1. INTRODUCTION

An electron in a laser field experiences an oscillating electromagnetic force. For high-intensity lasers, the energy of the quivering electron motion can be many times larger than the photon energy. By way of example, the energy for an electron in a laser field with a photon energy of ${\sim}1\; {\rm{eV}}$ and intensity of ${\sim}{10^{19}} \;{\rm{W}}/{{\rm{cm}}^2}$ is a million times the photon energy [1,2]. For a photoelectron created in the laser field, the first half-cycle of the oscillation is near the parent ion and can result in the energetic electron being driven back into the parent ion via a process called rescattering [3].

One may capitalize on rescattering and use; in essence, atomic and molecular ionization in strong laser fields as a laser-driven electron accelerator with the parent ion as the collision target. The rescattering collision has been observed to ionize or otherwise excite the remaining bound electrons [3,4] and image molecular wave functions in the parent ion [5]. The application of rescattering to create XUV and x-ray sources is revolutionary and resulted in the new area of attosecond science [6,7]. The use of kiloelectron volt (keV) photon energy high harmonic (HH) radiation [8,9] allows the dynamics of the inner shell processes in atomic and molecular systems to now be measured directly [10–12]. The impact of laser-driven rescattering has only begun with these pioneering measurements. Laser-driven rescattering and recollision induced ionization offers the opportunity to investigate fundamental physics and potential new applications. Attosecond rescattering, for example, is one of the only methods to directly measure the workings of the inner shell working within atoms and molecules, including complex electron correlation and attosecond dynamics within the ionization process.

Traveling wave laser beams [13–15] have an upper energy for recollision limited by relativistic effects that occur when the relativistic deflection parameter ${\Gamma _R} = \sqrt {2{E_{\rm{IP}}}} E_0^3/(16{\omega ^4}/{c^2})$ exceeds one [13] for ionization from a potential ${E_{\rm{IP}}}$ [16] in a laser field of strength ${E_0}$ and frequency $\omega$. In this work, we address two outstanding questions about laser driven rescattering: How far can laser rescattering push into the high-energy limit? And, is it possible to describe the optimal laser parameters to drive the rescattering processes, such as radiation via HH generation, atomic excitation, or high-energy elastic scattering?

There are a few key characteristics of laser-driven rescattering. To begin, the time scale for the external field (${10^{- 15}}\;{\rm s}$ at optical frequencies) is long compared to that of an electron in an atomic ground state, which can be estimated using the Bohr orbit time $2\pi\! {n^3}/{Q^2}$ in atomic units for a principle quantum number $n$, parent ion charge $Q$, and atomic unit of time $\hbar /{E_H}$ ($2.4 \times {10^{- 17}}$ s), where ${E_H}$ is the hartree energy unit. Because the external field slowly varies in time, the strong field interaction with a ground state atom or ion is typically a quasistatic, DC-type response. One impact of the quasistatic interaction is that the external field predominantly interacts with the least tightly bound electron [17]. Early measurements of recollision ionization or excitation of electrons in the parent ion were consistent with other strong field interactions insofar as the outermost electrons were first observed to recollisionally ionize via an (e,2e) process [18,19]. Further study provided insight into the recollision electron relativistic deflection [20,21], and an impressive agreement of strong field rescattering (e,ne) processes with field-free (e,ne) collision cross sections [22–24].

Here, we present the results for K-shell and ${{\rm{L}}_{\rm{I}}}$-shell ionization of atoms [25,26] with rescattering driven by an external, linearly polarized laser field that is illustrated in Fig. 1. These inner shells are among the highest energy excitation possible in atoms. While the field interaction with the bound electron is quasistatic [27], the ionization is actually time dependent, as it occurs in bursts at the maximum amplitude of the oscillating laser field. In the figure, we show rescattering projected along a time axis as: (1) ionization at the peak laser field, (2) acceleration and spread of the photoelectron, and (3) rescattering K-shell and L-shell excitation. Our work shows that laser-driven rescattering inner shell ionization [shortened to rescattering shell ionization (RSI)], is possible for all naturally occurring elements.

