Compact Acoustic Rainbow Trapping in a Bioinspired Spiral Array of Graded Locally Resonant Metamaterials

Acoustic rainbow trappers, based on frequency selective structures with graded geometries and/or properties, can filter mechanical waves spectrally and spatially to reduce noise and interference in receivers. These structures are especially useful as passive, always-on sensors in applications such as structural health monitoring. For devices that face space and weight constraints, such as microelectromechanical systems (MEMS) transducers and artificial cochleae, the rainbow trapping structures must be compact as well. To address this requirement, we investigated the frequency selection properties of a space-saving design consisting of Helmholtz resonators arranged at sub-wavelength intervals along a cochlear-inspired spiral tube. The height of the Helmholtz resonators was varied gradually, which induced bandgap formation at different frequencies along the length of the spiral tube. Numerical simulations and experimental measurements of acoustic wave propagation through the structure showed that frequencies in the range of 1–10 kHz were transmitted to different extents along the spiral tube. These rainbow trapping results were achieved with a footprint that was up to 70 times smaller than the previous structures operating at similar bandwidths, and the channels are 2.5 times of the previous structures operating at similar bandwidths.


Introduction
Frequency selective structures, which act as passive spectral filters for electromagnetic and mechanical waves, play an important role in many engineering applications. For instance, they are often employed to isolate desired frequencies in the multimodal Lamb wave to improve the accuracy of structural health monitoring of thin plates [1][2][3][4], and reduce noise and interference in radio frequency (RF) receivers in electronic and biomedical devices [5,6]. In recent years, it has been shown that the use of frequency selective structures with graded geometric structures or properties can be used to filter waves spectrally and spatially [7][8][9][10][11][12][13][14]. This technique has come to be known as rainbow trapping and was originally applied on electromagnetic waves to control optical delays and allow temporary storage of light [15][16][17][18][19]. However, it has been extended to acoustic waves as well [20].
In applications with space and/or weight constraints, such as microelectromechanical systems (MEMS) acoustic transducers [21][22][23] and artificial cochleae [24][25][26][27][28], the rainbow trapping structures need to be compact without altering the operational bandwidth. Meanwhile, a requirement that current studies continue to overlook is needed. For instance, in their proposed design for an artificial cochlear that realizes rainbow trapping of acoustic frequencies up to 10 kHz, Foucaud et al. [29] made use of a long (~1 m), straight plate of varying width, which was fabricated using traditional machining techniques that cannot be easily extended to produce miniature structures. On the other hand, White et al. [30] employed microfabrication techniques to manufacture a device, consisting of polyimide membranes and silicon nitride beams, which was less than 1 cm 3 in volume. However, its operational frequency was extended to 35 kHz, which is too high for the use in an artificial human cochlear.
To resolve this issue, we propose an acoustic rainbow trapping design that was inspired by the coiled shape of the cochlear, which had evolved naturally from the need to perform spatial-spectral isolation of acoustic waves in a tight space [28,31,32]. Helmholtz resonators [33][34][35][36][37][38][39][40], a locally resonant metamaterial that can be positioned at subwavelength intervals to enhance the transmission loss of specific frequency bands [35,[41][42][43], were arranged in this bioinspired spiral formation to further minimize the form factor of the design. The rainbow trapping function of the structure was realized by varying the height of the Helmholtz resonators, as shown in the numerical simulations and experiments in the following sections.

Numerical Model
In this study, we implement the rainbow trapping of acoustic waves using 40 individual Helmholtz resonators (Figure 1a,b) attached to a hollow tube in an Archimedean spiral configuration (Figure 1c), which can be described by the following equation using polar coordinates (r, θ): x = rθ cos (θ) and y = rθ sin (θ). In this paper, r = 20 mm, and θ changes from 0 to 2.1π. The total length of the spiral is l = 0.3 m, coiled within an area l a × w a = 0.1 m × 0.1 m. The periodical length of the Helmholtz resonators is a = 7 mm. All the Helmholtz resonators have the same cylinder inner radius r i = 2.5 mm, cylinder outer radius r o = 3.5 mm, cylinder top and bottom thickness H = 0.5 mm, neck inner radius r ni = 0.7 mm, neck outer radius r no = 1 mm, neck length L = 4 mm, duct inner radius R ni = 1 mm and duct outer radius R no = 1.5 mm. The cylinder height, h, is increased in discrete steps of δ = 0.359 mm from 1 mm (for the first cylinder (n = 1)) to 15 mm for the last cylinder (n = 40) (Figure 1d).
For the dispersion curve analysis, the structure is assumed to be infinite and periodic in the direction of wave propagation (x) with the period a and. According to the Floquet-Bloch theorem, the relation for the pressure distribution (p) can be expressed as [44]: where k is the wavenumber of the acoustic wave. For the spatial-spectral analysis, the equation used to analyze the acoustic wave problems is expressed as [44]: where p = p 0 e iωt , the input pressure amplitude is p 0 = 1 Pa, ρ is the density of air (ρ = 1.225 kg/m 3 ), c is the acoustic wave speed in air (c = 343 m/s). For the element size used in these studies, we choose the "physics-controlled mesh" type with "finer" element size.

