equation (5) may be regarded as the relativistic extension to its non-relativistic variant^{33,34,37,67,73} for \(\Omega _{c0} = 0\) and \(\gamma = 1\). When \(\varvec{E}_{sc} = \omega _\mathrm {M}^2\varvec{r}\) and \(\gamma \approx 1\), it gives harmonic oscillator frequency \({\omega }_{\mathrm {eff}}^2 [\varvec{r}

(6)

$$\begin{aligned} \varvec{B}_l (t’)&= \hat{\mathbf {z}} \times \varvec{E}_l (t’) / c \end{aligned}$$

(7)

where \(c_1=1/2, c_2=c_3= -1/4, \omega _1 = \omega , \omega _2 = (1+1/n)\omega\), and \(\omega _3 = (1-1/n)\omega\). For \(R\ll \lambda\), the dipole approximation \(z/\lambda \ll 1\) may be assumed.

*The cluster.* A deuterium cluster with number of atoms \(N=2176\) and \(R\approx 2.05\) nm is irradiated by above laser pulse for \(n=5\), \(\tau = nT\approx 13.5\) fs (\(\tau _{fwhm}\sim 5\) fs), unless explicitly mentioned. Cluster is \(\rho _i/\rho _\mathrm {c}\approx 27.1\) times overdense with \((\omega _\mathrm {M}/\omega )^2 \approx 9.1\), where \(\rho _\mathrm {c}\approx 1.75\times 10^{27} m^{-3}\) is the critical density at 800 nm. Equations (1)â€“(3) using Eqs. (6)â€“(7) are numerically solved by the Velocity Verlet method (VVM).

#### Regeneration of previous RSM results: single electron dynamics with laser field only

We begin with energy absorption and associated electronâ€™s dynamical variables as a conventional case of LCI without \(\varvec{B}_{ext}\). Figure 1 shows (column-wise) normalized co-ordinates (\(\overline{x}= x/R, \overline{y}= y/R, \overline{z}= z/R\)), squared effective frequency \(\overline{\omega }_{\mathrm {eff}}^2={\omega }_{\mathrm {eff}}^2/\omega ^2\), total energy \(\overline{\mathcal {E}} = ((\gamma -1)m_0 c^2 + q\phi )/U_\mathrm {p}\) in units of \(U_\mathrm {p}\) and corresponding momenta (\(\overline{p}_x= p_x/c, \overline{p}_y= p_y/c, \overline{p}_z= p_z/c\)) vs time *t*/*T* of an initially bound electron \([\,\varvec{r}(0)=\varvec{0}, \varvec{p}(0)=\varvec{0}\,]\) in the RSM potential, when driven by 5-cycle pulse of \(I_0 = 7.13\times 10^{16}\,\text{ W/cm}^{2}\). Associated fields \(E_x = \varvec{\hat{x}}.(\varvec{E}_l + \varvec{E}_{sc}), E_y = \varvec{\hat{y}}.\varvec{E}_{sc}\), \(E_z = \varvec{\hat{z}}.\varvec{E}_{sc}\) and \(E_l = \varvec{\hat{x}}.\varvec{E}_l\) vs *t*/*T* are also plotted, purpose of which will be evident when we consider \(\varvec{B}_{ext}\) later. Left panels (a1,b1,c1) and right panels (a2,b2,c2) are the cases with \(\varvec{B}_l = \varvec{0}\) and \(\varvec{B}_l \ne \varvec{0}\) respectively.

