Introduction
Multiphoton processes in the continuum are attracting increasing attention within the attosecond community. Recently, several theoretical frameworks have been developed to address this problem. Without loss of generality, these approaches can be categorized as either numerical or analytical methods. Building on the integration of established methodologies (Boll et al, 2025), we present analytical expressions for the matrix elements governing three-photon transitions, with applications to angularly resolved photoemission time delays. We benchmark our results against numerically exact solutions of the Time-Dependent Schrödinger Equation (TDSE) for hydrogen atoms.

Methods
To pursue our goal, we combine analytical and numerical methods to study angularly resolved time delays in simple atomic systems undergoing three-photon transitions.

Results and Discussion
Our findings show a transition from qualitative to quantitative agreement between analytic and exact numerical results for angularly resolved time delays at increasing photoelectron energy. Furthermore, differences observed at lower kinetic energies may be ascribed to the asymptotic description of intermediate states.

Conclusions
In summary, we demonstrate that analytic (radial) matrix elements, which convey information about the angular quantum numbers of final states, are accurate enough to theoretically describe angularly resolved time delays in simple atomic systems. This assertion is valid for photoelectron energies above ~7 eV, with the quality of the analytic results improving at higher energies.