## 2 Design and Performance of Unit Cells and AMs  
As schematically shown in Figure 1a, we propose a low-frequency ultrathin coiled unit cell that is made of epoxy resin. The parameters \(e\), \(w\), \(d\), \(l\) and \(h\) represent the wall thickness, the channel width, the length of the horizontal inner plate, and the total length and thickness of the unit cell, respectively. The designed unit cell can reflect incident sound energy, leading to phase delay. Here, the parameters \(l\), \(w\), \(e\) and \(h\) are selected as 70.0 mm, 6.0 mm, 1.5 mm and 28.5 mm, respectively, and the parameter \(d\) is tunable, which means it can be used to modulate the reflected phase delay of the unit cell. Throughout this work, we use the software of COMSOL Multiphysics to numerically simulate low-frequency low-reflection BSI through the tunnel; the settings of numerical models are listed in Table 1. Additionally, the thickness of the thermoviscous acoustic boundary layer is \(d_{v}=\sqrt{2\mu/\rho\omega}\)[25], in which the parameters \(\mu\), \(\rho\) and \(\omega\) are the coefficient of dynamic viscosity, the density of air and the angular frequency, respectively. The material parameters of epoxy resin and air in the numerical models are shown in Table 2. Figure 1b shows the reflected phase delays created by the unit cells with different values of \(d\) at 370 Hz. We can see that the reflected phase delay can cover almost the whole range of \(2\pi\) by changing \(d\), and therefore, a low-frequency reflected AM with an arbitrary phase profile can be constructed based on the proposed unit cell. Figure 2a shows the designed AM composed of six unit cells, in which the values of \(d\) are selected as 16.1 mm, 14.8 mm, 13.4 mm, 11.7 mm, 9.4 mm, and 6.0 mm for the unit cells from left to right. The phase gradient \(\mathrm{d}\varphi/\mathrm{d}x=-k\), where \(k=2\pi/\lambda\), is the wave number in air, and \(\lambda\) is the wavelength. The theoretical continuous and discrete phase delays of the AM are shown in Figure 2b, and the photograph of the AM fabricated with epoxy resin \begin{table}
\begin{tabular}{c c} \hline \hline
**Numerical Model** & **Setting** \\ \hline Region out of the unit cell & Module of Acoustic Pressure \\ Region in the unit cell & Module of Thermoviscous Acoustic-Solid \\ Surfaces of the unit cell & Interaction \\ Interfaces between regions in and out of & Thermoviscous acoustic boundary layers \\ unit cells & Acoustic-thermoviscous acoustic \\  & coupling boundary \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Settings of numerical models. Figure 1: **(a)** Schematic of a low-frequency ultrathin unit cell with a coiling space structure. **(b)** Phase delays of a normally incident wave created by the unit cell as a function of parameter \(d\) at 370 Hz. Six red open circles represent the selected units, with \(d=6.0\) mm, 9.4 mm, 11.7 mm, 13.4 mm, 14.8 mm and 16.1 mm from left to right. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Epoxy Resin** & **Air** \\ \hline Density (\(\rho\)) & \(1180\) kg/m\({}^{3}\) & \(p_{0}M/RT\) \\ Longitudinal wave velocity (\(c_{\mathrm{l}}\)) & 2720 m/s & \(\sqrt{\gamma RT/M}\) \\ Transversal wave velocity (\(c_{\mathrm{l}}\)) & 1460 m/s & / \\ \multicolumn{2}{c}{Molar mass (\(M\))} & / & \(28.97\times 10^{-3}\) kg/mol \\ Ratio of the molar heat capacities (\(\gamma\)) & / & 1.4 \\ \multicolumn{2}{c}{Molar gas constant (\(R\))} & / & 8.31 J/(mol/K) \\ \multicolumn{2}{c}{Air pressure (\(p_{0}\))} & / & 101.325 kPa \\ \multicolumn{2}{c}{Air temperature (\(T\))} & / & 293 K \\ Coefficient of dynamic viscosity (\(\mu\)) & / & \(1.56\times 10^{-5}\) Pa-s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters of numerical models. based on a 3D printing technology is exhibited in the upper inset. The AM can modulate the phase delay of reflected waves based on its phase gradient.