X-ray emission from layered media irradiated by an x-ray free-electron laser

This article presents a computational study of the x-ray fluorescence induced by the irradiation of thin layered media by intense, short x-ray pulses. The treatment is based on a numerical solution of the Helmholtz wave equation both for the pump and for the fluorescence signal. Consistently with a possible heating of the medium during the x-ray pulse, complex refractive indices are calculated at each time step from the results of an underlying treatment of atomic physics. In the context of an important core-hole production as a result of photoionization, we discuss the peculiarities of the resulting amplified fluorescence grazing emission and of the Bragg diffraction which can be realized at some angles inside a multilayer material or even in a perfect crystal.

Figure 1

Schematic diagram of a multilayered material irradiated by a plane wave in the x-ray range. ${\vec{E}}_i$ is the incident electric field (excitation field or pump). ${\vec{k}}_i$ is the corresponding wave vector. A reflected and a transmitted field also exist (not indicated in the figure). ${\vec{E}}_{\text{out}}$ is the outgoing (fluorescent) electric field. By virtue of the reciprocity theorem, it is calculated in a similar way to the incident field but at the fluorescence wavelength. ${\theta }_{\text{out}}$ is the glancing angle of detection. An electric field separates into two components: $S$ (along the $z$ axis) and $P$ (in the plane $xy$).

Figure 2

Sketch of a stack of bilayers (of two different elements) excited by x rays above an absorption edge ($K, L$, or $M$) in one of the two elements. This results in a $K\alpha ,\beta , L\alpha ,\beta ,\gamma ,...,$ or $M\alpha$ fluorescence emission which can be observed as a function of a glancing angle ${\theta }_{\text{out}}$. Thicknesses of a bilayer are $e_1$ and $e_2$, respectively, while refractive indices are $n_1$ and $n_2$, respectively. The period is $\mathrm{\Lambda }=e_1+e_2$.

Figure 3

Main levels (of solid Mg) involved in a pumping-fluorescence scheme. Other collisional-radiative couplings (not shown) are taken into account in the modeling. In solid density Mg (cold or hot), $\left[3s^2\right]$ electrons are delocalized free electrons.

Figure 4

Calculated angular scan for the Mg $K\alpha$ line emitted by a stack ${(\mathrm{Mg}/\mathrm{Co})}_{30}$ ($e_1=5.45$ nm and $e_2=2.55\mathrm{nm}$) irradiated by x rays above the Mg $K$ edge. Kossel patterns are labeled by their Bragg order $n$.

Figure 5

Calculated angular scan for the Co $K\alpha$ line emitted by two stacks ${(\mathrm{Mg}/\mathrm{Co})}_{50}$ and ${(\mathrm{Mg}/\mathrm{Co})}_{100}$ irradiated by x rays above the Co $K$ edge.

Figure 6

Calculated angular scan around the critical angle for the Co $K\alpha$ line emitted by slabs of different thicknesses. A preliminary uniform excitation ($K$-shell hole production) has been supposed over the whole slabs.

Figure 7

Calculated angular scan for the Mg $K\alpha$ line emitted by a stack ${(\mathrm{Mg}/\mathrm{vacuum})}_{30}$. A uniform excitation ($K$-shell hole production) has been supposed through the whole sample.

Figure 8

Square of the electric field at the Mg $K\alpha$ energy (1253.6 eV) inside the ${(\mathrm{Mg}/\mathrm{Co})}_{30}$ stack ($e_1=5.45\mathrm{nm}, e_2=2.55\mathrm{nm}$), for different imposed values of the imaginary part of the refractive index in the Mg layers. Here, ${\tilde{n}}_{\text{Mg}}=1–1.52149\times {10}^{-4}-i\beta$ while ${\tilde{n}}_{\text{Co}}=1–9.99187\times {10}^{-4}-i3.97076\times {10}^{-4}$. Results are normalized to an incident intensity $|E_o{|}^2$.

