Achromatic acoustic generalized phase-reversal zone plates

We report an achromatic acoustic generalized phase-reversal zone plate by harnessing the response of dipole and monopole, which eliminate the chromatic aberration of conventional zone plates. The focusing properties of the proposed metalens are compared with that of the conventional Soret-type Fresnel zone plate (FZP) in both experiments and simulations. Due to the combination of the phase-reversal characteristic and the tunable transmission phase induced by dipole and monopole, an achromatic high efficient focusing is confirmed by experiment in the frequency range from 3350 to 3950 Hz, with the focal intensity of achromatic metalens being approximately twice that of Soret-type FZP. The proposed achromatic metalens has potential applications in the broad field of acoustics, such as imaging and energy harvesting.

Zone plate as another type of planar diffractive lens has found many applications in optics [29], x-ray microscopy [30,31], THz optics [32] and microwaves [33], and it can also be used to focus sound waves [34][35][36][37][38]. To overcome the high chromatic aberration of Fresnel zone plate (FZP), several aperiodic zone plates such as fractal zone plates [39][40][41][42] and Thue-Morse zone plates [43,44] have been proposed, which produce an extended depth of field by the fractal structure of their foci. However, their focal intensity is lower than that of FZPs because part of the energy is distributed among the multiple subsidiary foci [42].
In this work, we propose a concept of achromatic acoustic generalized phase-reversal zone plate in figure 1(b), which combines the advantages of metasurface (locally tailoring phase) and the phase-reversal FZPs (without the requirement of the full 2π transmission phase tuning). Different from the conventional phase-reversal zone plates, in which a π-phase jump occurs abruptly from 0 to π and from π to 0, here, we manipulate the transmission phase gradually from 0 (or 2π) to π by a combination of dipole and monopole, and then an approximate π-phase jump occurs from π to 0 (or 2π). It is stressed that the frequency-dependent response of dipole and monopole plays a role to reduce chromatic aberration. On the other hand, different from the full 2π transmission phase tuning in optics to achieve achromatic focusing, the proposed achromatic metalens requires only the π transmission phase tuning, which can be satisfied by a transmitted metalens in acoustics. Achromatic metalens consists of a series of complex zones, in each zone the response of unit cell changes gradually from the dipole to monopole. (c) Dipole: the first unit cell marked in (b), where the width of unit cell is Λ, the thickness of lens being l = l 1 + l 2 + l 3 + l 4 = 1.4Λ with l 1 = 0.2Λ, l 2 = 0.5Λ, l 3 = 0.5Λ and l 4 = 0.2Λ, and for each Helmholtz resonator, the following parameters are fixed: the size of throat a = 0.125Λ, the length of throat b = 0.05Λ and the height of cavity h = 0.3Λ. (d) Monopole: the seventh unit cell marked in (b), with l = 1.4Λ. Note that the global coordinate system xOz is marked in (a) and (b) for the Soret-type FZP and achromatic metalens, respectively. In addition, the incident wave (p i ) comes from the negative direction of the z-axis, which has been marked in (c) as an example. Table 1. The optimal parameters w 0 , w 1 , w 2 and w 3 for 16 unit cells.