Dispersion Analysis
Since the cylinder height is designed to increase linearly along the length of the duct for the graded Helmholtz resonators and the increment δ = 0.359 mm between two consecutive cylinders are reasonably small, we can consider the n-th cylinder with the height hn in the graded Helmholtz resonators as a cylinder with the same height hn in a series of the periodic Helmholtz resonators. Therefore, we could use the dispersion characteristics of the periodic Helmholtz resonators to approximate those of the n-th cylinder in a set of the graded Helmholtz resonators [20].
Since the bandgap of a Helmholtz resonator is strongly dependent on its geometry, the cylinders at different locations have different dispersion curves. To demonstrate the spatial evolution of dispersion curves in the graded Helmholtz resonators, Figure 2a-d presents the frequencywavenumber dispersion curves in the first Brillouin zone for cylinders with different heights: h7 = 3.1 mm (the 7th cylinder), h15 = 6.0 mm (the 15th cylinder), h24 = 9.2 mm (the 24th cylinder), h32 = 12.1 mm (the 32nd cylinder). From Figure 2a-d, we can see there is a bandgap for each Helmholtz resonator, for which acoustic waves in those frequencies are not allowed to propagate. Figure 2e shows the frequency variation of acoustic wave bandgap (shaded in blue) with respect to cylinder height in the graded Helmholtz resonators. With an increase of cylinder height, the width of the bandgap becomes larger, and the bandgap gradually shifts to lower frequencies. Hence, if acoustic waves that carry frequencies in the range of 1614 Hz-6083 Hz transmit the graded Helmholtz resonators from cylinders n =1 to n = 40, the different frequency components will propagate to different extents. For instance, f1 = 1 kHz will be able to propagate to all the Helmholtz resonators,

Dispersion Analysis
Since the cylinder height is designed to increase linearly along the length of the duct for the graded Helmholtz resonators and the increment δ = 0.359 mm between two consecutive cylinders are reasonably small, we can consider the n-th cylinder with the height h n in the graded Helmholtz resonators as a cylinder with the same height h n in a series of the periodic Helmholtz resonators. Therefore, we could use the dispersion characteristics of the periodic Helmholtz resonators to approximate those of the n-th cylinder in a set of the graded Helmholtz resonators [20].
Since the bandgap of a Helmholtz resonator is strongly dependent on its geometry, the cylinders at different locations have different dispersion curves. To demonstrate the spatial evolution of dispersion curves in the graded Helmholtz resonators, Figure 2a-d presents the frequency-wavenumber dispersion curves in the first Brillouin zone for cylinders with different heights: h 7 = 3.1 mm (the 7th cylinder), h 15 = 6.0 mm (the 15th cylinder), h 24 = 9.2 mm (the 24th cylinder), h 32 = 12.1 mm (the 32nd cylinder). From Figure 2a-d, we can see there is a bandgap for each Helmholtz resonator, for which acoustic waves in those frequencies are not allowed to propagate. Figure 2e shows the frequency variation of acoustic wave bandgap (shaded in blue) with respect to cylinder height in the graded Helmholtz resonators. With an increase of cylinder height, the width of the bandgap becomes larger, and the bandgap gradually shifts to lower frequencies. Hence, if acoustic waves that carry frequencies in the range of 1614 Hz-6083 Hz transmit the graded Helmholtz resonators from cylinders n = 1 to n = 40, the different frequency components will propagate to different extents.
For instance, f 1 = 1 kHz will be able to propagate to all the Helmholtz resonators, while f 2 = 2 kHz can only propagate to resonators with h < 10.69 mm. Similarly, f 3 = 3 kHz and f 4 = 4 kHz can only propagate to resonators with h < 4.95 mm and h < 2.79 mm respectively (Figure 2e). In other words, the different frequencies in an acoustic wave will be spatially filtered into 1 of the 40 Helmholtz resonators, each acting as a spatial-spectral channel.
while f2 = 2 kHz can only propagate to resonators with h < 10.69 mm. Similarly, f3 = 3 kHz and f4 = 4 kHz can only propagate to resonators with h < 4.95 mm and h < 2.79 mm respectively (Figure 2e). In other words, the different frequencies in an acoustic wave will be spatially filtered into 1 of the 40 Helmholtz resonators, each acting as a spatial-spectral channel.

