Energy gain by laser-accelerated electrons in a strong magnetic field

This paper deals with electron acceleration by a laser pulse in a plasma with a static uniform magnetic field $B_{*}$. The laser pulse propagates perpendicular to the magnetic field lines with the polarization chosen such that $({\mathit{E}}_{\mathrm{laser}}·{\mathit{B}}_{*})=0$. The focus of the work is on the electrons with an appreciable initial transverse momentum that are unable to gain significant energy from the laser in the absence of the magnetic field due to strong dephasing. It is shown that the magnetic field can initiate an energy increase by rotating such an electron, so that its momentum becomes directed forward. The energy gain continues well beyond this turning point where the dephasing drops to a very small value. In contrast to the case of purely vacuum acceleration, the electron experiences a rapid energy increases with the analytically derived maximum energy gain dependent on the strength of the magnetic field and the phase velocity of the wave. The energy enhancement by the magnetic field can be useful at high laser amplitudes, $a_0\gg 1$, where the acceleration similar to that in the vacuum is unable to produce energetic electrons over just tens of microns. A strong magnetic field helps leverage an increase in $a_0$ without a significant increase in the interaction length.

Figure 1

Schematic representation of the considered setup where a plane electromagnetic wave irradiates a relativistic electron in a uniform magnetic field that is perpendicular to the wave propagation and to the wave electric field.

Figure 2

Electron trajectories in a uniform magnetic field with ${\omega }_{ce}/\omega =2.085$. The solid line is for an electron with an initial transverse momentum $p_y=-50m_{e}c$ that is irradiated by a laser pulse with $a_0=50$. The dotted line is for the same electron but without the laser pulse.

Figure 3

Longitudinal and transverse electron momentum along the trajectory shown in Fig. 2 with a solid curve. The circles mark the turning point from Fig. 2.

Figure 4

Relativistic factor $\gamma$, angle $\theta$, and the dephasing rate $R$ of the laser-irradiated electron in a uniform magnetic field. The circles mark the turning point from Fig. 2.

Figure 5

Scan over the phase offset for an electron with ${\mathit{p}}_0=(75m_{e}c,0,0)$ irradiated by a wave with $a_0=50$ in a magnetic field with ${\omega }_{ce}/\omega =1$. (d) the initial amplitude of the laser electric field. (a)–(c) The maximum relativistic factor ${\gamma }_{max}$ and the maximum displacement that the electron achieves while $E_y$ remains negative. The dashed lines are the estimates given by Eqs. (37), (42), and (43).

Figure 6

Scan over the phase offset $\psi$ and initial longitudinal momentum $p_0$ for an electron irradiated by a wave with $a_0=50$ ($\delta u=0$) in a magnetic field with ${\omega }_{ce}/\omega =1$. The color shows the maximum relativistic factor ${\gamma }_{max}$ that the electron achieves during acceleration (while $E_y$ remains negative). The dashed curves show ${\gamma }_{\max }=200$, 400, 600, and 800.

Figure 7

Scan over $a_0$ and ${\omega }_{ce}/\omega$ at $\psi =-0.65$ and $\delta u=0$. The initial longitudinal momentum is set at $p_0=a_0{m}_{e}c$. The color shows a relative difference between the calculated maximum relativistic factor ${\gamma }_{max}$ that the electron achieves during acceleration and $\mathrm{\Delta }\gamma$ predicted by Eq. (37).

Figure 8

Scan over the magnetic field strength for an electron irradiated by a wave with $a_0=50$ and $\delta u=0.01$. The phase offset is $\psi =-0.7$, and the initial longitudinal momentum is $p_0=10m_{e}c$. The solid curves are the calculated ${\gamma }_{max}$ and the corresponding longitudinal displacement $x_{max}$. The dotted curves are $\mathrm{\Delta }\gamma$ and $\mathrm{\Delta }x$ predicted by Eqs. (37) and (42). $\mathrm{\Delta }{\gamma }_{\mathrm{mag}}$ and $\mathrm{\Delta }{\gamma }_{\mathrm{ph}}$ are given by Eqs. (38) and (39). $\mathrm{\Delta }{x}_{\mathrm{mag}}$ is given by Eq. (42) with $\theta ={\theta }_{\mathrm{mag}}$.

Figure 9

Electron energy gain with and without the applied magnetic field. In all three cases we have $a_0=50$, and the electron starts its motion at $E_y=E_0$. The blue and red dotted curves are for the acceleration without the magnetic field with an initial longitudinal and an initial transverse momentum, respectively. The solid curve is for an electron with an initial transverse momentum $p_0=-75m_{e}c$ accelerated in a magnetic field, ${\omega }_{ce}/\omega =1.01$.

Figure 10

Parameter scan for a laser-irradiated electron that starts its motion with a transverse momentum. The laser amplitude is $a_0=50$ and the initial momentum is $p_0/m_{e}c=75-a_{0}sin\left(\psi \right)$. The color in panel (a) shows the maximum relativistic factor along the trajectory similar to that shown in Fig. 2, whereas the color in panel (b) shows $p_x/m_{e}c$ at the turning point.

Figure 11

Parameter scan for an electron irradiated by a superluminal electromagnetic wave with $\delta u=0.01$ and $a_0=50$. The electron starts its motion with a transverse momentum $p_0/m_{e}c=75-a_{0}sin\left(\psi \right)$. The color shows the maximum relativistic factor along the trajectory similar to that shown in Fig. 2.