Issue 
Acta Acust.
Volume 6, 2022



Article Number  12  
Number of page(s)  10  
Section  Acoustic Materials and Metamaterials  
DOI  https://doi.org/10.1051/aacus/2021058  
Published online  08 April 2022 
Scientific Article
2D phononiccrystal Luneburg lens for allangle underwater sound localization
^{1}
Ocean College, Zhejiang University, Zhoushan, Zhejiang 316021, PR China
^{2}
Research Center for Intelligent Sensing, Zhejiang Lab, Hangzhou, Zhejiang 311121, PR China
^{*} Corresponding author: Liangxu@zju.edu.cn
Received:
6
September
2021
Accepted:
16
December
2021
Phononic crystals are well known for acoustic wave manipulation which may have potential application in an underwater acoustic detection system. In this work, we design and simulate a twodimensional Luneburg lens based on gradientindex (GRIN) phononic crystal that is composed of PLAAir inclusion, and a novel application of GRIN phononic crystals is proposed to sound localization. The Luneburg lens has a broadband working range, from 1500 Hz to 7500 Hz, for acoustic wave focusing with sensitive directivity and signaltonoise improvement. By searching maximum wave intensity’s position of the focusing beam, the propagating direction of an unknown sound wave can be directly recognized covering 360°. Besides, we redesign the conventional squarelattice Luneburg lenses using annular lattices for better performance. The annularlattice Luneburg lens overcomes the weakness of configuration defect due to the square lattice. The numerical results show that the redesign Luneburg lenses have high accuracy for distance measurement from 5 m to 35 m through the triangulation location. In a word, this work tries to explore a novel application of phononic crystals in underwater acoustic positioning and navigation technology.
Key words: Underwater metamaterials / Phononic crystal / Luneburg lens / Acoustic detection / Sound localization
© The Author(s), Published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Recently, underwater acoustic metamaterial [1–5] attracts a lot of attention because its function of wave manipulation has potential applications in underwater technology, and a new crossing field may come out. As an initial version of acoustic metamaterial, the phononic crystal that is a periodic structure made of multiple mediums or a singlephase artificial microstructure enables negative refraction [6], wave bending [7], and splitting [8] to be realized. Due to the excellent performance of waveguide, phononic crystal is used for acoustic wave focusing [9–11], and the fabricated device is always named phononiccrystal lens. Acoustic Luneburg lens [12–15] is a gradient refraction index lens that can converge wave energy even though the acoustic wave enters from different directions covering 360°. This feature has great advantages for detecting an object which produces sound waves in a water environment, like underwater beacons and underwater vehicles. However, the conventional Luneburg lens is difficult to manufacture because the refractive index profile is not a uniform distribution [16] until the gradientindex (GRIN) phononic crystal [17] appears.
The design of a Luneburg lens lies in the arrangement of unit cells according to the discretized refraction index profile. In order to realize the effect of acoustic wave convergence, Kim et al. [13, 18] and Xie et al. [12] designed 2D and 3D Luneburg lenses based on the filling ratio of scatterer which is a function of effective refraction index, and Kim et al. [19] firstly utilized the discretized Luneburg lens for signal amplification in the ocean. Since the effective velocity of a periodic structure is proposed to be calculated based on the band structure [17], the phononic crystals as a more efficient and precise way than the filling ratio is used to design the Luneburg lens [15]. When the working environment is air, the fluidsolid coupling is always ignored [20, 21], so Yu et al. [22] redesigned an underwater Luneburg lens for enhancing the focused signal at a center frequency of 180 kHz. Allam et al. [23] and Lu et al. [24] provided a 3Dprinted approach to replace the metallic structure for acoustic wave focusing using the Luneburg lens [25]. 3Dprinted is the fastest way to fabricate the Luneburg lens if the hydrostatic pressure is ignored. For better performance, Sun et al. [26, 27] offered an optimized method to design the acoustic lens in the water environment. These researches lay a foundation for the application of phononiccrystal Luneburg lens in ocean technology and engineering.
Finding an unknown sound in the ocean is difficult for human beings. The traditional approach for localizing a sound source is hydrophone arrays according to the time difference of receiving signal [28, 29]. To overcome that the accuracy of sound localization is limited by the number of the microphones and the array scale [30], researchers attempt to use the metamaterial to develop new sound localization techniques for more application scenarios [31, 32]. In this work, we designed squarelattice and annularlattice Luneburg lenses based on GRIN phononic crystals whose unit cell is made of PLA and air inclusion in a water environment, and we proposed a novel application of GRIN phononic crystals to sound direction identification and sound localization covering 360°. The frequency response, sound directivity and sound amplification of the Luneburg lens within a range of 1500–7500 Hz are all studied, and the distance measurement via triangulation location is simulated from 5 m to 35 m.
