Evolution of the ionic polarization in multiple sequential ionization: General equations and an illustrative example

Modern free-electron lasers generate a highly intense polarized radiation which initiates a sequence of ionization and decay events. Their probability depends on the polarization of each involved state as a function of time. Its complete account is limited by the fact that a state can be formed in various ways. Here we present an equivalent of rate equations for population that completely accounts for polarization of radiation formulated in terms of statistical tensors. To illustrate our approach we theoretically consider the sequential photoionization of krypton by an intense extreme ultraviolet femtosecond pulse for photon energies below the $3d$-shell excitation threshold. The calculations of the ion yields, photoelectron spectra, and ionic polarization for various photon fluences are presented and the role of the polarization is discussed.

Figure 1

The scheme of transitions in the sequential multiphoton ionization of Kr. The experimental ionization thresholds are taken from [52] and averaged over a multiplet.

Figure 2

(a) The alignment of the ${\mathrm{Kr}}^{+}4s^{2}4p^{52}P$ ion calculated as a function of time at $\omega =50$, 65, and 80 eV. The solid, dashed, and dotted lines correspond to fluence $F=3000$, 1000, and $400\mathrm{ph}/{\AA }^2$, respectively. (b) Conventional alignment of the same ion as a function of energy (accounting for the difference in definitions these data are in accordance with [58]). Here “1-2” and after “a-b” indicates the transition between the states labeled according to Table 1.

Figure 3

The stationary alignment as a function of energy for the different ionization pathways: (a) from ions with the $4s^{n}4p^{m2S_i+1}S$ term, (b) from the $4s^{n}4p^{m2S_i+1}P$ term, and (c) from the $4s^{n}4p^{m2S_i+1}D$ term.

Figure 4

Population of the different ion charge states and configurations for the fluence $F=1000$ $\mathrm{ph}/{\AA }^2$ at the photon energies 65 eV (a), (c), (e) and 78 eV (b), (d), (f) calculated with (solid lines) and without (dashed lines) accounting for the polarization of the ionic states. The populations of the neutral Kr and ${\mathrm{Kr}}^{+}$ are presented in the first row (a), (b); the population of ${\mathrm{Kr}}^{2+}$ is presented in the second row (c), (d). Black lines, yield of neutral Kr; thick red lines, yield of the ions with $4s^{2}4p^{6-n}$ configuration, where $n$ is the ion charge; thin blue lines, the ions with $4s^{1}4p^{6-n+1}$ configuration; circled green lines, the ions with $4s^{0}4p^{6-n+2}$ configuration. The third row (e), (f) presents populations of ${\mathrm{Kr}}^{3+}$ species provided that there is no next step; therefore circled green lines show the low limit for the species with double $4s$ vacancy in ${\mathrm{Kr}}^{3+}$ and higher states, and thick red lines show the upper limit for species without vacancy in the $4s$ shell in ${\mathrm{Kr}}^{\ge 3+}$. The pulse envelope (gray line) is indicated in the upper panels. The missed green lines in panels (c) and (d) are below the plot region.

Figure 5

The charge-state yields for three fluences: $F=100$ $\mathrm{ph}/{\AA }^2$ (a), (b), $F=400$ $\mathrm{ph}/{\AA }^2$ (c), (d), and $F=1000$ $\mathrm{ph}/{\AA }^2$ (e), (f) for the photon energies 65 eV (a), (c), (e) and 78 eV (b), (d), (f). The shaded areas show the results for unpolarized radiation.

Figure 6

Photoelectron spectrum for different photon energies: $\omega =65\mathrm{eV}$ [(a) for $F=100\mathrm{ph}/{\AA }^2$ and (c) for $F=1000\mathrm{ph}/{\AA }^2$] and $\omega =78\mathrm{eV}$ [(b) for $F=100\mathrm{ph}/{\AA }^2$ and (d) for $F=1000\mathrm{ph}/{\AA }^2$]. The solid lines correspond to calculations for linearly polarized radiation; the dashed lines correspond to unpolarized radiation. The spectra are normalized in such a way that $1/2$ of the main line A equals unity. The spectral features are indicated by capital letters in accordance with Table 1. The difference of peak E for the dashed line compared to the data presented in [51] is because of the corrected cross section of the (2-6) channel.

Figure 7

Intensity dependence of the different photoelectron lines on the fluence $F$ at the photon energy $\omega =65\mathrm{eV}$. The vertical dashed lines indicate fluences related to Figs. 4, 5, 6: 100 and $400\mathrm{ph}/{\AA }^2$. The solid lines correspond to linear polarization of the field; the dashed lines correspond to the unpolarized radiation.