Homotopic Vortex Lasing Mediated by a Resonant Geometric Phase in a Bound States in the Continuum Cavity.

Geometric phases are fundamental to wave physics and underpin a broad range of optical phenomena. The most prominent example, the Pancharatnam-Berry (PB) phase, originates from polarization evolution during wave propagation. In contrast, geometric phases associated with optical resonances and cavities remain largely unexplored. Here, we uncover the relation between the generalized PB phase and the far-field polarization evolution in bound states in the continuum (BIC) cavities. We demonstrate the continuous homotopic evolution of vortex lasing modes through geometric-phase engineering of BIC resonances. To experimentally validate this concept, we realize metasurface lasers composed of titanium dioxide hole arrays integrated with CdSe/CdxZn1-xS core/shell colloidal quantum wells as the gain medium. Our results reveal a previously unexplored form of spin-orbit interaction in BIC cavities under oblique incidence and provide a versatile route for on-chip control of polarization and orbital angular momentum in nanolasers.