Influence of chirp and carrier-envelope phase on non-integer high-harmonic generation

High harmonic generation (HHG) is a versatile technique for probing ultrafast electron dynamics. While HHG is sensitive to the electronic properties of the target, HHG also depends on the waveform of the laser pulse. As is well known, (peak) positions, $\omega$, in the high-harmonic spectrum can shift when the carrier envelope phase (CEP), $\varphi$ is varied. We derive formulae describing the corresponding parametric dependencies of CEP shifts; in particular, we have a transparent result for the (peak) shift, $d\omega/d\varphi = {-} 2 \bar{\mathfrak f}' \omega/\omega_0$, where $\omega_0$ describes the fundamental frequency and $\bar{\mathfrak f}'$ characterizes the chirp of the driving laser pulse. We compare the analytical formula to full-fledged numerical simulations finding only 17 % average relative absolute deviation in $d\omega/d\varphi$. Our analytical result is fully consistent with experimental observations.