Author Affiliations
^{1}Department of Plasma Physics and Fusion Engineering, University of Science and Technology of China, Hefei, China^{2}Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA^{3}Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, Chinashow less
Fig. 1. Electric and magnetic field components of an LPLG laser beam before it encounters the plasma. Panels (a) and (d) show
; panels (b) and (e) show
; panels (c) and (f) show
. The lefthand column ((a)–(c)) shows the field structure in the
plane at
. The righthand column ((d)–(f)) shows the field structure in the
plane at the
position indicated with the dashed line in panels (a)–(c). All the snapshots are taken at
fs from the simulation with parameters listed in Table
1.
Fig. 2. Structure of electron bunches shortly after laser reflection off the plasma (
fs). (a) Electron density on a logscale, with the color representing
. The blue, red and green contours denote
,
and
, respectively. The dashed rectangle marks the third bunch, whose additional details are provided in the remaining panels. (b) Electron areal density
in the third bunch. (c) Cellaveraged electron divergence angle
in the third bunch. (d), (e) 3D rendering of the electron density in the third bunch using different viewpoints.
Fig. 3. (a) Areal density of the electrons in the third bunch at time
fs. (b) Three groups of electrons (blue, green and red markers) selected from the third bunch at
fs for tracking. The electrons in each group are selected randomly. (c) Transverse positions of the three groups of electrons from (b) at
fs. (d)–(f) Trajectories of the three groups of electrons in the transverse plane over the duration of the simulation. The line color shows electron energy. The markers show the electron locations at
fs. (g)–(i) Time evolution of the longitudinal position for the same three groups of electrons, with (g) showing ‘blue’ electrons, (h) showing ‘green’ electrons and (i) showing ‘red’ electrons. The line color shows electron energy.
Fig. 4. Electric and magnetic fields after reflection of the LPLG laser beam off the plasma. (a) Longitudinal profiles of the transverse electric field
(red curve) and longitudinal magnetic field
(blue line) at
fs. Here,
is plotted along the axis of the beam (
,
), whereas
is plotted at an offaxis location (
,
) where its amplitude has the highest value. (b) Frequency spectra of
(red line) and
(blue line) from panel (a).
Fig. 5. Result of the longterm electron acceleration in the reflected LPLG laser beam close to the beam axis. (a) Electron energy distribution as a function of
at
fs for electrons with
. The inset shows the third bunch that is marked with the dashed rectangle in the main plot. (b) Time evolution of the electron distribution over the divergence angle
in the third bunch (
). (c) Time evolution of the electron energy spectrum in the third bunch. The black dashed curve is the prediction obtained from
Equation (3) with
. The start time of the acceleration is used as an adjustable parameter. (d) Electron energy versus the divergence angle in the third bunch shown in the inset of panel (a).
Fig. 6. (a) Areal density
and (b) cellaveraged divergence angle
in the crosssection of the third bunch at
fs and
. (c)–(e) Snapshots of the longitudinal electric field
in the crosssection of the laser beam at
,
fs (c),
,
fs (d) and
,
fs (e). Here,
is calculated using the analytical expression
Equation (C28) given in Appendix C and
is the amplitude of
at
,
.
Parameters for linearly polarized Laguerre–Gaussian laser 

Peak power (period averaged)  0.6 PW  Radial and twist index 
$p = 0,\ l = 1$
 Wavelength 
${\lambda}_0 = 0.8\ \unicode{x3bc} \mathrm{m}$
 Pulse duration (
${\sin}^2$
electric field) 
${\tau}_{\textrm{g}} = 20$
fs  Focal spot size (
$1/\mathrm{e}$
electric field) 
${w}_0=3\ \unicode{x3bc} \mathrm{m}$
 Location of the focal plane 
$x = 0\ \unicode{x3bc}$
m  Laser propagation direction 
$x$
 Polarization direction 
$y$
 Other simulation parameters  Position of the foil and the preplasma 
$1.0\kern0.24em \mathrm{to}0.3\ \unicode{x3bc} \mathrm{m}$
and
$0.3$
to 0.0
$\unicode{x3bc} \mathrm{m}$
 Density distribution of preplasma 
${n}_{\textrm{e}} = 180.0{n}_{\textrm{c}}\exp \left[20\left(x+0.3\ \unicode{x3bc} \mathrm{m}\right)/{\lambda}_0\right]$
 Electron and ion (C
${}^{6+}$
) density in foil 
${n}_{\textrm{e}} = 180.0{n}_{\textrm{c}}$
and
${n}_{\textrm{i}} = 30.0{n}_{\textrm{c}}$
 Gradient length 
$L = {\lambda}_0$
/20  Simulation box (
$x\times y\times z$
)  10
$\unicode{x3bc} \mathrm{m}$
$\times$
20
$\unicode{x3bc} \mathrm{m}\times 20\ \unicode{x3bc} \mathrm{m}$
 Cell number (
$x\times y\times z$
)  800 cells
$\times$
1600 cells
$\times$
1600 cells  Macroparticles per cell for electrons  100 at
$r<$
2.5
$\unicode{x3bc} \mathrm{m}$
, 18 at
$r\ge 2.5\ \unicode{x3bc} \mathrm{m}$
 Macroparticles per cell for C
${}^{6+}$
 12  Order of electromagnetic field solver  4 

Table 1. 3D PIC simulation parameters. Here,
m
is the critical density corresponding to the laser wavelength
. The initial temperatures for electrons and ions are set to zero.
Sim. No.  Cell size  Cell number  Macroparticles per cell  Order of electromagnetic 

  (window size is the same)  e (
$r<2.5\;\unicode{x3bc} \mathrm{m}$
), e (
$r>2.5\;\unicode{x3bc} \mathrm{m}$
), C
${}^{6+}$
 field solver 

#1  1/40
$\unicode{x3bc} \mathrm{m}$

$400\times 800\times 800$
 200, 36, 24  2  #2  1/40
$\unicode{x3bc} \mathrm{m}$

$400\times 800\times 800$
 400, 72, 48  4  #3  1/50
$\unicode{x3bc} \mathrm{m}$

$500\times 1000\times 1000$
 200, 36, 24  4  #4  1/80
$\unicode{x3bc} \mathrm{m}$

$800\times 1600\times 1600$
 100, 18, 12  4 

Table 2. Parameters used for the four simulations depicted in Figure 7.
 #1  #2  #3  #4  #5 

${\varepsilon}_{\textrm{e}}$
[GeV] (
$\Delta {\varepsilon}_{\textrm{e}}/{\varepsilon}_{\textrm{e}}$
)  0.02–0.1  0.02–0.28  0.29 (10%)  0.22 (6%)  0.1 (15%) 
${\tilde{\varepsilon}}_{\textrm{rms}, yz}\ [\unicode{x3bc}$
m]  0.95  0.88  1.5  0.64  0.92 
$W$
[mJ]  0.06  1.5  2.2  1.3  0.06 
$Q$
[pC]  1.4  8  9  6.8  0.7 
$\Delta t$
[as]  300  360  270  260  540 

Table 3. Parameters of all five electron bunches at
= 261 fs.