Influence of zero-crossing return timing on focal-point behavior of backscattered H-CQSFA trajectories

Determine whether and how the electron’s return timing relative to a laser-field zero crossing controls the focal-point behavior (including the observed parity of focal points) along the short and long backscattered trajectories that form the rescattering ridge within the hybrid forward-boundary Coulomb Quantum-Orbit Strong-Field Approximation (H-CQSFA) for a hard-core Coulomb potential, and establish the underlying physical mechanism for this dependence.

Background

In the H-CQSFA analysis of the primary rescattering ridge for a hard-core Coulomb potential, the authors identify two branches of solutions (short and long backscattered orbits) that coalesce at the ridge. They observe that the short trajectory exhibits an odd number of focal points, while the long trajectory exhibits an even number, with each focal point occurring at a time when the Jacobian vanishes.

Focal points are directly tied to caustics in the photoelectron momentum distribution because the stability factor diverges when the Jacobian determinant vanishes. Understanding what controls the number and occurrence of focal points is therefore essential for characterizing the structure and formation of rescattering ridges and associated caustics.

The authors hypothesize that whether an electron returns before or after a field zero crossing may influence its focal-point dynamics, but they state that they cannot yet make concrete statements about this mechanism. Clarifying this connection would explain the observed parity difference between short and long backscattered trajectories and refine the catastrophe-theory interpretation of the mapping between initial and final momenta.