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Broadband high-efficiency controllable asymmetric propagation by pentamode acoustic metasurface
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Physics Letters A ••• (••••) ••••••
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Broadband high-efficiency controllable asymmetric propagation by pentamode acoustic metasurface Yangyang Chu a , Zhaohong Wang a,∗ , Zhuo Xu a,b a
Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Faculty of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi Xi’an, 710049, People’s Republic of China b Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education & International Center for Dielectric Research, Xi’an Jiaotong University, Shaanxi Xi’an, 710049, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 12 August 2019 Received in revised form 9 December 2019 Accepted 27 December 2019 Available online xxxx Communicated by M. Wu Keywords: Pentamode metasurface Broadband High-efficiency Asymmetric transmission
a b s t r a c t We utilise the pentamode metasurface to realise broadband high-efficiency and controllable asymmetric transmission. The designed metasurface can manipulate the acoustic waves, as expected from the generalised Snell’s law, and exhibits unique characteristics such as extraordinary broadband acoustic control, apparent negative refraction, and conversion from the propagating wave to surface mode. The asymmetric transmission features of positive refraction for the forward incidence (FI) and negative refraction for the reverse incidence (RI) can be realised within the range of 2600 Hz to 5600 Hz by controlling the incident angle from 0◦ to 35◦ with the transmission efficiency higher than 85.4% for the FI and RI. In addition, by further adjusting the angle of incidence in the range of 25◦ to 90◦ , asymmetric transmission characteristics can be expressed as surface wave transmission for the FI and transmitted wave transmission for the RI within the same frequency ranges. © 2020 Elsevier B.V. All rights reserved.
1. Introduction The relationship between the angle of incidence and the angle of refraction is certainly determined while light waves propagate from one medium to another under a different refractive index due to momentum conservation along the tangential boundary direction; this phenomenon exemplifies Snell’s law. Following this law, Federico Capasso’s team at Harvard University achieved arbitrarily controlled electromagnetic wave reflection (transmission) properties by introducing resonant elements at the material interface to generate localised momentum and design phase mutations at the interface; their discovery resulted in the generalised Snell’s law (GSL) [1]. The acoustic metasurface is a 2D artificial material with subwavelength thickness, which can provide unique local phase shifts or amplitude modulation. Introducing the concept of the acoustic metasurface into materials science and physics using GSL presents new possibilities for manipulating acoustic waves and producing a series of unconventional acoustic phenomena. Various advanced acoustic wave–manipulating functionalities have since been made possible, such as subwavelength imaging [2,3], focusing [4,5], near-perfect absorption [6,7], negative refraction of
*
Corresponding author. E-mail address: [email protected] (Z. Wang).
https://doi.org/10.1016/j.physleta.2019.126230 0375-9601/© 2020 Elsevier B.V. All rights reserved.
acoustic waves [8,9], and invisibility cloaking [10,11]. Yet the GSL is frequency-dependent [12,13], such that the metasurface based on GSL only works within a selected narrow band. A method using the gradient rate (rather than the gradient phase) to obtain a frequency-independent GSL has been proposed [14], making it possible for the metasurface to manipulate acoustic waves over a broad frequency range. Acoustic asymmetric transmission, as another fascinating metasurface phenomenon, has also been widely explored. Acoustic asymmetric transmission can generally be achieved in two ways, either by disrupting time-reversal symmetry using a medium with strong nonlinearity [15,16] or by breaking spatial inversion with the asymmetric geometry profile [17,18]. Recently, some scholars have realised asymmetric transmission through a device consisting of an acoustic gradient-index metasurface (GIM) and another structure [19–22]. This type of structure can perform functions similar to acoustic diodes. The GIM is realised via phase engineering, enabling the device to produce efficient unidirectional acoustic transmission within a narrow bandwidth. In addition, introducing judiciously tailored losses into the GIM has been found to achieve acoustic asymmetric transmission [23,24]. Acoustic waves can then be strongly attenuated for the reverse incidence and freely transmitted for the forward incidence. Due to the introduction of losses in the GIM, acoustic waves transmitted for FI will generate a large energy loss. The loss is usually considered
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a disadvantage in manipulating acoustic waves, which limits the exploration of new possibilities. Pentamode metamaterials (PMs) can achieve impedance matching over broadband, and the effective constitutive parameters can be tailored independently [25,26]. Thus, PMs provide a good solution for achieving broadband and high-efficiency acoustic asymmetric transmission. 2. Theory of acoustic asymmetric diffraction transmission When an acoustic wave impinges on the metasurface, the relationship between the incident and transmitted waves should follow the GSL:
k0 (sin θt − sin θi ) = d∅(x)/dx
(1)
where θt is the angle of refraction, θi is the angle of incidence, k0 is the wavenumber of incident wave, ∅(x) is the phase accumulation across the metasurface, and x is the transversal coordinate along the metasurface. The k0 in Formula (1) is frequency-dependent, resulting in GSL-based metasurfaces with inherent narrowband limitations. Drawing upon previous research [14], the transversal gradient velocity was used to replace the gradient phase by substituting the phase accumulation ∅ (x) = lk0 c 0 /c (x) into Formula (1); thus, a frequency-independent GSL is obtained:
sin θt − sin θi = lc 0 d[1/c (x)]/dx
Fig. 1. Schematic of proposed microstructures, where the thickness t and side length m of hexagonal PMs are adjustable.
