Laser-driven electron acceleration in nanoplate array targets

This paper proposes a model of the laser-driven electron acceleration that occurs when a high-intensity laser interacts with a nanoplate target. It shows that quasistatic electric ${\mathbf{E}}_{\mathrm{qs}}$ and magnetic ${\mathbf{B}}_{\mathrm{qs}}$ fields can be formed when the laser, polarized normal to the nanoplates, extracts electrons from the nanoplates. Considering the physical natures of ${\mathbf{E}}_{\mathrm{qs}}$ and ${\mathbf{B}}_{\mathrm{qs}}$, the amplitude of ${\mathbf{E}}_{\mathrm{qs}}$ is relatively larger than ${\mathbf{B}}_{\mathrm{qs}}$. Such a residual between static electric and magnetic field is shown to be crucial for the electron acceleration beyond the ponderomotive scaling, as it can cause onset of stochastic electron motion. The analysis demonstrates that the maximum electron energy in units of ponderomotive scaling depends on a single universal parameter, which is composed of laser amplitude, spacing between nanoplates, and electron initial conditions. The analytical results are confirmed by a series of two-dimensional particle-in-cell simulations using epoch code.

Figure 1

Schematic view of the simulation setup. The black region corresponds to where the target is located in the domain. The nanoplates are separated by vacuum gaps of width $D$. Each nanoplate has the width $d$ along the $y$-axis and length $L$ along the $z$-axis. Nanoplates are connected on the right side at $z=L$ with the bulk of width $\lambda$. Top and bottom boundaries of the simulation domain are periodic, left and right boundaries are open. The laser pulse arrives from the left boundary. The electric field of the laser at time $t=0$ is shown by the red curve at $z$ between $-19\lambda$ and 0.

Figure 2

Unperturbed trajectories of energetic electrons (17) for $\widehat{H}=50, W_{\max }^{(-)}=1$, and various values of $C$. Red lines that cross at $\widehat{D}/2$ show the trajectory with $C=1/2\widehat{H}+W_{\max }^{(-)}=1.01$, separating bounded and unbounded trajectories. Dots show the stroboscopic portrait of the motion.

Figure 3

Frequency $\widehat{\mathrm{\Omega }}=(d\widehat{H}/d\mathcal{I})$ of unperturbed electron motion (17) for $\widehat{H}=50, W_{\max }^{(-)}=2, \widehat{D}=8\pi$, and various æ values. The bifurcation at æ=0 corresponds to the separatrix, where $\widehat{\mathrm{\Omega }}=0.$

Figure 4

Poincaré cross section for the electron with $C=W_{\max }^{(-)}=2, W_{\max }^{(+)}=5, \widehat{y}\left(0\right)=0.4\widehat{D}, {\widehat{P}}_y\left(0\right)=3.0, \widehat{D}=500$. Parameters are chosen so that $\zeta =1000$, and assumptions of analysis are satisfied. The motion is stochastic; however, a stability island exists for energies above 600. Red line at $\widehat{H}\sim 100$ shows initial energy of the electron, and blue line at $\widehat{H}=7$ shows $W_{\max }^{(+)}+C$.

Figure 5

Scaling of maximum stochastic energy for cases $C\sim W_{\max }^{(-)}$ [blue line corresponding to $\max \left({\widehat{H}}_2\right)$] and $C\ll W_{\max }^{(-)}$ [red line corresponding to $\max \left({\widehat{H}}_3\right)$], for various values of $\zeta , C$, and $W_{\max }^{(-)}$. Dashed lines show scalings (26) and (32) for reference.

Figure 6

The $x$-component of the electric field ${\widehat{E}}_{\mathrm{x}}$ inside the gap of size $\widehat{D}=4\pi$, obtained from PIC simulation (top) with an $x$-polarized laser and analytical model from Sec. 2a (bottom).

Figure 7

The $z$-component of the magnetic field ${\widehat{B}}_{\mathrm{z}}$ inside the gap of size $\widehat{D}=4\pi$, obtained from the PIC simulation (top) with the $x$-polarized laser and analytical model from Sec. 2a (bottom).

Figure 8

Electron number density, overlaid with the components of electric $\widehat{E_{\mathrm{x}}}$ and magnetic ${\widehat{B}}_{\mathrm{z}}$ fields for the case of an $x$-polarized pulse.

Figure 9

Electron number density, overlaid with the components of electric $\widehat{E_{\mathrm{y}}}$ and magnetic ${\widehat{B}}_{\mathrm{x}}$ fields for the case of the $y$-polarized pulse.

