A geometric approach to decoding molecular structure and dynamics from photoionization of isotropic samples.

We propose a geometric approach to the description and analysis of photoelectron angular distributions resulting from isotropic samples in the case of few-photon ionization by electric fields of arbitrary polarization. This approach formulates the standard photoionization observables - the expansion coefficients of the photoelectron angular distribution, in terms of geometrical properties of the vector field () ≡ 〈||0〉 describing the electronic transition from a bound state |0〉 into a scattering state |〉 - the photoionization transition dipole. Besides revealing selection rules for the enantio-sensitivity of coefficients in multiphoton ionization, our approach yields very compact expressions for both chiral and achiral molecules revealing how the molecular rotational invariants couple to the rotational invariants of the setup defined by the electric field polarization and the arrangement of photoelectron detectors. We apply this approach to one-photon ionization and find that the forward-backward asymmetry parameter , emerging exclusively in chiral molecules and encoded in the field () ≡ *() × (), is sensitive only to the components of () perpendicular to , while the regular asymmetry parameter emerging in chiral and achiral molecules is sensitive only to the component of () parallel to . Next, we analyze resonantly enhanced two-photon ionization and show that and can be written in terms of an effectively stretched (), and how and can be used to probe ().