Spin and orbital angular momenta of acoustic beams

We analyze spin and orbital angular momenta in monochromatic acoustic wave fields in a homogeneous medium. Despite being purely longitudinal (curl-free), inhomogeneous acoustic waves generically possess nonzero spin angular momentum density caused by the local rotation of the vector velocity field. We show that the integral spin of a localized acoustic wave vanishes in agreement with the spin-0 nature of longitudinal phonons. We also show that the helicity or chirality density vanishes identically in acoustic fields. As an example, we consider nonparaxial acoustic Bessel beams carrying well-defined integer orbital angular momentum, as well as nonzero local spin density, with both transverse and longitudinal components. We describe the nontrivial polarization structure in acoustic Bessel beams and indicate a number of observable phenomena, such as nonzero energy density and purely circular transverse polarization in the center of the first-order vortex beams.

Figure 1

Schematics of the acoustic Bessel beams. (a) The momentum (plane-wave) spectrum of the beam is a circle with fixed polar angle ${\theta }_0$. The mutual phases of the plane waves (color-marked) have an azimuthal gradient and the $2\pi \ell$ increment around the circle ($\ell =2$ is shown here). (b) The real-space field forms a cylindrically symmetric vortex beam possessing the helical phase front and carrying the orbital AM $\langle \mathbf{L}\rangle \propto \ell \overline{\mathbf{z}}$. This angular momentum is produced by the spiraling canonical momentum density $\mathbf{p}$ in the beam (shown in cyan). Although all plane waves in the spectrum (a) are longitudinally polarized (i.e., the Fourier components of the velocity $\tilde{\mathbf{v}}\left(\mathbf{k}\right)\parallel \mathbf{k}$), the local polarization in real space, $\mathbf{v}\left(\mathbf{r}\right)$, becomes elliptical, which produces a nonzero spin AM density $\mathbf{S}\propto \mathrm{Im}\left({\mathbf{v}}^{*}\times \mathbf{v}\right)$ (shown by red arrows).

Figure 2

Distributions of the energy-density $W\left(x,y\right)$ (greyscale background plots) and polarization ellipses of the velocity field $\left(v_x,v_y\right)$ in the transverse cross sections of acoustic Bessel beams with ${\theta }_0=\pi /4$ and different orders $\ell$. Magenta and cyan colors correspond to right-handed ($S_z>0$) and left-handed $(S_z<0)$ elliptical polarizations, respectively. The nonvortex $\ell =0$ beam is radially polarized, i.e., $S_z\equiv 0$, while vortex beams with $\ell \ne 0$ have nonzero longitudinal spin density $S_z$ and purely circular polarization in the center. Flipping the sign of $\ell$ flips the handedness of the polarization, i.e., $S_z$. One can also see a nonzero energy density in the center of the $\ell =1$ beam.

Figure 3

Distributions of the energy density $W\left(r,z\right)$ (greyscale background plots) and polarization ellipses of the velocity field $\left(v_r,v_z\right)$ in the longitudinal cross sections of acoustic Bessel beams with ${\theta }_0=\pi /4$ and different orders $\ell$. Magenta and cyan colors correspond to right-handed ($S_{\phi }>0$) and left-handed ($S_{\phi }<0$) elliptical polarizations, respectively. Flipping the sign of $\ell$ does not change the handedness of the polarization, i.e., does not affect the transverse spin $S_{\phi }$.