Double Compton scattering in a constant crossed field

Two-photon emission of an electron in an electromagnetic plane wave of vanishing frequency is calculated. The unpolarized probability is split into a two-step process, which is shown to be exactly equal to an integration over polarized subprocesses, and a one-step process, which is found to be dominant over the formation length. The assumptions of neglecting spin and simultaneous emission, commonly used in numerical simulations, are discussed in light of these results.

Figure 1

Feynman diagram for single Compton scattering.

Figure 2

Total rate of polarized single Compton scattering $R_{\gamma }$. The closed (open) markers on solid (dashed) lines refer to a photon scattered into a polarization state parallel (perpendicular) to the external field where the marker-free solid line refers to the unpolarized rate.

Figure 3

Feynman diagram for double Compton scattering. The prescription on the exchange term $\left(k\leftrightarrow k^{\prime }\right)$ also applies to swapping the polarization vectors.

Figure 4

Dynamical part of the two-step probability ${\mathcal{I}}_{\gamma \gamma }^{\left(2\right)}$ with the approximation of using unpolarized electrons ${\overline{\mathcal{I}}}_{\gamma \gamma }^{\left(2\right)}$ and the asymptotic limits ${\tilde{\mathcal{I}}}_{\gamma \gamma }^{\left(2\right)}$.

Figure 5

Absolute dynamical part of the one-step probability ${\mathcal{I}}_{\gamma \gamma }^{\left(1\right)}$ compared to equivalent two-step ${\mathcal{I}}_{\gamma \gamma }^{\left(2\right)}$ and single Compton scattering ${\mathcal{I}}_{\gamma }^{\left(1\right)}=R_{\gamma }^{\left(1\right)}$ quantities when an infrared cutoff ${\Delta }_k={\Delta }_{k^{\prime }}={10}^{-5}$ is included.

Figure 6

Few-cycle pulse with field strength equivalent to $500\text{TeV}$ focused to a focal width of $10\mu \text{m}$ with a $910\text{-nm}$ wavelength with peak $\xi =10$ and pulse duration of $5\text{fs}$ counterpropagating with $10\text{-GeV}$ seed electrons.

Figure 7

Logarithm to base 10 of the differential rate for the two-step process ${\partial }^2{R}_{\gamma \gamma }^{\left(2\right)}/d{k}_{y}d{k}_y^{\prime }$ when (a) ${\chi }_p=0.1$ and (b) ${\chi }_p=10$.

Figure 8

Logarithm to base 10 of the differential rate ${\partial }^2(-R_{\gamma \gamma }^{\left(1\right)})/d{k}_{y}d{k}_y^{\prime }$ for the one-step process for (a) ${\chi }_p=0.1$ and (b) ${\chi }_p=10$.

Figure 9

Comparison of the integrated differential rate $\int d{k}_y^{\prime }[{\partial }^2(-R_{\gamma \gamma }^{\left(1\right)})/d{k}_{y}d{k}_y^{\prime }]$ of the one-step process (solid line) compared with the differential rate for single Compton scattering $\partial R_{\gamma }/\partial k_y$ (dashed line) for (a) ${\chi }_p=0.1$ and (b) ${\chi }_p=10$.