Extreme-ultraviolet-initiated High-harmonic Generation in Ar$^{+}$

We employ the R-matrix with time-dependence method to investigate extreme-ultraviolet-initiated high-harmonic generation (XIHHG) in Ar$^{+}$. Using a combination of extreme-ultraviolet (XUV, $92\textrm{ nm}$, $3\times 10^{12}\,\textrm{Wcm}^{-2}$) and time-delayed, infrared (IR, $800\textrm{ nm}$, $3\times 10^{14}\,\textrm{Wcm}^{-2}$) laser pulses, we demonstrate that control over both the mechanism, and timing, of ionization can afford significant enhancements in the yield of plateau, and sub-threshold, harmonics alike. The presence of the XUV pulse is also shown to alter the relative contribution of different electron emission pathways. Manifestation of the Ar$^{+}$ electronic structure is found in the appearance of a pronounced Cooper minimum. Interferences amongst the outer-valence $3p$, and inner-valence $3s$, electrons are found to incur only a minor suppression of the harmonic intensities, at least for the present combination of XUV and IR laser light. Additionally, the dependence of the XIHHG efficiency on time delay is discussed, and rationalized with the aid of classical trajectory simulations.