The physical principle to design an achromatic metalens
For an achromatic transmitted metalens in figure 1(b), where only half of metalens is given due to symmetry, the transmission phases of the nth unit cell and the first unit, φ n and φ 1 , follow a simple relation, where F is the focal length, x cn is the center position of the nth unit cell, and k = ω/c 0 = 2π/λ is the wavenumber in air. However, for the conventional Soret-type FZP in figure 1(a), the radius of each zone is determined by r n = nλF + ( nλ 2 ) 2 , (n = 1, 2, . . .), and the focal length F has a dependence on frequency for the given radius r n . To reduce the chromatic aberration, a dipole in figure 1(c) and a monopole in figure 1(d) are substituted for the transparent zone and the opaque zone, respectively, where the dipole is achieved by Fano resonances [11,45,46], and it can also be obtained by coiled-up space [47]. By the position of the first opaque zone in figure 1(a), the seventh unit cell in figure 1(b) is set to be the monopole, and its transmission phase is approximate to π when Fabry-Pérot resonance is satisfied, l = c 0 /2f 0 , with l being the thickness of lens, the sound speed c 0 = 343 m s −1 , f 0 = 0.355c 0 /Λ a reference frequency, and Λ the width of unit cell. Once the seventh unit cell is given, the required transmission phases of all unit cells are determined by equation (1) with the given focal length F = 4c 0 /f 0 = 4Λ/0.355. For detailed parameters of each unit cell, equation (1) is adopted by a particle swarm optimization algorithm to obtain the width of slit and widths of cavities of three Helmholtz resonators, which are marked by w 0 , w 1 , w 2 , and w 3 in figure 1(c), respectively. Through manipulating these four widths, the response of dipole can be smoothly transferred to the response of monopole, and the corresponding optimized values for each unit cell are listed in table 1. Note that a high transmission coefficient |t 0 | > 0.85 is achieved for each unit cell with a relative bandwidth of 16.9%, lager than the bandwidth (11.5%) of achromatic metalens in the visible [48]. Therefore, the designed metalens is called as achromatic acoustic metalens.
By the theoretical derivations given in appendix A, figures 2(a)-(c) gives the theoretical transmission phases φ of 16 unit cells, which are extracted from the transmission coefficient t 0 of a normally incident plane wave. It is found that the transmission phase gradually changes from 2π (or 0) to π for the number of unit cells from 1 to 8 in (a), from 9 to 12 in (b), and from 13 to 16 in (c), which look like the phase-reversal zone plates, but the transmission phase has a frequency-dependent characteristic, resulting in the reduction of chromatic aberration. By calculating the volume velocity U 0 at the inlet (z = 0) of each unit and the volume velocity U l at the outlet (z = l), the contribution of monopole and dipole can be described by two quantities U l −U 0 2 and U l +U 0 2 , respectively, and we then define a volume velocity ratio between the dipole and monopole R = | U l +U 0 U l −U 0 |. It is seen in figures 2(d)-(f) that the volume velocity ratios have peak points or valley points for different unit cells. The peak point represents that the contribution of dipole dominates the transmission, having a phase near 2π. However, for the valley point, the monopole plays a role in tuning the transmission phase close to π. Therefore, the proposed achromatic metalens generalizes the conventional phase-reversal zone plates by introducing the dipole and monopole as a substitute, which not only has the phase-reversal characteristic, but also has abilities to tune the frequency dependence of transmission phase, resulting in an achromatic focusing.
To validate theoretical predictions, the finite element simulation is carried out by using the software COMSOL MULTIPHYSICS, where the viscous friction on the wall of apertures is modeled using the thermoviscous acoustic module. It is shown in figure 3(a) that the theoretical transmission coefficients t 0 for the first unit cell and the seventh unit cell are consistent with the numerical simulation, except for the simulated transmission peak of the first unit cell at the frequency near the bandgap induced by Helmholtz resonances. The corresponding volume velocity ratios between the dipole and monopole are also compared in figure 3(b). It is illustrated that the peak point (dipole) and the valley point (monopole), adapted to design the metalens, are robust to the thermoviscous effect. In addition, the simulated distributions of normalized pressure and velocity for the dipole and monopole are given in figures 3(c) and (d), respectively. It is shown that the vibration of dipole at inlet and outlet of the unit cell are in phase, while an anti-phase motion is observed for the monopole.