2 Luneburg lens based on phononic crystals
The Luneburg lens is designed by a discrete refractionindex profile using the GRIN phononic crystals [33]. In this work, the unit cell is composed of PLA (polylactic acid) and air inclusion in the water, as the inset picture illustrated in Figure 1. The PLA annulus is a commonly used 3Dprinted material [23] whose Young’s modulus, density and Poisson’s ratio are E = 3.2 Gpa, ρ = 1250 kg/m^{3} and μ = 0.3, respectively. The density and sound speed of air are 1.205 kg/m^{3} and 343 m/s, and those of water are 1025 kg/m^{3} and 1500 m/s. As depicted in Figure 3, the inner and outer radii of the PLA annulus are r_{1} and r_{2} respectively, and the lattice constant is a = 4 cm. Return to Figure 1, the shaded area in the inset picture is the reduced Brillouin zone of phononic crystal with high symmetry points Γ, X, and M. The solid lines are the first energy band of longitudinal waves with respect to r_{2} = 7, 10, and 13 mm. The band structure is calculated by COMSOL Multiphysics 5.5, including the acoustic module and mechanical module with an acousticsolid coupling connection. The straight dashed black line is the energy band of water which indicates that the traveling waves don’t have any dispersion during propagation. There are many references [34, 35] and websites^{1}^{,}^{2} where readers can download the examples, available to calculate the band structure based on the BlochFloquet theory, so the detailed process of calculation is not given in this paper.
Figure 1 Band structures in Γ − X direction of the PLAAir phononic crystals with different scatterer’s radii, r_{2} = 7, 10 and 13 mm. 
The target frequency for wave focusing is 3500 Hz. Figure 1 shows that the three energy bands are approximately straight lines around 3500 Hz. Therefore, the phase velocity which is nearly equal to the group velocity [17, 24, 36] can be directly calculated by v_{p} = 2πf/k along the Γ − X direction, where k is the reciprocal wavenumber. The effective refractive index is defined as n_{eff} = v_{w}/v_{p}, where v_{w} is the sound velocity in water. Compared with the steel inclusion whose radius is equal to r_{2}, the PLAAir unit cell has a higher effective refraction index, as demonstrated in Figure 2a, so PLAAir inclusion is a better choice for designing acoustic lens in water. The red points are the selected refraction indexes for designing the Luneburg lens.
Figure 2 (a) Effective refraction index as a function of scatterer’s radius. (b) Ideal and discretized refractionindex profile of Luneburg lens. 
The refractionindex profile of the Luneburg lens is , where r is the distance from center, and R = 0.32 m is the radius of Luneburg lens in this work. As shown in Figure 2b, the black line is the ideal refraction index profile of the Luneburg lens, but a material satisfying the ideal refraction index profile doesn’t exist in nature. Therefore, the refraction index profile is discretized as the red dashed line depicted in Figure 2b. Then a discretized Luneburg lens can be designed by finding matched unit cells in Figure 2a. Figure 3 demonstrated the acoustic Luneburg lens which is designed based on the refractionindex profile, and the number of unit cells is 177. The orange, blue and white parts are PLA, water and air, respectively.
Figure 3 Acoustic Luneburg lens designed by square lattice and its unit cell. 
3 Underwater sound direction identification
The difference between the Luneburg lens and other metalenses [5, 37, 38] is that the acoustic wave focusing can be achieved even though the acoustic wave comes from 360° directions, so it can be used to recognize the propagating direction of a sound wave. Figures 4a–4d presents the intensity field of focusing beam when a plane wave comes from the left side. The incident angles between the plane wave direction and xaxis are θ_{i} = 0°, 15°, 30° and 45°, and θ_{i} = 0°–45° is decided to be studied because the phononiccrystal Luneburg lens is a symmetry structure which can be divided into 8 parts. In the four cases, the maximum mesh size doesn’t exceed 1/10 wavelength. To eliminate the wave reflection on the boundary of computational domain, the PML (perfectly matched layer) and wave radiation boundary are set around the whole region to simulate an infinite area. The incident pressure amplitude is 1 Pa which is applied by a background pressure field on water area. The interfaces between PLA and water, including PLA and air, are connected by an acousticstructure coupling boundary. These simulations are carried out in the frequency domain. The results show that the designed Luneburg lens has the functionality of acoustic wave focusing, and the focusing beam is always on the opposite side of the incident plane.
Figure 4 (a)–(d) Wave intensity field of acoustic wave focusing when the incident angles θ_{i} are 0°, 15°, 30° and 45°. (e) Rotation correction procedure of squarelattice Luneburg lens. 