Without \(\varvec{B}_l\) (in many works \(\varvec{B}_l\) was neglected^{21,27,31,76} since \(\max \vert \varvec{B}_l\vert =E_0/c \ll 1\)), it is shown that electron starts (Fig. 1a1) with the binding energy \(\overline{\mathcal {E}} = -1.5\omega _\mathrm {M}^2 R^2/U_\mathrm {p}\) at \(t/T=0\), it oscillates in the potential with increasing amplitude in *x* (while \(y=0,z=0\)) as the total field \({E}_x\) oscillates in time (while \(E_y = 0, E_z=0\)) and approaches to the peak value \(E_0\) of \(\varvec{E}_l\) (Fig. 1c1) around \(t/T\approx 2\). Inside the potential, for \(r/R\le 1\), \(E_x\) is suppressed due to opposite phase of \(\varvec{\hat{x}}.\varvec{E}_l\) and \(\varvec{\hat{x}}.{\varvec{E}_{sc}}\). As long as \(r/R\le 1\), \(\overline{\omega }_{\mathrm {eff}}^2\) continues (Fig. 1a1) at its initial value \(\omega _\mathrm {M}^2/\omega ^2 \approx 9.1\). When \({E}_x\) increases sufficiently strong (due to reduced phase mismatch between \(\varvec{\hat{x}}.\varvec{E}_l\) and \(\varvec{\hat{x}}.{\varvec{E}_{sc}}\)) leading to increasing \(r/R>1\); \(\overline{\omega }_{\mathrm {eff}}^2\) falls rapidly, it meets the AHR condition \(\overline{\omega }_{\mathrm {eff}}^2 = 1\) around \(t/T\approx 2.1\) (marked by horizontal dashed line and vertical shaded bar) and then electron leaves the cluster forever (Fig. 1a1) with \(\overline{\mathcal {E}}>0\) associated with non-zero transverse momentum \(\overline{p}_x\) (in Fig. 1b1). After the AHR, \(\varvec{E}_x\) follows \(\varvec{E}_l\). Though LR can not happen, AHR is dynamically met here leading to the electronâ€™s removal from the cluster with \(\overline{\mathcal {E}}>0\) and non-zero \(\overline{p}_x\) eventually. Similar results (neglecting \(\varvec{B}_l\)) are shown in Refs.^{31,32,33,34,37,41}.

Considering \(\varvec{B}_l\) now, Fig. 1a2,b2,c2 show indistinguishable variation of (\(\overline{x},\overline{y},\overline{\omega }_{\mathrm {eff}}^2, \overline{\mathcal {E}}, \overline{p}_x, \overline{p}_y, E_x, E_y\)) with respective Fig. 1a1,b1,c1. Also \(\overline{z}\approx 0, \overline{p}_z\approx 0\) (in a2, b2) before the occurrence of AHR near \(t/T\approx 2.1\), since \(\varvec{v}\times \varvec{B}_l\) field along *z* is much weaker and leads to a negligible \(E_z\) [Fig. 1c2, clearly seen in zoomed inset plots in (b2,c2)]. As the electron is liberated (Fig. 1a2) via AHR around \(t/T\approx 2.1\) with dominant velocity in *x* (Fig. 1b2), the \(\varvec{v}\times \varvec{B}_l\) field imparts a forward momentum \(p_z\) along the laser propagation (Fig. 1b2) and its *z* co-ordinate sharply increases (Fig. 1a2) by many times *R*. Electron is now emitted in the \(z-x\) plane with an angle \(\theta \approx \arctan (p_x/p_z)\) in contrast to Fig. 1a1,b1 where electron is emitted *only* along the polarization axis. Though \(I_0 < 10^{18}\,\text{ W/cm}^{2}\), the liberated electron via the AHR process around \(t/T\approx 2.1\) is *self-injected* into the remaining laser pulse with some forward momentum \(\overline{p}_z>0\) and transverse momentum \(\overline{p}_x\); and from this time onward electronâ€™s acceleration resembles the standard DLA. Clearly, inclusion of \(\varvec{B}_l\) here yields (Fig. 1a2,b2,c2) *different electron dynamics* (see also Mulser et al.^{31}) for LCI than neglecting it^{21,27,31,41,76,77,78} in previous works.