The performance of achromatic focusing
With the optimized parameters in table 1, it is illustrated in figure 4(a) that the designed metalens can focus sound waves at the predetermined focusing position. To demonstrate the performance of achromatic focusing, the transmitted intensity along the axis of lens (x = 0 m) is given in figure 4(b) for a normally incident plane wave of unit amplitude. For comparison, the Soret-type FZP (shown in figure 1(a)) is also designed with its size and the focusing position being the same as the metalens. Figure 4(c) shows that the focusing intensity of Soret-type FZP is lower than that of metalens in figure 4(a) at the reference frequency f 0 = 3550 Hz, and its focusing position is slightly deviated from the predetermined value. This is because that the value of the phase retardation difference has a little deviation from 2π for sound waves propagating from the center of the neighbouring transparent zone of the Soret-type FZP to the predetermined focal point. In addition, an obvious chromatic aberration is observed in figure 4(d) when the frequency f is away from its reference value f 0 . By comparing between the transmitted intensities along the axis of lens in figures 4(b) and (d) (the same color bar range), it is concluded that an achromatic high efficient focusing can be obtained by the proposed metalens. Figure 5 shows the focusing performance comparison under the oblique incidence. It is found that the achromatic focusing can also be maintained at the incident angle θ i = 10 • and θ i = 30 • in figures 5(a) and (c), respectively. Owing to the design of achromatic metalens at normal incidence, the focusing intensity decreases with the incident angle (see the difference between the color bar range in (a) and (c)), which means that the proposed achromatic metalens cannot be worked at very large incident angles. For comparison, the transmitted intensities through FZP at the incident angles θ i = 10 • and θ i = 30 • are also illustrated in figures 5(b) and (d), respectively, where an obvious chromatic aberration and a lower focusing efficiency are observed.

Experimental measurement
Then an experiment is carried out to confirm the theoretical prediction. Figure 6(b) shows a photo of achromatic metalens, which is fabricated by 3D printing with the optimized parameters given in table 1. This metalens consists of 32 unit cells (Λ = 3.43 cm), where the aperture of the lens is L = 1.0976 m (along the x-axis), its thickness l ≈ 4.8 cm (along the z-axis), and its height 3 cm (along the y-axis). In addition, two walls perpendicular to the y-axis are added to the fabricated lens and their thickness is 2 mm. Figure 6(a) illustrates the fabricated Soret-type FZP with its size and the predetermined focusing position being the same as the achromatic metalens. Figure 6(c) shows a schematic diagram of experimental measurement, where the lens is sandwiched in between two Plexiglas plates of dimensions 4 m × 2 m × 8 mm, the edge of which is closed by anechoic cottons. A loudspeaker is placed at a distance 2.17 m away from the lens to radiate sine waves in the frequency range from 3300 to 4000 Hz. A probe microphone is moved in 1 cm step by a stepping motor to detect acoustic field along the measurement line marked in figure 6(c). All the receiving acoustic signals are achieved by the National Instruments PXI-6733 and PXI-4496. Since sound radiation from a point source in the two-dimensional free space is attenuated with the distance away from it, the method of insertion-loss measurement is always adopted to eliminate the effect of distance attenuation, where the insertion loss is defined as the ratio of sound pressure being measured with and without sample in dB [49]. Similar to the concept of insertion loss, the normalized transmitted intensity |p(x, z)| 2 /|p free (x, z)| 2 is introduced here to describe the focusing performance, where p(x, z) and p free (x, z) represent the measured sound pressure with and without the lens, respectively. Since p free (x, z) represents the pressure of the incident wave, the ratio |p(x, z)| 2 /|p free (x, z)| 2 can be considered as the power transmission coefficient, which eliminates the effect of distance attenuation, especially when the point source is located in the near-field zone. On the other hand, the introduction of this ratio has an advantage that we do not need to precisely control the amplitude of incident wave. Figure 7(a) gives the experimental normalized transmitted intensity of achromatic metalens. It is found that the focal length for the incidence of a point source is larger than that of plane wave incidence. This is because that the point source is located in the near-field zone, L 2 /λ > 2.17 m. For the Soret-type FZP in figure 7(c), an obvious chromatic aberration for the focusing of acoustic field is observed. For clarity, the range of color bar for Soret-type FZP is half that for the achromatic metalens, which means that the focal intensity of achromatic metalens is approximately twice that of Soret-type FZP. In addition, the simulated normalized transmitted intensities of achromatic metalens and FZP are given in figures 7(b) and (d), respectively. By comparison, it is concluded that the experimental measurement is consistent with the theoretical prediction.