The wave intensity corresponding to the four cases in Figures 4a–4d is extracted at r = R + 0.04 m (solid circle) where the total number of mesh elements is 1600. As shown in Figure 5a, the directivity of the focusing beams simultaneously varies with the incident angles. It implies that the sound direction can be directly detected by measuring the position of maximum intensity. In Figures 4a–4d and 5a, the angles between the focusing beam and xaxis are 0°, 14.93°, 30.15° and 45°. The focusing angles, 0° and 45°, are exact because the squarelattice Luneburg lens is symmetric at 0° and 45°. However, the focusing angles, 14.93° and 30.15°, are not accurate enough. The error can be contributed to that the discretized lens is not a complete circular lens with configuration defect. In order to improve the accuracy, as demonstrated in Figure 4e, the Luneburg lens is rotated according to the first measured angles (θ_{a} = 14.93° and 30.15°), and then the focusing angles are measured again. The secondly measured focusing angles are θ_{b} = 15.01° and 30°. This correction treatment obviously improves the precision of recognizing the sound direction, because the rotation correction procedure avoids the configuration defect for some incident angles. In practice, the hydrophone for measuring the maximum intensity can be installed on the symmetry axis of the Luneburg lens and rotated together with the Luneburg lens.
Figure 5 Wave intensity of acoustic wave focusing beam with different incident angles (a) and frequencies (b). 
In this work, the designed frequency is 3500 Hz, but the acoustic wave focusing doesn’t fail immediately when the working frequency isn’t equal to the designed frequency. Hence, the bandwidth of acoustic wave focusing is studied. The focusing effect will disappear when the wavelength is much bigger than the size of Luneburg lens. In ultralow frequency, due to acoustic wave diffraction, the wave can directly pass through the Luneburg lens. On the contrary, when the wavelength is much smaller than the size of Luneburg lens, irregular multiple scattering caused by unit cells will occur inside the phononiccrystal lens and generate band gaps. In order to study the bandwidth of the designed Luneburg lens, frequency from 1500Hz to 7500 Hz is simulated. Figure 5b is a directivity graph when the frequency is from 1500 Hz to 7500 Hz. The results suggest that the Luneburg has a narrow focusing beam and a broad range of working frequency for sound amplification. The focused intensity becomes higher when the frequency increases. In higher frequency, the wavelength is shorter, which leads more energy into the Luneburg lens while the size of Luneburg lens is unchanged. Moreover, as the frequency increases, the slope of the energy band in Figure 1 gradually becomes flat, so the effective velocity v_{p} = 2πf/k is smaller. Then the effective refraction index of phononic crystal relative to water, n_{eff} = v_{w}/v_{p}, is greater. It also causes higher focused intensity when the frequency increases.
Furthermore, Table 1 provides the numerical results of sound direction identification for random incident angles with a frequency range of 1500–7500 Hz. The error is defined as ξ = (X_{2} − X_{1})/1° × 100%, where X_{2} and X_{1} are the numerical results and true value respectively. In this table, the biggest error of the first focusing angle θ_{a} is 0.3° (ξ_{1} = 30%), and the smallest error of θ_{a} is 0.08° (ξ_{1} = 8%). By the rotation correction procedure in Figure 4e, the maximum angle error is reduced to 0.05° (ξ_{2} = 5%), and the minimum error is decreased to 0° (ξ_{2} = 0%). The accuracy of wave direction recognition is raised utilizing the symmetry axis of the squarelattice Luneburg lens. The rotation operation makes the symmetry axis approximately parallel to the direction of incident wave at different incident angles as much as possible. At the symmetry axis, the errors of direction identification are close to zero such as θ_{i} = 0° and 45° in Figure 4. In addition, compared with the traditional way (hydrophone array) to measure the sound direction, the Luneburg lens has a higher ratio of signal to noise because it can converge wave energy.
Sound propagating direction identification with rotation correction.