Figure 10

Profiles of the electrostatic potential ${\widehat{U}}^{\mathrm{PIC}}$ (red solid lines) and vector potential ${\widehat{A}}_{\mathrm{B}}^{\mathrm{PIC}}{\mathbf{e}}_{\mathrm{z}}$ (blue dotted lines). Each ${\widehat{U}}^{\mathrm{PIC}}$ and ${\widehat{A}}_{\mathrm{B}}^{\mathrm{PIC}}{\mathbf{e}}_{\mathrm{z}}$ is shown from $\widehat{y}=0$ to $\widehat{y}=\widehat{D}$, where $\widehat{D}$ is the gap size from corresponding PIC simulation. All PIC simulations shown had the same laser with $a_0=2.4$.

Figure 11

Maximum values of scalar $max\left({\widehat{U}}^{\mathrm{PIC}}\right)$ and vector $max\left({\widehat{A}}_B^{\mathrm{PIC}}\right)$ agree with the estimate $max\left({\widehat{A}}_{\mathrm{B}}^{\mathrm{PIC}}\right)=\langle v_{\mathrm{z}}\rangle max\left({\widehat{U}}^{\mathrm{PIC}}\right)$, where $\langle v_{\mathrm{z}}\rangle$ is the average $v_z$ velocity component of the electrons in the gap.

Figure 12

Top panel shows scalar potential ${\widehat{U}}^{\mathrm{PIC}}$ inside the gap from a PIC simulation with $a_0=2.4, \widehat{D}=8\pi$, at time $\widehat{t}=48\pi$. Bottom panel shows positions of the electrons from the random subset (light gray dots) sampled in the same PIC simulation at time $\widehat{t}=48\pi$. Extracted electrons fill the region of the gap with $\widehat{z}$ from $12\pi$ to $32\pi$; the same region has the ${\widehat{U}}^{\mathrm{PIC}}$ and ${\widehat{A}}_{\mathrm{B}}^{\mathrm{PIC}}$ independent of $\widehat{z}$. Green line marked (a) shows the extraction angle estimate (6). In the bottom panel, red points (dark gray if grayscale) at $\widehat{D}=0$ and $8\pi$ show the initial position $\vec{\widehat{r}}(\widehat{t}=0)$ of the electrons that are located between $\widehat{D}=0$ and $4\pi$ at time $\widehat{t}=48\pi$.

Figure 13

Distribution of an electron's Lorentz factor $\gamma$ normalized by the total number of particles in the target ($\sim {10}^{10}$ taking the size along $x$ as $\mathrm{\Delta }x\sim \lambda$ and size along $y$ as $\mathrm{\Delta }y\sim d+D$) for PIC simulations with $a_0=2.4$ and different $\widehat{D}$. The corresponding values of $\widehat{D}$ are $2\pi$ (green), $4\pi$ (black), $6\pi$ (red), $8\pi$ (blue). Data shown for the $x$-polarized $\widehat{D}=8\pi$ verify that $x$-polarized pulse heats up electrons less efficiently, in agreement with Sec. 2a.

Figure 14

The frequency of unperturbed electron motion $\widehat{\mathrm{\Omega }}$ for scalar $\widehat{U}$ and vector ${\widehat{A}}_{\mathrm{B}}{\mathbf{e}}_{\mathrm{z}}$ potentials measured in PIC simulations with $a_0=2.4$ and $\widehat{D}=2\pi$ (green), $4\pi$ (black), $6\pi$ (red), $8\pi$ (blue). Dotted lines show $C\approx W_{\max }^{(-)}$, where $\widehat{\mathrm{\Omega }}$ drops to zero.

Figure 15

The maximum energy (dashed curves) and energy below stability arms (solid curves) from single-particle simulations with initial $\widehat{H}\sim 1$ and for scalar $\widehat{U}$ and vector ${\widehat{A}}_{\mathrm{B}}{\mathbf{e}}_{\mathrm{z}}$ potentials from PIC simulations with $a_0=2.4$ and $\widehat{D}=2\pi$ (green), $4\pi$ (black), $6\pi$ (red), $8\pi$ (blue).

Figure 16

The energy scaling in the PIC simulations with $W_{\max }^{(-)}\sim 1$. Data are labeled with blue numbers, corresponding to the values of $a_0$ in PIC simulations. The fitted slope is 0.98. The black star that does not fit the scaling corresponds to the simulation with $a_0=2.4$ and $\widehat{D}=12\pi$, where $\widehat{L}=20$ turned out to be insufficient for the electrostatic potentials to form.