Empirical formula for over-barrier strong-field ionization

A modified empirical ionization formula is proposed to extend Tong and Lin's ionization formula [X. M. Tong and C. D. Lin, J. Phys. B 38, 2593 (2005)] to a laser field up to 4.5 times barrier-suppression strength $\left(E_b\right)$ in the over-barrier regime. The validity of the formula is checked by solving the Schrödinger equation within the single-active-electron approximation with an excellent agreement. This modified empirical formula still keeps its simplicity and can calculate ionization rates in the deep over-barrier regime.

Figure 1

The ratio between the ionization rates obtained with the Schrödinger equation and ADK formula in logarithmic scale (square) and the linear and parabolic fitting using Eq. (8) (dashed and solid line).

Figure 2

Same as Fig. 1, but the target is ${\mathrm{Ar}}^{+}$.

Figure 3

The ionization rates obtained with the Schrödinger equation, ADK [Eq. (1)] formula, WFAT [Eq. (2)] formula, Tong and Lin's formula [Eq. (3)], and modified empirical formula [Eq. (8)] in (a) logarithmic scale and (b) linear scale. The upper axis shows the corresponding laser intensity in units of $1\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$.

Figure 4

Same as Fig. 3, but the target is ${\mathrm{Ar}}^{+}$.

Figure 5

Relative error between the ionization rates calculated with the Schrödinger equation and those calculated with Eqs. (3) and (8). The targets are He, Ne, Ar, Kr, Xe, and ${\mathrm{Ar}}^{+}$, respectively.

Figure 6

Ionization probabilities of the Ar atom in the 2-fs laser pulses with varying peak intensities.