The phononiccrystal Luneburg lens can be designed to be of different sizes. Figure 6a shows the focusing intensity with different radii of the Luneburg lens, R = 0.32–0.64 m. It can be concluded that the focused intensity steadily increases when the radius of lens has an increase because the Luneburg lens with a larger radius has a wider receiving area for the incident wave, and the size of the acoustic lens doesn’t have much influence on the relative width of focusing beam. In Figure 6b, the sound pressure level, SPL (dB) = 20 log(P_{rms}/P_{ref}), as the function of frequency is analyzed. The rootmeansquare pressure and the reference pressure in water are denoted by P_{rms} and P_{ref} = 1 μPa, respectively. The background sound pressure level is 116.99 dB when the incident pressure is 1 Pa. All the results are bigger than the background sound pressure level due to the acoustic wave focusing. With the increase of frequency, the sound amplification level is higher, but the SPL begins going down after around 6500 Hz because the frequency is close to the bandgap [39] that a frequency band of the acoustic wave is forbidden propagating in the periodic structure. In this work, the bandgap of the longitudinal wave is above the first energy band. As shown in Figure 1, the starting frequency of the bandgap is near 9.43 kHz when the radius r_{2} = 13 mm. If the working frequency exceeds 9.4 kHz into the bandgap, the phononiccrystal Luneburg lens will be disabled. In the first energy band, owing to the acoustic wave focusing, the Luneburg lens has the advantage of receiving a signal with intensity amplification to recognize the sound wave direction. Besides, it can use only one hydrophone or piezoelectric film around the lens to determine the direction of arbitrary incidence by measuring the position of maximum intensity, which would reduce the complexity of computation and integrated circuit than current positioning systems.
Figure 6 (a) Wave intensity of the acoustic wave focusing with different radii R of the Luneburg lens. (b) Sound pressure level as a function of frequency when the radius R is from 0.32 m to 0.64 m. 
4 Underwater sound source localization
As a result of the direction identification, an unknown sound source can be detected through the triangulation location. In the far field, the wavefront of spherical wave and cylindrical wave generated by a point source can be regarded as a plane wave, so the wave energy can also be converged, which has been proved by experiments in our previous work [25]. As illustrated in Figure 7, the localization of the underwater object can be divided into three steps: firstly, measure the sound direction as the correction method described in the third section, and assume that the first measured direction is yaxis; secondly, perpendicularly move the Luneburg lens by d, and set the moving direction as xaxis; finally, measure another angle θ_{2} and calculate the distance from the sound source to the center of Luneburg lens by l_{1} = d tan(θ_{2}). The first angle θ_{1} is equal to 90° because the moving direction of the Luneburg lens is always perpendicular to the first measured direction. If the minimum angle which can be measured by sensors is 0.1° in reality, the distance l_{1} is 35.80 m when d = 1 m and θ_{2} = 88.4°. The distance l_{1} is 38.19 m when d = 1 m and θ_{2} = 88.5°. The deviation is (38.19–35.80)/38.19 = 6.26% that is greater than 5%. To guarantee the accuracy, the distance l_{1} from 5 m to 35 m is studied as a short baseline positioning system in this work. The computation domain for sound localization is 4 × 2 m, and the sound source is applied by the background pressure field.
Figure 7 Schematic view of the underwater sound source localization. 
Table 2 compares the results of the underwater object localization with frequency from 1500 Hz to 7500 Hz. The movement of Luneburg lens d is 1 m, and the real distance is l = 10 m. From this table, the localization of the sound source has a minimum error of 0.19% when the frequency is around the central frequency of 3500 Hz. Moreover, the real distance with a range of 5–35 m is given in Table 3. It shows that the errors have a growth when the distance of sound source increases. Some cases in Tables 2 and 3 are not precise enough, more than 1.00%, when the working frequency is off the designed frequency 3500 Hz or the distance of sound source increases. Hence, to improve the localization precision, we change the movement d from 1.5 m to 2.5 m, as listed in Table 4. From the first three rows, it can be seen that the measurements become more accurate with the increase of the movement d. The errors are reduced from 1.74% to 0.19% in the case of f = 3500 Hz and d = 2.5 m. About the cases of 1500 Hz and 7500 Hz in the Table 2, the errors are decreased to 0.64% by setting d = 2.5 as shown in Table 4. Therefore, by means of the triangulation location, the designed Luneburg lens can be used to search for an unknown sound source within 35 m precisely.
Underwater sound localization when the frequency f is 1500–7500 Hz.
Underwater sound localization when the distance l is 5–35 m.
Underwater sound localization when movement d is 1.5–2.5 m.