 

Fig. 1. Sequence of (1) tunneling ionization (red) of the outermost electron, (2) laser acceleration of the photoelectron over length scale of ${E_0} {\lambda ^2}$, (3) electron return on a time scale of $1/\omega$ to the parent ion where recollision creates a K-shell (green) and an L-shell RSI (blue).

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2. RESULTS

Laser ionization, the ensuing relativistic acceleration dynamics, and the ultimate limit of rescattering was recently described theoretically using semiclassical and quantum models [13]. In this work, we use the semiclassical trajectory ensemble method [28,29]. Briefly, bound state ionization is calculated by tunneling [30] (Fig. 1), and the space–time relativistic photoelectron is modeled semiclassically with a ${10^4}$ trajectory Monte-Carlo ensemble using a position and momentum spread-matched to the tunneling wave function probability as it appears in the continuum. Trajectories that return to the parent ion are counted as the rescattering fluence, $F$ (dimension of charge/area) with the tunneling rate and the energy resolved probability of return calculated from the trajectories [31] comprising the energy-resolved rescattering electron fluence ${\rm d}F/{\rm d}E$ [charge/(area energy)].

We consider one-electron laser interactions with atomic and ion species where the ionization probability is ${\sim}10\%$ per optical cycle, which is comparable to experimental yields near saturation. With this constraint, $\omega$, ${E_0}$, and ${\Gamma _R}$ are related and can be expressed as one variable. We use ${\Gamma _R}$ as it clearly indicates the onset and end of excitation by rescattering. To aid the presentation, ${\rm d}F/{\rm d}E$ is divided by the ionization fraction that is of order 10% per optical cycle to give the energy-resolved rescattering fluence ${\rm d}\tilde F/{\rm d}E$ normalized for the amount of ionization.

The energy-resolved rescattering electron fluence ${\rm d}F/{\rm d}E$ and field-free K-shell and ${{\rm{L}}_{\rm{I}}}$-shell cross sections [32] are used to calculate RSI. Figure 2 displays two examples of the energy-resolved rescattering fluence and the K-shell cross section. The overlap gives the yield for the ejected inner shell electron as well as the excess energy shared between the outgoing electrons. Figure 2(a) typifies results at the edge of the nonrelativistic strong field (${\Gamma _R} = 1$) with a $3.2 {U_p}$ cutoff of 110 hartree (3 keV). This energy is sufficient to ionize the carbon K-shell with a cross section threshold at ${\sim}10$ hartree (284 eV). The created photoelectron yield $({\rm{d}}F/{\rm{d}}E) \sigma$ ranges from ${10^{- 12}}$ electrons/hartree near threshold (${\sim}0\;{\rm{eV}}$) to ${10^{- 9}}$ electrons/hartree for photoemission with kinetic energies near 100 hartree. K-shell creation for ${\rm{C}}{{\rm{u}}^{3 +}}$ is a case well into the relativistic limit (${\Gamma _R} = 90$), where “short trajectories” [33] are deflected less and predominantly contribute [13] at the ultimate cutoff. The rescattering cutoff energy of ${\sim}800$ hartree is just sufficient to excite the copper K-shell with a peak yield of ${10^{- 19}}$ electrons/hartree in an energy spectrum stretching from threshold (0 hartree) to 500 hartree; i.e., the maximum rescattering cutoff energy minus the K-shell threshold.

 

Fig. 2. Energy-resolved recollision flux, ${\rm d}F/{\rm d}E$ (red), for (a) ${{\rm{C}}^{2 +}}$ at $4.9\;\unicode{x00B5}{\rm m}$ and a peak intensity of $4 \times {10^{14}} \;{\rm{W}}/{{\rm{cm}}^2}$ and (b) ${\rm{C}}{{\rm{u}}^{3 +}}$ at $10\;\unicode{x00B5}{\rm m}$, $1 \times {10^{15}} \;{\rm{W}}/{{\rm{cm}}^2}$. Also plotted are the inelastic cross sections $\sigma$ (green) [32]. The ${\rm d}F/{\rm d}E\;\sigma$ product, which contributes K-shell RSI (blue), is shown tied to the far right y axis.

