Fig. 1.

Geometry and spin of arbitrary waves. The strong relations between polarization profile and wave vector reflect the spin–orbit couplings: ${\mathit{k}}_L\times {\mathit{u}}_L=0$ and ${\mathit{k}}_T\cdot {\mathit{u}}_T=0$, i.e., the spin–momentum lockings. (A) For the combinations of two longitudinal waves with different wave vectors ${\mathit{k}}_{L,1}\hspace{0.17em}\ne \hspace{0.17em}{\mathit{k}}_{L,2}$, ${\mathit{u}}_L={\mathit{u}}_{L,1}+{\mathit{u}}_{L,2}$, the total elastic field carries nontrivial spin angular momentum density ${\mathit{s}}_L\hspace{0.17em}\ne \hspace{0.17em}0$ due to the wave interference, $⟨{\mathit{u}}_{L,1}\left|\widehat{\mathit{S}}\right|{\mathit{u}}_{L,2}⟩$. (B) The transverse waves in the same settings, ${\mathit{u}}_T={\mathit{u}}_{T,1}+{\mathit{u}}_{T,2}$, also induce nontrivial spin density ${\mathit{s}}_T\hspace{0.17em}\ne \hspace{0.17em}0$. (C) For the total elastic waves that contain longitudinal and transverse plane waves simultaneously, $\mathit{u}={\mathit{u}}_L+{\mathit{u}}_T$, the total elastic spin density is attributed to the hybrid spin density $\mathit{s}={\mathit{s}}_h$, which reflects the major geometrical difference between longitudinal and transverse waves. The spheres in the figure are $\mathit{k}$ spheres, and the real parts of displacement fields are plotted.