Kinetics of ultrashort relativistic electron pulses emitted from solid targets

Interaction of ultrashort high-intensity laser pulses with solid targets generates relativistic electrons which escape from the target. The kinetics of these ultrashort electron pulses is governed by self-fields generated by the charge of the electron cloud. In this paper an analytical theory is developed which allows calculation of electron trajectories, electron fluxes, and electron spectra at any distance from the target. The theory is exact for two limiting cases: (a) a monoenergetic electron pulse with an arbitrary temporal shape; (b) an infinitely short electron pulse with an arbitrary energy spectrum. These results have applications in high-intensity irradiation experiments, e.g., in experiments irradiating samples with ultrashort electron or x-ray pulses, in developing optics for fourth-generation light sources, and in work relating to x-ray lasers.

Figure 1

Electron trajectories for a monoenergetic “Rayleigh” pulse for ${\tau }_{\max }=2$ (drawn dotted) with the Lagrangian coordinate $\xi$ as a parameter. The initial electron energy is $500\mathrm{keV}$. The space and time coordinates are dimensionless. Curves are drawn for $\xi =0.001$, 0.1, $0.2,\dots$ up to 0.9. The curve with $\xi =0.8$ is seen to cross the curve with $\xi =0.9$, indicating breakdown of the Lagrangian coordinate description.

Figure 2

Electron trajectories for an infinitely short electron pulse with an exponential energy distribution. The space and time coordinates are dimensionless. Electron temperature $k{T}_e=500\mathrm{keV}$. The electrons are released at the rear of a foil with a normalized thickness $D=5$. Curves are drawn for $\xi =0.001$, 0.1, $0.2,\dots$ up to 0.8. The curve with $\xi =0.7$ is seen to cross the curve with $\xi =0.8$, indicating breakdown of the Lagrangian coordinate description.

Figure 3

Fraction of electrons crossing a gap. The curves are drawn for real coordinates and an areal electron density of ${10}^{15}{\mathrm{cm}}^{-2}$. Dotted curves are for monoenergetic electrons with electron energies of 200, 500, and $1000\mathrm{keV}$ (from bottom). These curves are seen to originate at a small distance from the target, a consequence of the breakdown of the Lagrangian coordinate description. Solid curves are for exponential electron energy distributions with $k{T}_e=200$, 500, and $1000\mathrm{keV}$ (from bottom).

Figure 4

Electron spectra at various distances from the target. The curves are drawn for real coordinates and an areal electron density of ${10}^{15}{\mathrm{cm}}^{-2}$. The input spectrum (at $x=0$) is exponential with a temperature of $500\mathrm{keV}$. Further spectra are shown (from top) at distances $x=5$, 10, 25, 50, and $100\mu \mathrm{m}$ from the target.