Conclusion
In summary, we have proposed an achromatic acoustic generalized phase-reversal zone plate, the focusing performance of which is demonstrated numerically and confirmed experimentally. Although the working bandwidth of the proposed achromatic metalens is narrower compared to that designed by using a bottom-up inverse-design paradigm [28], the proposed metalens is designed based on an intuitive and physical approach, and it has the advantages of higher efficient focusing (transmission coefficient |t 0 | > 0.85 for each unit cell) and thinner thickness (one half of central wavelength). Therefore, the proposed metalens has potential application in acoustic energy harvesting and imaging.

Acknowledgments
We wish to acknowledge the support of the National Science Foundation of China under Grant No. 11674293.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.

Appendix A. Theoretical method
By the transfer matrix method and Floquet's theory [50], we derive the volume velocities U 0 and U l at the inlet (z = 0) and outlet (z = l) of a metasurface, respectively. Then, the transmission coefficient t 0 is obtained. Without loss of generality, a metasurface made up of unit cells shown in figure 1(c) is adopted in theoretical derivation. Note that we will always omit the temporally harmonic factor e −jωt as understood throughout this paper, and since the higher-order modes in the cavity of Helmholtz resonator are being adopted, the viscothermal loss is neglected in the theoretical derivations for simplification, however, the viscothermal loss is considered in FEM simulations.

A.1. Input acoustic impedance of a Helmholtz resonator grafted to the wall of a slit
In this subsection, with consideration of acoustic end correction, the input acoustic impedance of a Helmholtz resonator grafted to the wall of a slit is derived. For convenience, a local coordinate system x O z is introduced in figure 8.
Let the sound pressure in the throat ( w 0 2 < x < w 0 2 + b, − a 2 < z < a 2 ) be expressed as, and the sound pressure in the cavity where the eigenmode ϕ n z = √ 2 − δ 0n cos nπ h z + h 2 satisfies the rigid boundary condition at z = ±h/2, and k zn = nπ h . By the continuity of sound pressure and normal velocity at the boundary between the throat and cavity (x = w 0 2 + b), we obtain a relationship between the coefficients A and B in equation (A1), Substituting equation (A3) into equation (A1), and using the equation of motion jkρ 0 c 0 u x (x , z ) = ∂p/∂x , we obtain the input acoustic impedance at x = w 0 2 + b, A.2. The transfer matrix between the input and output of a metasurface By considering Helmholtz resonators as point-like resonators [51], the transfer matrix is derived in this subsection. For a metasurface made up of unit cells shown in figure 1(c) under a global coordinate system xOz, let the sound pressure in the slit (0 < x < w 0 , 0 < z < l 1 ) be expressed as, (A10) Using the equation of motion along the z-axis, jkρ 0 c 0 u z (x, z) = ∂p(x, z)/∂z, we obtain where the matrix is M 1 = ⎡ ⎣ cos(kl 1 ) −jρ 0 c 0 sin(kl 1 ) −j sin(kl 1 ) ρ 0 c 0 cos(kl 1 ) ⎤ ⎦ , and u z the velocity along the z-axis.
At the position of the first Helmholtz resonator (x = w 0 , z = l 1 ), we have the continuity of sound pressure, and the continuity of volume velocity where aū x represents the volume velocity at the inlet of the first Helmholtz resonator, and Z b1 denotes the input acoustic impedance of the first Helmholtz resonator being grafted to the wall of a slit, given by equation (A9). Equations (A12) and (A13) can be written in a matrix form as with the matrix N 1 = Similar to the derivations adopted in equations (A10)-(A14), we finally get a transfer matrix between the input and output of the metasurface as, A.3. The derivation of transmission coefficient t 0 We assume a unit incident sound pressure,