# Ultra-Broadband Bending Beam and Bottle Beam Based on Acoustic Metamaterials
## 2 Materials and Methods  
We will first explain the design of the acoustic meta-structure for ultra-broadband bending beams. The structure consists of an array of straight tubes with different spatially modulated depths \(h\) (Figure 1). [6] Each tube of the structure has an identical width much smaller than the working wavelength, i.e., \(w<<\lambda\) with \(w\) and \(\lambda\) being the width of the tubes and the working wavelength, respectively, (see Inset in Figure 1). In addition, the wall of each tube is designed to be sufficiently thin to avoid the impedance mismatch between the structure and the background medium (the thickness of the wall is \(0.1p\) with \(p\) being the period of the structure). Hence, the incident sound waves can easily enter the tubes with little reflections on the interface. After the reflection by the bottom of each tube, the reflected sound wave will possess a depth-dependent phase shift, i.e., \(\phi=k\times(2h)\), with \(k=2\pi/\lambda\) being the wavenumber. Consequently, the desired phase profile on the interface of the structure can be achieved by carefully engineering the depth profile of the tubes, which suggests the ability to generate a ultra-broadband bending beam desired here. In the current study, the the proposed scheme will be utilized to generate an ultra-broadband bending beam (Figure 1). To realize an arbitrary bending trajectory \(y=f(x)\), the phase profile can be expressed as [3]  
\[\frac{d\phi(y)}{dy}=-k_{0}\sin\theta_{r}=-k_{0}\frac{f^{\prime}(x)}{\sqrt{1+[f ^{\prime}(x)]^{2}}} \tag{1}\]  
where \(f^{\prime}(x)=\tan\theta_{r}\) is the slope of the trajectory with \(\theta_{r}\) being the reflected angle. For a half circular trajectory \(f(x)=\sqrt{R^{2}-(x-R)^{2}}\) with the center at \((x,y)=(R,0)\), the  
Figure 1: Illustration of the ultra broadband structure. Black arrows: incident waves; red arrows: reflected waves. Inset: zoom-in plot of a single tube within the structure. The depth profile of the tubes, \(h\), is a function of the spatial location \(y\). \(p=\lambda/8\) is the period of the structure, and \(w=0.8p\) is the width of each tube. desired phase profile can be obtained as \(\phi(y)=-k_{0}(y-2R\sqrt{y/R})\), and then the depth profile of the tube can be simply expressed as:  
\[h(y)=C-(y/2-R\sqrt{y/R}), \tag{2}\]  
where a positive number \(C\) is used to make sure the depth of each tube, \(h(y)\), is non-negative. It is worth mentioning that the depth profile of the tubes in Equation (2) does not explicitly rely on the working frequency or the wavelength, suggesting the possibility to achieve a bending beam in a broad band of different frequencies with the same structure. Here, the numerical simulations for bending beams were conducted in the two-dimensional pressure acoustic module using the commercial software, COMSOL MULTIPHYSICS; see Figure 2 for the sketch of the numerical setup. The size of the computation domain is \(1.8\) m \(\times\) 1.8 m. Radiation boundaries were applied to the outer boundaries of the computation domain to eliminate the reflection. The whole length of the structure is 1m, and the period of the structure is \(p=1.2\) cm. Maximum mesh size was set to be \(\lambda/20\) so as to obtain the precise and accurate sound fields. The background medium is air with the density of 1.21 kg/m\({}^{3}\) and the sound speed of 343 m/s. For the simulations with the acoustic meta-structure (considered to be rigid) (Figure 2a), the background pressure field was used as the source to generate an incident plane wave propagating along the \(-x\) direction, and the scattered fields were examined to check the performance. However, for the simulation with the ideal source (Figure 2b), the source is located at \(x=0\) to generate the ideal bending field directly. The simulations for the three-dimensional bottle beams were conducted with the same parameters as mentioned above but in a two-dimensional symmetry model with \(y=0\) being the symmetrical axis.