(2)
where l is the metasurface thickness; c 0 is wave velocity in the background medium; and c (x) is the wave velocity in the structure. As shown in Formula (2), there is no frequency-dependent term; accordingly, the transmitted wave can be manipulated by 1/c (x) over a wide frequency band. In this study, we present a broadband controllable asymmetric transmission metasurface scheme based on the frequency-independent GSL. In designing the metasurface, we adopt two-dimensional hexagonal PMs whose schematic diagram is illustrated in Fig. 1. The framework is a regular hexagonal honeycomb lattice composed of six metallic arms with length a and thickness t, and six additional weights (an equilateral triangular terminal with side length m) are at the hexagon vertices. Theoretical analysis reveals that the thickness t mainly controls the effective bulk modulus of the PMs. The side length m of the equilateral triangle primarily controls the effective density, which is approximately equal to the average bulk density in the long-wavelength limitation 26]. The PM structure is made of titanium (density ρTi = 4500 kg/m3 ; Young’s modulus E Ti = 108 GPa; Poisson’s ratio c Ti = 0.34) or lead (density ρ L = 11400 kg/m3 ; Young’s modulus E L = 16.4 GPa; Poisson’s ratio c L = 0.44), and the hollow part is filled with air (density ρa = 1.21 kg/m3 ; velocity ca = 343 m/s). In using the proposed PMs to design a broadband and controllable asymmetric transmission metasurface, the key factor is to maintain acoustic impedance that matches the background medium (water with density ρ0 = 1000 kg/m3 ; velocity c 0 = 1490 m/s) at the metasurface interface. In this study, we design eight pentamode units to construct the metasurface given a distribution of effective velocities from c 0 to c 0 /7. The metasurface structure and required spatial distribution of 1/c (x) are depicted in Fig. 2(a). Geometric parameters of the eight unit cells are labelled S1–S8, as listed in Table 1. Considering that the 1/c (x) introduced in the metasurface is directional, the transmission relationship between FI and RI will differ when the metasurface is tilted at the incident angle θi (θi = 0◦ ) as indicated in Fig. 2(b). The pentamode units change from S1 to S8 along the incident direction for FI, the corresponding 1/c (x) changes from 1/c 0 to 7/c 0 , and the transversal gradient velocity becomes positive. In this case, Formula (2) can be rewritten as Formula (3)(a). However, for RI, the pentamode units change from S8 to S1 along
Fig. 2. (a) The spatial distribution of 1/c (x) for ideal metasurface (black dashed line) and pentamode metasurface (red solid line). (b) Sketch of FI and RI with angle θi .
Table 1 Geometric parameters of the eight unit cells as shown in Fig. 2(a). Unit cells
a (mm)
t (mm)
m (mm)
S1 S2 S3 S4 S5 S6 S7 S8
12 12 12 12 12 12 12 12
0.434 0.148 0.800 0.566 0.394 0.268 0.184 0.120
3.894 6.704 3.526 5.144 6.272 7.066 7.908 8.562
Fig. 3. Relationship between θt and θi for FI and RI. Black and red dashed lines represent theoretical calculations; black diamonds and red circles represent simulated results.
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Fig. 4. Calculated acoustic pressure fields for (a) FI and (b) RI at an angle of 10◦ at 3600 Hz.
the incident direction, the corresponding 1/c (x) changes from 7/c 0 to 1/c 0 , and the transversal gradient velocity becomes negative, therefore Formula (2) can be rewritten as Formula (3)(b):
sin θt + = sin θi + lc 0 d[1/c (x)]/dx sin θt − = sin θi − lc 0 d[1/c (x)]/dx
(a) (b)
(3)
where θt + and θt − are the transmitted angles of FI and RI, respectively. As seen from the previous definition, d[1/c (x)]/dx is always greater than 0 in this study. Therefore, Formulas (3)(a) and (3)(b) indicate that the forward and reverse transmitted angles differ. In addition, sin θt + is always greater than 0 and sin θt − is always less than 1. We can obtain sin θt − < 0 by adjusting the incident angle, at which point the transmitted waves demonstrate negative refractive properties. In this case, the asymmetric transmission characteristic exhibits negative refraction for RI and positive refraction for FI. In addition, by further adjusting the incident angle to sin θt + ≥ 1, the conversion from a propagating wave to surface wave can be realised; that is, the asymmetric transmission characteristics can be expressed as surface wave transmission for FI and transmitted wave transmission for RI. Furthermore, the PMs can achieve broadband impedance matching, and the characteristics of broadband and controllable acoustic asymmetric transmission can be obtained.
