Riemann-Silberstein geometric phase for high-dimensional light manipulation

Geometric phases provide a powerful mechanism for light manipulation. In particular, the Pancharatnam-Berry (PB) phase has enabled optical metasurfaces with broad applications. However, the PB phase is based on polarization evolution in a two-dimensional space, which fails to account for other polarization degrees of freedom. Here, we generalize the concept of geometric phase to a four-dimensional (4D) Riemann-Silberstein (RS) space that characterizes the complete electromagnetic polarization, including electric, magnetic, and hybrid polarizations. We show that the 4D polarization evolution in the RS space can give rise to a new geometric phase-the RS phase-in addition to the PB phase. The PB phase depends on optical spin and usually manifests in circularly polarized light, whereas the RS phase depends on optical linear momentum and can manifest in arbitrarily polarized light. Their synergy provides a unified geometric framework for light propagation at interfaces and enables unprecedented high-dimensional light control. As a proof of principle, we propose and demonstrate RS metasurfaces capable of multiplexed wavefront shaping, which can reconfigure up to twelve distinct outputs via switching incident 4D polarization. Our work uncovers a new class of optical geometric phases, with promising applications in high-capacity optical communication, parallel information processing, and multifunctional nanophotonic design.