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RSI across the periodic table is a complex process. For this reason we calculated K-shell and ${{\rm{L}}_{\rm{I}}}$-shell excitation for different ion species interacting with many laser conditions. Using this exhaustive approach, we were able to find maximum yields for atoms at different intensities, wavelengths, and ionization states. To maintain the integrity of the inner shell state and cross sections, we impose a limit on the ionization progression. For K-shell species, typical ion final states reside in the $2p$ subshell, with noted exceptions including the $2{s^1}$ electron on ${\rm{L}}{{\rm{i}}^ +}$ and ionization of the $3{d^9}$ electron on ${\rm{K}}{{\rm{r}}^{9 +}}$ and ${{\rm{U}}^{65 +}}$. The ${{\rm{L}}_{\rm{I}}}$ final ion core configurations lie between a full Ne core (with an exception of the $3{s^1}$ for ${\rm{N}}{{\rm{a}}^ +}$) and a full Ar core (the $3{d^1}$ of ${\rm{C}}{{\rm{u}}^{11 +}}$ and ${\rm{X}}{{\rm{e}}^{36 +}}$). In general, the species chosen for this analysis favors final ion states with configurations in the upper $p$ sub-shell of the next full shell; e.g.,  ${\rm{n}} = {{3}}$ for the L-shell.

RSI is limited from a low-energy side whenever the energy is insufficient to reach the minimum required by the cross section threshold; e.g.,  284 eV (10 hartree) in Fig. 2(a) to remove the 1s electron for the carbon K-shell. On the high-energy side, production is limited by the relativistic rescattering deflection (${\Gamma _R} \gt 1$). The K-shell for xenon (${k_{\alpha 2}} = 29{,}458\; {\rm eV}$) is at the ${\sim}1{,}000$ hartree limit, beyond which rescattering yields drop when ${\Gamma _R} \gt 1$. For heavier nuclei ($54 \lt Z \lt 92$), it is still possible to extend the rescattering energy by a factor of a few (e.g.,  from 1,000 hartree to 3,500 hartree) to reach uranium (${k_{\alpha 2}} = 94{,}665\; {\rm eV}$) by a careful choice of the laser wavelength and intensity.

We present our energy-resolved rescattering K-shell and ${{\rm{L}}_{\rm{I}}}$-shell results as 3D “island” plots. The island plots indicate concisely the location of the peak yield in the midst of the threshold and upper energy limits for the rescattering. Laser parameters are shown on the vertical axis expressed as ${\Gamma _R}$. Wavelength information is also superimposed on the 3D island plot as an aid to the reader. The K-shell emission is shown in the 3D color scale. A line-out of this 3D plot at one particular ${\Gamma _R}$ is comparable to the $({\rm{d}}F/{\rm{d}}E) \sigma$ plotted in Fig. 2. Figure 3(a) is the ionization of neutral Ne to ${\rm{N}}{{\rm{e}}^ +}$. K-shell ionization begins at $\lambda = 2.05 \;\unicode{x00B5}{\rm m}$ (${\Gamma _R} = 0.05$, laser intensity $8 \times {10^{14}} \;{\rm{W}}/{{\rm{cm}}^2}$), peaks at $2.6 \;\unicode{x00B5}{\rm m}$ (${\Gamma _R} = 0.12$, $7.5 \times {10^{14}} \;{\rm{W}}/{{\rm{cm}}^2}$), and then decreases as the interaction enters relativistic territory for longer wavelengths (${\Gamma _R} \gt 1$), disappearing from the figure as the recollision energy exceeds 1,000 hartree at $\lambda = 8 \;\unicode{x00B5}{\rm m}$ (${\Gamma _R} \sim 10$, $7 \times {10^{14}} \;{\rm{W}}/{{\rm{cm}}^2}$). Besides the island plot, the K-shell spectra integrated over recollision energy are shown in Fig. 3(e) to give the K-shell hole yield as a function of ${\Gamma _R}$; i.e., the laser parameters. Figure 3(e) shares the same ${\Gamma _R}$ y axis and imbedded y axis wavelength tags as the adjacent Fig. 3(a). For example, the energy-integrated RSI island plot in Fig. 3(a) is shown in Fig. 3(e). The maximum value for the energy-integrated RSI hole yield can be seen to occur at ${\Gamma _R} = 0.1$, which occurs at a wavelength just longer than 2.1 µm. One may convert between ${\Gamma _R}$ and $\lambda$ since at the ionization saturation field strength ${E_0}$ for a species with an ionization potential of ${E_{\rm{IP}}}$, the relationship between wavelength and ${\Gamma _R}$ follows directly from the definition of ${\Gamma _R}$ and the laser frequency $\omega$. The same is true for the coupled Figs. 3(b)–3(f), Figs. 3(c)–3(g), and Figs. 3(d)–3(h). Figures 3(b) and 3(f) is an example of an ionization channel (${\rm{K}}{{\rm{r}}^{8 +}} \to {\rm{K}}{{\rm{r}}^{9 +}}$) at high ${\Gamma _R}$ that has a narrow excitation window between $\lambda = 525\; {\rm{nm}}$ and $\lambda = 2,000\; {\rm{nm}}$ for an interaction intensity at ${\sim}{10^{17}} \;{\rm{W}}/{{\rm{cm}}^2}$ and a maximum of $5 \times {10^{- 14}}$ of the one-electron ionization process.

