Orbital Angular Momentum Reversal and Asymmetry in Acoustic Vortex Beam Reflection

We present acoustic modeling, measurements, and interpretation of the angular momentum carried in an ultrasonic vortex beam that is obliquely reflected off a flat water-air interface. The experimental measurements observe the theoretically predicted reversals of phase rotation, topological charge, and orbital angular momentum in a reflected vortex beam in direct analogy to optical phenomena. The spatial and temporal evolution of the linear and angular momentum during the reflection are determined by calculating the velocity field from two-dimensional scanned pressure fields. A conversion of the angular momentum indicates a radiation torque along the oblique reflecting surface. We understand this radiation torque originates from the break of rotational symmetry with respect to the incident plane for normal components of the energy flux and linear momentum density at the reflecting surface. Our study provides mechanical evidence on the effect of a flat surface on the reflection of vortex beams and gains insight into the underlying physics, impacting non-contact manipulation of objects and communication.

Figure 1

Schematic illustration of the total reflection of an acoustic vortex beam. A counterclockwise (CCW) vortex beam (with linear momentum $L_i$ and angular momentum $J_i$) generated by a source array $S$ (consisting of a circle of $N$ point sources, each with $2\pi /N$ vortex phase increment defined in our code), is obliquely incident to the interface at an angle $\theta$. Following from the method of images, the reflected beam (with $L_r$ and $J_r$) extended from the virtual source $S^{\prime }$ rotates clockwise (CW). During the reflection, change of linear momentum $\mathrm{\Delta }L=L_r-L_i$ yields an acoustic radiation force $F$, and change of angular momentum $\mathrm{\Delta }J=J_r-J_i$ exerts a torque $T$ at the interface. The inset shows numerically modeled acoustic pressure amplitude (normalized) and wave phase at planes normal to the propagating direction for incident and reflected vortex beams (the source number is $N=16$).

Figure 2

Experimental setup. (a) Photo of the experimental setup. The scan areas for the incident and reflected are both $20\mathrm{mm}\times 20\mathrm{mm}$ center at the depth of 10 mm. (b) Acoustic transducer with the 16 segment vortex lens. (c) Example of a received acoustic signal, where the time is relative to the source. (d) Frequency spectrum of the signal in (c), indicating a central frequency of 0.9 MHz.

Figure 3

Comparison of experimental measurement (denoted by Exp.) with numerical model for the normalized acoustic pressure amplitude (top panels) and phase (center panels) at the scanning planes for both the incident and reflected beams. Bottom panels: projection to planes normal to propagation direction (see text).

Figure 4

Experimental results of the temporal evolution of (a) vertical linear momentum $L_z$ and (b) horizontal angular momentum $J_x$ contained in the scanned planes, exhibiting reversals from the incident (left) to reflected (right) field. The values have been normalized to their own maximum.

Figure 5

Symmetry and asymmetry features of a vortex beam at the obliquely reflecting interface: (a) pressure amplitude $|p|$, (b) total linear momentum density $L$, (c) vertical component of linear momentum density $L_z$, (d) projection angle $\beta =\arccos (L_z/L)$. The solid lines in each plot depict associated values along $y$ axis ($x=0\mathrm{mm}$) to highlight the symmetry in (a),(b) and asymmetry in (c),(d). The incident angle is 45°. (e) Illustration of the asymmetry feature of a symmetric vortex vector field due to an oblique projection.