Retrieval of complex angle-dependent transition dipoles by using macroscopic high harmonics generated from aligned N₂ molecules

The high-order harmonic generation (HHG) by aligned molecules is the consequence of coherent summation of the complex laser-induced dipoles according to the alignment distribution of the molecules. Using the single-molecule quantitative rescattering (QRS) model with the known alignment distribution and the strong-field ionization of fixed-in-space molecules, the complex angle-dependent photorecombination transition dipoles between the ground and ionic states of molecules can be retrieved from the single-molecule HHG. However, such retrieval does not consider the real experimental conditions, such as the propagation of HHG in the medium and the inaccurate measurement of the probe intensity and the alignment degree. Here we propose a method to retrieve the complex angle-dependent molecular frame transition dipoles between the highest occupied molecular orbital (HOMO) and the ionic states of N₂ molecules from the macroscopic HHG of aligned molecules. In this method, the complex transition dipole is expanded with Legendre polynomials, and the HHG is expressed as a function of the modulus and phase of the complex expansion coefficients. The complex expansion coefficients can be determined by fitting the HHG data, then the angle-dependent amplitude and phase of the complex transition dipole can be obtained. We show that the angle-dependent amplitude and phase of the transition dipoles for each individual harmonic can be successfully retrieved at both low and high probe intensity and degrees of alignment. Then we show the inaccuracy of the retrieved angular dependence amplitude and phase due to the uncertainty of the probe intensity and alignment degree. As the probe intensity increases, which is usually not measured precisely in the experiment, HOMO-1 may also contribute to HHG, we thus compare the retrieved amplitude and phase of the transition dipole with or without HOMO-1’s contribution to HHG.