Acoustic collimation based on an ultra-thin metasurface

How to modulate acoustic waves at small scales has been an area of intense investigation. In this paper, an artificial ultra-thin structure with a series of zigzag-shaped grooves located in the center and with bilateral symmetry is designed to realize ultra-strong directional collimated acoustic beams. The simulations agree well with the theoretical analysis, and show that the acoustic collimated structure has high directivity at the resonant frequency, with a beam length exceeding 40 wavelengths. The structure has deep subwavelength scales and a simple design, and is expected to have applications in fields such as directional acoustic radiation, medical ultrasound detection, etc.

How to modulate acoustic waves at small scales has been an area of intense investigation.In this paper, an artificial ultra-thin structure with a series of zigzag-shaped grooves located in the center and with bilateral symmetry is designed to realize ultra-strong directional collimated acoustic beams.The simulations agree well with the theoretical analysis, and show that the acoustic collimated structure has high directivity at the resonant frequency, with a beam length exceeding 40 wavelengths.The structure has deep subwavelength scales and a simple design, and is expected to have applications in fields such as directional acoustic radiation, medical ultrasound detection, etc. © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd H ow to manipulate acoustic waves efficiently has been a research hotspot of great interest.Acoustic metamaterials have novel properties that natural materials do not possess, and their high degree of freedom in cell design provides more possibilities for acoustic wave modulation to branch out in many directions, such as acoustic focusing, [1][2][3][4][5][6] acoustic absorption, [7][8][9][10][11][12] directional acoustic radiation, [13][14][15][16][17][18] etc.
Coiling up space [19][20][21][22][23][24] is a common unit of acoustic artificial structures. Cmpared with a straight channel, the propagation path of acoustic waves in the coiling up space is prolonged and the phase is delayed, and the equivalent refractive index of its unit is increased, which provides a new idea for acoustic wave modulation.Liang et al. introduced spatial curl structures to obtain negative refraction and super-transmission phenomena.25) Li et al. designed an acoustic lens by controlling the length of the Fabry-Pérot (FP) channels to change the phase of the acoustic waves, and realized acoustic focusing.26) Cai et al. presented an ultra-thin acoustic plate based on FP resonance by convoluting the channels, and achieved sound absorption at low frequencies.27) Zhang et al. demonstrated a ventilating noise barrier consisting of several curled spaces and hollow pipes.Based on the interference between the resonant scattering of discrete states and the background scattering of continuous states, Zhang et al. achieved Fano resonance to implement sound dissipation.28) Hu et al. proposed an efficient directional artificial structure by folding the central channel to downscale the size of the structure, and a high-efficiency collimating acoustic beam was detected with a thickness of 1/15 of the incident wavelength.29) A lot of works have been done on coiling up channels, and researchers have utilized them to achieve acoustic wave modulation while further reducing the size of the structure.However, in practical applications, especially in the low-frequency range, it is our eternal pursuit to realize the regulation of acoustic waves on a much smaller scale.
In this paper, zigzag channels are applied to realize acoustic collimation at a longitudinal scale of just 1/33 of the incident wavelength.Compared with the structure in existence, the structure has a smaller size and is easy to fabricate, which is more favorable for integration in loudspeakers, medical ultrasound imaging, and other applications.
A schematic illustration of the proposed structure is shown in Fig. 1.A rigid plate etching a number of convolute units is presented in this paper: one in the center location running through both ends, and the others periodically distributed on both sides.
By folding the channels in both the center and bilateral sides, the acoustic waves pass through an elongated path in a zigzag manner with much thinner longitudinal size, equivalent to a slower velocity in a straight channel with the same height and a smaller effective refractive index, which shifts the resonant frequency to a lower frequency in an ultra-thin structure.
As for the central aperture, most acoustic waves are reflected back because of impedance mismatch due to the slim width of the propagation path.Meanwhile, at FP resonant frequency, the transmission efficiency reaches a peak where the relationship between the working wavelength at resonant frequency and the aperture length can be obtained from Eq. (1): 30) where k 0 is the wave vector and l 1 is the length of the channel, which is set to be mλ/2 to guarantee the resonance.Therefore, the length of the slit needs to be at least λ/2 to ensure the generation of FP resonance.And based on the shortest slit length, the corresponding resonant frequency shifts to the lowest which makes the thickness of the curly aperture much smaller than the height of the straight aperture to realize the same way of resonating.
On the other hand, as the bilateral slits are evenly and symmetrically perforated on the upper and lower sides, the incident waves will be bounced back as well unless the resonance condition is satisfied.As shown in Fig. 1, the resonant frequency of the reflected aperture versus the slit length l 2 is concluded from Eq. ( 2 If the relationship between the length of the reflective slit and the incident sound frequency satisfies Eq. ( 2), it is obvious that the length of the channel should be at least λ/4 in order to support the lowest FP resonant mode, which ensures a deep subwavelength size in the propagation direction when the channel is coiled up.
Here we set the parameters of the structure as the overall thickness of the structure L = 3.4 mm, the distance between two adjacent grooves w = 40 mm, the width of each channel d = 1 mm, the central aperture length l 1 = 60 mm, and the length of the engraved slits on both sides l 2 = 30 mm.Then, the corresponding lowest FP resonance frequencies in the bilateral slits and the central channel are calculated to be the same, at 2858 Hz.
As the acoustic waves go through the structure with the same frequency as the resonant ones, there exists sympathetic vibration in the region of the curled arrays on the lower side of the rigid flat surface, which binds the perpendicular acoustic energy from the bottom and converts the waves into surface mode.High transmission is realized by coupling the FP resonance in the middle with the bilateral surface mode.
