Attosecond pulses at kiloelectronvolt photon energies from high-order-harmonic generation with core electrons

High-order harmonic generation (HHG) in simultaneous intense near-infrared (NIR) laser light and brilliant x rays above an inner-shell absorption edge is examined. A tightly bound inner-shell electron is transferred into the continuum. Then, NIR light takes over and drives the liberated electron through the continuum until it eventually returns to the cation leading in some cases to recombination and emission of a high-order harmonic (HH) photon that is upshifted by the x-ray photon energy. We develop a theory of this scenario and apply it to $1s$ electrons of neon atoms. The boosted HH light is used to generate a single attosecond pulse in the kiloelectronvolt regime. Prospects for nonlinear x-ray physics and HHG-based spectroscopy involving core orbitals are discussed.

Figure 1

Schematic of the modified three-step model of HHG with core electrons ionized by x rays. (a) X-ray absorption ejects a core electron (b) which is subsequently driven through the continuum by the NIR light; (c) upon returning to the parent ion it may recombine with the core hole and release its excess energy in terms of a high-order harmonic of the NIR laser upshifted by the x-ray photon energy. The NIR laser has only a noticeable influence on valence and continuum electrons and barely influences core electrons. The converse holds true for the interaction with the x rays.

Figure 2

Two exemplary SASE x-ray FEL pulses (black and red) for an x-ray photon energy of ${\omega }_{\mathrm{X}}=569{\omega }_{\mathrm{L}}=881.8\mathrm{eV}$. The electric field envelope $E_{0\mathrm{X}}\left(t\right)$ is specified in units of the peak electric field $E_{0\mathrm{X}}^{\prime }$ for the peak intensity of $2.2\times {10}^{16}\mathrm{W}/{\mathrm{cm}}^2$ and is represented by solid lines whereas the phase ${\phi }_{\mathrm{X}}\left(t\right)$ is depicted as dashed lines with CEP ${\phi }_{\mathrm{X},0}=0$. The average FWHM pulse duration of an ensemble of SASE pulses is 1.5 optical cycles with a cosine square pulse envelope [35]. The average pulse spectrum is taken to be Gaussian with a FWHM bandwidth of 8 eV which corresponds to a coherence time of 0.24 fs.

Figure 3

Harmonic photon number spectra (HPNS) [Eq. (16)] for $1s$ core electrons of neon using the SASE pulses of Fig. 2. The solid, black line is from the black SASE pulse and the dashed, red line from the red pulse. The x-ray intensity (caption of Fig. 2) was chosen such that the ionization probability is the same as the tunnel ionization probability by the NIR laser. The NIR laser vector potential ${\vec{A}}_{\mathrm{L}}\left(t\right)$ has a cosine square pulse envelope [35] with a FWHM duration of 1.5 optical cycles, CEP ${\phi }_{\mathrm{L},0}=\pi /2$, and a peak intensity of $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ for 800 nm wavelength, i.e., the photon energy is ${\omega }_{\mathrm{L}}=1.55\mathrm{eV}$.

Figure 4

Difference plot [Eq. (21)] of the time-frequency analysis [Eq. (19)] of HHG with $1s$ core electrons of neon for the black and the red SASE pulses of Fig. 2. The plotted values are unity for $|{\check{\stackrel{̃}{D}}}_1(t,\omega )|=|{\check{\stackrel{̃}{D}}}_2(t,\omega )|$ and they are zero in the limit that the absolute value of the difference between both dipole moments goes to infinity. For the plotting, we set values ${\log }_{10}{\left|{\check{\stackrel{̃}{D}}}_1(t,\omega )\right|}^2<-22$ to $-80$ and values ${\log }_{10}{\left|{\check{\stackrel{̃}{D}}}_2(t,\omega )\right|}^2<-22$ to $-40$ in Eq. (21).

Figure 5

Attosecond pulses from x-ray-boosted HHG for harmonics with $\omega \ge 608{\omega }_{\mathrm{L}}$. The pulse shapes are approximated by Gaussian functions with a FWHM duration of $220\mathrm{as}$. Line styles as in Fig. 3.

Figure 6

Schematic of (a) propagation of a NIR laser pulse, FEL x rays, and the generated HH light in a macroscopic medium. (b) Longitudinal phase matching condition of HH emission by two atoms. Here, $L$ is the length of the macroscopic medium and $\theta$ is the opening angle of the cone of coherent HH emission.