Tracking the ultrafast XUV optical properties of x-ray free-electron-laser heated matter with high-order harmonics

We present measurements of photon absorption by free electrons as a solid is transformed to plasma. A femtosecond x-ray free-electron laser is used to heat a solid, which separates the electron and ion heating time scales. The changes in absorption are measured with an independent probe pulse created through high-order-harmonic generation. We find an increase in electron temperature to have a relatively small impact on absorption, contrary to several predictions, whereas ion heating increases absorption. We compare the data to current theoretical and numerical approaches and find that a smoother electronic structure yields a better fit to the data, suggestive of a temperature-dependent electronic structure in warm dense matter.

Figure 1

The experimental arrangement for the measurement of XUV absorption in an XFEL heated aluminum foil. A 40 fs IR pulse at a wavelength of 800 nm and intensity $\sim {10}^{15}$ W ${\mathrm{cm}}^{-2}$ is focused into a gas cell of 1.5 cm in length and filled with argon (Ar) at a pressure of 30 mbar to generate high-order harmonics in the XUV. The XUV beam is then focused by a multilayer mirror (ML) near the target plane. The XUV pulse is partially transmitted through the foil, and the remaining XUV light is absorbed. An adjustable slit is placed after the target to block the XUV light that does not pass through the plasma. The absorption of three high-order harmonics through the foil target is measured using an XUV spectrometer. A typical spectrum transmitted through the cold (blue) and heated (noisy red) aluminum foil is shown in the inset. By dividing the red by the blue, we can extract the transmission relative to cold at three photon energies. Spectra transmitted through the foil were measured at different times with respect to the peak of the XFEL. The measured relative transmission time histories are plotted in Figs. 4 and 5. The (a) zero order and (b) spectrally resolved regions are separated for clarity in the inset. The noise in the plasma shot in (b) is due to scattered hard x rays. The harmonic order (HO) of the three photon energies is shown in (b).

Figure 2

(a) The smoothed time-integrated XUV plasma emission measurement (solid noisy thick black line) with error bars (dashed thin black lines), and the best fit with a peak electron temperature of $T_e=13\pm 3.25$ eV (solid green line). (b) The measured fluence profile of the XFEL spot on target (blue circles) and fit (dashed blue line) which is used in the TTM to calculate the temperature profile across the target (solid red line, a 0.5-$\mu \mathrm{m}$-width line out). The solid black lines indicate the borders of the probed region $R_1$ in (c). (c) The calculated two-dimensional peak electron temperature profile using the measured beam fluence profile. The dashed lines in (c) enclose the area probed by the XUV pulse, $R_1$, and the area that dominates the XUV emission spectrum, $R_2$, in (a). (d) The calculated spatial average of the electron (solid blue line) and ion (dashed red line) temperatures in time of the probed area, $R_1$, from the TTM. The three time scales of interest are shown in (d) (arrows and dash-dotted lines): $\mathrm{\Delta }{\tau }_c$ is the cold solid foil, $\mathrm{\Delta }{\tau }_e$ is the time scale of electron heating only, and $\mathrm{\Delta }{\tau }_w$ is the time scale of electron-ion coupling.

Figure 3

The electron-phonon coupling coefficient used in the TTM, $G_{ep}$, as a function of time from the peak of the heating pulse for three functions that give a melting time of $\sim 1$.5 ps, where the maximum value of $G_{ep}$ is given as $G_{max}=3.47\times {10}^{17}$ W ${\mathrm{m}}^{-3}$ ${\mathrm{K}}^{-1}$ (broken black lines): the theoretical value of $G_{ep}$ at an electron temperature of 5 eV (red line) [21], theoretical value of $G_{ep}$ (green line) [22], experimental value of $G_{ep}$ (blue line) [23], experimental value of $G_{ep}$ (solid black line) [24], theoretical values of the coupled mode calculation in the range of $T_e$ in this study and $T_i=0.08$ eV (shaded black region) [25, 26].

