Extended particle-in-cell schemes for physics in ultrastrong laser fields: Review and developments

We review common extensions of particle-in-cell (PIC) schemes which account for strong field phenomena in laser-plasma interactions. After describing the physical processes of interest and their numerical implementation, we provide solutions for several associated methodological and algorithmic problems. We propose a modified event generator that precisely models the entire spectrum of incoherent particle emission without any low-energy cutoff, and which imposes close to the weakest possible demands on the numerical time step. Based on this, we also develop an adaptive event generator that subdivides the time step for locally resolving QED events, allowing for efficient simulation of cascades. Further, we present a unified technical interface for including the processes of interest in different PIC implementations. Two PIC codes which support this interface, picador and elmis, are also briefly reviewed.

Figure 1

Schematic representation of the typical EM-energy deposition.

Figure 2

Spectra of coherent and incoherent emission obtained with PIC simulations for the process of giant attosecond pulse generation in the regime of relativistic electronic spring (see [17] for details).

Figure 3

Feynman diagrams for the two basic processes included in the PIC approach. Time flows from left to right. A straight line indicates an electron or positron interacting with an external field (e.g., an intense laser), and a wavy line is a photon. Left: nonlinear Compton scattering. Right: stimulated pair production.

Figure 4

Feynman diagrams for the two basic processes neglected in the PIC approach: pair annihilation to one photon (left) and single photon absorption (right). Time flows from left to right.

Figure 5

Pair production from a seed electron. The intermediate photon may be real or virtual.

Figure 6

A (greatly simplified) illustration of cascade formation from a seed electron in a PIC simulation. At each time step, fermions can emit photons (or not) and photons can produce pairs (or not). All particles are real.

Figure 7

Extension of the PIC approach for taking into account novel channels of energy transformation that could be triggered by laser fields of extreme intensity.

Figure 8

Regions of qualitatively different effect of electron individual emission on its motion. Fundamental values of Schwinger limit, Compton wavelength, and intensity threshold of relativistic motion of electrons are shown with solid lines. Some experimental capabilities are shown with red (ultraintense lasers) and green (laser pulse collision with external electron beam, SLAC experiment E144 [61]) lines.

Figure 9

The emission spectrum for $\gamma =100, \chi =1$, obtained from the exact expression (21) (gray) and from running various event generators: the traditional event generator with a cutoff (green), the alternative event generator (blue and purple), and our modified event generator (red). Clearly, the modified event generator gives excellent agreement with the analytic result. The absence of a low-energy cutoff allows us to model the entire range of spectral emission.

Figure 10

Cascade development: comparison of the number of particles produced in a cascade, under the conditions of the first test problem described in the text and in [7, 76]. The plot shows the number of produced particles with energy exceeding 0.1% of the initial electron's energy, as a function of time. We show the results of [76] (gray heavy curve), [7] (dark gray curve), and of our implementation for the adaptive event generator (AEG) with small (thin red curve) and large (red squares) time steps. As one can see, for this benchmark we find excellent agreement of our results with the published ones.

Figure 11

Distribution functions: particle distribution functions; comparison of our AEG implementation (thin red curves) with the results of [9] (gray heavy curves). Excellent agreement is clearly seen for this benchmark.

Figure 12

Avalanche-type cascade: number of particles produced in an avalanche-type cascade, as a function of time. The avalanche is initiated by an electron ($x=\lambda /8$ at $t=0$) with $p_y=1000mc$ in a plane standing wave defined in the form $E_y(t=0)={10}^{-2}{E}_{S}sin(2\pi x/\lambda )$ ($\lambda =1\mu \mathrm{m}$). The results are obtained from an AEG implementation with $dt=0.17t_{\mathrm{rad}}$ (thin red curve) and $dt=173t_{\mathrm{rad}}$ (red squares). The agreement between these results indicates that the time step and particles and photons subdivision procedures allow much more efficient use of computational resources without affecting the physical results.

Figure 13

Description of the PIC data access structure of the PIC-MDK interface. FP is a type for floating point operations (can be either float or double). FP3 and Int3 are types for arrays of three variables.