Fig. 5. Relationship between the refraction angle θt and f for FI and RI at an angle of 10◦ . Black and red dashed lines represent theoretical calculations; black diamonds and red circles represent simulated results.
3. Numerical simulations Based on the preceding analysis, one metasurface is designed using the eight unit cells (S1–S8) in Table 1. The adopted metasurface is composed of 48 × 4 microstructures. Thus, the metasurface thickness is l√= 6a, and the spatial velocity variation is d[1/c (x)]/dx = 1/6 3ac 0 . Using this metasurface to achieve asymmetric transmission of positive refraction for FI and negative refraction for RI, it is necessary to make sin θt − < 0, where
√
sin θt − = sin θi − 1/ 3 < 0
(4)
Because sin θi is always greater than 0, Formula (4) shows that 0◦ < θi < 35◦ . Therefore, the asymmetric transmission characteristic of negative refraction for RI and positive refraction for FI can be realised when the incident angle of the acoustic wave ranges from 0◦ to 35◦ . Then, the relationship between θt and θi within the range of [0◦ , 35◦ ] can be analysed, as shown in Fig. 3. This metasurface can achieve asymmetric transmission of positive refraction (θt + > 0) for FI and negative refraction (θt − < 0) for RI within the range of [0◦ , 35◦ ]. We randomly chose θi = 10◦ for numerical simulation, and θt + = 48.7◦ and θt − = −23.8◦ can thus be obtained using Formulas (3)(a) and (3)(b). The finite element method is used to demonstrate the unique performance of the proposed metasurface. Figs. 4(a) and 4(b) show acoustic transmission through the PM metasurface
Fig. 6. Transmission efficiency for pentamode metasurface for FI (black line) and RI (red line) at an angle of 10◦ .
for FI and RI at f = 3600 Hz, respectively. The black arrows indicate the direction of the acoustic wave. The incident waves for FI and RI can be freely transmitted through the metasurface. Fig. 4(a) reveals that positive refraction occurs when the FI with an angle of 10◦ is transmitted through the metasurface. The propagation direction of the transmitted wavefronts follows the black line, and the angle of refraction is 48.5◦ , consistent with the theoretical prediction. Unlike the case of Fig. 4(a), negative refraction appears when the RI with an angle of 10◦ is transmitted through the metasurface (see Fig. 4(b)). The refraction angle in this case is −23.6◦ , which also coincides with the previous theoretical calculation. In addition, a series of numerical simulations were conducted for the FI and RI in the range of 2600 Hz ≤ f ≤ 5600 Hz. The relationship between the θt + (θt − ) and f is displayed in Fig. 5. As indicated, θt + remains around 48.7◦ , and θt − remains around −23.8◦ when the FI and RI are at an angle of 10◦ , coinciding with the previous theoretical calculation of the entire study bandwidth. Energy transmittance spectra of the metasurface was also analysed with partial simulation results presented in Fig. 6. The transmis-
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Fig. 7. Calculated acoustic pressure fields for (a) FI and (b) RI with an angle of 40◦ at 3600 Hz.
sion efficiency of FI and RI each exceed 85.4% over the range of 2600 Hz–5600 Hz; therefore, the metasurface can be considered highly transparent to acoustic waves. This kind of metasurface thus makes the acoustic wave highly transparent over broadband in addition to achieving broadband asymmetric transmission. Apart from the above qualities, the acoustic characteristics of surface wave transmission for FI can be achieved by merely adjusting the incident angle. Using the same metasurface structure as earlier, the sin θt + ≥ 1 can be expressed as
√
sin θt + = sin θi + 1/ 3 ≥ 1
(5)
In Formula (5), sin θt + ≥ 1 indicates that the transmission angle is 90◦ ; that is, the acoustic wave will be converted from a propagating wave to surface mode after passing through the metasurface. As sin θi is always less than 1, the angle of incidence can be obtained from Formula (5) such that 25◦ ≤ θi ≤ 90◦ . Accordingly, the asymmetric transmission characteristics can be expressed as surface wave transmission for the FI and transmitted wave transmission for the RI at an incident angle greater than 25◦ . Here, we chose θi = 40◦ for the numerical simulation; θt + = 90◦ and θt − = 3.8◦ can be obtained from Formulas (3)(a) and (3)(b). Figs. 7(a) and 7(b) demonstrate the pressure field distribution of the metasurface for the FI and RI of f = 3600 Hz, respectively. As shown in Fig. 7(a), the transmitted wave propagates along the metasurface when the FI with an incident angle of 40◦ is transmitted through the metasurface. The acoustic wave can thus be converted from a propagating wave to surface wave mode through the metasurface. However, transmission characteristics differ from Fig. 7(a) when the RI with an incident angle of 40◦ is transmitted through the metasurface, as displayed in Fig. 7(b). The transmitted wave propagates at an angle of 3.5◦ from the normal (the black dotted line), corroborating the previous theoretical calculation. The acoustic transmission characteristics of FI and RI in the range of 2600 Hz ≤ f ≤ 5600 Hz were then simulated to study broadband characteristics. Fig. 8 illustrates the relationship between the θt + (θt − ) and f . In this case, θt + remains around 90◦ for the FI with an angle of 40◦ ; that is, surface wave transmission can be achieved after the incident acoustic wave passes through the metasurface. However, θt + is around 3.8◦ for the RI with an angle of 40◦ . The metasurface structure is the same as in Fig. 4, and the transmission efficiency remains above 85.4% as similar to Fig. 5. Therefore, this metasurface can achieve broadband asymmetric transmission characteristics of surface wave transmission for FI and transmitted wave transmission for RI. 4. Conclusion To conclude, a broadband and controllable acoustic metasurface is developed in this study based on the frequency-independent GSL and PMs. By utilising the impedance matching characteristics of PMs and the directionality of the gradient 1/c (x), the metasurface can realise acoustic asymmetric transmission over broadband ranges from 2600 Hz to 5600 Hz, and the transmission ef-
Fig. 8. Relationship between θt and f for FI and RI at an angle of 40◦ . Red dashed lines represent theoretical calculations; black diamonds and red circles represent simulated results.
