As schematically shown in Fig. 1(a), the acoustic focusing lens consists of two-layer annuluses made of brass cylinders immersed in air, and its cross section in the x–y plane is exhibited in Fig. 1(b) for convenience. The incident cylindrical source is located on the left side of the model. The inner (R₁) and outer (R₂) radii of the focusing lens (black dashed line region) are 13.0 mm and 22.5 mm, respectively, and the diameters of the inner (r₁) and outer (r₂) cylinders are 2.0 mm and 3.0 mm, respectively. Besides, the angle (θ) between two adjacent cylinders in both annuluses is 9° as indicated in Fig. 1(b).

Fig. 1.

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	Fig. 1. (a) Schematic diagram of acoustic focusing lens and (b) cross section in the x–y plane.

Throughout this work, the focusing characteristics of the focusing lens are numerically simulated by the finite element method based on COMSOL Multiphysics software, and the material parameters are listed as follows: the density ρ_b = 8400 kg/m³, the longitudinal wave velocity c_l = 4400 m/s, and the transversal wave velocity c_s = 2200 m/s for brass; ρ_w= 1.21 kg/m³ and c_w = 344 m/s for air. In the simulations, the acoustic-structure boundary conditions are imposed on the boundaries of each brass cylinder.

Figures 2(a) and 2(b) exhibit the amplitude distributions (|p/p₀|) of the pressure fields for the free space and the focusing lens at 42.63 kHz, respectively. The incident cylindrical source is located at (−40 mm, 0) in Cartesian coordinates, and its initial amplitude |p₀| is 1.0 Pa. As shown in Fig. 2(a), in the free space, the cylindrical wave is attenuated rapidly with the increase of the propagation distance. However, with the focusing lens, the cylindrical wave is focused into a perfect point without diffraction at the center of the annuluses, which is shown in Fig. 2(b). Moreover, the focusing effect also exists at many other frequencies, such as 29.50 kHz and 55.32 kHz, which are shown in Figs. 2(c) and 2(d). It also presents great focusing performance at these frequencies. By comparing the results in Figs. 2(b)–2(d), we find that the size of the focal spot becomes smaller with the increase of the frequency due to the wavelength of the cylindrical wave. Figures 2(e) and 2(f) show the focusing characteristics in the lens with different layer numbers, and the other parameters are the same as those in Fig. 2(b). As shown in Figs. 2(e) and 2(f), the focusing effect also exists. Compared with the scenario in Fig. 2(b), the focusing effect is much weaker for the single layer lens, but is almost the same for the three layer lens. The maximum values of |p/p₀| at the focal spot are 0.43, 1.19, and 1.22 in Figs. 2(e), 2(b), and 2(f), respectively. Therefore, we design two-layer annuluses in the acoustic focusing lens.

Fig. 2.

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	Fig. 2. Pressure–field amplitude patterns for (a) free space at 42.63 kHz and focusing lenses with two-layer annuluses at (b) 42.63 kHz, (c) 29.50 kHz, and (d) 55.32 kHz, with (e) single layer annulus at 42.51 kHz, and with (e) three-layer annuluse at 42.59 kHz, respectively.

Figures 3(a) and 3(b) present the longitudinal and transverse profiles of the pressure amplitudes passing through the focal spots in Fig. 2(b), and the pressure profiles in the free space are plotted there for comparison. Note that the focusing effects exist in both the x and the y directions, and the width of the focal spot in the x direction is λ/2, which is the same as that in the y direction. Moreover, compared with the results in the free space, the amplitude at the focal spot is enhanced by about 12 times, with using the annuluses. The aforementioned results indicate that the great focusing effect is realized by the two-layer annuluses in air.

Fig. 3.

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	Fig. 3. Pressure amplitude distributions along a line passing through focal spot (a) in the propagation direction and (b) in a direction transverse to the propagation direction (shown as lines “1” and “2” in Fig. 2(b)).

In order to verify the performance of the focusing lens, we establish the simplified model composed of a cylinder structure immersed in air, which is shown in Fig. 4(a), and the parameters R₁ and R₂ are the same as those in Fig. 1(b). The absolute value of the acoustic velocity |v_eff| and the real part of the density ρ_eff of the effective medium are shown in Fig. 4(b), and the two parameters are numerically calculated from the complex reflection and transmission coefficients.^[32] Figure 4(c) presents the amplitude distribution of the pressure field for the simplified model at 42.63 kHz, and the focusing characteristics are very similar to those in the focusing lens [Fig. 2(b)], indicating the validity of the proposed focusing lens.

Fig. 4.

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	Fig. 4. (a) Schematic diagram of the simplified model, and (b) the real part of effective density ρeff and the absolute value of effective acoustic velocity |veff| each as a function of frequency. (c) Pressure field amplitude pattern for the simplified model at 42.63 kHz.

Figures 5(a)–5(c) show the phase distributions of the pressure fields, which correspond to Figs. 2(a), 2(b), and 4(c), respectively. We find that the phase waveforms are a series of undisturbed circular lines in the free space [Fig. 5(a)]. However, it is found from Figs. 5(b) and 5(c) that the phase distributions are reconstructed inside the annuluses, and the phase waveforms are transformed into a series of concentric circles with the same values and eventually focus into a point in the centre of the annuluses.

