Nonlinear single Compton scattering of an electron wave packet

Nonlinear single Compton scattering has been thoroughly investigated in the literature under the assumption that the electron initially has a definite momentum. Here, we study a more general initial state and consider the electron as a wave packet. In particular, we investigate the energy spectrum of the emitted radiation and show that, in typical experimental situations, some features of the spectra shown in previous works are almost completely washed out. Moreover, we show that, at comparable relative uncertainties, the one in the momentum of the incoming electron has a larger impact on the photon spectra at a fixed observation direction than the one on the laser frequency.

Figure 1

Representation of the frame of reference employed.

Figure 2

The function $\psi \left(\eta \right)$ for a two-cycle laser pulse ($n_C=2$) and two choices of the carrier-envelope phase ${\eta }_0$. The solid curve corresponds to ${\eta }_0=0$, while the dotted curve corresponds to ${\eta }_0=\pi /2$.

Figure 3

Energy emission spectrum along the negative $z$ direction for an incoming electron with definite initial momentum $\mathit{p}=(0,0,-4.2\mathrm{GeV})$ interacting with a laser of intensity $I\approx 4.3\times {10}^{20}\text{W}/{\text{cm}}^2$.

Figure 4

Change of emission spectrum for an electron with definite initial momentum $(0,0,p_z)$ as a function of $|p_z|$ (Fig. 3 corresponds to a section of the upper part of this figure for $p_z=-4.2\mathrm{GeV}$). In the range considered, the position of the peaks increases linearly with $p_z$, albeit with different slopes depending on the position of the peak. Some of these slopes were computed numerically and are shown in the bottom part of the plot (blue dots), together with the same quantity computed analytically for a monochromatic pulse (red continuous line).

Figure 5

Emission spectra along the negative $z$ direction for an electron wave packet with $\overline{\mathit{p}}=(0,0,-4.2\text{GeV})$ interacting with a laser pulse of peak intensity $I\approx 4.3\times {10}^{20}\text{W}/{\text{cm}}^2$ for different values of the spread of the longitudinal momentum.

Figure 6

Shift of the emission frequencies ${\omega }_n^{\prime }$ along the negative $z$ direction for different values of $n$ as a function of $p_T$ (vertical axis). The numerical parameters are $p_z=-4.2\mathrm{GeV}$ and $I\approx 1.1\times {10}^{20}\text{W}/{\text{cm}}^2$.

Figure 7

Emission spectra in the negative $z$ direction for electrons having initially $p_z=-4.2\mathrm{GeV}$ and either $p_y=0$ or $p_x=0$, after the interaction with a short laser pulse with $I\approx 1.1\times {10}^{20}\text{W}/{\text{cm}}^2$.

Figure 8

Energy emission spectrum in the negative $z$ direction, for some different initial electron states. Here, $\overline{\mathit{p}}=(0,0,-4.2\mathrm{GeV}), {\sigma }_{p_T}=3\times {10}^{-4}\left|\overline{\mathit{p}}\right|$, and ${\sigma }_{p_z}=6\times {10}^{-2}\left|\overline{\mathit{p}}\right|$. The intensity of the laser field is $I\approx 1.1\times {10}^{20}\text{W}/{\text{cm}}^2$.

Figure 9

Energy emission spectrum on a direction in the $xz$ plane forming an angle $\theta =m\xi /2\overline{\varepsilon }$ with the negative $z$ axis, for some different initial electron states. The numerical parameters are the same as in Fig. 8.

Figure 10

Energy emission spectrum for an electron wave packet in the quantum regime ($\chi \approx 0.85$), in the direction that lies on the laser polarization plane and forms an angle $m\xi /(2\overline{\varepsilon )}$ with the negative $z$ axis. The numerical parameters are the same as in Fig. 8, except that $I\approx 1.2\times {10}^{21}\text{W}/{\text{cm}}^2$.

Figure 11

Distribution of the total emitted energy by an electron in a Volkov state or in a Gaussian wave packet as a function of the frequency of the emitted photon. All the numerical parameters for this figure are the same as in Fig. 10.

Figure 12

Angular distribution of the total energy emitted by an electron in a Volkov state (upper panel) or in a Gaussian superposition of them (lower panel) after interacting with a strong laser field. The numerical parameters used here are the same as in Fig. 10.