  Symmetry-driven phenomena arising in nonlocal metasurfaces supporting
quasi-bound states in the continuum (q-BICs) have been opening new avenues to
tailor enhanced light-matter interactions via perturbative design principles.
Geometric phase concepts - observed in many physical systems - are particularly
useful in nonlocal metasurfaces, as they enable to locally pattern the q-BIC
scattering rate and phase across the metasurface aperture without affecting the
delocalized nature of the q-BIC resonance. However, this control typically
comes with stringent limitations in terms of efficiency and/or of polarization
operation. Here, we unveil a new form of geometric phase control, accumulated
along a continuous contour of geometric perturbations that parametrically
encircle a singularity associated with a bound state. This response is obtained
regardless of the chosen polarization state, which can be rationally specified
by a judicious choice of the perturbation geometry. Our findings extend
geometric phase concepts to arbitrary polarizations, including linear and
elliptical states, with near-unity scattering efficiency. This
polarization-agnostic geometric phase offers new opportunities for wavefront
manipulation, multifunctional operation and light emission, with applications
in augmented reality, secure communications, wireless systems, imaging, lasing,
nonlinear and quantum optics.