Fig. 5.

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	Fig. 5. Pressure field phase patterns for (a) free space, (b) focusing lens, and (c) the simplified model at 42.63 kHz, and (d) deformation of total displacement field pattern in annuluses, with red arrows representing vibration directions of displacements.

To obtain an insight into the mechanism of the focusing effect, we simulate the deformation of the total displacement distributions in the annuluses in Fig. 2(b), of which the result is shown in Fig. 5(d). It is found from Fig. 5(d) that the Mie-resonance mode ^[30,31] is excited in the annuluses at an eigenfrequency of 43.62 kHz, and the vibration directions (indicated by red arrows) pointing to the focal spot, especially those in the inside annulus. It is deduced that the propagation directions of the cylindrical wave are restructured by the Mie-resonance mode, and the cylindrical waves are finally focused into a perfect point in the center of the annuluses.

We simulate the amplitude distributions of the pressure fields by magnifying all the sizes of the outer annulus (R₂ and r₂) and inner annulus (R₁ and r₁) n times. From Figs. 6(a)–6(h), we find that the focusing frequencies are almost the same for different size outer annuluses [Figs. 6(a)–6(d)], but are inversely proportional to the size of the inner annulus [Figs. 6(e)–6(h)]. It indicates that the focusing frequency is closely related to the size of the inner annulus.

Fig. 6.

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	Fig. 6. Pressure–field amplitude patterns for focusing lens by magnifying the parameters ((a)–(d)) R2 and r2 and ((e)–(h)) R1 and r1 n times.

Furthermore, we find that the focusing effect exists in different shaped annulus structures. To verify the feasibility of this case, we design four types of annuluses with different shapes immersed in air, which are shown in Figs. 7(a)–7(d). As shown in Figs. 7(e)–7(p), the cylindrical waves are focused into the four different-annulus structures at 17.19, 42.50, 37.42, and 29.55 kHz, respectively, and note that the incident cylindrical waves are located at different positions outside the annuluses.

Fig. 7.

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Fig. 7. Schematic diagrams of (a) square annuluses (R1 = 24.00 mm, R2 = 36.00 mm, D1 = 12.00 mm, D2 = 18.00 mm, r1 = 2.00 mm, and r2 = 3.00 mm), (b) square and circular annuluses (θ = 9°, R1 = 14.00 mm, R2 = 18.00 mm, D1 = 12.00 mm, D2 = 18.00 mm, r1 = 2.00 mm, and r2 = 3.00 mm), (c) rhomboic annuluses (θ = 60°, d1 = 7.22 mm, d2 = 12.50 mm, D1 = 10.83 mm, D2 = 18.75 mm, r1 = 2.00 mm, and r2 = 3.00 mm), and (d) triangular annuluses (R1 = 26.27 mm, R2 = 39.40 mm, r1 = 2.00 mm, r2 = 3.00 mm, and θ = 60°). Pressure–field amplitude patterns for ((e)–(g)) square annuluses, ((h)–(j)) square and circular annuluses, ((k)–(m)) rhomboic annuluses, and ((n)–(p)) triangular annuluses with different incident positions.

The focusing effect on the annuluses has substantial practical significance in the acoustic beam splitter in air. As is well known, the energies of acoustic waves are divided into several portions and reduce greatly in the traditional beam splitters. Here, we propose an acoustic beam splitter with the energy enhancement by using annulus structures shown in Fig. 7(a), but the incident cylindrical wave is located in the center of the annulus structure, and its initial amplitude | p₀| is also 1.0 Pa. Besides, the amplitude distribution of the pressure field in free space is also calculated for comparison as shown in Fig. 8(a), in which the white circular lines represent the contour lines of the amplitude (|p/p₀|). It is found from Fig. 8(b) that the output waves are divided into four acoustic beams. Compared with the result in Fig. 8(a), the amplitude at the exciting point is enhanced by about 40 times, and the amplitude of each beam through the annuluses is increased by about 15 times (Fig. 8(b)). To clearly exhibit the characteristics of the output waves, we magnify the pressure field outside the annuluses with different color bar and plot it in Fig. 8(c), and the amplitudes of the output waves are larger than those in free space.

Fig. 8.

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Fig. 8. Pressure–field amplitude patterns for (a) free space and (b) square annuluses at 17.19 kHz, and (c) partial zoom in panel (b). (d) Schematic diagram of acoustic directional transmitter (R1 = 23.50 mm, and D = 2.00 mm). (e) Pressure–field amplitude pattern for acoustic directional transmitter at 17.19 kHz, and (f) partial zoom in panel (e). White circular lines represent contour lines of amplitude.

Furthermore, we consider the annulus structure [Fig. 7(a)] surrounded by a square annulus structure with an opening on an edge, which is schematically shown in Fig. 8(d). Figure 8(e) shows the amplitude distribution of the pressure field for this structure, where the energies of the output waves through the opening are concentrated in one direction. Compared with Fig. 8(b), figure 8(e) shows that the amplitude of the output waves is magnified 1.5 times, which is also clearly shown in Fig. 8(f). This effect may have potential applications in acoustic directional transmitters.