Fig. 1. Spectrum of the squared modulus of the pulse Fourier transform divided by the total signal energy ϵ_tot; see the text. The blue, green, and red curves represent the calculations for regular spikes, the simulations for random spikes, and the expectation values, respectively. The parameters of the pulse are T = 20 fs, τ = 0.1 fs, ω_c = 1560 eV, and n = 20.

Fig. 2. Total probability of K-shell ionization of solid aluminum for different carrier frequencies ω_c as a function of the pulse duration τ of a single spike. Spikes are equidistant with zero phase, the envelope characteristic time T = 15 fs, and the number of spikes n = 24. The K-shell ionization threshold is at ℏω_ion = 1559.6 eV.

Fig. 3. Energy dependence of the K-shell absorption cross-section of solid aluminum for different temperatures near threshold.

Fig. 4. Ionization probability of K-shell absorption in solid aluminum as a function of carrier frequency for different durations of regular spikes for T = 20 fs, n = 20, and Θ = 0.025 eV.

Fig. 5. Ionization probability as a function of carrier frequency. Solid lines correspond to single Gaussian pulses of duration τ_G = 20 fs at different temperatures Θ = 2.4 eV (black) and 6.2 eV (blue). Dot-dashed lines correspond to x-ray pulses composed of regular spikes at room temperature for T = 20 fs, n = 50, and spike duration τ = 0.1 fs (red) and 0.05 fs (magenta).

Fig. 6. Probability of K-shell ionization of solid aluminum as a function of carrier frequency for T = 20 fs, n = 50, Θ = 0.025 eV, and different durations of random spikes. Solid lines are simulations for random spikes (with different colors corresponding to different runs of random parameters), dashed lines are simulations for regular spikes, and crosses represent the results of the expectation value approach.