5 Phononiccrystal Luneburg lens by annular lattice
However, we always worry about the phononiccrystal Luneburg lens’s unsmooth outline that may bring about a greater discrepancy when the source direction and position are unknown. As the results are shown in Table 1, the approximatively circular shape by square lattice reduces the reliability of sensitive directivity, and the rotation correction procedure for sound direction recognition and localization is really clumsy and slow. In addition, the Luneburg lens designed by square lattice costs a lot of time for geometry modeling. As sketched in Figure 8a, the Luneburg lens with square lattice needs to arrange the unit cell one by one in the 1/8 symmetry model, because the radii of unit cells are different from center to border of the Luneburg lens. If the radius of Luneburg lens is bigger, it will take more time for geometry modeling. For example, when the radius of Luneburg lens is R = 0.64 m, the total number of cylinders is 751 which leads the geometry modeling to be a timeconsuming task. Inspired by Maxwell’s fisheye lens which is composed of annular arrays [9], in this section, an alternative method is proposed to design the Luneburg lens. It will save a lot of time to design the Luneburg lens, and the focusing wave is more isotropy for sound direction recognition and positioning. Figure 8b shows the Luneburg lens with an annular arrangement of phononic crystals. The geometry modeling of annularlattice Luneburg lens just needs a rotation array at each radius. There are Round[2πr/a] cylinders at each radius r, and the total number of unit cells is equal to the squarely arranged one in Figure 8a, so this kind of design doesn’t increase the material consumption. Different from the square lattice, the annularlattice Luneburg lens with rotation array doesn’t need to align the unit cell one by one, which significantly simplifies the geometric modeling. Figure 9 demonstrates the focusing beam of the second Luneburg lens. Compared with the intensity field in Figure 4, it means that the redesigned Luneburg doesn’t change the focusing effect.
Figure 8 (a) Luneburg lens designed by square lattice. (b) Luneburg lens designed by annular lattice. 
Figure 9 Wave intensity field of the Luneburg lens with an annular arrangement. 
To further verify the functionality of the second Luneburg lens, the wave focusing beam, frequency response of sound pressure level and lens size are all analyzed. As shown in Figure 10a, the black and red lines are the intensity distribution along the arc at r = 0.36 m concerning the Luneburg lenses with square and annular lattice. Two solid lines show a small discrepancy when the incident angle θ_{i} is 0° where wave direction parallels the symmetry axis of squarelattice Luneburg lens. It implies that the annularlattice Luneburg lens doesn’t change the focusing property based on GRIN phononic crystal. For the annularlattice Luneburg lens, the intensity distribution with incident angles θ_{i} = 3°, 14°, 22°, and 41° are also calculated in Figure 10b. The focusing angles of these four cases are 3°, 14.03°, 22.05°, and 40.95°, respectively. The errors of sound direction recognition don’t exceed 5%, so the accuracy is better than the focusing angle that is directly calculated by the squarelattice Luneburg lens, as ξ_{1} (8–30%) demonstrated in Table 1. The annularlattice Luneburg lens has smaller errors than the squarelattice one, because the annularlattice one is more isotropic for incident waves. If the width of focusing beam is defined as the interval between positions at I_{max}/, where I_{max} is the maximum intensity. In the four cases, the largest width is 23.85°, and the smallest width is 23.78°. The width of focusing beam has little fluctuation when the incident angle changes, so the annularlattice Luneburg lens can be regarded as a nearly isotropic lens for incident waves. Figure 10c is the sound pressure level as a function of frequency. The sound pressure level is extracted from the focusing intensity at r = 0.36 m too. It shows that the annularlattice Luneburg lens has a similar frequency response to the squarelattice Luneburg lens. Furthermore, the annularlattice one also doesn’t change the focusing effect when the lens size is redesigned as demonstrated in Figure 10d. Therefore, all the results indicate that the redesigned Luneburg lens with annular arrays has little impact on the focusing effect, but the annular arrays make the configuration more isotropy.
Figure 10 (a) Wave intensity along the arc when r = 0.36 m when θ_{i} = 0°. (b) Wave intensity profile when θ_{i} = 3°, 14°, 22°, 41°, respectively. (c) Sound pressure level as a function of frequency. (d) Sound pressure level with the radius of Luneburg lens increased. 
The redesigned Luneburg lens can improve the accuracy of sound localization because it is nearly isotropic for any incident angles. To clearly illustrate the advantages of annularlattice Luneburg lens, an example is demonstrated in Figure 11. The incident angle and the working frequency are 5° and 5500 Hz respectively. The solid lines in the picture are the wave intensity of focusing beams, and the dashed lines are drawn at the peak intensity with respect to square lattice and annular lattice. After the sound wave converges, the acquired focusing angle of squarelattice Luneburg lens is 5.17°, and the error of focusing angle is 17%. By comparison, the focusing angle of annularlattice Luneburg lens is 4.95°, and the error is only 5 < 17%. Besides the example, in our simulation, the accuracy of annular lattice is all higher than the square lattice without rotation correction procedure in Figure 4e. Only the lattice form is changed in these cases, so we think the error is caused by the configuration defect of squarelattice Luneburg lens. As listed in Table 5, the symbols ξ_{1}, ξ_{2} and ξ_{3} represent the errors of squarelattice Luneburg lens without rotation correction procedure, squarelattice one with rotation correction procedure and annularlattice one without rotation correction procedure, respectively. The computation domain and mesh elements for sound localization simulation are consistent with the cases in the third section. The squarelattice one without rotation correction procedure has the worst precision because of the configuration defect. The largest error is 9.42%, and the smallest value is 4.56%. The squarelattice one with rotation correction procedure makes the accuracy better. The largest error is 3.91%, and the smallest value is 1.53%. However, the rotation measurements are clumsy and slow. By calculating the annularlattice Luneburg lens which doesn’t need a rotation correction procedure, the largest error is 2.09%, and the smallest error is only 0.02%. Better results are obtained by the redesigned Luneburg lens. Therefore, making a comparison of geometrical modeling and accuracy, the annulararranged Luneburg lens is more applicable than the squarelattice one obviously.
