## 4 Mechanisms of Low-Frequency Low-Reflection BSI  
To provide insight into the mechanisms of low-frequency BSI, we present the schematic diagrams of propagation paths in the tunnel for the left and right incidences of sound based on wave modulations of both AMs. As shown in Figure 4a, for the incident wave I1 from the left side with the angle \(\theta_{\mathrm{i}}\) = 90\({}^{\circ}\), the sound energy was reflected by the upper AM with \(\theta_{\mathrm{r}}\) = 0\({}^{\circ}\) and then was reflected back by the bottom AM with \(\theta_{\mathrm{r}}\) = \(-90^{\circ}\). Beyond that, for the other incident waves with the arbitrary angle in the range 0\({}^{\circ}\)\(<\)\(\theta_{\mathrm{i}}\)! 90\({}^{\circ}\) (I2 in Figure 4b), the incident wave was reflected by the upper AM with the angle \(\theta_{\mathrm{r}}\)! 0\({}^{\circ}\), and therefore, the sound energy was reflected to the bottom AM, which was transformed into the surface evanescent wave owing to the ABA. Figure 4c,d show the propagation paths of the incident waves I3 and I4, respectively, from the right side with the arbitrary angle in the range \(-90^{\circ}\)\(\leq\)\(\theta_{\mathrm{i}}\)! 0\({}^{\circ}\). We can see that I3 and I4 could be directly transformed into the surface evanescent wave by the AMs on both sides of the tunnels, owing to the ABA. Therefore, based on the sound reflections and the ABA created by the gradient phase of the AMs, the incident sound energy on both sides could not transmit through the tunnel. In addition to the low-frequency BSI, we also simulated the low-reflection characteristic of the tunnel. Figure 5a shows the simulated total pressure distributions and the corresponding reflected pressure distributions through the tunnel by separately placing a cylindrical sound source at O and O\({}^{\prime}\) at 343 Hz. We can see that the incident sound energy almost could not transmit through the tunnel in both cases, showing the characteristic of low-frequency BSI. However, as shown in the middle two insets, there still existed a certain amount of reflected sound energy for the sound incidences from both sides. It is obvious that the sound reflection from the sound source at O\({}^{\prime}\) was much weaker than that at O,  
Figure 3: **(a)** Schematic of a tunnel structure with low-frequency low-reflection BSI. A cylindrical sound source is separately placed at the point O or O\({}^{\prime}\). The tunnel width \(H\) is 343 mm. The distance between the sound source and adjacent tunnel entrance is 100 mm. We simulated (**b**) transmittance and (**c**) reflectance spectra through the tunnel by separately placing a cylindrical sound source at the points O and O\({}^{\prime}\). Black shaded regions show the sound transmittances below \(-10\) dB. showing the typical characteristics of low-reflection sound for the left and right incidences, especially for the right incidence, which agrees well with that in Figure 3c. Next, we also simulated the distributions of thermoviscous loss density in the unit cells of both AMs for the left and right incidences at 343 Hz, which are shown in Figure 5b,c, respectively. For the left incidence (Figure 5b), the thermoviscous energy loss was mainly in the coiled channels of the left three unit cells, owing to the sound reflections and the existence of the ABA (Figure 4a,b). However, for the right incidence (Figure 5c), the thermoviscous energy loss was almost distributed in the coiled channels of all unit cells. This is because the incident sound energy from the right side was transformed into the surface evanescent wave, propagating on the surfaces of both AMs (Figure 4c,d). Thus, the total thermoviscous energy loss in the unit cells of both AMs for the left incidence was obviously weaker than that for the right incidence, which was consistent with the reflected sound energy on both sides of tunnel (Figure 5a).