Hybrid Spin and Anomalous Spin-Momentum Locking in Surface Elastic Waves

Transverse spin of surface waves is a universal phenomenon which has recently attracted significant attention in optics and acoustics. It appears in gravity water waves, surface plasmon polaritons, surface acoustic waves, and exhibits remarkable intrinsic spin-momentum locking, which has found useful applications for efficient spin-direction couplers. Here we demonstrate, both theoretically and experimentally, that the transverse spin of surface elastic (Rayleigh) waves has an anomalous sign near the surface, opposite to that in the case of electromagnetic, sound, or water surface waves. This anomalous sign appears due to the hybrid (neither transverse nor longitudinal) nature of elastic surface waves. Furthermore, we show that this sign anomaly can be employed for the selective spin-controlled excitation of symmetric and antisymmetric Lamb modes propagating in opposite directions in an elastic plate. Our results pave the way for spin-controlled manipulation of elastic waves and can be important for a variety of areas, from phononic spin-based devices to seismic waves.

Figure 1

Transverse spin density in surface water (a), electromagnetic (plasmon-polariton) (b), acoustic (c), and elastic Rayleigh (d) waves. Shown are: the $(x,z)$-plane elliptical polarizations of the relevant vector fields for the $x$-propagating surface waves and the corresponding spin densities $S_y\equiv S$. (a) Gravity water waves are described by the divergenceless and curl-less velocity field. (b) Surface plasmon-polaritons at the interface between the vacuum and a negative-permittivity medium (metal) have a transverse (divergence-less) electric field. (c) Surface acoustic waves at the interface between air and a negative-density metamaterial have a longitudinal (curl-less) velocity field. (d) Elastic Rayleigh waves at the surface of an isotropic solid are described by a hybrid displacement field which has both longitudinal and transverse contributions, Eq. (2).

Figure 2

(a) Rayleigh-wave-induced deformations of an elastic medium caused by the purely longitudinal and transverse displacement field parts ${\mathbf{u}}_l$ and ${\mathbf{u}}_t$, as well as by the mixed fields ${\mathbf{u}}_{tl}=(u_{tz},u_{lx})$ and ${\mathbf{u}}_{lt}=(u_{lz},u_{tx})$ (the latter ones are not real displacement fields but rather visualized contributions to the total spin density). The corresponding field polarizations are shown by ellipses, whereas the spin densities are indicated by the red-blue color scheme. (b) The experimental setup for the measurements of the Rayleigh-wave spin (see explanations in the text). (c) The measured versus calculated normalized spin density $s_y\equiv s$ of the $x$-propagating Rayleigh wave. The Rayleigh wavelength is ${\lambda }_R\approx 8\mathrm{cm}$ at 35 kHz. The background colors highlight the areas of positive and negative spin, cf. Fig. 1.

Figure 3

Transverse spin density in the lowest-order antisymmetric (A0) and symmetric (S0) modes in electromagnetic (a), acoustic (b), and elastic (c) thin-plate or slit systems, cf. Figs. 1. The A0 elastic Lamb mode has a reversed spin sign compared to its electromagnetic and acoustic counterparts.

Figure 4

(a) The pure, $s_p=s_{ll}+s_{tt}$, and hybrid, $s_h=s_{lt}+s_{tl}$, contributions to the normalized spin $s$ of the $x$-propagating A0 and S0 Lamb modes in an elastic aluminium plate $z\in (-d,d)$, see Fig. 3. The parameters used here are $f=20\mathrm{kHz}$ and $d=3\mathrm{cm}$. (b) Schematics of the experimental setup for the spin-induced directional excitation of the Lamb modes. A circularly polarized source with spin $s=+1$, placed at the edge of an aluminum plate, $x=0$, $z=-d$, excites oppositely propagating A0 and S0 modes. (c) The experimentally measured Fourier $(\omega ,k_x)$ spectra of the excited waves in the $x>0$ and $x<0$ zones of the plate versus the calculated spectra of the A0 and S0 Lamb modes. One can clearly see the spin-induced counterdirectional propagation of the symmetric and antisymmetric modes.