ficiency exceeds 85.4% for FI and RI. Adjusting the incident angle can modify anomalous asymmetric transmission characteristics, including converting the propagating wave into surface mode or negative refraction characteristics. This metasurface can also be applied to achieve efficient anomalous transmission of acoustic waves over an ultrawide frequency range. Furthermore, this work outlines a means of broadband-tunable asymmetric transmission with substantial design flexibility, revealing promising applications for acoustic cloaks, acoustic energy collection, novel acoustic function devices, and so on. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. References [1] N. Yu, P. Genevet, M.A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science 334 (2011) 333–337. [2] J. Li, L. Fok, X. Yin, G. Bartal, X. Zhang, Experimental demonstration of an acoustic magnifying hyperlens, Nat. Mater. 8 (2009) 931. [3] H. Zhang, X. Zhou, G. Hu, Shape-adaptable hyperlens for acoustic magnifying imaging, Appl. Phys. Lett. 109 (2016) 224103. [4] Y. Li, B. Liang, X. Tao, X. Zhu, X. Zou, J. Cheng, Acoustic focusing by coiling up space, Appl. Phys. Lett. 101 (2012) 233508. [5] W. Wang, Y. Xie, A. Konneker, B.-I. Popa, S.A. Cummer, Design and demonstration of broadband thin planar diffractive acoustic lenses, Appl. Phys. Lett. 105 (2014) 101904. [6] Y. Li, B.M. Assouar, Acoustic metasurface-based perfect absorber with deep subwavelength thickness, Appl. Phys. Lett. 108 (2016) 063502. [7] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound, Nat. Commun. 3 (2012) 756.
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[8] Z. Liang, J. Li, Extreme acoustic metamaterial by coiling up space, Phys. Rev. Lett. 108 (2012) 114301. [9] B. Liu, B. Ren, J. Zhao, X. Xu, Y. Feng, W. Zhao, Y. Jiang, Experimental realization of all-angle negative refraction in acoustic gradient metasurface, Appl. Phys. Lett. 111 (2017) 221602. [10] H. Chen, C.T. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials, Appl. Phys. Lett. 91 (2007) 183518. [11] S. Zhang, C. Xia, N. Fang, Broadband acoustic cloak for ultrasound waves, Phys. Rev. Lett. 106 (2011) 024301. [12] Y. Xie, W. Wang, H. Chen, A. Konneker, B.I. Popa, S.A. Cummer, Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface, Nat. Commun. 5 (2014) 5553. [13] X. Ni, N.K. Emani, A.V. Kildishev, A. Boltasseva, V.M. Shalaev, Broadband light bending with plasmonic nanoantennas, Science 335 (2012) 427. [14] Y. Tian, Q. Wei, Y. Cheng, X. Liu, Broadband manipulation of acoustic wavefronts by pentamode metasurface, Appl. Phys. Lett. 107 (2015) 221906. [15] B. Liang, B. Yuan, J.C. Cheng, Acoustic diode: rectification of acoustic energy flux in one-dimensional systems, Phys. Rev. Lett. 103 (2009) 104301. [16] B. Liang, X. Guo, J. Tu, D. Zhang, J. Cheng, An acoustic rectifier, Nat. Mater. 9 (2010) 989. [17] Z. He, S. Peng, Y. Ye, Z. Dai, C. Qiu, M. Ke, Z. Liu, Asymmetric acoustic gratings, Appl. Phys. Lett. 98 (2011) 083505. [18] S. Xu, C. Qiu, Z. Liu, Acoustic transmission through asymmetric grating structures made of cylinders, J. Appl. Phys. 111 (2012) 094505.
