# Ultrathin Acoustic Metasurface Holograms with Arbitrary Phase Control
## 2 Results and Discussion  
We have first designed a planar hologram performance for a single spot focusing effect. As illustrated in Figure 1, the incident acoustic waves impinge on the hologram plane, and a circular focal pattern is obtained by manipulating phases of the reflected waves. Knowing the desired focal image, the calculation of the acoustic field distribution on the hologram plane can be achieved by using the Rayleigh-Sommerfeld diffraction theory [37; 38]. \[U_{0}=\iint U_{1}\frac{\exp(\mathrm{j}kd)}{2\pi d}\cdot\frac{h}{d}\cdot\left( \frac{1}{d}-\mathrm{j}\mathrm{k}\right)ds, \tag{1}\]  
with \(U_{0}\) and \(U_{1}\) standing for the acoustic fields at point \(P_{0}(x_{0},y_{0},z_{0})  
As depicted in Figure 2a, the focal image we designed is a circular spot located in the center of the image plane with a radius of 10 mm. The distance between the two planes is 20 mm. The calculated amplitude and phase distribution on the hologram plane are shown in Figure 2b,c, respectively. Applying the calculated field information to Equation (2), the recovered focal image is acquired, as shown in Figure 2d. Calculation results show a good consistency with the ideal focal image, proving the validity of the adopted equations. Figure 1: Scheme of the acoustic focusing with a hologram. A series of acoustic waves are incident onto the hologram plane vertically and reflected to focus on the image plane, which is parallel to the hologram plane. The different colors represent the different reflection phases for the incident acoustic wave. During the actual design, we made the whole hologram plane a flat square with a side length of 200 mm. A general view of the hologram plane is displayed in Figure 3a, with each row and each column comprising 20 unit cells. Helmholtz-like elements were chosen to be the unit cells of our acoustic metasurface. As demonstrated in Figure 3b, the sampled unit cell made of copper, contains a center-positioned cavity, whose geometries are adjustable. In our design, each unit cell's side length \(l\) and thickness \(t\) were fixed to be 10 mm and 2.5 mm, respectively. Since the working frequency selected in this paper was 7.0 kHz, the side length and thickness of these unit cells were only about 1/5 and 1/20 of the operating wavelength (49 mm), respectively. Acoustic waves are incident vertically to the holograms and come back with various reflection phases determined by the dimensions of the cavities. Here, 8 different unit cells were designed to obtain 8 different phases, and the corresponding reconstructed amplitudes and phases of the 8 structures were simulated using commercial finite elements package COMSOL Multiphysics 5.3, as illustrated in Figure 3c. It can be clearly seen that under 7.0 kHz, a set of 8 unit cells provided a gradual phase shift from \(-\pi\) to \(\pi\) with \(\pi/4\) intervals, while the reflection amplitudes were all kept constant. Detailed variable geometry parameters of each unit cell are listed in Table 1. It is worth noting that, since we tuned the structures of the Helmholtz-like unit cells with more degrees of freedom, these 8 different geometry parameters were subtly adjusted, to have almost linear phase dispersion around the working frequency of 7 kHz. As shown in the shadowed region of Figure 3c, the adjacent unit cells had almost the same phase difference in a frequency band from 6.9 kHz to 7.1 kHz. Although with only a narrow band, it shows the possibility to realize hologram not only at a single frequency but also for a certain frequency range. In real scenarios, to ensure exact phase shift, accurate reflection, and minor acoustic loss, potential airborne thermoviscous effects should be taken into consideration [39]. The viscous property of our structure was directly decided by the geometries of the cavity, which manipulated the interior air path. For these elements, once the side length and thickness of each unit cell were fixed, the other 3 variable parameters (i.e., the neck side length \(a\), the inner cavity side length \(b\), and neck thickness \(h\).