Water-Wave Pancharatnam-Berry Phase Induced by 4D Spin-Orbit State Evolution.

Geometric phase is a far-reaching concept in quantum and classical physics, revealing deep connections between topology and wave dynamics in diverse systems. The Pancharatnam-Berry (PB) geometric phase, which arises from 2D optical polarization evolution, has revolutionized light manipulation by enabling optical metasurfaces. The PB phase is commonly regarded as a unique property of light waves. Generalizing this concept to other classical waves can uncover new geometric phase properties and enable novel wave control mechanisms. Here, the first observation of the PB phase in surface water (gravity) waves is reported, achieved through the interaction with symmetry-engineered scatterers. In contrast to the optical PB phase, the water-wave PB phase is an intrinsic higher-order geometric phase induced by 4D evolution of spin-orbit states. It exhibits remarkable properties, including unbounded phase values, spin-momentum-locked unidirectional channels, and a Poincaré hypersphere representation. PB metasurfaces are further realized for versatile manipulation of water wavefronts, including steering and focusing. This work reveals the universality of the PB phase across different wave systems and generalizes it to higher-dimensional state evolutions. The results open new avenues for exploring the geometric and topological properties of water waves, with potential applications in coastal protection and wave energy harvesting.