However, both the cases in Fig. 1 show maximum attainable energy \(\max {\overline{\mathcal {E}}} = 8\) (marked by upper horizontal dashed line) near the laser peak at \(t/T= 2.5\); but the electron retains only a lower value of energy \(\overline{\mathcal {E}}_A = \overline{\mathcal {E}}(\tau ) \approx 2.1\) in the end. We may compare these two limits of \(\overline{\mathcal {E}}\) with the laser-driven electron-atom re-collision model^{50,51,52,79,80} of harmonic generation where \(\max {\overline{\mathcal {E}}}\) of electron may go up to \(\approx 8\) during the pulse, but the returned electron when re-collides with the parent ion has a lower \(\overline{\mathcal {E}}\approx 3.17\) which is often manifested as a harmonic cut-off energy. In laser-cluster experiments, an electronâ€™s final energy is reported to be less than the above mentioned laser-atom interaction case^{81,82,83} and the final absorbed energy limit \(\overline{\mathcal {E}}_{A}^{max} = \max {\overline{\mathcal {E}}_A} \lesssim 3.17\) seems to obey^{8,14,40} herein. Particle simulations^{33,34,37,41,42,45,46,47,48,49,67} and simple models^{24,31,32,33,34,37,41,67} employed so far for LCI also indicate \(\overline{\mathcal {E}}_{A}^{max} \lesssim 3.17\) in the collision-less case. Thus, though the role of \(\varvec{B}_l\) can not be neglected for altering the electron dynamics (Fig.1a2,b2,c2) at a \(I_0>7.13\times 10^{16} \text{ W/cm}^{2}\) (where peak magnetic field can be substantial \(> 2.44\) kT), the average \(\overline{\mathcal {E}}_{A}^{max} \lesssim 3.17\) seems to follow (see Table 1) for the traditional LCI. The aim of the paper is to increase this limit far beyond \(\overline{\mathcal {E}}^{max}_{A} \sim 3.17\) with an ambient \(\varvec{B}_{ext}\).

### PIC simulation

We also study LCI with/without \(\varvec{B}_{ext}\) using three-dimensional PIC simulation code^{33,34,41,76,78,84,85}. The same deuterium cluster with number of atoms \(N = 2176\) is considered. Atoms are placed in a cubical computational box according to the Wigner-Seitz radius \(r_w\approx 0.17\) nm (giving cluster radius \(R=r_w N^{1/3} \approx 2.05\) nm) so that center of the cluster coincides the center of the computational box of side \(L = 24.6 R\). Initially laser \(\varvec{E}_l

(8)

where \(\varvec{p}_{j \vert k} = m_{j\vert k} \varvec{v}_{j\vert k}/\sqrt{1-v_{j\vert k}^2/c^2}, \varvec{v}_{j\vert k}, \varvec{r}_{j \vert k}, m_{j\vert k}, q_{j\vert k}\) are relativistic momentum, velocity, position, mass, and charge of a PIC electron/ion respectively. In the present case, \(m_j = m_0 = 1\), \(m_k = M_0 = 2\times 1386\), \(q_j = -1\) and \(q_k = 1\) in a.u.. Poissonâ€™s equation \(\nabla ^2\phi _G = -\rho _G\) is solved for \(\phi _G\) on the numerical grid (subscript *G* indicates grid values of potential and charge density) with time-dependent monopole boundary condition. Interpolating \(\phi _G\) to the particle position corresponding potential \(\phi (\varvec{r}_{j\vert k},t)\) is obtained. Field \(\varvec{E}_{sc}(\varvec{r}_{j\vert k}) = -\varvec{\nabla } \phi (\varvec{r}_{j\vert k})\) in (8) is obtained by analytical differentiation^{76} of interpolated \(\phi (\varvec{r}_{j\vert k})\) locally at \(\varvec{r}_{j\vert k}\). Equation (8) is solved by VVM using laser fields (6)â€“(7). Total absorbed energy \(\mathcal {E}

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