  Exploring physics and potential applications in vectorially structured light
with propagation-invariant transverse structures has benefited many areas of
modern optics and photonics. In this paper, we focus on non-eigen vector modes
of paraxial light fields, and systemically investigate propagation variations
and revivals of their transverse structures, including both amplitude and
polarization profiles. We show that the deeper mechanism behind the evolution
of the transverse structure is the variations and revivals of intramodal phases
within spin-orbit coupled space and spatial-mode subspace, which originates
from fractional Gouy phases, or rather, Geometric-phase difference between
spatial modes under a same unitary transformation. This underlying principle,
provides a general guideline for shaping vectorially structured light with
custom propagation-evolution properties, and may also inspire many other new
applications based on structured light.