 

Fig. 3. 3D island plots of the $\sigma {\rm d}\tilde F/{\rm d}E$ cross section and rescattering product as a function of ${\Gamma _R}$ for a 10% ionization/cycle with (a)${\rm{N}}{{\rm{e}}^ +}$, (b) ${\rm{K}}{{\rm{r}}^{9 +}}$, (c) ${\rm{K}}{{\rm{r}}^{27 +}}$, and (d) ${{\rm{U}}^{83 +}}$. A color legend indicates the energy resolved “holes/hartree.” A hard cutoff due to the threshold can be seen on the left side while the right of the island is caused by the rescattering energy cutoff. The island bottom is due to high ${\Gamma _R}$. The energy integrated probability of the K-shell RSI (e–h) is adjacent right to the respective island plots. The peak corresponds to the maximum RSI for that ion.

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In the search for optimal laser parameters for RSI, we consider extreme wavelengths and highly charged ions well beyond the confines of traditional strong field interactions. Especially at shorter wavelengths, rescattering may hold surprises [34,35]. Studies with high-intensity x-ray sources have demonstrated multiphoton, x-ray processes [36,37], including intermediate resonances in multiphoton excitations [38,39]. These “few photon” x-ray processes are similar to what one would expect for three- to five-photon processes at optical frequencies, which would also fall into a multiphoton description versus quasistatic tunneling [40]. At this time, a strong field quasistatic x-ray response has not been experimentally found. That said, we aim to present all results arguably within the strong field response based on scaling known processes.

The interaction of 800 nm radiation with He to create ${\rm{H}}{{\rm{e}}^ +}$ has been measured as in the strong field at saturation with the tunneling ionization rate eclipsing the multiphoton ionization rate [19,41]. For helium, the bound state classical orbit time is 6% of the laser period; helium responds “instantaneously” to the field magnitude by tunneling when the electron samples the potential barrier greater than 15 times per optical cycle. The parameter ${\gamma _K} = \frac{{\omega \sqrt {2{E_{\rm{IP}}}}}}{{{E_0}}}$ in [42] is often used to gauge such adiabaticity, and, in this case, is 0.5.

