Modeling of the interaction of an x-ray free-electron laser with large finite samples

We describe a model for the study of the interaction of short x-ray free-electron laser (XFEL) pulses with large finite samples. Hydrodynamics is solved in one-dimensional planar geometry with consideration of the electron-ion energy exchange and of the possible elastoplastic behavior. From a time-dependent calculation of the complex refractive index and of the underlying atomic physics, XFEL energy deposition is modeled through a calculation of the radiation field in the material. In the case of hard x-ray irradiation, energetic electrons induced by the XFEL absorption can propagate and deposit their energy outside the interaction region. Simulations of the interaction of hard x-ray ultrashort pulses with solid materials Ru and Si at different grazing incidence angles are presented and discussed. The results obtained demonstrate the potential of this approach to predict damage dynamics for materials of interest for x-ray optics.

Figure 1

Sketch of the 1D material irradiated by a plane wave in the x-ray range. The material is discretized into different layers, the thicknesses of which may vary over time. ${\vec{E}}_i, {\vec{E}}_r$, and ${\vec{E}}_t$ are the incident, reflected, and transmitted electric fields, respectively. ${\vec{k}}_i, {\vec{k}}_r$, and ${\vec{k}}_t$ being the corresponding wave vectors. $\theta$ is the grazing incidence angle. Components of the electric field separate into two components: component S (along the $z$ axis) and component P (in the plane $xy$).

Figure 2

Calculated absorption coefficient $N\kappa$ vs photon energy for cold solid ($\rho =12.2$ g ${\mathrm{cm}}^{-3}$) Ru (black) and for warm solid Ru (red).

Figure 3

Calculated real part of the atomic scattering factor $f_1$ vs photon energy for cold solid Ru (black) and for warm solid Ru (red). The inset is a magnification in the region around 3.2 keV.

Figure 4

Calculated reflectivity for Mo $K\alpha$ radiation (17.5 keV) of a Ru-coated Si substrate (50 nm of Ru on 1-$\mu \mathrm{m}$-thick Si).

Figure 5

Calculated electron temperature profiles at different times in a Si/Ru target without transport of energetic photoelectrons (purely local absorption and diffusive transport). The intensity is of $1.55\times {10}^{16}$ W/${\mathrm{cm}}^2$, the (Gaussian) pulse duration is 20 fs (FWHM), and the photon energy is 7 keV. XFEL radiation comes from the right with a grazing incidence of 4.5 mrad. The thick black curve represents the initial density profile of the sample.

Figure 6

Calculated electron temperature profiles in a Si/Ru target with transport of energetic photoelectrons. Irradiation conditions of Fig. 5. The thick black curve represents the initial density profile of the sample.

Figure 7

Calculated electron temperature profiles at different times in a Si/Ru target with transport of energetic photoelectrons. The intensity is of $2.35\times {10}^{17}$ W/${\mathrm{cm}}^2$, the pulse (Gaussian) duration is 20 fs (FWHM), and the photon energy is 12 keV. XFEL radiation comes from the right with a grazing incidence of 3 mrad. The thick black curve represents the initial density profile of the sample.

Figure 8

Calculated ion temperature profiles in a 1-mm-thick bulk Si target with transport of energetic photoelectrons. The intensity is of $3.1\times {10}^{15}$ W/${\mathrm{cm}}^2$, the pulse (Gaussian) duration is 20 fs (FWHM), and the photon energy is 10 keV. XFEL radiation comes from the right at normal incidence.

Figure 9

Calculated electron temperature profiles in a 1-mm-thick bulk Si target with transport of energetic photoelectrons. Irradiation conditions of Fig. 8.

Figure 10

Calculated electron temperature profiles in a 1-mm-thick bulk Si target with transport of energetic photoelectrons. The intensity is of $9\times {10}^{15}$ W/${\mathrm{cm}}^2$, the pulse (Gaussian) duration is 20 fs (FWHM), and the photon energy is 10 keV. XFEL radiation comes from the right at normal incidence.