Capturing Photoelectron Motion with Guiding Centers

Electron motion in combined strong laser and Coulomb fields is central to laser-matter interactions. By mapping this problem onto the motion of a guiding center, we derive a reduced model which naturally embeds important Coulomb effects such as focusing and asymmetry, and clearly distinguishes direct versus rescattered electron ionization processes. We demonstrate the power of this tool by unraveling the bifurcation in photoelectron momentum distributions seen in experiments.

Figure 1

(a) ATI spectrum as a function of ellipticity, computed using CTMC from Hamiltonian (1). The color scale indicates the probability distribution of photoelectron energies. The gray curve is the prediction of our model. (b) The $T$ trajectory (see text) final momentum $\mathbf{P}=P_x\widehat{\mathbf{x}}+P_y\widehat{\mathbf{y}}$ as a function of the laser ellipticity. The solid colored curves and circles are computed using our model and Hamiltonian (1), respectively. The shaded area is where the $T$ trajectory is rescattered in both our model and Hamiltonian (1). The solid and dashed black curves are computed using the SFA and the perturbed SFA, respectively (the lower curves correspond to $P_x$ and the upper curves to $P_y$). (c),(d) The $T$ trajectory for $\xi =0.25$ (c) and $\xi =0.5$ (d) shown for $f=1$. The blue and cyan solid curves are the $T$ trajectory of Hamiltonian (1) and our model, respectively. The dashed cyan curve is the guiding-center trajectory of the $T$ trajectory. The black crosses show the initial position of the $T$ trajectory. Distances, momenta, and energies are scaled by $E_0/{\omega }^2$, $E_0/\omega$, and $U_p$, respectively.

Figure 2

Polar plots of the PADs for $\xi =0.25$ (left panel) and $\xi =0.5$ (right panel) computed using CTMC. The computation from our model (cyan) is in agreement with the one from Hamiltonian (1) (blue). Also shown are computations using the SFA (solid black line) and the perturbed SFA (dashed black line).

Figure 3

Scattering angle of the electron of Hamiltonian (1) as a function of the initial conditions $(t_0,p_{\perp })$, for $\xi =0.25$ (upper panel) and $\xi =0.5$ (lower panel), for a field envelope $f$ with an eight laser-cycle plateau and a two laser-cycle ramp down. The final electron energy is negative in gray areas. The black lines show where the guiding-center energy is $\mathcal{E}=0$ according to our model (3). The crosses show the initial conditions of the $T$ trajectory. The dashed lines are the level curves of the ADK and DK ionization rate normalized by the maximum. The momentum $p_{\perp }$ is in atomic units.