Towards Laser Intensity Calibration Using High-Field Ionization

Abstract

We present an approach for direct measurement of ultrahigh laser intensities in the range $10^{20}$–$10^{24}$ W/cm$^2$. The method is based on the use of multiple sequential tunneling ionization of heavy atoms with adequately high ionization potentials. We show that, due to the highly nonlinear dependence of tunneling ionization rates on the electromagnetic field strength, an off-set in the charge distribution of ions appears to be clearly sensitive to the peak value of intensity in the laser focus. Based on the tunnel-ionization mechanism, a simple analytic theory helps in estimating the maximal charge state produced at a given laser intensity. Our theory also allows for calculating qualitatively a distribution in charge states generated in different zones of the laser focus. These qualitative predictions are in excellent agreement with numerical simulations of the tunneling cascades, developed in the interaction of a short tightly focused laser pulse with low-density noble gas targets. The method could be particularly useful and of instrumental demand in view of the expected commissioning of several new laser facilities, capable of delivering ultra-powerful light pulses in the above mentioned domain of intensities.

Appendices

Appendix A. Ionization Cascades

Starting from a given ground state of an atom or an ion, ionization can proceed along different pathways. Clearly, for multi electron atoms the total number of such paths grows very quickly with the atomic number. Most of them do not give any considerable contribution into the production of ionic states owing to the structure of the tunneling rate. The tunneling exponent in (8.9a) is maximal for the outermost electron which has the minimal ionization potential. This makes ionization of inner shells highly improbable before the outer shells have been stripped out, so that such ionization pathways can be safely discarded. However, when electrons are being removed from the same shell, ionization of lower s-levels may proceed with comparable or even higher probability than that of p-levels with lower ionization potentials. The reasons for that is (a) a smaller value of the asymptotic coefficients $C^2_{\nu l}$ in (8.9b) and (b) the factor $F^{|m|}$ in (8.9a), which is small for nonzero magnetic quantum numbers, owing to the condition $F\ll 1$. As an example, for $\mathrm{Kr}^{26+}$ with the ground state configuration $1s^22s^22p^6$, ionization of a p-electron with $I_p\approx \!2929$ eV proceeds with probability comparable to that for an s-electron whose ionization potential is by $\Delta I_p\approx \!235$ eV higher [44]. Indeed, taking $F\simeq F^*=0.05$, one obtains that the tunneling exponent for the p-electron is $\exp (\Delta I_p/I_pF)\approx \!5$ times higher than that for the s-electron. At the same time, for the latter $C_{\nu 0}^{2}=1.088, ~B_{00}=1$, while for the p-electron $C_{\nu 1}^{2}=0.315$ and $B_{10}=3$, $B_{1\pm 1}=3/2$, and finally the factor $F^{|m|}$ gives 0.05 for $m=\pm 1$. Thus, the p-state rate averaged over the magnetic quantum number appears only 5 times greater than that of the s-state. For partially stripped p-shells the difference in ionization potentials appears to be even smaller, so that for configurations $2s^22p$ or $2s^22p^2$, the s- and p-rates are almost equal. These estimates show that the described sub-manifold of pathways may play an essential role in the ionization dynamics.

An example of the structure of levels is shown on Fig. 8.9 for the initial configuration $1s^22s^22p^6$, which corresponds to the neutral neon, $\mathrm{Ar}^{10+}$, $\mathrm{Kr}^{26+}$, etc.

Fig. 8.9

Substructure of levels and ionization pathways making the main contribution into sequential multiple ionization of atomic systems prepared in the ground $1s^22s^22p^6$ state. Ground and excited states are shown by blue solid and dashed lines correspondingly. Most of the excited states consist of several sub-levels with different values of the full angular momentum J. The ionization pathway which involves only the ground states is shown by red arrows, all other pathways—by black arrows

For the initial configuration shown on Fig. 8.9 the total number of pathways is equal to 28. For the $1s^22s^22p^2$ configuration only 5 relevant pathways left; starting from the $1s^22s^2$ configuration ionization proceeds along the unique pathway.

Appendix B. Systems of Rate Equations

We solve numerically the system of rate equations for argon in the interval of intensities $\mathcal{I}_m=10^{19}$–$10^{22}$ W/cm$^2$ and for krypton between $\mathcal{I}_m=10^{19}$–$10^{23}$ W/cm$^2$ using an adaptive stepsize Runge-Kutta scheme [43]. We start with the simplest configuration $1s^22s^2$ for argon ($\mathrm{Ar}^{14+},~I_p\approx \!855$ eV). The value of $I_p$ is well below that of 8.14 for $10^{20}$W/cm$^2$ which is $I_p^*\approx \!1420$ eV. For this initial configuration, only one relevant pathway contributes (see Fig. 8.9 and Appendix A). The resulting system of rate equations is therefore particularly simple and reads:

$$\begin{aligned} \frac{{\text {d}}c_{14}}{{\text {d}}t}=-2c_{14}w(\nu _{14},0,0;t), \end{aligned}$$

