Nonconstant ponderomotive energy in above-threshold ionization by intense short laser pulses

We analyze the contribution of the quiver kinetic energy acquired by an electron in an oscillating electric field of a short laser pulse to the energy balance in atomic ionization processes. Due to the time dependence of this additional kinetic energy, a temporal average is assumed to preserve a stationary energy conservation rule, which is used to predict the position of the energy peaks observed in the photoelectron (PE) spectra. For a plane wave and a flattop pulse, the mean value of the quiver energy over the whole pulse leads to the concept of ponderomotive energy $U_p$. However, for a short pulse with a fast changing intensity, the stationary approximation loses its validity. We check these concepts by studying first the PE spectrum within the semiclassical model (SCM) for multiple-step pulses. The SCM offers the possibility to establish a connection between emission times and the PE spectrum in the energy domain. We show that PE substructures stem from ionization at different times mapping the pulse envelope. We also analyze the PE spectrum for a realistic sine-squared envelope within the Coulomb-Volkov and ab initio calculations solving the time-dependent Schrödinger equation. We found that the electron emission amplitudes produced at different times interfere with each other producing, in this way, a new additional pattern that modulates the above-threshold ionization (ATI) peaks.

Figure 1

Vector potential (left panels) and quiver kinetic energy (right panels) as a function of time (in units of the number of cycles $m=\omega t/2\pi$, for three envelopes: flattop, step, and sine-squared envelope (top, middle, and lower panels, respectively). The laser pulse duration is equivalent to eight cycles. Here $F_1=F_0/\sqrt{2}, F_2=F_0$, and $t_1=t_2=\tau /2$. Left column: Vector potential amplitude $A\left(t\right)$ in thin (blue) line and envelope of the laser field thick (red) line in units of $F_0/\omega$. Right column: Quiver kinetic energy $A^2\left(t\right)/2$ in thin (green) line and cycle-averaged quiver kinetic energy $U_p\left(m\right)$ in thick (blue) line in units of ${(F_0/2\omega )}^2$.

Figure 2

H($1s$) ATI PE spectra obtained within the SCM by laser pulses with envelope given by Eq. (16b) with $\omega =0.25$ a.u., $N_1=N_2=8$ cycles, and $F_1=0.2$ a.u. Left column (red lines): two identical pulses case with $F_1=F_2$. Right column (blue lines): $F_2=0.225$ a.u. Vertical dashed lines indicate the position of the ATI peaks following Eq. (1) with $U_{p1}$ and $U_{p2}$ values. The upper labels $n=3$ or 4 indicate the number of absorbed photons. In (a),(b), (c),(d), and (e),(f) cases the time delay between the two parts of the pulse are $0,4$ and 8 cycles, respectively. In (b) PE spectrum for a flattop with eight-cycle pulse with thin (green) line. In (d) CVA PE spectrum with thin (black) line.

Figure 3

CVA H($1s$) ATI PE spectra with $F_0=\omega =0.25$ a.u. and envelope shown in the inset. (a) PE spectrum for five-level envelope. (b) PE spectra for one step of amplitude $F_0$ with (red) line with dots, for two steps of amplitude $0.9F_0$ delayed eight cycles with (blue) thick line, and for two steps of amplitude $0.8F_0$ delayed 24 cycles with (green) filled line.

Figure 4

H($1s$) CVA ATI PE spectra for ${sin}^2$ envelope with $F_0=0.1$ a.u., $\omega =0.15$ a.u., and (a) $N=24$, (b) $N=48$ cycles. In the insets we show the quiver (in thin green line) and the cycle-dependent ponderomotive energy (in thick blue line) as functions of time.

Figure 5

H($1s$) TDSE (black thick line), CVA (blue thin line), and SFA (red dashed line) PE spectra for sinusoidal envelope with $F_0=0.07$ a.u., $\omega =0.114$ a.u., and $N=16$ cycles.