Figure 11 The focusing angles of squarelattice and annularlattice Luneburg lens when the incident angle and frequency are 5° and 5500 Hz, respectively. 
The errors of rearranged Luneburg lens for sound localization.
6 Conclusions
In conclusion, we designed a phononiccrystal Luneburg lens in a water environment which can be used to recognize wave direction and locate a sound source with sound amplification. The numerical results show that the Luneburg lens has a high accuracy of direction identification for 360° incident angles with a broadband frequency range between 1500 Hz and 7500 Hz. Meanwhile, the acoustic wave focusing improves the signaltonoise ratio, so it enables only one hydrophone to be used for sound direction recognition and locating the position of an unknown sound source from 5 m to 35 m, with errors below 1% by increasing the movement d properly. To improve the functionality and reliability, a Luneburg lens composed of annular lattices is redesigned based on the gradient phononic crystals. The comparison results show that the annularlattice Luneburg lens not only guarantees accuracy but also reduces complexity for geometric modeling. The phononiccrystal Luneburg lens may have an application prospect of nextgeneration sonar in underwater technology, such as underwater acoustic positioning systems and acoustic wireless communication [13, 19, 40]. In future, multiple sources, 3D detecting, moving targets and a silent object may also be developed. In addition, the experimental validation also needs to be carried out for further application.
Data availability
The data and programs that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 51879231 and 51679214), the Hightech Ship Research Project of the Ministry of Industry and Information Technology, the Primary Research and Development Plan of Zhejiang Province (Grant Nos. 2019C03115 and 2019C02050), the Zhejiang Daishan No. 4 Project of China General Nuclear Power Group, the Marine Innovation and Development Demonstration Project of State Oceanography Bureau (Grant No. PDHY002). Ruan thanks Chuanjie Hu from Xiamen University for the helpful discussion.
Conflict of interest
The authors have no conflicts to disclose.
References
 Y. Zhang, K. Chen, X. Hao, Y. Cheng: A review of underwater acoustic metamaterials. Kexue Tongbao/Chinese Science Bulletin 65, 15 (2020) 1396–1410. [CrossRef] [Google Scholar]
 S. Huang, L. Peng, H. Sun, Q. Wang, W. Zhao, S. Wang: Frequency response of an underwater acoustic focusing composite lens. Applied Acoustics 173 (2021) 107692. [CrossRef] [Google Scholar]
 M. Duan, C. Yu, F. Xin, T.J. Lu: Tunable underwater acoustic metamaterials via quasiHelmholtz resonance: From lowfrequency to ultrabroadband. Applied Physics Letters 118, 7 (2021) 071904. [CrossRef] [Google Scholar]
 Y. Shen, C. Qiu, X. Cai, L. Ye, J. Lu, M. Ke, Z. Liu: Valleyprojected edge modes observed in underwater sonic crystals. Applied Physics Letters 114, 2 (2019) 023501. [CrossRef] [Google Scholar]
 J. Chen, J. Rao, D. Lisevych, Z. Fan: Broadband ultrasonic focusing in water with an ultracompact metasurface lens. Applied Physics Letters 114, 10 (2019) 104101. [CrossRef] [Google Scholar]
 A. Sukhovich, L. Jing, J.H. Page: Negative refraction and focusing of ultrasound in twodimensional phononic crystals. Physical Review B 77, 1 (2008) 014301. [CrossRef] [Google Scholar]
 L. Wu, L. Chen: An acoustic bending waveguide designed by graded sonic crystals. Journal of Applied Physics 110, 11 (2011) 114507. [CrossRef] [Google Scholar]
 B. Li, J. Guan, K. Deng, H. Zhao: Splitting of selfcollimated beams in twodimensional sonic crystals. Journal of Applied Physics 112, 12 (2012) 124514. [CrossRef] [Google Scholar]
 B. Yuan, Y. Tian, Y. Cheng, X. Liu: An acoustic Maxwell’s fisheye lens based on gradientindex metamaterials. Chinese Physics B 25, 10 (2016) 104301. [CrossRef] [Google Scholar]
