Adapted Gouy phase and multi-topological reactions in a strongly focusing system with a parabolic mirror

In this paper, we present a systematic investigation into the intrinsic correlations among Gouy phase dynamics, wavefront spacings, and topological reactions in a high-numerical-aperture (NA) parabolic mirror system. Since the NA can theoretically approach 2 in such a system, a paradox is observed between the behaviors of Gouy phase and wavefront spacing by using the conventional definition when the semi-aperture angle is greater than 90^∘ (i.e., NA >1). To resolve this issue, we analyze the underlying physical mechanism of the paradox and subsequently propose an adapted redefinition of the Gouy phase. When a vortex is embedded in the incident beam, abrupt phase transitions and anomalous wavefront spacings are observed. These phenomena are attributed to the topological reactions of two-dimensional phase singularities in the vicinity of the optical axis, with close associations to the propagation properties of three-dimensional singularities (i.e., spin density phase singularities and spin density vector singularities). Furthermore, we demonstrate that strategic modulation of the semi-aperture angle and beam size parameter enables programmable control over wavefront spacing from 0.9λ to 5λ. These findings may establish parabolic mirrors as versatile platforms for topological photonics and quantum metrology, bridging singular optics with phase-front engineering to advance light-matter interaction control and nanoscale phase gradient applications.