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[19] C. Shen, Y. Xie, J. Li, S.A. Cummer, Y. Jing, Asymmetric acoustic transmission through near-zero-index and gradient-index metasurfaces, Appl. Phys. Lett. 108 (2016) 223502. [20] B. Liu, Y. Jiang, Controllable asymmetric transmission via gap-tunable acoustic metasurface, Appl. Phys. Lett. 112 (2018) 173503. [21] X. Wang, L. Wan, T. Chen, Q. Liang, A. Song, Broadband acoustic diode by using two structured impedance-matched acoustic metasurfaces, Appl. Phys. Lett. 109 (2016) 044102. [22] Song, T. Chen, X. Wang, L. Wan, Waveform-preserved unidirectional acoustic transmission based on impedance-matched acoustic metasurface and phononic crystal, J. Appl. Phys. 120 (2016) 085106. [23] Y. Li, C. Shen, Y. Xie, J. Li, W. Wang, S.A. Cummer, Y. Jing, Tunable asymmetric transmission via lossy acoustic metasurfaces, Phys. Rev. Lett. 119 (2017) 035501. [24] F. Ju, Y. Tian, Y. Cheng, X. Liu, Asymmetric acoustic transmission with a lossy gradient-index metasurface, Appl. Phys. Lett. 113 (2018) 121901. [25] T. Bückmann, M. Thiel, M. Kadic, R. Schittny, M. Wegener, An elasto-mechanical unfeelability cloak made of pentamode metamaterials, Nat. Commun. 5 (2014) 4130. [26] M. Kadic, T. Bückmann, R. Schittny, P. Gumbsch, M. Wegener, Pentamode metamaterials with independently tailored bulk modulus and mass density, Phys. Rev. Appl. 2 (2014) 054007.
[m5G; v1.261; Prn:15/01/2020; 11:57] P.1 (1-5)
Physics Letters A ••• (••••) ••••••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Broadband high-efficiency controllable asymmetric propagation by pentamode acoustic metasurface Yangyang Chu a , Zhaohong Wang a,∗ , Zhuo Xu a,b a
Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Faculty of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi Xi’an, 710049, People’s Republic of China b Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education & International Center for Dielectric Research, Xi’an Jiaotong University, Shaanxi Xi’an, 710049, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 12 August 2019 Received in revised form 9 December 2019 Accepted 27 December 2019 Available online xxxx Communicated by M. Wu Keywords: Pentamode metasurface Broadband High-efficiency Asymmetric transmission
a b s t r a c t We utilise the pentamode metasurface to realise broadband high-efficiency and controllable asymmetric transmission. The designed metasurface can manipulate the acoustic waves, as expected from the generalised Snell’s law, and exhibits unique characteristics such as extraordinary broadband acoustic control, apparent negative refraction, and conversion from the propagating wave to surface mode. The asymmetric transmission features of positive refraction for the forward incidence (FI) and negative refraction for the reverse incidence (RI) can be realised within the range of 2600 Hz to 5600 Hz by controlling the incident angle from 0◦ to 35◦ with the transmission efficiency higher than 85.4% for the FI and RI. In addition, by further adjusting the angle of incidence in the range of 25◦ to 90◦ , asymmetric transmission characteristics can be expressed as surface wave transmission for the FI and transmitted wave transmission for the RI within the same frequency ranges. © 2020 Elsevier B.V. All rights reserved.
1. Introduction The relationship between the angle of incidence and the angle of refraction is certainly determined while light waves propagate from one medium to another under a different refractive index due to momentum conservation along the tangential boundary direction; this phenomenon exemplifies Snell’s law. Following this law, Federico Capasso’s team at Harvard University achieved arbitrarily controlled electromagnetic wave reflection (transmission) properties by introducing resonant elements at the material interface to generate localised momentum and design phase mutations at the interface; their discovery resulted in the generalised Snell’s law (GSL) [1]. The acoustic metasurface is a 2D artificial material with subwavelength thickness, which can provide unique local phase shifts or amplitude modulation. Introducing the concept of the acoustic metasurface into materials science and physics using GSL presents new possibilities for manipulating acoustic waves and producing a series of unconventional acoustic phenomena. Various advanced acoustic wave–manipulating functionalities have since been made possible, such as subwavelength imaging [2,3], focusing [4,5], near-perfect absorption [6,7], negative refraction of
*
Corresponding author. E-mail address: [email protected] (Z. Wang).
https://doi.org/10.1016/j.physleta.2019.126230 0375-9601/© 2020 Elsevier B.V. All rights reserved.
