XUV superfluorescence from helium gas in the paraxial three-dimensional approximation

We present the results of a theoretical study of XUV superfluorescence from doubly excited states of helium resonantly pumped by free-electron laser (FEL) pulses. Our model allows us to predict both the spectrum and angular distribution of emitted XUV radiation in a wide range of experimentally accessible parameters. This is achieved by going beyond two key deficiencies of most previous models: The one-dimensional treatment in space is upgraded to three dimensions with electromagnetic fields treated in the paraxial approximation and spontaneous emission is modeled by a recently developed approach that avoids the unrealistic delayed response but preserves the expected characteristics of the emitted field in the spontaneous emission limit. The case study of $3a{}^1{P}^o$ resonance in helium with $63.66\mathrm{eV}$ excitation energy is presented for realistic parameters of seeded light pulses from the FERMI FEL facility and a recently developed high-pressure gas cell. Results of numerical simulations show that both the spectral width and angular divergence of emitted radiation vary with gas pressure and pump pulse intensity in a complex way.

Figure 1

Schematic representation of the level scheme. A helium atom in the ground state 0 is resonantly photoexcited to the selected doubly excited state $i$, which is coupled to the continuum and decays via fluorescence to a singly excited state $f$ or via autoionization $a$ to ${\mathrm{He}}^{+}1s$. The latter is coupled to ${\mathrm{He}}^{+}2p$ ($b$) by the emitted radiation.

Figure 2

Number of emitted photons as a function of gas pressure and pump pulse energy. Dashed lines denote the corresponding number of pump photons impinging on the target.

Figure 3

Spectral intensity of radiation emitted from the target exit integrated over transverse dimensions for different backing pressures and $40\mu \mathrm{J}$ pump pulse energy. The vertical dashed line denotes the $|i\rangle \text{−}|f\rangle$ transition energy.

Figure 4

Single-repetition and average time-integrated angular spectra of radiation emitted from the target exit for 5 mbar (top row) and 25 mbar backing pressure (bottom row) and $10\mu \mathrm{J}$ pump pulse energy.

Figure 5

(a) Scaled angular spectra of emitted radiation at ${\theta }_y=0$ for different backing pressures and $10\mu \mathrm{J}$ pump pulse energy. (b) Angular divergence of the emitted field (see the text) as a function of backing pressure for two different pump pulse energies. The gray dashed lines denote the (a) angular spectrum and (b) angular divergence of the pump pulse.

Figure 6

Results of a single repetition of the simulation for $25\mathrm{mbar}$ backing pressure and $10\mu \mathrm{J}$ pump pulse energy. The left-hand side of the figure shows the number of emitted photons as a function of propagation distance, along with the scaled longitudinal density profile of gas inside the microfluidic cell. The transverse $(x,y)$ dependence of the gas density is assumed to be constant. Plots on the right-hand side show the scaled emitted field intensity, population inversion in neutral helium ${\rho }_{ii}-{\rho }_{ff}$, and ${\mathrm{He}}^{+}$ population inversion ${\rho }_{bb}-{\rho }_{aa}$ at $y=0\mu \mathrm{m}$ for two propagation distances, i.e., $z=-2.8\mathrm{mm}$ (bottom row) and $z=2.8\mathrm{mm}$ (top row), indicated by black dots on the left-hand-side plot. Time $\tau =0\mathrm{fs}$ coincides with the peak pump pulse intensity.

Figure 7

(a) Shown on the left is the temporal profile of the pump pulse intensity $I_{\mathcal{F}}$, excited-state population ${\rho }_{ii}$, and emitted field intensity $\mathrm{Re}\left(I_{\mathcal{E}}\right)$ at the target exit integrated over transverse dimensions. On the right is the spectral intensity of emitted radiation at the target exit (num.), along with the expected spectral profile (an.) and Lorentzian profile of width $\gamma =9\mathrm{meV}$ (Lorentz). (b) Numerically and analytically calculated numbers of emitted photons as a function of propagation distance.