Retrieval of full angular- and energy-dependent complex transition dipoles in the molecular frame from laser-induced high-order harmonic signals with aligned molecules

Abstract

High-order harmonic signals generated in molecules are the consequence of coherent summation of complex laser-induced transition dipoles $d(\ensuremath{\theta},\ensuremath{\omega})$ with each fixed-in-space molecule; here $\ensuremath{\theta}$ is the angle of the molecular axis with respect to the laser polarization axis and $\ensuremath{\omega}$ is the harmonic energy. In the so-called rotational coherent spectroscopy, it is proposed to extract the fixed-in-space $d(\ensuremath{\theta},\ensuremath{\omega})$ in the molecular frame by measuring harmonics generated by a probing laser from the rotational molecular wave packets that have been prepared by a prior pump laser. By varying the time delay between the two lasers, methods have been utilized to extract the $\ensuremath{\theta}$ dependence of both the amplitude and phase of each individual harmonic, but the relative phase between harmonics cannot be retrieved. Here we report that this limitation can be removed. It requires the additional measurement of harmonic spectra versus the pump-probe angles at one fixed time delay. The two-dimensional input harmonic data (time-delay and pump-probe angle) are then used to retrieve the full complex transition dipole $d(\ensuremath{\theta},\ensuremath{\omega})$ using a retrieval method based on machine learning algorithms. We demonstrate this method on ${\mathrm{N}}_{2}$ and ${\mathrm{CO}}_{2}$ molecules.