Figures 3(c) and 3(d) gives island plots for Kr and U at XUV and near x-ray wavelengths. For ${\rm{K}}{{\rm{r}}^{27 +}}$ the bound state energy is 107.7 hartree for a classical orbit time of 0.0029 $\hbar /{E_h}$. An equivalent, scaled wavelength for the electron in this bound state meeting the 6% quasistatic tunneling criteria would have a laser wavelength of 10 nm. Excitation of ${\rm{K}}{{\rm{r}}^{26 +}}\;[{\rm{He}}]2{s^2}2{p^6}$ to ${\rm{K}}{{\rm{r}}^{27 +}}[{\rm{He}}]2{s^2}2{p^5}$ at this wavelength would require a minimum of 18 photons, an intensity of $8 \times {10^{20}} \;{\rm{W}}/{{\rm{cm}}^2}$ and have a ${\gamma _K} = 0.4$. For ${{\rm{U}}^{82 +}}$ to ${{\rm{U}}^{83 +}}$ (ionization energy of 929 hartree) shown in Figs. 3(f) and 3(l) a similar scaling analysis gives a classical orbit time of $0.025 \hbar /{E_h}$ and a laser wavelength of 0.8 nm. While the photon energy of 1.5 keV and would normally be considered as inordinate for a tunneling interaction, the removal of the 2p electron from neon like uranium at $5 \times {10^{23}} \;{\rm{W}}/{{\rm{cm}}^2}$ has a Keldysh parameter of ${\gamma _k} = 0.7$, which would place it at the edge of a quasistatic interaction.

Validity can be assessed by comparing to experiments [43–45]. The recent results from [44] used a 12 fs pulse of $1.8\;\unicode{x00B5}{\rm m}$ laser light focused to a beam waist of $65\;\unicode{x00B5}{\rm m}$ across a $350\;\unicode{x00B5}{\rm m}$ diameter pulsed valve operating at 20 Hz. The reported peak intensity for this experiment was $3 \times {10^{15}} \;{\rm{W}}/{{\rm{cm}}^2}$ at the laser focus, which exceeds the $1 \times {10^{15}} \;{\rm{W}}/{{\rm{cm}}^2}$ saturation intensity for ionization of the neutral ground state Ne to form ${\rm{N}}{{\rm{e}}^ +}$ [46] and ensures the entire focal volume of approximately ${10^6}\;\unicode{x00B5} {{\rm{m}}^3}$ at ${\sim}1$ bar is ionized and contributes to the measured K-shell holes. When the volume integrated across the laser focus and includes a 0.015 fluorescence yield for a K-shell hole in neon, we calculate a yield of $0.35\;{{\rm{k}}_\alpha}$ photons per laser shot, which is consistent with the observation of [44]. A similar agreement holds for the L-shell. Beyond the current experimental studies, it is possible to extend to lower $Z$ species that offer higher RSI yields. Modern high repetition rate laser systems extend the count and wavelength range for RSI measurements. With a factor of one-thousand improvement in the signal range over previous studies [44], the RSI in a $Z \sim 30$ species could be experimentally accessed. X-ray free electron laser experiments should be able to further extend the RSI to the $Z \gt 30$.

From island plots of atoms and ions across the periodic table, we have determined the laser fields that give a maximum in the K- and ${{\rm{L}}_{\rm{I}}}$-shell yields, shown as a function of $Z$ in Fig. 4. We have superimposed on the calculated points a simple scaling as both yields demonstrate an empirical ${Z^{- 8}}{(\lambda)^{- 1.4}}$ dependence. As the atomic species studied cross the ${\Gamma _R} \gt 1$ boundary highlighted in the figure, the RSI yields continue to trend similar to the nonrelativistic interaction. This results from the optimal field solutions emphasising the yield and trading off other physical options (such as $\omega ,Q,{E_{\rm{IP}}}$) to limit high ${\Gamma _R}$ solutions. At these ideal laser conditions for RSI, one can observe the yields equal the peak inner shell cross section times a fluence that is independent of $Z$. For the K-shell, the yield is ${10^{- 4}} \sigma$ and for the ${{\rm{L}}_{\rm{I}}}$-shell, it follows ${10^{- 3}} \sigma$. These scaled $\sigma$ values are shown in the Yield - Z back plane of Fig. 4 with tie-bars to our calculated values. Example of optimized laser-produced K-shell values for various species are shown in Table 1.

