The reflected AF lens in this work is based on the Bessel-like beam owing to its features of line focusing and non-diffraction. Assuming an acoustic wave with the normal incidence propagates along the x direction and passes through a reflected phased array, the reflected angle θ_r (measured from the x direction) can be derived according to the generalized Snell’s law^[52] 1 where k₀ = 2πf/c₀ is the wave number, c₀ is the acoustic velocity of air at 300 K, f is the frequency, and φ(y) is the phase delay distribution of the reflected phase array along the y direction.

To realize a reflected acoustic Bessel-like beam, the distribution of the phase delays along the y direction is expressed as^[39] 2 where β is the base angle of the Bessel-like beam. In this work, we propose a reflected AF lens consisting of a thermoacoustic phased array of the Bessel-like beam. As shown in Fig. 1(a), the incident acoustic wave (black arrows) is placed at the right side. The reflected acoustic wave is formed as the Bessel-like beam, focusing on the red line AB. The relationship between the length of the focus line (D) and the base angle β is 3 where H is the half height of the AF lens.

Fig. 1.

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	Fig. 1. (color online) (a) Schematic diagram of reflected AF lens based on Bessel-like beam. (b) Discrete phase distribution of 81 units (red hollow circles) of designed AF lens. Blue lines in panel (b) represent theoretical continuous phase delays.

In the reflected thermoacoustic phased array, the parameters are selected as y = 81 cm, β = π/20, f = 5.0 kHz, and c₀ = 343 m/s. As shown in Fig. 1(b), the theoretical continuous phase delays [blue solid lines] of the Bessel-like beam can be obtained by Eq. (2), and 81 red hollow dots represent discrete phase delays of the thermoacoustic phased array.

To realize thermoacoustic phased array, we first design a reflected thermoacoustic unit with phase manipulation. Assuming that air is the ideal fluid, the acoustic velocity c and the density ρ of air are determined by the temperature T, which are shown as follows^[47] 4 5 where γ = 1.4 is the ratio of the molar heat capacities of air, M = 28.97 × 10^–3 kg/mol is the molar mass of air, R = 8.31 J/(mol/K) is the molar gas constant, and p₀ = 101.325 kPa is the pressure at 273 K.

The reflected thermoacoustic unit is constructed by three rigid thermal insulation boundaries (blue solid lines) and a thermal insulation film (red dashed line) immersed in air with the temperature T, which is shown in Fig. 2(a). The thickness of the rigid thermal insulation boundary is d, and the length and width of the unit are l and h, respectively. To obtain the reflected phase delays, we design a model shown in Fig. 2(b). The acoustic wave is horizontally incident from the right side and passes through the unit with the rigid thermal insulation boundaries. At the left boundary, the acoustic wave is reflected again and returns to the outside.

Fig. 2.

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	Fig. 2. (color online) (a) Reflected unit of thermoacoustic phase manipulation, and (b) numerical model for simulation of reflected phase delays. (c) Reflected phase delays of units with different temperatures.

Throughout this work, the acoustic characteristics are simulated by the finite element method based on COMSOL Multiphysics software, and the structure parameters of the unit are selected as l = 8 cm, h = 1 cm, d = 0.05 cm, and the ambient temperature T₀ = 300 K. Besides, we introduce the influence of temperature on the thermal parameters of air, such as the thermal conductivity k = –0.00227583562 + (1.15480022 × 10^–4) × T – (7.90252856 × 10^–8) × T² + (4.11702505 × 10^{– 11}) × T³ – (4.11702505 × 10^–15) × T⁴ W·m^–1·K^–1 and the thermal capacity C_p = 1047.63657–0.372589265×T +(9.45304214×10^–4) × T² - (6.02409443 × 10^–7)× T³ + (1.2858961 × 10^–10) × T⁴ J·(kg·K)^–1. The parameters c and ρ of air can be calculated from Eqs. (4) and (5), respectively. Moreover, the heat flux of the external boundaries is continuous, and the thermal convection inside the model is neglected in the thermal simulations. Figure 2(c) shows the reflected phase delays φ of the unit with different air temperatures, in which the parameter φ is shown in the range from 0 to 2π owing to its periodicity. Note that the reflected phase delays could cover a whole 2π range in the temperature range 560 K–2450 K (shown in shaded region).

