Compton process in intense short laser pulses

The spectra of Compton radiation emitted during electron scattering off an intense laser beam are calculated using the framework of strong-field quantum electrodynamics. We model these intense laser beams as finite length plane-wave-fronted pulses, similar to Neville and Rohrlich [Phys. Rev. D 3, 1692 (1971)], or as trains of such pulses. Expressions for energy and angular distributions of Compton photons are derived such that a comparison of both situations becomes meaningful. Comparing frequency distributions for both an isolated laser pulse and a laser pulse train, we find a very good agreement between the results for long pulse durations which breaks down, however, for ultrashort laser pulses. The dependence of angular distributions of emitted radiation on a pulse duration is also investigated. Pronounced asymmetries of angular distributions are found for very short laser pulses, which gradually disappear with increasing the number of laser field oscillations. Those asymmetries are attributed to asymmetries of the vector potential describing an incident laser beam.

Figure 1

Laser field shape functions [defined by Eqs. (55) and (56)] normalized according to Eq. (57). Each curve relates to a different number of laser field oscillations, namely, $N_{\mathrm{osc}}=2$ (solid blue line), $N_{\mathrm{osc}}=3$ (dashed red line), and $N_{\mathrm{osc}}=4$ (dash-dotted black line).

Figure 2

Energy spectra, Eqs. (34) and (54), for Compton photons emitted in the direction ${\mathit{n}}_{\mathit{K}}$ determined by the polar and azimuth angles ${\theta }_{\mathit{K}}=0.2$ and ${\varphi }_{\mathit{K}}=0$ (in radians). The initial electron is at rest and the laser photon frequency is such that $\hslash {\omega }_{\mathrm{L}}=3\times {10}^{-6}{m}_{\mathrm{e}}{c}^2$. The intensity of the laser beam is determined by $\mu =1$. The solid (blue) line corresponds to scattering by a single laser pulse, Eq. (54), the dashed (green) line to scattering by a train of pulses, Eq. (34), whereas the red bullets indicate the Compton photon frequencies, ${\omega }_{\mathit{K}}=c{K}^0$ in Eq. (26), for integer $N$. The black pentagrams represent the energy spectrum (34) for a plane wave, when the shape function in Eq. (55) is proportional to $\sin (k_{\mathrm{L}}·x)$. In all cases $N_{\mathrm{osc}}=16$. The lower panel shows the enlarged part of the upper one in order to prove a very good agreement between the results obtained for a single pulse and a train of such pulses.

Figure 3

The same as in Fig. 2 but for larger frequencies (upper panel). In the upper panel, we clearly see the “blue-shift” of the spectrum for Compton photons scattered by a train of pulses, as compared to those scattered by an individual pulse. This shift is absent, however, when the spectra are shown as functions of $N_{\mathrm{eff}}$ or $N$ (lower panel).

Figure 4

The same as in Fig. 2 but for very short pulses with $N_{\mathrm{osc}}=2$ (upper panel) and $N_{\mathrm{osc}}=1$ (lower panel). Let us note that pentagrams, representing the results for the monochromatic plane wave, in both these panels and in Fig. 2 provide the same results up to multiplication by the number of oscillations $N_{\mathrm{osc}}$.

Figure 5

Angular distributions of the Compton radiation energy (green lines), Eq. (54), as functions of the angle ${\Theta }_{\mathit{K}}$ in the plane spanned by the laser field propagation direction and the polarization vector (i.e., in the $xz$ plane) for an ultrashort ($N_{\mathrm{osc}}=2$, upper panel) and a relatively long ($N_{\mathrm{osc}}=8$, lower panel) laser pulses. The laser field central frequency ${\omega }_{\mathrm{L}}$ is such that $\hslash {\omega }_{\mathrm{L}}=3\times {10}^{-4}{m}_{\mathrm{e}}{c}^2$ whereas $\mu =1$. The initial electron is at rest and the Compton photon energy equals $\hslash {\omega }_{\mathit{K}}={10}^{-2}{m}_{\mathrm{e}}{c}^2$. The smooth blue lines represent the averaged distributions, as described by Eq. (60).

Figure 6

Polar plots of the (normalized to 1 and window-averaged) Compton photon energy distribution, Eq. (60), as functions of ${\Theta }_{\mathit{K}}$ for different-cycle laser pulses with $N_{\mathrm{osc}}=2,4,8$, and 16, as indicated under the plots. The other parameters are exactly the same as in Fig. 5.

Figure 7

Polar plots of the normalized to one Compton radiation energy distribution (54) ($\hslash {\omega }_{\mathit{K}}=100m_{\mathrm{e}}{c}^2)$ as a function of ${\Theta }_{\mathit{K}}$ for two- and 32-cycle laser pulses (left and right panels, respectively). Here, $\hslash {\omega }_{\mathrm{L}}=m_{\mathrm{e}}{c}^2$ (in the electron rest frame) and $\mu =10$. For visual purposes, the rates have been raised to power 0.01.