(B25)

$$\begin{aligned} \frac{{\text {d}}c_{15}}{{\text {d}}t}=2c_{14}w(\nu _{14},0,0;t)-c_{15}w(\nu _{15},0,0;t), \end{aligned}$$

(B26)

$$\begin{aligned} \frac{{\text {d}}c_{16}}{{\text {d}}t}=c_{15}w(\nu _{15},0,0;t)-2c_{16}w(\nu _{16},0,0;t), \end{aligned}$$

(B27)

$$\begin{aligned} \frac{{\text {d}}c_{17}}{{\text {d}}t}=2c_{16}w(\nu _{16},0,0;t)-c_{17}w(\nu _{17},0,0;t), \end{aligned}$$

(B28)

$$\begin{aligned} \frac{{\text {d}}c_{18}}{{\text {d}}t}=c_{17}w(\nu _{17},0,0;t). \end{aligned}$$

(B29)

Coefficients 2 in (B25)–(B28) are due to the presence of two equivalent electrons in the sub-shell.

For the interval of intensities used for Kr, the system of rate equations has to include p-states of the 2p shell. In order to simplify the calculations, we consider two cases: (a) the $1s^22s^22p^6$ state as initial configuration, with only the most probably pathway accounted for (shown on Fig. 8.9 by red arrows) and (b) the $1s^22s^22p^2$ as initial configuration with all relevant pathways accounted for (shown on Fig. 8.9 by red and black arrows). For case (a) the system explicitly reads:

$$\begin{aligned} \frac{{\text {d}}c_{26}}{{\text {d}}t}=-2c_{26}\{w(\nu _{26},1,0;t)+2w(\nu _{26},1,\pm 1;t)\}, \end{aligned}$$

(B30)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{27}}{{\text {d}}t}&=2c_{26}\{w(\nu _{26},1,0;t)+2w(\nu _{26},1,\pm 1;t)\} \\&\quad -\frac{5}{3}c_{27}\{w(\nu _{27},1,0;t)+2w(\nu _{27},1,\pm 1;t)\}, \end{aligned} \end{aligned}$$

(B31)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{28}}{{\text {d}}t}&=\frac{5}{3}c_{27}\{w(\nu _{27},1,0;t)+2w(\nu _{27},1,\pm 1;t)\}\\&\quad -\frac{4}{3}c_{28}\{w(\nu _{28},1,0;t)+2w(\nu _{28},1,\pm 1;t)\}, \end{aligned} \end{aligned}$$

(B32)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{29}}{{\text {d}}t}&=\frac{4}{3}c_{28}\{w(\nu _{28},1,0;t)+2w(\nu _{28},1,\pm 1;t)\}\\&\quad -c_{29}\{w(\nu _{29},1,0;t)+2w(\nu _{29},1,\pm 1;t)\}, \end{aligned} \end{aligned}$$

(B33)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{30}}{{\text {d}}t}&=c_{29}\{w(\nu _{29},1,0;t)+2w(\nu _{29},1,\pm 1;t)\}\\&\quad -\frac{2}{3}c_{30}\{w(\nu _{30},1,0;t)+2w(\nu _{30},1,\pm 1;t)\}, \end{aligned} \end{aligned}$$

(B34)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{31}}{{\text {d}}t}&=\frac{2}{3}c_{30}\{w(\nu _{30},1,0;t)+2w(\nu _{30},1,\pm 1;t)\} \\&\quad -\frac{1}{3}c_{31}\{w(\nu _{31},1,0;t)+2w(\nu _{31},1,\pm 1;t)\}, \end{aligned} \end{aligned}$$

(B35)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{32}}{{\text {d}}t}&=\frac{1}{3}c_{31}\{w(\nu _{31},1,0;t)\\&\quad +2w(\nu _{31},1,\pm 1;t)\}-2c_{32}w(\nu _{32},0,0;t), \end{aligned} \end{aligned}$$

(B36)

$$\begin{aligned} \frac{{\text {d}}c_{33}}{{\text {d}}t}=2c_{32}w(\nu _{32},0,0;t)-c_{33}w(\nu _{33},0,0;t), \end{aligned}$$

(B37)

$$\begin{aligned} \frac{{\text {d}}c_{34}}{{\text {d}}t}=c_{33}w(\nu _{33},0,0;t)-2c_{34}w(\nu _{34},0,0;t). \end{aligned}$$

(B38)

The system can be safely truncated by (B38), as the ionization potential of $\mathrm{Kr}^{34+}$, $I_p=17296$ eV, is too high to expect any considerable ionization below $10^{23}$ W/cm$^2$ (see Figs. 8.1 and 8.2). The effective principal quantum numbers $\nu _{z}$ are calculated using data from [44].

