Generalized Geometric Phase for Coupled Meta-Atoms

The geometric phase, such as the Pancharatnam-Berry (PB) phase, is widely employed in metasurfaces for broad functionalities. Conventional studies dominantly rely on the linear dependence of the PB phase on the orientation angle of the meta-atoms. Although some previous studies indicate the breakdown of such a linear dependence, its underlying mechanisms remain largely unknown. In this study, we provide a detailed analysis of the evolution of PB phases, unveiling the nonnegligible role of coupling among adjacent meta-atoms. From the perspective of quasi-normal modes (QNMs), we demonstrate that the PB phase is solely related to the far-field radiation polarizations of the QNMs excited, decided by both meta-atom orientations and their mutual coupling. We experimentally demonstrate these effects using a meticulously designed grating, aligning closely with our theoretical predictions. This study establishes a systematic framework for rigorous PB phase analysis, broadening the horizon for the high-precision characterization of meta-optical devices.