 X. Hu, C.T. Chan, J. Zi: Twodimensional sonic crystals with Helmholtz resonators. Physical Review E 71, 5 (2005) 055601. [CrossRef] [PubMed] [Google Scholar]
 S.S. Lin, B.R. Tittmann, T.J. Huang: Design of acoustic beam aperture modifier using gradientindex phononic crystals. Journal of Applied Physics 111, 12 (2012) 123510. [CrossRef] [PubMed] [Google Scholar]
 Y. Xie, Y. Fu, Z. Jia, J. Li, C. Shen, Y. Xu, H. Chen, S.A. Cummer: Acoustic imaging with metamaterial Luneburg lenses. Scientific Reports 8, 1 (2018) 16188. [CrossRef] [PubMed] [Google Scholar]
 S. Kim: Sound focusing by acoustic Luneburg lens (2014). ArXiv preprint [arXiv: 1409.5489] [Google Scholar]
 Y. Fu, J. Li, Y. Xie, C. Shen, Y. Xu, H. Chen, S.A. Cummer: Compact acoustic retroreflector based on a mirrored Luneburg lens. Physical Review Materials 2, 10 (2018) 105202. [CrossRef] [Google Scholar]
 S. Tol, F.L. Degertekin, A. Erturk: Phononic crystal Luneburg lens for omnidirectional elastic wave focusing and energy harvesting. Applied Physics Letters 111, 1 (2017) 013503. [CrossRef] [Google Scholar]
 N.J. Whitehead, S.A.R. Horsley, T.G. Philbin, V.V. Kruglyak: A Luneburg lens for spin waves. Applied Physics Letters 113, 21 (2018) 212404. [CrossRef] [Google Scholar]
 S.S. Lin, T.J. Huang, J. Sun, T. Wu: Gradientindex phononic crystals. Physical Review B 79, 9 (2009) 094302. [CrossRef] [Google Scholar]
 J. Kim, S. Lee, J. Jo, S. Wang, S. Kim: Acoustic imaging by threedimensional acoustic Luneburg metalens with lattice columns. Applied Physics Letters 118, 9 (2021) 091902. [CrossRef] [Google Scholar]
 S. Kim, ByeongWon, KyungMin, G.S. Lim: Sound reception system by an acoustic Luneburg lens (2019). ArXiv preprint [arXiv: 1906.07174] [Google Scholar]
 C.M. Park, S.H. Lee: Acoustic Luneburg lens using orificetype metamaterial unit cells. Applied Physics Letters 112, 7 (2018) 074101. [CrossRef] [Google Scholar]
 L. Zhao, E. Laredo, O. Ryan, A. Yazdkhasti, H. Kim, R. Ganye, T. Horiuchi, M. Yu: Ultrasound beam steering with flattened acoustic metamaterial Luneburg lens. Applied Physics Letters 116, 7 (2020) 071902. [CrossRef] [Google Scholar]
 R. Yu, H. Wang, W. Chen, C. Zhu, D. Wu: Latticed underwater acoustic Luneburg lens. Applied Physics Express 13, 8 (2020) 84003. [Google Scholar]
 A. Allam, K. Sabra, A. Erturk: 3Dprinted gradientindex phononic crystal lens for underwater acoustic wave focusing, Physical Review Applied 13, 6 (2020) 064064. [CrossRef] [Google Scholar]
 C. Lu, R. Yu, Q. Ma, K. Wang, J. Wang, D. Wu: GRIN metamaterial generalized Luneburg lens for ultralong acoustic jet. Applied Physics Letters 118, 14 (2021) 144103. [CrossRef] [Google Scholar]
 Y. Ruan, X. Liang, Z. Wang, T. Wang, Y. Deng, F. Qu, J. Zhang: 3D underwater acoustic wave focusing by periodic structure. Applied Physics Letters 114, 8 (2019) 081908. [CrossRef] [Google Scholar]
 H. Sun, S. Wang, S. Huang, L. Peng, Q. Wang, W. Zhao, J. Zou: 3D focusing acoustic lens optimization method using multifactor and multilevel orthogonal test designing theory. Applied Acoustics 170 (2020) 107538. [CrossRef] [Google Scholar]
 H. Sun, S. Wang, S. Huang, L. Peng, Q. Wang, W. Zhao: Design and characterization of an acoustic composite lens with highintensity and directionally controllable focusing. Scientific Reports 10, 1 (2020) 1469. [CrossRef] [PubMed] [Google Scholar]
 H.C. Song, G. Byun: Localization of a distant ship using a guide ship and a vertical array. The Journal of the Acoustical Society of America 149, 4 (2021) 2173–2178. [CrossRef] [PubMed] [Google Scholar]
 H.C. Song, C. Cho: Array invariantbased source localization in shallow water using a sparse vertical array. The Journal of the Acoustical Society of America 141, 1 (2017) 183–188. [CrossRef] [PubMed] [Google Scholar]