For case (b) we take into account all relevant pathways (see red and black arrows in Fig. 8.9) up to the same ionic state as the one used in (a). As a result, excited states of two types, $1s^22s2p^n$ and $1s^22p^n$ with $n=1,2$ enter in the calculation. We denote the values corresponding to these two sets of excited states by one and two primes, respectively. Then the system of rate equations reads:

$$\begin{aligned} \frac{{\text {d}}c_{30}}{{\text {d}}t}=-c_{30}\{\frac{2}{3}[w(\nu _{30},1,0;t)+2w(\nu _{30},1,\pm 1;t)]+2w(\nu '_{30},0,0;t)\}, \end{aligned}$$

(B39)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{31}}{{\text {d}}t}&=-c_{31}\{\frac{1}{3}[w(\nu _{31},1,0;t)+ \\ \quad&+2w(\nu _{31},1,\pm 1;t)]+2w(\nu '_{31},0,0;t)\}+c_{30}\frac{2}{3}[w(\nu _{30},1,0;t)+2w(\nu _{30},1,\pm 1;t)], \end{aligned} \end{aligned}$$

(B40)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c'_{31}}{{\text {d}}t}&=-c'_{31}\{\frac{2}{3}[w(\nu '_{31},1,0;t) \\&\quad +2w(\nu '_{31},1,\pm 1;t)]+w(\nu ''_{31},0,0;t)\}+2c_{30}w(\nu '_{30},0,0;t), \end{aligned} \end{aligned}$$

(B41)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{32}}{{\text {d}}t}&=-2c_{32}w(\nu _{32},0,0;t)\\&\quad +c_{31}\{\frac{1}{3}[w(\nu _{31},1,0;t)+2w(\nu _{31},1,\pm 1;t)]\}, \end{aligned} \end{aligned}$$

(B42)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c'_{32}}{{\text {d}}t}&=-c'_{32}\{\frac{1}{3}[w(\nu '_{32},1,0;t)+2w(\nu '_{32},1,\pm 1;t)]+w(\nu ''_{32},0,0;t)\} \\&\quad +2c_{31}w(\nu '_{31},0,0;t)+c'_{31}\{\frac{2}{3}[w(\nu '_{31},1,0;t)+2w(\nu '_{31},1,\pm 1;t)]\}, \end{aligned} \end{aligned}$$

(B43)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c''_{32}}{{\text {d}}t}&=-\frac{2}{3}c''_{32}[w(\nu ''_{32},1,0;t) \\&\quad +2w(\nu ''_{32},1,\pm 1;t)]+c'_{31}w(\nu ''_{31},0,0;t), \end{aligned} \end{aligned}$$

(B44)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{33}}{{\text {d}}t}&=-c_{33}w(\nu _{33},0,0;t)+2c_{32}w(\nu _{32},0,0;t)\\&\quad +\frac{1}{3}c'_{32}[w(\nu '_{32},1,0;t)+2w(\nu '_{32},1,\pm 1;t)], \end{aligned} \end{aligned}$$

(B45)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c''_{33}}{{\text {d}}t}&=-\frac{1}{3}c''_{33}[w(\nu '_{33},1,0;t)+2w(\nu '_{33},1,\pm 1;t)]\\&\quad +c'_{32}w(\nu ''_{32},0,0;t)+\frac{2}{3}c''_{32}[w(\nu ''_{32},1,0;t)+2w(\nu ''_{32},1,\pm 1;t)], \end{aligned} \end{aligned}$$

(B46)

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}c_{34}}{{\text {d}}t}&=c_{33}w(\nu _{33},0,0;t)+\frac{1}{3}c''_{33}[w(\nu '_{33},1,0;t)\\&\quad +2w(\nu '_{33},1,\pm 1;t)]-2c_{34}w(\nu _{34},0,0;t). \end{aligned} \end{aligned}$$

(B47)

Table 8.1 Parameters for the first ionization cascade of Kr

Appendix C. Parameters for Krypton and Xenon

Here the values of $I_p$, $\nu$, $C_{\nu l}^2$ and $B_{lm}$ are given for the two ionization cascades in krypton described by (B30)–(B38) and (B39)–(B47) and for the xenon case, that can be simulated using the same set of rate equations (B25)–(B29), changing the respective ionic state populations and their associated parameters.

Fig. 8.10

Same as on Fig. 8.9 but for the initial ground state $1s^22s^22p^2$. See the text for details about the numeration of transitions and their parameters

The first case of Kr corresponds to ionization of the outermost orbitals (red arrows on Fig. 8.9) and starts from the $1s^22s^22p^6$ state of $\mathrm{Kr}^{26+}$. The parameters of the first ionization cascade are shown in the Table 8.1. The second one starts from the $1s^22s^22p^2$ state of $\mathrm{Kr}^{30+}$ and accounts all pathways (shown by red and black arrows on Fig. 8.9).

For this case one should take into account that several transitions may occur from/to a given state. These transitions are depicted and numbered in Fig. 8.10. The respective parameters are presented in Table 8.2.

Table 8.2 Parameters for the second ionization cascade of Kr (see text and Fig. 8.10 for more details)

Finally, we run the same system of rate equations (B25)–(B29) for xenon starting from the ground state configuration $1s^22s^2$ of the $\mathrm{Xe}^{50+}$ ion. The parameters are shown the Table 8.3.

Table 8.3 Parameters for Xe