Abstract
The trajectories of electrons and ions when a hot plasma expands under vacuum are studied in detail from a theoretical point of view and with the aid of numerical simulations. Exact analytic solutions are obtained in multidimensions, starting from the solution for the expansion of a quasineutral, Gaussian, collisionless plasma in vacuum [D. S. Dorozhkina and V. E. Semenov, Phys. Rev. Lett. 81, 2691 (1998)]. Focusing of laseraccelerated ions with concave targets is investigated with the hybrid particleincell code Lsp. For a given laser energy and pulse duration, a larger laser focal spot is found to be beneficial to focus the ion beam to a smaller focal spot, due both to a geometric effect and to the decrease in the transverse gradient of the hot electron pressure.
C.B. acknowledges useful discussions with A. Kemp. This work was partially supported by the U. S. Department of Energy Grant DESC0001265.
I. INTRODUCTION
II. ELECTRON AND ION TRAJECTORIES—ANALYTIC SOLUTIONS
A. Adiabatic solutions and plasma dynamics
B. Electron trajectories
C. Ion trajectories
D. Ion trajectories for concave targets
III. 2D CARTESIAN SIMULATIONS WITH INJECTION OF HOT ELECTRONS
IV. CONCLUSIONS
Key Topics
 Plasma expansion
 18.0
 Protons
 16.0
 Electric fields
 9.0
 Hot carriers
 9.0
 Speed of sound
 8.0
H05H1/02
Figures
Spatial distribution of the electric field (a) and temporal variation of the electron density at two fixed positions () (b), during the expansion of a Gaussian 1D plasma in vacuum (see text for details). Note that in (a), only half of the problem is shown.
Spatial distribution of the electric field (a) and temporal variation of the electron density at two fixed positions () (b), during the expansion of a Gaussian 1D plasma in vacuum (see text for details). Note that in (a), only half of the problem is shown.
(a) Onedimensional electron position (and velocity, insert) as a function of time, for and ; the final rectilinear trajectory corresponds to a velocity (in arbitrary units). (b) Twodimensional electron trajectories for two different initial positions (blue and red lines, the starting position is denoted with a filled circle) and for in the x direction and in the y direction. Black broken lines represent isodensity contours, assuming the same speed of sound in the two directions.
(a) Onedimensional electron position (and velocity, insert) as a function of time, for and ; the final rectilinear trajectory corresponds to a velocity (in arbitrary units). (b) Twodimensional electron trajectories for two different initial positions (blue and red lines, the starting position is denoted with a filled circle) and for in the x direction and in the y direction. Black broken lines represent isodensity contours, assuming the same speed of sound in the two directions.
Electron trajectories for particles starting at rest, for the same simulation as in Fig. 1. Note that the symmetry condition at x = 0 has the effect of reflecting the particles at the boundary.
Electron trajectories for particles starting at rest, for the same simulation as in Fig. 1. Note that the symmetry condition at x = 0 has the effect of reflecting the particles at the boundary.
(a) Proton trajectories from the expansion of a Gaussian proton plasma, obtained from Eq. (15). Blue, red, and green lines correspond to particles of increasing energy and black lines are isoenergy contours (ellipsoids). Parameters: m, m m. The blue and red trajectories are used in (b) to plot the asymptotic angle of divergence as a function of the initial radial position of the particle. The inset in (b) shows traces of the same protons, if an RCF were positioned at a distance m from the origin.
(a) Proton trajectories from the expansion of a Gaussian proton plasma, obtained from Eq. (15). Blue, red, and green lines correspond to particles of increasing energy and black lines are isoenergy contours (ellipsoids). Parameters: m, m m. The blue and red trajectories are used in (b) to plot the asymptotic angle of divergence as a function of the initial radial position of the particle. The inset in (b) shows traces of the same protons, if an RCF were positioned at a distance m from the origin.
Proton trajectories after 60 ps, from the expansion of a flat foil in vacuum, with initial temperature keV and for a thickness (a) m and (b) m. Broken lines denote the initial position of the foils, black and red trajectories originate from different positions in x (see text for details).
Proton trajectories after 60 ps, from the expansion of a flat foil in vacuum, with initial temperature keV and for a thickness (a) m and (b) m. Broken lines denote the initial position of the foils, black and red trajectories originate from different positions in x (see text for details).
Proton trajectories from the expansion of a Gaussian proton plasma, obtained from Eq. (22) in cylindrical geometry. Blue and red lines correspond to particles of increasing energy. Parameters: m, ms m, and ms^{− 1}.
Proton trajectories from the expansion of a Gaussian proton plasma, obtained from Eq. (22) in cylindrical geometry. Blue and red lines correspond to particles of increasing energy. Parameters: m, ms m, and ms^{− 1}.
Proton trajectories resulting from the expansion of a partial, hollow hemisphere with radius of curvature of m, thickness of m and (a) keV and (b) keV. Test protons in (a) and (b) are initialized at the same positions, at a distance of m (black lines) and m (red lines) on the right of the central arc passing through m. Note that the higher electron temperature in (b) results in a larger focal spot.
Proton trajectories resulting from the expansion of a partial, hollow hemisphere with radius of curvature of m, thickness of m and (a) keV and (b) keV. Test protons in (a) and (b) are initialized at the same positions, at a distance of m (black lines) and m (red lines) on the right of the central arc passing through m. Note that the higher electron temperature in (b) results in a larger focal spot.
Snapshots of the spatial distribution of the protons at different times (in red, units are in ps) for different energies ( MeV and MeV). The target is a partial hollow hemisphere with radius of curvature m, the hot electrons are excited at the target front side for 500 fs, with either temperature keV and m (cases a and b) or MeV and m (cases c and d).
Snapshots of the spatial distribution of the protons at different times (in red, units are in ps) for different energies ( MeV and MeV). The target is a partial hollow hemisphere with radius of curvature m, the hot electrons are excited at the target front side for 500 fs, with either temperature keV and m (cases a and b) or MeV and m (cases c and d).
D _{50} diameter (see text for details) vs. longitudinal direction for different energy ranges (blue: 35 MeV, red: 1012 MeV, and green: 1520 MeV) and different target and heating conditions (in terms of FWHM, duration and temperature of the hot electrons): (a) m, m, and fs  corresponding to cases a and b in Fig. 8; (b) m, m, and fs  corresponding to cases c and d in Fig. 8; (c) m, m, and fs; (d) m, m, and fs. For cases a, c, and d, the hot electron temperature keV and for case b, MeV.
D _{50} diameter (see text for details) vs. longitudinal direction for different energy ranges (blue: 35 MeV, red: 1012 MeV, and green: 1520 MeV) and different target and heating conditions (in terms of FWHM, duration and temperature of the hot electrons): (a) m, m, and fs  corresponding to cases a and b in Fig. 8; (b) m, m, and fs  corresponding to cases c and d in Fig. 8; (c) m, m, and fs; (d) m, m, and fs. For cases a, c, and d, the hot electron temperature keV and for case b, MeV.
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