Broadening of cyclotron resonance conditions in the relativistic interaction of an intense laser with overdense plasmas

The interaction of dense plasmas with an intense laser under a strong external magnetic field has been investigated. When the cyclotron frequency for the ambient magnetic field is higher than the laser frequency, the laser's electromagnetic field is converted to the whistler mode that propagates along the field line. Because of the nature of the whistler wave, the laser light penetrates into dense plasmas with no cutoff density, and produces superthermal electrons through cyclotron resonance. It is found that the cyclotron resonance absorption occurs effectively under the broadened conditions, or a wider range of the external field, which is caused by the presence of relativistic electrons accelerated by the laser field. The upper limit of the ambient field for the resonance increases in proportion to the square root of the relativistic laser intensity. The propagation of a large-amplitude whistler wave could raise the possibility for plasma heating and particle acceleration deep inside dense plasmas.

Figure 1

(a) Temporal evolution of plasma kinetic energies in the fiducial model for the electrons ${\mathcal{E}}_e$ (red), ions ${\mathcal{E}}_i$ (green), and total ${\mathcal{E}}_{\mathrm{kin}}={\mathcal{E}}_e+{\mathcal{E}}_i$ (black), which are normalized by the incident laser energy ${\mathcal{E}}_0$. The gray area denotes the interaction duration of the laser with the plasma foil. (b, c) Phase diagram of the position $x$ and momentum in the $x$ direction $p_x$ for (b) the electrons and (c) ions, which are taken at the end of calculation $t=1$ ps. The color shows the particle number. The initial location of the target foil is from $x=0$ to 1 $\mu \mathrm{m}$.

Figure 2

(a) Energy absorption evaluated by the gain of the total plasma energy $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{kin}}$ as a function of the external field strength $B_{\mathrm{ext}}$ for the CP laser cases of $I_0={10}^{21}$ (black), ${10}^{20}$ (red), ${10}^{19}$ (green), and ${10}^{18}\mathrm{W}/{\mathrm{cm}}^2$ (blue). The critical field strength $B_c$ is indicated by the dotted line. For the purpose of comparison, the results of the ion immobile runs for $I_0={10}^{21}$ and ${10}^{20}\mathrm{W}/{\mathrm{cm}}^2$ are also shown by the open squares. (b) The same figure as panel (a) for the LP laser cases.

Figure 3

Upper limits of the external field strength $B_{\mathrm{ext}}$ (filled circles) for the range where the absorption rate is enhanced over a threshold value, which are shown as a function of the normalized intensity of the incident laser $a_0$. The different colors indicate different model assumptions. The red and green marks are for the cases with the RCP and LP laser, respectively. The blue ones are with the RCP laser, but the ions are assumed to be immobile. The solid curve shows the theoretical prediction for ${\tilde{B}}_{\mathrm{upper}}$ given by Eq. (5) assuming ${\tilde{B}}_{\mathrm{res}}=1.7$. The lower limits obtained numerically are also plotted by the open squares.

Figure 4

(a–c) Parallel and perpendicular velocity diagram of electrons, $v_{\parallel }\text{−}{v}_{\perp }$, for the fiducial model observed at (a) $t=0$, (b) 0.05, and (c) 0.1 ps. The color denotes the particle number. The cyclotron resonance condition given by Eq. (6) is shown by the solid and dashed curves. The dotted curve denotes the upper limit of $\left|\text{v}\right|=c$. (d) The same figure as panel (c) for a lower intensity $I_0={10}^{19}\mathrm{W}/{\mathrm{cm}}^2$. (e) Energy spectrum of electrons $\epsilon f_{\epsilon }\left(\epsilon \right)$ in the fiducial (solid) and lower intensity models (dashed) taken at $t=0.05$ (red) and 0.1 ps (black). The gray curve is the initial Maxwellian spectrum of $T_e=0.05$ keV. The resonance condition $\gamma \gtrsim {\gamma }_{\mathrm{res}}$ is indicated by the gray zone.

Figure 5

Trajectories of four representative electrons in the fiducial model shown by (a) the time-energy and (b) position-energy diagrams. The same color in the both panels means the track of an identical particle, which is drawn form $t=0$ [open circles in panel (b)] to 150 fs (filled circles). Gray areas denote (a) the duration of 30 fs when the laser pulse is hitting the foil target and (b) the initial target location of 1 $\mu \mathrm{m}$ thickness. As a reference, the resonance condition $\gamma ={\gamma }_{\mathrm{res}}$ and the quiver energy $\gamma ={\gamma }_q$ are depicted by the horizontal dashed and dotted lines, respectively.

Figure 6

Energy spectrum of ions $\epsilon f_{\epsilon }\left(\epsilon \right)$ in the fiducial model (black) as well as the different $B_{\mathrm{ext}}$ runs with 20.0 kT (red) and nothing (green). The RCP laser with $I_0={10}^{21}\mathrm{W}/{\mathrm{cm}}^2$ is used in these models, and all the spectra are taken at $t=1$ ps. The gray curve is the initial Maxwellian spectrum of $T_e=0.5$ keV. The quiver kinetic energy of electrons ${\epsilon }_q$ and the free-electron ponderomotive limit ${\epsilon }_p$ are indicated by arrows.

Figure 7

(a) Dependence of the external field strength $B_{\mathrm{ext}}$ on the maximum energy of ions ${\epsilon }_{i,max}$ in the models of $I_0={10}^{21}$ (black) and ${10}^{20}\mathrm{W}/{\mathrm{cm}}^2$ (red). The LP laser cases are also shown by the open circles. The other parameters are identical to the models shown in Fig. 2. The dotted line indicates the critical field strength ${\tilde{B}}_{\mathrm{ext}}=1$. (b) The maximum ion energy for the models of a fully ionized carbon target. The LP laser is used and the other parameters are the identical to the fiducial hydrogen model.

Figure 8

(a) Energy absorption in the density and field strength diagram constructed from the results of $50\times 50$ simulations. The thickness of the hydrogen target is assumed to be 3 $\mu \mathrm{m}$ for these calculations. The RCP laser is used with the intensity $I_0={10}^{19}\mathrm{W}/{\mathrm{cm}}^2$. The solid and dashed lines denote the R cutoff (${\tilde{n}}_e=1-{\tilde{B}}_{\mathrm{ext}}$) and the cyclotron resonance (${\tilde{B}}_{\mathrm{ext}}=1$), respectively. (b) The same figure as panel (a) for the immobile ion simulations.