However, the gathered acoustic wave reaches the transmitting end through the center aperture and will spread around, which hinders the formation of collimation.The slit array on the upper side of the structure prevents the scatter waves from escaping and forces the waves to concentrate in the center.As a result, an ultra-strong collimated acoustic beam with high energy is formed.
Meanwhile, by folding the straight channel, the total length of the structure is compressed to a great extent, which downscales the size of the device significantly.Compared to previous work such as in Refs.14, 29 etc., the design in this paper takes the reflected FP resonance into account, which produces a stronger collimating beam, and the labyrinth structure further compresses the size of the physical dimensions.
To further determine the effectiveness of the proposed mechanism, here we perform a series of numerical simulations as shown in Figs.2-6.Throughout the manuscript, the simulations are performed using the finite element method.During the simulation, the number of bilateral grooves is designed to be 33 with 8 slots on each side to ensure that there are enough of them to collect the acoustic energy.For the narrow space in the zigzag channel, the thermal viscosity is taken into consideration during simulation.
The transmittance versus frequency in Fig. 2 shows that the transmittance reaches a peak at 3050 Hz, where abnormal transmission occurs and most of the acoustic energy (>75%) is transmitted.At this frequency, the incident waves with similar operating frequency to the FP resonant frequency excite the FP resonance in both the central aperture and the arrays on lateral sides.Then, because of the interaction with the resonator arrays in the bottom, the redistribution of incident plane waves couples with the central FP resonance and highly enhances the transmission, despite the large contrast in the interface.Moreover, the slits decorated in the upper sides virtually eliminate the diffraction of the transmitted waves, and a subsequent collimating beam can be found in the outer field.The discrepancy in the working frequency for the theory versus simulation may be due to the following reasons: the path length we calculate differs slightly in theory and simulation; in simulation, there exists not only FP resonance, but also resonance coupling, which is responsible for the gap in frequency between theory and simulation.
In order to demonstrate the impact of the upper slit arrays on the collimation efficiency, we remove the slit arrays on the upper side of the structure so that it becomes a rigid wall.As seen in Fig. 2, a weak peak close to 0 can be observed at the resonance point.Compared to Fig. 3(b), there is almost no acoustic wave transmission at the transmissive end in Fig. 3(a), which confirms the effectiveness of the upper side slit arrays.
Theoretically, the more slits on each side of the structure, the more scattered waves can be converted into acoustic surface waves, which is conducive to increasing the transmittance as well as producing better collimation.We parameterize the number of slits N on each side from 4-10, as shown in Fig. 4, and the structural resonance frequency remains unchanged regardless of the change in N. When N is less than 8, the transmittance is below 0.55, and when N is large enough, i.e. 8 or 10, the transmittance is quite close and more than 0.75, which indicates that the ability to gather the  014003-2 © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd scattered waves and convert them into acoustic surface waves weakens as the number of slits decreases.However, increasing the number of slits will enlarge the lateral dimension, which in turn affects its applicability range.Taking all factors into consideration, here we choose N = 8 as the number of slits on each side of the device.
Figure 3(b) depicts the radiated field at the frequency of 3050 Hz.The bilateral slits prevent the high-order waves from escaping into the far-field region, and a strong collimating beam exceeding 40 incident wavelengths is formed through the central aperture with nearly 75% incident acoustic energy.The reason the transmission doesn't reach 100% is that the number of bilateral slits isn't big enough to catch all the diffracted waves in free space, and the viscosity effect occurring in the narrow tunnels leads to a waste of energy.
For better analysis of the radiation properties of our structure, here we add a simulated polar diagram of the farfield sound pressure at 3050 Hz.As shown in Fig. 5, the main lobe stretches out with a narrow angle of approximately 15°, and the side lobes are quite small compared with the main one.The incident plane waves converted to a highly collimated beam by an extremely thin flat plate may contribute to a diverse range of applications, such as novel transducers and underwater communication, etc.
Consider that in practical scenarios, we sometimes need to adjust the outer angle of the collimating beam without moving the fixed structure according to the actual requirements.To directly address the flexibility and robustness of the structure, the incidence of sound waves from different angles is further investigated.
Here we calculate the distribution of sound pressure at the transmissive end at different angles (30°, 45°, and 60°, respectively) in Figs.6(a , which demonstrates that even if it is oblique incidence, a good collimating effect is still maintained.
In this paper, we designed an ultra-thin artificial structure based on FP resonance to modulate the acoustic waves by adjusting the length of the curled channels to lower their resonant frequencies and decrease the thickness of the structures.The bilateral zigzag slits of the structure convert the high mode acoustic waves to the surface mode to couple with the FP resonant mode in the central aperture to generate a highly effective collimating beam.The theoretical analysis and simulation show great consistency in that the proposed structure implements a highly efficient acoustic collimation beam exceeding 40 wavelengths on the deep subwavelength scale with a longitudinal length only 1/33 of the incident wavelength at FP resonant frequency with strong directivity.Our designs open a route to the unconventional manipulation of acoustic waves with compact dimensions and easy fabrication that may have far-reaching impacts in various branches of acoustics, ranging from acoustic communications to acoustic imaging.014003-3 © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd

Method
Throughout the manuscript, the simulations were performed using the finite element method.The background medium was air, for which the mass density and sound speed were 1.21 kg m −3 and 343 m s −1 , respectively.The material of the ultra-thin structure in the simulation was chosen to be steel, whose mass density and sound speed were 7800 kg m −3 and 5800 m s −1 , respectively.Plane wave radiation boundary conditions were imposed on the boundaries to eliminate the reflected waves by the outer boundaries in Figs. 3 and 5.

Fig. 2 .
Fig. 2. Simulated frequency dependence of the transmission coefficient.The red line and the black line indicate the transmissivity of the structure with slits on both sides and on the lower side only, respectively.
)-6(c).When the acoustic waves are transmitted through the structure at a frequency of 3050 Hz under different incident angles, the emergent angle coincides well with the incident one, as seen in the figure.The far-field polar diagrams indicate that the focusing angles of the outer inclined acoustic beams are restricted within 20°[ Figs.6(d)-6(f)]

Fig. 3 .
Fig. 3.The simulated spatial distribution of the acoustic pressure on the transmitted side.(a) The acoustic pressure distribution of the structure with slit arrays on lower surface; (b) The acoustic pressure distribution of the structure with slit arrays on both upper and lower surface.

Fig. 4 .
Fig. 4. The effect of the number of slits on each side on the transmittance.

Fig. 5 .
Fig. 5. Simulated polar diagram of the far-field sound pressure at 3050 Hz.