Scaling laws for positron production in laser–electron-beam collisions

Showers of $\gamma$ rays and positrons are produced when a high-energy electron beam collides with a superintense laser pulse. We present scaling laws for the electron-beam energy loss, the $\gamma$-ray spectrum, and the positron yield and energy that are valid in the nonlinear, radiation-reaction-dominated regime. As an application we demonstrate that by employing the collision of a $>\mathrm{GeV}$ electron beam with a laser pulse of intensity $>5\times {10}^{21}\mathrm{W}{\mathrm{cm}}^{-2}$, today's high-intensity laser facilities are capable of producing $O\left({10}^4\right)$ positrons per shot via light-by-light scattering.

Figure 1

Ultrarelativistic electrons (blue) collide with a counterpropagating laser pulse (magenta, green) and lose energy by emitting photons (yellow). Positrons (red) are created when photons undergo the nonlinear Breit-Wheeler process.

Figure 2

Number of positrons, $N_{+}$, produced in a collision between a beam of $N_{\gamma }$ photons with energy $\omega$ and a linearly polarized laser pulse that has peak intensity $I_{22}\times {10}^{22}\mathrm{W}{\mathrm{cm}}^{-2}$, wavelength $\lambda$, and FWHM $\tau$. Results from our scaling law in Eq. (4) (black) are compared with numerical integration of the full pair-creation rate (dashed yellow).

Figure 3

${\varphi }_c$, the phase at which $\chi$ is maximized, as given by Eqs. (7) and (8) for electrons colliding with laser pulses that have FWHM 30 fs, wavelength 800 nm (1.55 eV), and peak intensity (a) $1\times {10}^{23}\mathrm{W}{\mathrm{cm}}^{-2}$ (yellow) and (b) $1\times {10}^{21}\mathrm{W}{\mathrm{cm}}^{-2}$ (blue). Solid lines are calculated including $g\left(\chi \right)$; dashed lines have been calculated in the classical limit $g\left(\chi \right)=1$.

Figure 4

Top: Color scale, the probability density $P_{\chi ,\varphi }$ that a stochastically radiating electron reaches a maximum quantum parameter $\chi$ at phase $\varphi$; solid blue curve, the $\chi$ of an electron that loses no energy; dashed blue curve, the $\chi$ of an electron that loses energy according to Eq. (6); and circle, the ${\chi }_c$ and ${\varphi }_c$ given by Eqs. (7) and (8). Observe that the region of maximum emission probability is correctly identified by the predicted ${\varphi }_c$. Bottom: Color scale, the probability density $P_{\omega ,\varphi }$ that a photon is emitted with energy $\omega$ at phase $\varphi$ and (vertical line) ${\varphi }_c$. (See text for collision parameters.)

Figure 5

For a collision between an electron beam with energy $E_0$ and a linearly polarized laser pulse with peak intensity $I_{21}\times {10}^{21}\mathrm{W}{\mathrm{cm}}^{-2}$, wavelength $\lambda$, and FWHM $\tau$, left, the electron quantum nonlinearity parameter $\chi$ as a function of phase $\varphi$, as predicted by Eq. (17) (dashed yellow curves), solution to equation of motion [Eq. (6)], and in the absence of radiation reaction (dotted red curves); right, energy spectra (normalized per electron) of the emitted photons, as predicted by our scaling [Eq. (20), solid black curves] and Monte Carlo simulation with stochastic radiation reaction (dashed yellow curves) and no radiation reaction (dotted red curves).

Figure 6

Number of positrons, $N_{+}$, produced in a collision between a beam of $N_e$ electrons with energy $E_0$ and a linearly polarized laser pulse that has peak intensity $I_{21}\times {10}^{21}\mathrm{W}{\mathrm{cm}}^{-2}$, wavelength $\lambda$, and FWHM $\tau$. Results from our scaling law [Eq. (24), solid black curve] and simulations using the full pair-creation rate (dashed yellow curve).

Figure 7

Color scale, energy spectra of positrons emerging from a collision between an electron beam with energy $E_0$ and a laser pulse with peak intensity $I_{21}\times {10}^{21}\mathrm{W}{\mathrm{cm}}^{-2}$, wavelength $\lambda$, and FWHM $\tau$; black lines, the characteristic energy predicted by Eqs. (23) and (25); and black circles, the mean energy of the simulated spectra.

Figure 8

Left: ${log}_{10}$-scaled yield (per electron). Right: Typical energy in MeV of positrons produced in the collision of an electron beam with energy $E_0$ and a laser pulse with peak intensity $I_0$, wavelength 1 $\mu \mathrm{m}$, and FWHM 30 fs, as predicted by Eq. (24).

Figure 9

The energy spectra of photons (solid blue curve), positrons (dashed yellow curve), and beam electrons (dotted green curve) emerging from a collision between an electron beam with energy 2 GeV and a laser pulse with peak intensity $5\times {10}^{21}\mathrm{W}{\mathrm{cm}}^{-2}$ (see Sec. 4c for details). The positron energy predicted by Eqs. (25) and (23) is indicated by a yellow arrow.