acoustic waves [8,9], and invisibility cloaking [10,11]. Yet the GSL is frequency-dependent [12,13], such that the metasurface based on GSL only works within a selected narrow band. A method using the gradient rate (rather than the gradient phase) to obtain a frequency-independent GSL has been proposed [14], making it possible for the metasurface to manipulate acoustic waves over a broad frequency range. Acoustic asymmetric transmission, as another fascinating metasurface phenomenon, has also been widely explored. Acoustic asymmetric transmission can generally be achieved in two ways, either by disrupting time-reversal symmetry using a medium with strong nonlinearity [15,16] or by breaking spatial inversion with the asymmetric geometry profile [17,18]. Recently, some scholars have realised asymmetric transmission through a device consisting of an acoustic gradient-index metasurface (GIM) and another structure [19–22]. This type of structure can perform functions similar to acoustic diodes. The GIM is realised via phase engineering, enabling the device to produce efficient unidirectional acoustic transmission within a narrow bandwidth. In addition, introducing judiciously tailored losses into the GIM has been found to achieve acoustic asymmetric transmission [23,24]. Acoustic waves can then be strongly attenuated for the reverse incidence and freely transmitted for the forward incidence. Due to the introduction of losses in the GIM, acoustic waves transmitted for FI will generate a large energy loss. The loss is usually considered
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AID:126230 /SCO Doctopic: General physics
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a disadvantage in manipulating acoustic waves, which limits the exploration of new possibilities. Pentamode metamaterials (PMs) can achieve impedance matching over broadband, and the effective constitutive parameters can be tailored independently [25,26]. Thus, PMs provide a good solution for achieving broadband and high-efficiency acoustic asymmetric transmission. 2. Theory of acoustic asymmetric diffraction transmission When an acoustic wave impinges on the metasurface, the relationship between the incident and transmitted waves should follow the GSL:
k0 (sin θt − sin θi ) = d∅(x)/dx
(1)
where θt is the angle of refraction, θi is the angle of incidence, k0 is the wavenumber of incident wave, ∅(x) is the phase accumulation across the metasurface, and x is the transversal coordinate along the metasurface. The k0 in Formula (1) is frequency-dependent, resulting in GSL-based metasurfaces with inherent narrowband limitations. Drawing upon previous research [14], the transversal gradient velocity was used to replace the gradient phase by substituting the phase accumulation ∅ (x) = lk0 c 0 /c (x) into Formula (1); thus, a frequency-independent GSL is obtained:
sin θt − sin θi = lc 0 d[1/c (x)]/dx
Fig. 1. Schematic of proposed microstructures, where the thickness t and side length m of hexagonal PMs are adjustable.
(2)
where l is the metasurface thickness; c 0 is wave velocity in the background medium; and c (x) is the wave velocity in the structure. As shown in Formula (2), there is no frequency-dependent term; accordingly, the transmitted wave can be manipulated by 1/c (x) over a wide frequency band. In this study, we present a broadband controllable asymmetric transmission metasurface scheme based on the frequency-independent GSL. In designing the metasurface, we adopt two-dimensional hexagonal PMs whose schematic diagram is illustrated in Fig. 1. The framework is a regular hexagonal honeycomb lattice composed of six metallic arms with length a and thickness t, and six additional weights (an equilateral triangular terminal with side length m) are at the hexagon vertices. Theoretical analysis reveals that the thickness t mainly controls the effective bulk modulus of the PMs. The side length m of the equilateral triangle primarily controls the effective density, which is approximately equal to the average bulk density in the long-wavelength limitation 26]. The PM structure is made of titanium (density ρTi = 4500 kg/m3 ; Young’s modulus E Ti = 108 GPa; Poisson’s ratio c Ti = 0.34) or lead (density ρ L = 11400 kg/m3 ; Young’s modulus E L = 16.4 GPa; Poisson’s ratio c L = 0.44), and the hollow part is filled with air (density ρa = 1.21 kg/m3 ; velocity ca = 343 m/s). In using the proposed PMs to design a broadband and controllable asymmetric transmission metasurface, the key factor is to maintain acoustic impedance that matches the background medium (water with density ρ0 = 1000 kg/m3 ; velocity c 0 = 1490 m/s) at the metasurface interface. In this study, we design eight pentamode units to construct the metasurface given a distribution of effective velocities from c 0 to c 0 /7. The metasurface structure and required spatial distribution of 1/c (x) are depicted in Fig. 2(a). Geometric parameters of the eight unit cells are labelled S1–S8, as listed in Table 1. Considering that the 1/c (x) introduced in the metasurface is directional, the transmission relationship between FI and RI will differ when the metasurface is tilted at the incident angle θi (θi = 0◦ ) as indicated in Fig. 2(b). The pentamode units change from S1 to S8 along the incident direction for FI, the corresponding 1/c (x) changes from 1/c 0 to 7/c 0 , and the transversal gradient velocity becomes positive. In this case, Formula (2) can be rewritten as Formula (3)(a). However, for RI, the pentamode units change from S8 to S1 along
Fig. 2. (a) The spatial distribution of 1/c (x) for ideal metasurface (black dashed line) and pentamode metasurface (red solid line). (b) Sketch of FI and RI with angle θi .
