Interaction of ultrashort laser pulses with metal surfaces: Impulsive jellium-Volkov approximation versus the solution of the time-dependent Schrödinger equation

Electron emission coming from the valence band of metal surfaces due to grazing incidence of high-frequency ultrashort laser pulses is studied. We introduce a distorted-wave method, named impulsive jellium-Volkov (IJV) approximation, in which the surface is represented by the jellium model while the interaction with the laser field is described by means of the Volkov phase. With the purpose of examining the proposed approach, we compare IJV results with values derived from the numerical solution of the corresponding time-dependent Schrödinger equation (TDSE). For Al(111) surfaces, double and single differential probability spectra are calculated considering different durations of the laser pulse. Very good agreement between IJV and TDSE results was found. The total probability dependence on the intensity and carrier-envelope phase of the pulse is also investigated.

Figure 1

Differential electron emission probability as a function of the electron energy. The parameters of the laser field are $F_0=0.1\mathrm{a.u.}$, $\omega =1\mathrm{a.u.}$, and $\tau =\left(\mathrm{a}\right)$ 4 and (b) $40\mathrm{a.u.}$ Lines, IJV results for different ejection angles ${\theta }_e$; full dots, TDSE data for ${\theta }_e=90°$.

Figure 2

Similar to Fig. 1 for the emission angle ${\theta }_e=90°$. Laser field with $F_0=0.05\mathrm{a.u.}$, $\tau =40\mathrm{a.u.}$, and frequency $\omega =\left(\mathrm{a}\right)$ 1 and (b) $0.5\mathrm{a.u.}$

Figure 3

Electron momentum distribution derived from IJV model for a laser pulse with $F_0=0.1\mathrm{a.u.}$, $\omega =1\mathrm{a.u.}$, and $\tau =40\mathrm{a.u.}$

Figure 4

Energy distribution of emitted electrons, as a function of the electron energy, for a laser field with $F_0=0.1\mathrm{a.u.}$ and $\omega =1\mathrm{a.u.}$ The duration of the pulse is $\tau =\left(\mathrm{a}\right)$ 4 and (b) $40\mathrm{a.u.}$ Solid line, IJV approach; full dots, TDSE results.

Figure 5

Total emission probability as a function of the maximum field strength of the laser pulse. Three different sets of laser parameters are shown. Solid line, IJV model; full dots, TDSE results, with lines as a guide to the eyes.

Figure 6

Results obtained within the IJV model for (a) the probability $P$ and (b) the relative probability ratio $R=(P-P_0)∕P_0$, as a function of the carrier-envelope phase $\phi$, with $P$ $\left(P_0\right)$ the total photoemission probability for a laser pulse with $\phi$ $({\phi }_0=-\omega \tau ∕2+\pi ∕2)$. The laser frequency is $\omega =1\mathrm{a.u.}$ Lines as defined in the figure.