Figures 3(a) and 3(b) show the distributions of the reflected intensity field through the theoretical continuous phase delays and the thermoacoustic phased array, respectively. It is obvious that the reflected acoustic wave is formed as the Bessel-like beam [shown in red diamonds]. The reflected acoustic energy is focused on a line at the right side of the lens almost without diffraction. Based on Eq. (3), the theoretical length of the focus D is calculated as 2.53 m, which agrees well with the simulated result in Fig. 3(b), and is larger than that of previous AF lenses.^[40–46] Furthermore, the AF characteristic induced by the thermoacoustic phased array agrees well with the theoretical result in Fig. 3(a). Moreover, it is found from Fig. 3(c) that the AF does not exist in the free space. Therefore, the reflected AF lenses with large focusing region and high performance is demonstrated based on the Bessel-like beam.

Fig. 3.

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	Fig. 3. (color online) Spatial distributions of reflected intensity field for Bessel-like beam induced by (a) theoretical continuous phase delays and (b) thermoacoustic phased array, and (c) for free space at 5.0 kHz. Red arrows in panel (b) refer to incident acoustic waves.

To clearly exhibit the AF performances, the transverse and longitudinal distributions of the acoustic intensities [lines I–IV in Figs. 3(b) and 3(c)] are simulated, which is shown in Fig. 4. Note that, with the thermoacoustic lens, the centre position of the focus is (124.1 cm, 0), the maximum intensity at the center of the focus is about 6 times than that in the free space. Besides, as shown in Fig. 4(a), the intensity along the line I (from 0.5 m to 2.5 m) is larger than 3.0 Pa², showing a high performance of line focusing. Furthermore, it is found from Fig. 4(b) that the acoustic energy is mainly focused at the position y = 0, and the diffraction is very weak, showing obvious non-diffracting characteristic of the AF lens.

Fig. 4.

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	Fig. 4. (color online) Distributions of reflected intensity through (a) lines I and III, and (b) lines II and IV in Fig. 3.

To show the bandwidth of the AF lens, we simulate the distributions of the reflected intensity field at different frequencies, which is shown in Fig. 5. It is found that, at three selected frequencies, the reflected acoustic energy is also focused on a line at the right side. Thus, the working bandwidth of the lens reaches more than 1.4 kHz, and the fractional bandwidth (the ratio of the bandwidth to the center frequency) is about 0.29. Besides, with the increase of the frequency, the focus position moves to the right. This characteristic is also shown in Fig. 6. Such a phenomenon results from the fact that the incident angle and the phase distribution of the lens are the same. By increasing the frequency, the reflected angle θ_r decreases based on Eq. (1), and therefore the focus length increases. Based on these results, we deduce that the proposed AF lens has the advantages of s broad bandwidth and an adjustable focus position.

Fig. 5.

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	Fig. 5. (color online) Spatial distributions of reflected intensity field induced by reflected AF lens at (a) 4.2 kHz, (b) 4.9 kHz, and (c) 5.6 kHz. Red arrows refer to incident acoustic waves.

Fig. 6.

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	Fig. 6. (color online) Transverse distributions of reflected intensity through focus induced by reflected AF lens at different frequencies.

We also study the influence of the base angle β of the Bessel-like beam on the AF performance. Figures 7(a)–7(c) show the theoretical continuous phase delays (blue line) and discrete phase delays (red hollow dots) of the reflected AF lenses with different β. The distributions of the acoustic intensity field induced by the reflected AF lenses with β = 6°, 12°, and 15° are shown in Figs. 7(d)–7(f), respectively. It is obvious that the Bessel-like beam and the AF effect still exist with different β. The focal length and the focus size decrease gradually with the increase of β, which can be explained by Eq. (3). Therefore, the focus energy is more concentrated and the focus position moves to the left gradually. The transverse distributions of the reflected intensity through the focus with different β are shown in Fig. 8.

Fig. 7.

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	Fig. 7. (color online) Theoretical continuous phase delays (blue line) and discrete phase delays (red hollow dots) of reflected AF lenses with base angles (a) 6°, (b) 12°, and (c) 15°. Spatial distributions of reflected intensity field induced by reflected AF lenses with base angles (d) 6°, (e) 12°, and (f) 15° at 5.0 kHz. Red arrows in panels (d), (e), and (f) refer to incident acoustic waves.