Figure 4

The experimental transmission of zero order (all harmonics) relative to cold of a 600 nm aluminum foil heated by an XFEL pulse (black squares). The time axis is relative to the peak of the XFEL pulse. Calculated transmission values for the electron-ion potential $V_1$ (dashed red line) and $V_2$ (solid blue line), for the average transmission of the probed region, are shown. A delay in electron-ion coupling is incorporated into the TTM to give a heating rate in agreement with other studies (detailed in Sec. 3). A semianalytic model by Iglesias et al. [9] predicting changes due to electron heating is shown in the $\mathrm{\Delta }{\tau }_e$ phase (boxed green star). The three time scales of interest are shown (arrows and dash-dotted lines): $\mathrm{\Delta }{\tau }_c$ is the cold solid foil, $\mathrm{\Delta }{\tau }_e$ is the time scale of electron heating only, and $\mathrm{\Delta }{\tau }_w$ is the time scale of electron-ion coupling.

Figure 5

Spectrally resolved relative transmission through the XFEL heated foil for the three photon energies of (a) 41.8 eV, (b) 38.8 eV, and (c) 35.7 eV used in this study. The calculated relative transmission for the temperatures in the probed region $[R_1$ in Fig. 2] for the electron-ion potentials $V_1$ (dashed line) and $V_2$ (solid line) is shown for each photon energy. The three time scales of interest are shown (arrows and dash-dotted lines): $\mathrm{\Delta }{\tau }_c$ is the cold solid foil, $\mathrm{\Delta }{\tau }_e$ is the time scale of electron heating only, and $\mathrm{\Delta }{\tau }_w$ is the the time scale of electron-ion coupling. The results of DFT+$U$ calculations are shown for the $\mathrm{\Delta }{\tau }_e$ and $\mathrm{\Delta }{\tau }_w$ time scales of interest and discussed in Sec. 6.

Figure 6

An electron absorbing energy from an optical field in the presence of an ion represented by a pseudopotential of the nucleus and bound electrons with (a) small and (b) large cutoff radius. Shielding of the nucleus by the bound electrons is represented by the cutoff radius, $r_c$, which determines the range of potentials the electron can interact with. (c) The pseudopotential approach to treating the electron-ion potential near the bound (core) electrons, for a sharp cutoff radius $V_1$ (sharp red curve) and a soft cutoff $V_2$ (smooth blue curve) according to Eq. (3).

Figure 7

The same as Fig. 4 with a different vertical scaling and including predictions using DFT calculations. The experimental transmission of zero order (all harmonics) relative to cold of a 600-nm aluminum foil heated by an XFEL pulse (black squares). The time axis is relative to the peak of the XFEL pulse. Calculated transmission values for the electron-ion potential, $V_1$ (dashed red line) and $V_2$ (solid blue line), for the average transmission of the probed region are shown. A delay in electron-ion coupling is incorporated into the TTM to give a heating rate in agreement with other studies (detailed in Sec. 3). A semianalytic model by Iglesias et al. [9] predicting changes due to electron heating is shown in the $\mathrm{\Delta }{\tau }_e$ phase (boxed green star). Standard DFT (solid purple circles) and DFT+$U$ (pink diamonds) are shown with the boxed area representing the uncertainties in temperature and temporal resolution. The three time scales of interest are shown (arrows and dash-dotted lines): $\mathrm{\Delta }{\tau }_c$ is the cold solid foil, $\mathrm{\Delta }{\tau }_e$ is the time scale of electron heating only, and $\mathrm{\Delta }{\tau }_w$ is the time scale of electron-ion coupling.

Figure 8

DFT calculations of the density of states (DOS) (solid lines), electron transitions for 39 eV photons (arrows), and transition probabilities $P_r$ (color-scale shading). The calculations correspond to the three temporal regions of interest ($\mathrm{\Delta }{\tau }_c$, $\mathrm{\Delta }{\tau }_e$, and $\mathrm{\Delta }{\tau }_w$) shown in Fig. 7 (top arrows). The $\mathrm{\Delta }{\tau }_c$ region corresponds to cold electrons in a cold lattice, the $\mathrm{\Delta }{\tau }_e$ phase corresponds to $T_e=6$ eV in a cold lattice, and $\mathrm{\Delta }{\tau }_w$ corresponds to $T_e=6$ eV and $T_i=2$ eV. (a)–(c) Standard DFT calculations corresponding to the DFT points in Fig. 7. (d)–(f) DFT+$U$ calculations corresponding to the DFT+$U$ points in Figs. 5 and 7.