 

Fig. 4. 3D plots of (a) the peak K-RSI and (b) the ${{\rm{L}}_{\rm{I}}}$-RSI as a function of $Z$ and laser parameters expressed as ${\Gamma _R}$. Empirical fits for the yields of (a) $2 \times {10^7}{Z^{- 8}}{(\lambda)^{- 1.4}}$ and (b) $1.9 \times {10^{10}}{Z^{- 8}}{(\lambda)^{- 1.4}}$ are shown. Yellow-shaded regions on the vertical planes delineate regions where relativistic effects occur (${\Gamma _R} \gt 1$; $Z \gt 54$). The peak K-shell and ${{\rm{L}}_{\rm{I}}}$-shell cross sections on the Yield - Z plane are scaled by (a) ${10^{- 4}}$ and (b) ${10^{- 3}}$.

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Table 1. Optimal Laser Parameters for Rescattering Driven K-Shell Ionization

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A simple physical picture emerges in the high-energy, strong field rescattering optimization. First is the parent ion charge state and ionization energy, which determines the peak laser field and (for the sake of simplicity) can be estimated as a classical barrier suppression ionization, ${E_{\rm{bsi}}} = \frac{{E_{\rm{IP}}^2}}{{4Q}}$. Second is matching the rescattering to the cross section. The rescattering fluence peaks at the maximum return energy (see Fig. 2), which for ${\Gamma _R} \le 1$ is ${\sim}3.2$ times the quiver, or ponderomotive, energy, ${U_p} = E_{\rm{bsi}}^2/(4{\omega ^2})$. Coincidentally, the characteristic inner shell cross section peaks at ${\sim}3$ times the cross section threshold for energies below the pair creation energy. The maximum yield occurs when the fluence and cross section peaks overlap; the greatest rescattering inner-shell ionization occurs when ${U_p}$ is approximately equal to the threshold for the inner shell predicted by Moseley’s law as $1/2(Z - {1)^2}$. Therefore, the ideal laser frequency is $\omega = {E_{\rm{IP}}}/(\sqrt {32} q (Z - 1))$. Finally, optimal RSI occurs for ${\Gamma _R} \le 1$. The criteria described above combine to give a way to determine the ideal laser conditions for high-energy rescattering. With an adjustment in the second step, the analysis can be extended to any physical process to find the optimized conditions for laser rescattering.

The scaling of RSI with $Z$ and the laser wavelength $\lambda$ also can be explained by analyzing the atomic inner shell cross section $Z$ dependence [16] and rescattering laser physics. The $Z$ dependence comes from two factors. First is the electron impact K-shell cross section; the K-shell cross section has a peak value that scales as ${\sim}1/{Z^4}$. The second factor comes from the rescattering fluence. The recollision yield dependence is not a simple function of the atomic parameters such as Z. However, there is a physical connection between the ponderomotive energy and the inner-shell cross section thresholds described above; i.e., the ponderomotive energy is approximately equal to the threshold for the inner shell cross section. By Moseley’s law, this results in a ${\sim}{Z^2}$ dependence for ${U_p}$. Non-relativistic laser–matter interactions have a rescattering fluence at the cutoff that scales as $1/U_p^2$ [13]. The result is rescattering dynamics contribute ${\sim}1/{Z^4}$ dependence to the RSI yield. Combining both factors results in a net ${\sim}1/{Z^8}$ scaling. The final dependence on $\lambda$ comes from the inverse dependence of the rescattering fluence on the wavelength ${\sim}1/{\lambda ^2}$ due to the rescattering electron wave function spread in the continuum [13]. This heuristic argument results in an inner shell yield dependence that scales roughly as $1/({Z^8}{\lambda ^2})$, which is comparable to the empirically observed $1/({Z^8}{\lambda ^{1.4}})$.

3. CONCLUSION

In conclusion, we show that laser rescattering can access high energy, $ {\rm K}- {{\rm{L}}_{\rm{I}}}$-shell excitation across the periodic table from lithium to uranium. The optimized laser conditions for RSI follow a simple scaling as a function of the laser wavelength and the atomic number. This result is promising because many laser rescattering processes across the periodic table may follow a predictable scaling for the ideal drive laser parameters.