Fig. 8.

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	Fig. 8. (color online) Transverse distributions of reflected intensity through the focus with different values of β.

We also find that, with the increase of β, the intensity at the focus increases, and the focus position moves to the left, which arises from the fact that the refraction angles of two transmitted beams increases, and the interference region moves to the left and becomes small. Therefore, the acoustic energy is more concentrated, which agrees with that in Fig. 7.

To demonstrate the robustness of a reflected AF lens, we separately adopt four, six, and eight types of basic reflected units with different air temperatures to design the AF lenses with 81 discrete phase delays (red hollow dots), in which the phase delay distributions are displayed in Figs. 9(a)–9(c). As shown in Figs. 9(d)–9(f), the Bessel-like beam and the AF effect still exist for three cases, and the reflected acoustic energy is also focused on a line at the right side. Note that with more basic reflected units, the maximum intensity at the focus increases slightly, and the characteristics of the focus almost remain unchanged.

Fig. 9.

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Fig. 9. (color online) Theoretical continuous phase delays (blue line) and discrete phase delays (red hollow dots) of reflected AF lenses based on (a) four, (b) six, and (c) eight types of reflected units. Spatial distributions of reflected intensity field induced by reflected AF lenses with (d) four, (e) six, and (f) eight types of reflected units at 5.0 kHz. Red arrows in panels (d), (e), and (f) refer to incident acoustic waves.

Figures 10(a)–10(c) show the distributions of the intensity field induced by the thermoacoustic phased array, in which the temperatures of all units increase 100T, 300T, and 500T, respectively, and the other parameters are the same as those in Fig. 3(b). It is obvious that, with the increase of the temperatures of all units, the lens also has high AF performance, and the focus position remains the same, but the intensity of the focus decreases slightly. The aforementioned results verify the high robustness of the proposed reflected AF lens.

Fig. 10.

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	Fig. 10. (color online) Spatial distributions of reflected intensity field induced by reflected AF lenses in Fig. 3(b) by increasing temperatures of units (a) 100T, (b) 300T, and (c) 500T at 5.0 kHz. Red arrows in panels (a), (b), and (c) refer to incident acoustic waves.

Based on the generalized Snell’s law, we also realize a reflected AF lens based on the Bessel-like beam for a cylindrical acoustic wave. As shown in Fig. 11, the position of the incident cylindrical wave is located at (25 cm, 0), and the reflected acoustic wave induced by the cylindrical acoustic wave is formed as Bessel-like beam, and is focused on a line at the right side. The phase delay φ(y) of the thermoacoustic phased array for the cylindrical acoustic wave satisfies 6 where L is the distance between the positions of the cylindrical acoustic source and the lens.

Fig. 11.

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	Fig. 11. (color online) Schematic diagram of reflected AF lens for cylindrical acoustic wave.

The theoretical continuous phase delays (blue lines) and the 81 discrete phase delays (red hollow dots) of the AF lenses for the cylindrical acoustic wave with different base angles are shown in Figs. 12(a)–12(c), in which the parameter L is selected as 0.25 m. Figures 12(d)–12(f) show the spatial distributions of the reflected intensity field induced by the designed AF lenses with β = 6°, 12°, and 15°, respectively. It is obvious that the reflected acoustic energy induced by the proposed AF lenses for the cylindrical acoustic wave is also formed as the Bessel-like beam and is focused on a line at the right side. With the increase of β, the focus size decreases, and the focus position moves to the left gradually; which are similar with those in Fig. 7.

Fig. 12.

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Fig. 12. (color online) Theoretical continuous phase delays (blue line) and discrete phase delays (red hollow dots) of reflected AF lenses for cylindrical acoustic wave with base angles (a) 6°, (b) 12°, and (c) 15°. Spatial distributions of reflected intensity field induced by reflected AF lenses for cylindrical acoustic wave with base angles (d) 6°, (e) 12°, and (f) 15° at 5.0 kHz. White dots in panels (d), (e), and (f) refer to cylindrical acoustic source.