Table 1 Geometric parameters of the eight unit cells as shown in Fig. 2(a). Unit cells
a (mm)
t (mm)
m (mm)
S1 S2 S3 S4 S5 S6 S7 S8
12 12 12 12 12 12 12 12
0.434 0.148 0.800 0.566 0.394 0.268 0.184 0.120
3.894 6.704 3.526 5.144 6.272 7.066 7.908 8.562
Fig. 3. Relationship between θt and θi for FI and RI. Black and red dashed lines represent theoretical calculations; black diamonds and red circles represent simulated results.
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Fig. 4. Calculated acoustic pressure fields for (a) FI and (b) RI at an angle of 10◦ at 3600 Hz.
the incident direction, the corresponding 1/c (x) changes from 7/c 0 to 1/c 0 , and the transversal gradient velocity becomes negative, therefore Formula (2) can be rewritten as Formula (3)(b):
sin θt + = sin θi + lc 0 d[1/c (x)]/dx sin θt − = sin θi − lc 0 d[1/c (x)]/dx
(a) (b)
(3)
where θt + and θt − are the transmitted angles of FI and RI, respectively. As seen from the previous definition, d[1/c (x)]/dx is always greater than 0 in this study. Therefore, Formulas (3)(a) and (3)(b) indicate that the forward and reverse transmitted angles differ. In addition, sin θt + is always greater than 0 and sin θt − is always less than 1. We can obtain sin θt − < 0 by adjusting the incident angle, at which point the transmitted waves demonstrate negative refractive properties. In this case, the asymmetric transmission characteristic exhibits negative refraction for RI and positive refraction for FI. In addition, by further adjusting the incident angle to sin θt + ≥ 1, the conversion from a propagating wave to surface wave can be realised; that is, the asymmetric transmission characteristics can be expressed as surface wave transmission for FI and transmitted wave transmission for RI. Furthermore, the PMs can achieve broadband impedance matching, and the characteristics of broadband and controllable acoustic asymmetric transmission can be obtained.
Fig. 5. Relationship between the refraction angle θt and f for FI and RI at an angle of 10◦ . Black and red dashed lines represent theoretical calculations; black diamonds and red circles represent simulated results.
3. Numerical simulations Based on the preceding analysis, one metasurface is designed using the eight unit cells (S1–S8) in Table 1. The adopted metasurface is composed of 48 × 4 microstructures. Thus, the metasurface thickness is l√= 6a, and the spatial velocity variation is d[1/c (x)]/dx = 1/6 3ac 0 . Using this metasurface to achieve asymmetric transmission of positive refraction for FI and negative refraction for RI, it is necessary to make sin θt − < 0, where
√
sin θt − = sin θi − 1/ 3 < 0
(4)
Because sin θi is always greater than 0, Formula (4) shows that 0◦ < θi < 35◦ . Therefore, the asymmetric transmission characteristic of negative refraction for RI and positive refraction for FI can be realised when the incident angle of the acoustic wave ranges from 0◦ to 35◦ . Then, the relationship between θt and θi within the range of [0◦ , 35◦ ] can be analysed, as shown in Fig. 3. This metasurface can achieve asymmetric transmission of positive refraction (θt + > 0) for FI and negative refraction (θt − < 0) for RI within the range of [0◦ , 35◦ ]. We randomly chose θi = 10◦ for numerical simulation, and θt + = 48.7◦ and θt − = −23.8◦ can thus be obtained using Formulas (3)(a) and (3)(b). The finite element method is used to demonstrate the unique performance of the proposed metasurface. Figs. 4(a) and 4(b) show acoustic transmission through the PM metasurface
Fig. 6. Transmission efficiency for pentamode metasurface for FI (black line) and RI (red line) at an angle of 10◦ .
for FI and RI at f = 3600 Hz, respectively. The black arrows indicate the direction of the acoustic wave. The incident waves for FI and RI can be freely transmitted through the metasurface. Fig. 4(a) reveals that positive refraction occurs when the FI with an angle of 10◦ is transmitted through the metasurface. The propagation direction of the transmitted wavefronts follows the black line, and the angle of refraction is 48.5◦ , consistent with the theoretical prediction. Unlike the case of Fig. 4(a), negative refraction appears when the RI with an angle of 10◦ is transmitted through the metasurface (see Fig. 4(b)). The refraction angle in this case is −23.6◦ , which also coincides with the previous theoretical calculation. In addition, a series of numerical simulations were conducted for the FI and RI in the range of 2600 Hz ≤ f ≤ 5600 Hz. The relationship between the θt + (θt − ) and f is displayed in Fig. 5. As indicated, θt + remains around 48.7◦ , and θt − remains around −23.8◦ when the FI and RI are at an angle of 10◦ , coinciding with the previous theoretical calculation of the entire study bandwidth. Energy transmittance spectra of the metasurface was also analysed with partial simulation results presented in Fig. 6. The transmis-
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Fig. 7. Calculated acoustic pressure fields for (a) FI and (b) RI with an angle of 40◦ at 3600 Hz.
