Topologically opposite spin with quantized spin wavenumber for Rayleigh–Lamb modes

Spin angular momentum in elastic waves provides a fundamental description of polarization dynamics and underpins spin–momentum locking and topological behavior. While spin in guided waves has been recently explored, its spatial structure across the thickness of a waveguide remains largely uncharacterized. In this paper, we demonstrate two key results: (i) symmetric (S) and antisymmetric (A) Rayleigh–Lamb modes exhibit intrinsically opposite spin polarities, and (ii) the spin angular momentum density itself exhibits a modal wave structure along the length and across the thickness of the waveguide, enabling the definition of an effective “spin wavenumber-vector,” arising from the unique nine spin interactions of longitudinal and vertically polarized shear wave potentials. Starting from a reduced formulation of guided wave potentials, we derive a compact expression for spin density and show that its spatial variation follows a quantized pattern analogous to Lamb mode shapes. Specifically, the fundamental modes (S0, A0) exhibit a half-wavelength spin distribution across the thickness, whereas higher modes (S1, A1) exhibit a full-wavelength distribution, indicating that spin density possesses its own modal spectrum. Numerical eigenmode solutions and finite-element simulations confirm both polarity reversal and the emergence of spin wavenumber. These findings provide a theoretical framework for describing the spatial organization of spin angular momentum density in guided waves and may facilitate future investigations of new phenomena utilizing spin degrees of freedom and spin–orbit interactions in elastic waveguides.