Author Affiliations
1Department of Plasma Physics and Fusion Engineering, University of Science and Technology of China, Hefei, China2Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA3Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, Chinashow less
Fig. 1. Electric and magnetic field components of an LP-LG laser beam before it encounters the plasma. Panels (a) and (d) show

; panels (b) and (e) show

; panels (c) and (f) show

. The left-hand column ((a)–(c)) shows the field structure in the

-plane at

. The right-hand 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 log-scale, 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) Cell-averaged 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 LP-LG 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 off-axis 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 long-term electron acceleration in the reflected LP-LG 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) cell-averaged divergence angle

in the cross-section of the third bunch at

fs and

. (c)–(e) Snapshots of the longitudinal electric field

in the cross-section 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 pre-plasma |
$-1.0\kern0.24em \mathrm{to}-0.3\ \unicode{x3bc} \mathrm{m}$
and
$-0.3$
to 0.0
$\unicode{x3bc} \mathrm{m}$
| Density distribution of pre-plasma |
${n}_{\textrm{e}} = 180.0{n}_{\textrm{c}}\exp \left[-20\left(x+0.3\ \unicode{x3bc} \mathrm{m}\right)/{\lambda}_0\right]$
| 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 | Macro-particles 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.