 M. Brandstein: Microphone arrays: Signal processing techniques and applications. Springer Science & Business Media, Germany, 2001. [CrossRef] [Google Scholar]
 J. Gu, W. Lin, C. Kan: Sound source localization using piezoelectric acoustic metasurfaces. Acoustics Australia 48, 3 (2020) 455–461. [CrossRef] [Google Scholar]
 X. Sun, H. Jia, Z. Zhang, Y. Yang, Z. Sun, J. Yang: Sound localization and separation in 3D space using a single microphone with a metamaterial enclosure. Advanced Science 7, 3 (2020) 1902271. [CrossRef] [Google Scholar]
 Y. Jin, B. DjafariRouhani, D. Torrent: Gradient index phononic crystals and metamaterials. Nanophotonics 8, 5 (2019) 685–701. [CrossRef] [Google Scholar]
 Y. Cao, Z. Hou, Y. Liu: Finite difference time domain method for bandstructure calculations of twodimensional phononic crystals. Solid State Communications 132, 8 (2004) 539–543. [CrossRef] [Google Scholar]
 G. Wang, J. Wen, Y. Liu, X. Wen: Lumpedmass method for the study of band structure in twodimensional phononic crystals. Physical Review B 69, 18 (2004) 184302. [CrossRef] [Google Scholar]
 Y. Tian, Z. Tan, X. Han, W. Li: Phononic crystal lens with an asymmetric scatterer. Journal of Physics D: Applied Physics 52, 2 (2019) 25102. [Google Scholar]
 D.C. Calvo, A.L. Thangawng, M. Nicholas, C.N. Layman: Thin Fresnel zone plate lenses for focusing underwater sound. Applied Physics Letters 107, 1 (2015) 014103. [CrossRef] [Google Scholar]
 D. TarrazóSerrano, S. PérezLópez, P. Candelas, A. Uris, C. Rubio: Acoustic focusing enhancement in Fresnel Zone Plate Lenses. Scientific Reports 9, 1 (2019) 7067. [CrossRef] [PubMed] [Google Scholar]
 Y. Pennec, J.O. Vasseur, B. DjafariRouhani, L. Dobrzyński, P.A. Deymier: Twodimensional phononic crystals: Examples and applications. Surface Science Reports 65, 8 (2010) 229–291. [CrossRef] [Google Scholar]
 X. Su, I. Ullah, X. Liu, D. Choi: A review of underwater localization techniques, algorithms, and challenges. Journal of Sensors 2020 (2020) 6403161. [Google Scholar]
Cite this article as: Ruan Y. & Liang X. 2022. 2D phononiccrystal Luneburg lens for allangle underwater sound localization. Acta Acustica, 6, 12.
All Tables
All Figures
Figure 1 Band structures in Γ − X direction of the PLAAir phononic crystals with different scatterer’s radii, r_{2} = 7, 10 and 13 mm. 

In the text 
Figure 2 (a) Effective refraction index as a function of scatterer’s radius. (b) Ideal and discretized refractionindex profile of Luneburg lens. 

In the text 
Figure 3 Acoustic Luneburg lens designed by square lattice and its unit cell. 

In the text 
Figure 4 (a)–(d) Wave intensity field of acoustic wave focusing when the incident angles θ_{i} are 0°, 15°, 30° and 45°. (e) Rotation correction procedure of squarelattice Luneburg lens. 

In the text 
Figure 5 Wave intensity of acoustic wave focusing beam with different incident angles (a) and frequencies (b). 

In the text 
Figure 6 (a) Wave intensity of the acoustic wave focusing with different radii R of the Luneburg lens. (b) Sound pressure level as a function of frequency when the radius R is from 0.32 m to 0.64 m. 

In the text 
Figure 7 Schematic view of the underwater sound source localization. 

In the text 
Figure 8 (a) Luneburg lens designed by square lattice. (b) Luneburg lens designed by annular lattice. 

In the text 
Figure 9 Wave intensity field of the Luneburg lens with an annular arrangement. 

In the text 
Figure 10 (a) Wave intensity along the arc when r = 0.36 m when θ_{i} = 0°. (b) Wave intensity profile when θ_{i} = 3°, 14°, 22°, 41°, respectively. (c) Sound pressure level as a function of frequency. (d) Sound pressure level with the radius of Luneburg lens increased. 

In the text 
Figure 11 The focusing angles of squarelattice and annularlattice Luneburg lens when the incident angle and frequency are 5° and 5500 Hz, respectively. 

In the text 
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