Figure 9

Maximum of the emerging intensity on the Co $K{\alpha }_1$ found in a glancing angular interval around the first Bragg angle (${\theta }_B=1.{025}^{\circ }$) as a function of the imaginary part of the refractive index of Co. Three stacks ${(\mathrm{Mg}/\mathrm{Co})}_N$ with $e_1=2.5\mathrm{nm}, e_2=2.5\mathrm{nm}$, and $N=10,30,50$ are considered. Here the pumping is uniform, i.e., arbitrarily $j_i=1$ for all cells $i$.

Figure 10

Snapshots of the outgoing Mg $K\alpha$ emission from a ${(\mathrm{Mg}/\mathrm{Co})}_{30}$ stack in real pumping conditions (1332 eV photons, ${10}^{16}\mathrm{W}/{\mathrm{cm}}^2$, FWHM 10 fs, normal incidence). Indicated instants correspond to the time elapsed from the moment the pulse enters the multilayer structure (the peak of the pulse being at 12.8 fs).

Figure 11

Spatial profile of electronic temperature at the end of the x-ray pulse, in a ${(\mathrm{Mg}/\mathrm{Co})}_{30}$ stack of period $\mathrm{\Lambda }=8\mathrm{nm}$, irradiated at normal incidence by a Gaussian XFEL pulse at ${10}^{16}\mathrm{W}/{\mathrm{cm}}^2$, of 1332 eV photons, and of 10 fs duration (FWHM). The x-ray pulse comes from the right.

Figure 12

Time integration (over the pulse) and angular integration of the outgoing Mg $K\alpha$ emissivity as a function the (XFEL) pump intensity for a 10 fs duration Gaussian pulse of 1332 eV photons. The sample is a ${(\mathrm{Mg}/\mathrm{Co})}_{30}$ stack with $e_1=5.45$ nm and $e_2=2.55$ nm. Solid line: integration on the interval $[2.5^{\circ }–5^{\circ }]$ (around the first Kossel pattern). Dashed line: integration on the interval $[{45}^{\circ }–47.5^{\circ }]$ (out of the Kossel directions).

Figure 13

Snapshots of the grazing outgoing Co $K{\alpha }_1$ emission from a ${(\mathrm{Mg}/\mathrm{Co})}_{50}$ stack ($e_1=e_2=2.5\mathrm{nm}$) in real pumping conditions (7720 eV photons, FWHM 10 fs, normal incidence). (a) $6\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$. (b) $3\times {10}^{19}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 14

Angular scan of the Si $K\alpha$ emission, at different times during the irradiation of the stack ${(\mathrm{Si}/\mathrm{vacuum})}_{1000}$ ($e_1=0.1309\mathrm{nm}, e_2=0.2541\mathrm{nm}$) by a ${10}^{17}\mathrm{W}/{\mathrm{cm}}^2$ pulse, 10 fs long, of 1900 eV photons. The inset is a zoom on the Kossel diffraction region around ${67}^{\circ }$.

Figure 15

Angular scan of the Si $K\alpha$ emission, at different times during the irradiation of the stack ${(\mathrm{Si}/\mathrm{vacuum})}_{10000}$ ($e_1=0.1309\mathrm{nm}, e_2=0.2541\mathrm{nm}$) by a ${10}^{17}\mathrm{W}/{\mathrm{cm}}^2$ Gaussian pulse, 10 fs duration (FWHM), of 1900 eV photons. The inset is a zoom on the Kossel diffraction region around ${67}^{\circ }$.

Figure 16

Angular scan of the Si $K\alpha$ emission, at different times during the transverse irradiation (along the $y$ axis; cf. Fig. 1) of the stack ${(\mathrm{Si}/\mathrm{vacuum})}_{5000}$ ($e_1=0.1309\mathrm{nm}, e_2=0.2541\mathrm{nm}$) by Gaussian pulses, 10 fs duration (FWHM), of 1900 eV photons. (a) ${10}^{16}\mathrm{W}/{\mathrm{cm}}^2$. (b) ${10}^{17}\mathrm{W}/{\mathrm{cm}}^2$. Thickness (along the $y$ axis) is less than 1 $\mu \mathrm{m}$.