sion efficiency of FI and RI each exceed 85.4% over the range of 2600 Hz–5600 Hz; therefore, the metasurface can be considered highly transparent to acoustic waves. This kind of metasurface thus makes the acoustic wave highly transparent over broadband in addition to achieving broadband asymmetric transmission. Apart from the above qualities, the acoustic characteristics of surface wave transmission for FI can be achieved by merely adjusting the incident angle. Using the same metasurface structure as earlier, the sin θt + ≥ 1 can be expressed as
√
sin θt + = sin θi + 1/ 3 ≥ 1
(5)
In Formula (5), sin θt + ≥ 1 indicates that the transmission angle is 90◦ ; that is, the acoustic wave will be converted from a propagating wave to surface mode after passing through the metasurface. As sin θi is always less than 1, the angle of incidence can be obtained from Formula (5) such that 25◦ ≤ θi ≤ 90◦ . Accordingly, the asymmetric transmission characteristics can be expressed as surface wave transmission for the FI and transmitted wave transmission for the RI at an incident angle greater than 25◦ . Here, we chose θi = 40◦ for the numerical simulation; θt + = 90◦ and θt − = 3.8◦ can be obtained from Formulas (3)(a) and (3)(b). Figs. 7(a) and 7(b) demonstrate the pressure field distribution of the metasurface for the FI and RI of f = 3600 Hz, respectively. As shown in Fig. 7(a), the transmitted wave propagates along the metasurface when the FI with an incident angle of 40◦ is transmitted through the metasurface. The acoustic wave can thus be converted from a propagating wave to surface wave mode through the metasurface. However, transmission characteristics differ from Fig. 7(a) when the RI with an incident angle of 40◦ is transmitted through the metasurface, as displayed in Fig. 7(b). The transmitted wave propagates at an angle of 3.5◦ from the normal (the black dotted line), corroborating the previous theoretical calculation. The acoustic transmission characteristics of FI and RI in the range of 2600 Hz ≤ f ≤ 5600 Hz were then simulated to study broadband characteristics. Fig. 8 illustrates the relationship between the θt + (θt − ) and f . In this case, θt + remains around 90◦ for the FI with an angle of 40◦ ; that is, surface wave transmission can be achieved after the incident acoustic wave passes through the metasurface. However, θt + is around 3.8◦ for the RI with an angle of 40◦ . The metasurface structure is the same as in Fig. 4, and the transmission efficiency remains above 85.4% as similar to Fig. 5. Therefore, this metasurface can achieve broadband asymmetric transmission characteristics of surface wave transmission for FI and transmitted wave transmission for RI. 4. Conclusion To conclude, a broadband and controllable acoustic metasurface is developed in this study based on the frequency-independent GSL and PMs. By utilising the impedance matching characteristics of PMs and the directionality of the gradient 1/c (x), the metasurface can realise acoustic asymmetric transmission over broadband ranges from 2600 Hz to 5600 Hz, and the transmission ef-
Fig. 8. Relationship between θt and f for FI and RI at an angle of 40◦ . Red dashed lines represent theoretical calculations; black diamonds and red circles represent simulated results.
ficiency exceeds 85.4% for FI and RI. Adjusting the incident angle can modify anomalous asymmetric transmission characteristics, including converting the propagating wave into surface mode or negative refraction characteristics. This metasurface can also be applied to achieve efficient anomalous transmission of acoustic waves over an ultrawide frequency range. Furthermore, this work outlines a means of broadband-tunable asymmetric transmission with substantial design flexibility, revealing promising applications for acoustic cloaks, acoustic energy collection, novel acoustic function devices, and so on. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. References [1] N. Yu, P. Genevet, M.A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science 334 (2011) 333–337. [2] J. Li, L. Fok, X. Yin, G. Bartal, X. Zhang, Experimental demonstration of an acoustic magnifying hyperlens, Nat. Mater. 8 (2009) 931. [3] H. Zhang, X. Zhou, G. Hu, Shape-adaptable hyperlens for acoustic magnifying imaging, Appl. Phys. Lett. 109 (2016) 224103. [4] Y. Li, B. Liang, X. Tao, X. Zhu, X. Zou, J. Cheng, Acoustic focusing by coiling up space, Appl. Phys. Lett. 101 (2012) 233508. [5] W. Wang, Y. Xie, A. Konneker, B.-I. Popa, S.A. Cummer, Design and demonstration of broadband thin planar diffractive acoustic lenses, Appl. Phys. Lett. 105 (2014) 101904. [6] Y. Li, B.M. Assouar, Acoustic metasurface-based perfect absorber with deep subwavelength thickness, Appl. Phys. Lett. 108 (2016) 063502. [7] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound, Nat. Commun. 3 (2012) 756.
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