Laser acceleration of proton bunches by petawatt chirped linearly polarized laser pulses

Detailed single- and many-particle calculations are carried out for the acceleration of protons employing linearly polarized plane-wave and tightly focused chirped laser pulses of several ten to several hundred femtosecond durations, petawatt peak powers, and relativistic peak intensities. Analytic and numerical methods of calculation are used in the single-particle cases (in vacuum), and particle-in-cell (PIC) simulations (underdense plasma) are employed in the many-particle investigations, without and with electromagnetic particle-particle interactions, respectively. The investigations cover a wide range of cases corresponding to upchirped as well as downchirped pulses.

Figure 1

A schematic illustrating the ionization of the target atoms and subsequent acceleration of the ions and electrons, as a result of interaction with a chirped pulse.

Figure 2

Shifted ($\overline{\eta }=4\sigma$) normalized electric field of the plane-wave Gaussian pulse of duration $\tau =25$ fs and waist radius $w_0=1.8{\lambda }_0$ [introduced here solely for the purpose of calculating the peak intensity from Eq. (5)]. (a) Unchirped field. (b) Chirped field with $E_0\left(\eta \right)=E_0(1+b\eta )$.

Figure 3

Single proton exit kinetic energy as a function of the dimensionless chirp parameter $b$. The proton is assumed initially at rest at the origin of coordinates and is subsequently accelerated by a laser system of peak power $P_0$ and peak intensity $I_0$, and modeled by a finite Gaussian pulse of duration $\tau$ and focused to a waist radius $w_0$. In the calculations, a shift of $\overline{\eta }=4\sigma$ and ${\eta }_f=8\sigma$, have been used. (b) Merely a zoom-in on the central part of (a).

Figure 4

Single-proton's kinetic energy evolution, as a function of $\eta /2\pi$, as it interacts with a chirped laser pulse. Initial conditions and all laser parameters are the same as in Fig. 3.

Figure 5

Normalized chirped electric field of the plane-wave Gaussian pulse of duration $\tau =25$ fs for optimal $b^{*}=0.035224$ (black) and for $b=0.03$ (red or light gray).

Figure 6

Maximum proton exit kinetic energy as a function of (a) the pulse duration $\tau$, and (b) the optimal chirp parameter $b^{*}$. Everywhere in this figure, the peak power, the waist radius at focus, and the initial conditions are the same as in Fig. 3. The continuous lines are best fits to the data points shown by solid circles.

Figure 7

Single-proton's kinetic energy evolution, as a function of $\eta /2\pi$, as it interacts with a chirped laser pulse with the zero-frequency component (black) and without it (red or light gray). The filter width (see appendix) is $\Delta =0.0002$, and the initial conditions and all laser parameters are the same as in Fig. 3.

Figure 8

Same as Fig. 3, but for a tightly focused laser pulse.

Figure 9

(a) Same as Fig. 4, but for a tightly focused laser pulse. (b) Normalized electric field component $E_x/E_0\left(\eta \right)$, where $E_0\left(\eta \right)=E_0(1+b\eta )$, sensed by the proton whose kinetic energy evolution is shown in (a) during interaction with the pulse.

Figure 10

Evolution in time of an ensemble of 3000 noninteracting protons, initially at rest and randomly distributed within a cylinder of length $L=0.2{\lambda }_0$ and radius $R=0.3{\lambda }_0$, centered at the origin of a Cartesian coordinate system and whose axis is oriented along $z$. (a)–(d) Snapshots of the projection of the evolving ensemble onto the $xz$ plane; (e)–(h) snapshots of the projection onto the $xy$ plane. Top to bottom, the pairs of snapshots are for times corresponding to values of the variable $\eta$ equal to $0\sigma$, $8\sigma$, $60\sigma$, and $100\sigma$, respectively.

Figure 11

Snapshots of the electron and proton density distributions during and after interaction with a plane-wave pulse whose parameters are the same as in Fig. 9, at times corresponding to values of $\eta$ equal to $2.5\sigma ,8\sigma ,60\sigma$, and $100\sigma$, respectively. The electron distribution is out of range and absent from (b)–(d). The initial distribution is two-dimensional (in the $xz$ plane). Note the change of scale in reading Figs. 10 and 11 together.

Figure 12

Evolution in $\eta$ of the proton kinetic energy during interaction with a 350-fs pulse represented employing the plane-wave model (black) and the tightly focused fields (red or light gray).