Classical Effects of Laser Pulse Duration on Strong-Field Double Ionization

We use classical electron ensembles and the aligned-electron approximation to examine the effect of laser pulse duration on the dynamics of strong-field double ionization. We cover the range of intensities ${10}^{14}–{10}^{16}\mathrm{W}/{\mathrm{cm}}^2$ for the laser wavelength 780 nm. The classical scenario suggests that the highest rate of recollision occurs early in the pulse and promotes double-ionization production in few-cycle pulses. In addition, the purely classical ensemble calculation predicts an exponentially decreasing recollision rate with each subsequent half cycle. We confirm the exponential behavior by trajectory back analysis.

Figure 1

Plots of electron energy versus time measured in laser cycles, showing a pair of two-electron trajectories, with the trapezoidal laser field envelope $f(t)$ superposed for reference. During laser turn-on the bound electrons exchange energy rapidly, and soon after laser turn-on is complete one (lighter red line) escapes and then returns in repeated $e\mathrm{\text{−}}e$ recollisions until the inner electron (darker blue line) also has enough energy to escape the nucleus, and thus the pair become doubly ionized. Notice that after each recollision, the energy of the inner electron is modified, and the outer electron carries different energies in each recollision.

Figure 2

Variation of “knee” structure in laser pulses with different numbers of plateau cycles ($N$). The probability of double ionization is the number of double-ionization events divided by 100 000, the total number of $2\mathrm{\text{−}}e$ trajectories used.

Figure 3

Number of recolliding electrons vs time, counted at 8 times in each half cycle, for 16 different intensities of 8-cycle pulses, from $4\times {10}^{13}$ to ${10}^{15}\mathrm{W}/{\mathrm{cm}}^2$. The pulses with intensities in this range, which includes the main knee region, remove electron pairs smoothly and steadily, leading to the almost exactly exponential decrease of peak numbers through the duration of the pulse.

Figure 4

(a)–(d) Total number of recolliding electron trajectories in the overall ensemble counted at eight half cycles of an 8-cycle trapezoidal pulse with a 2-cycle on-ramp and 2-cycle off-ramp. Data is shown for four laser intensities in the NSDI knee region, and intermediate intensities produce results in the same range. The solid line in each plot corresponds to the best fit exponential function characterized with an exponential rate $\gamma$ for the data points.

Figure 5

Number of recolliding electrons vs time, counted at 8 times in each half cycle, for eight different intensities of 8-cycle pulses, from $2\times {10}^{15}$ to $9\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$. The intensities in this range begin to be high enough to distort the NSDI process. The doubled peaks indicate that a major fraction of all members of the ensemble can be induced to leave the nuclear region in the same direction at the same time.