Near-Field Hierarchical Beam Training for Reconfigurable Holographic Surfaces

Reconfigurable holographic surfaces (RHS) are expected to play a key role in future mobile networks. However, the substantial increase in antenna aperture and operating frequency brings new challenges for near-field communication. We propose a near-field multi-user 3D hierarchical beam training scheme tailored for RHS-based multi-input multi-output (MIMO) systems, supporting both near-field and far-field user deployment, while considering hardware constraints. Since the hierarchical beam training scheme involves activating varying numbers of transmitting elements at each search layer, and RHS elements are densely packed, significant mutual coupling effects may arise. To mitigate this, we propose two element activation strategies: centered activation and sparse activation based on different RHS element positioning patterns within the hierarchical beam training framework. Furthermore, we design a practical beam training approach tailored to a hybrid digital–holographic architecture, optimized through an alternating algorithm that accounts for both binary and coupled amplitude-phase hardware constraints on RHS meta-elements. Simulation results demonstrate strong robustness under various hardware and channel state information (CSI) imperfections, achieving performance close to that of fully digital systems. Finally, we further analyse the asymptotic orthogonality of near-field beam focusing vectors under different RHS surface geometries. The results show that rectangular surfaces offer superior beam orthogonality for beams steered in the same direction but located at different distances.