Fig. 1. Schematic illustration of the notation in Eqs. (8)–(11).

Fig. 2. Normalized distributions of the effective intensity [Eq. (1)] in the planes (x, y, z = 0) (a), (x, y = 0, z) (c), and (x = 0, y, z) (d) for the NF beam with Δ = 0.178 corresponding to the parameters λ = 1 µm and w₀ = 3 µm of the Gaussian beam [Eq. (6)]. (b) shows the cuts x = 0 (solid red line) and y = 0 (solid blue line) of the (x, y, z = 0) distribution in (a). The dashed lines show the analogous cuts for the NF beam with Δ = 0.35.

Fig. 3. Number of ions vs peak laser intensity calculated from Eq. (5) for argon (a), krypton (b), and xenon (c) using the intensity distributions for the Gaussian (solid lines) and NF (dashed lines) beams. The beam parameters are the same as in Fig. 2. For argon and for Xe⁵¹⁺ and Xe⁵²⁺, the results corresponding to the tightly focused NF beam with Δ = 0.35 are shown by dash-dotted lines. The gray areas indicate the interval N(A^k+) = 10–100 where the number of ions grows by one order of magnitude. The values of I10 and I100 given in Table I are indicated by filled and open circles, respectively. The values of I0 where the numbers of two selected charge states become equal are shown by open squares for the Gaussian beam and by open triangles for the NF beam. For the pair of states Xe⁵¹⁺ and Xe⁵²⁺, the values of I0 for the two beams are difficult to distinguish visually, and so the corresponding triangle symbol is omitted. Vertical lines indicate the threshold intensities calculated from the analytic estimate in Eq. (13).

Fig. 4. Derivative d⁡log(N)/d⁡log(I) for Ar¹⁷⁺ (a) and Xe⁵²⁺ (b) calculated for the parameters of Fig. 3. Solid curves correspond to the Gaussian beam and dash-dotted curves to the NF beam. Δ = 0.178 for the blue curves and Δ = 0.35 for the red curves.

Table 1. Reference intensity values for ion yields shown in Fig. 3: ${\mathcal{I}}_{10}$ and ${\mathcal{I}}_{100}$ are the intensities corresponding to N(A^k+) = 10 and N(A^k+) = 100, respectively, ${\mathcal{I}}_0$ is the value of intensity such that N(A^k+) = N(A^(k+1)+), and ${\mathcal{I}}_{\ast }$ is given by the analytic estimate in Eq. (13). Intensities are given in units of 10²⁰ W/cm². The parameters of the laser beams are w₀ = 3 µm, λ = 1 µm, and Δ = 0.178. Values of ${\mathcal{I}}_{10}$ and ${\mathcal{I}}_{100}$ for the NF beam with Δ = 0.35 are shown in parentheses for Ar¹⁷⁺ and Xe⁵²⁺.