Spatial-Spectral Analysis
To ascertain the above expectation that different frequencies propagate to different extents in the spiral structure, finite element simulations are performed using COMSOL Multiphysics 5.3 [45]. An incident pressure field is applied for excitation starting from n = 1 using four different frequencies f 1 = 1 kHz, f 2 = 2 kHz, f 3 = 3 kHz and f 4 = 4 kHz.
Results presented in Figure 3a-d indicate that the acoustic wave can propagate through all the graded Helmholtz resonators at the frequency f 1 = 1 kHz, which is not within the bandgap of any cylinder. In contrast, the acoustic wave would not propagate beyond the 28-th, 12-th, and 6-th cylinders for the frequencies f 2 = 2 kHz, f 3 = 3 kHz and f 4 = 4 kHz, respectively. These results are markedly different from those obtained when the graded Helmholtz resonators are conjoined together graded [46]. In such structures, each resonator can only support a narrow range of frequencies and the tight mechanical coupling between adjacent resonators prevents complete rainbow trapping from being achieved i.e. waves of a single frequency will be "trapped" at the target resonator, as well as its neighbors. Sensors 2018, 18, x FOR PEER REVIEW 6 of 12   (Figure 4d-f). This clearly shows that only acoustic waves with the frequency of f1 = 1 kHz can reach the 37th cylinder, while waves with the frequencies f1 = 1 kHz and f2 = 2 kHz were present in the 17th cylinder, and the frequencies f1 = 1 kHz, f2 = 2 kHz and f3 = 3 kHz were found in the 7th cylinder. These results are consistent with the implications derived from the bandgap plot in Figure 2e and demonstrate the rainbow trapping potential of the spiral structure. A more comprehensive examination is conducted using a frequency sweep, f = 1 kHz-10 kHz, of the excitation source, which contains a broadband frequency information. The excitation signal used in this study is similar to the broadband pulse used in [47]. The frequency-space representation of normalized pressure amplitude, distributed along the Archimedean spiral shape, was plotted in Figure 3e. It clearly shows that acoustic waves of different frequencies stopped propagating forward and concentrated their wave energy at different locations along the spiral tube. The height of the Helmholtz resonators and the "trapped" wave frequencies at each of these locations correspond to the lower boundary of the bandgap as shown in Figure 2e. In addition, it is observed that with increasing excitation frequency, the wave propagation distance along the spiral tube became shorter and the Sensors 2019, 19, 788 6 of 11 location with concentrated energy gradually shifted towards the source. Note that the upper frequency range (> 8kHz) means that the signals can propagate through the entire structure, that's why the pressure are very high over the entire length range. This result is consistent with the bandgap plot in Figure 2e, because when the frequency over 8 kHz, it is a passband.
These observations are supported by the time dependent pressure wave results obtained from the 37th (Figure 4a), 17th (Figure 4b) and 7th (Figure 4c) cylinders, as well as their Fast Fourier transform (FFT) (Figure 4d-f). This clearly shows that only acoustic waves with the frequency of f 1 = 1 kHz can reach the 37th cylinder, while waves with the frequencies f 1 = 1 kHz and f 2 = 2 kHz were present in the 17th cylinder, and the frequencies f 1 = 1 kHz, f 2 = 2 kHz and f 3 = 3 kHz were found in the 7th cylinder. These results are consistent with the implications derived from the bandgap plot in Figure 2e and demonstrate the rainbow trapping potential of the spiral structure.

Experimental Setup
A proof-of-concept experiment is conducted on a sample with the same dimensions as the numerical model described in Figure 1. The sample is 3D printed with the Grey Resin, which has a high stiffness after curing [48], using a commercial stereolithography 3D printer, Form 2 (Formlabs Inc., USA) (Figure 5a). A single hole with diameter of 1 mm is drilled in the hollow tube at the location of the 7th, 17th and 37th cylinders and covered over with reflective tapes for acoustic wave measurements using a point laser Doppler vibrometer (Polytech GmbH, Germany). The scanning laser vibrometer is used to measure the particle velocity on the reflective tapes. The measured particle velocity amplitude on the reflective tapes is proportional to the pressure of sound waves, which is

Experimental Setup
A proof-of-concept experiment is conducted on a sample with the same dimensions as the numerical model described in Figure 1. The sample is 3D printed with the Grey Resin, which has a Sensors 2019, 19, 788 7 of 11 high stiffness after curing [48], using a commercial stereolithography 3D printer, Form 2 (Formlabs Inc., USA) (Figure 5a). A single hole with diameter of 1 mm is drilled in the hollow tube at the location of the 7th, 17th and 37th cylinders and covered over with reflective tapes for acoustic wave measurements using a point laser Doppler vibrometer (Polytech GmbH, Germany). The scanning laser vibrometer is used to measure the particle velocity on the reflective tapes. The measured particle velocity amplitude on the reflective tapes is proportional to the pressure of sound waves, which is used to reflect the sound wave amplitude at different frequency ranges. The vibrometer is connected to a decoder box for acquiring the particle velocity. A small loudspeaker is used to generate the acoustic wave at different frequencies. In this study, there are four tests for each hole at the frequencies of f 1 = 1 kHz, f 2 = 2 kHz, f 3 = 3 kHz and f 4 = 4 kHz.

Experimental Results
The experimental results are shown in Figure 6. They are very similar to those predicted in Figure 4, except that shorter peaks are observed for f2 = 2 kHz (Figure 6e and Figure 6f) and f3 = 3 kHz (Figure 6f). This is likely a combined result of attenuation along the spiral tube and background noise. The background noise is due to the broadband white noise exists everywhere and this experiment is performed without echoless chamber. Nevertheless, Figure 6 clearly indicates that the spiral structure with the Helmholtz resonators does indeed possess rainbow trapping capabilities, in accordance to the results obtained from numerical simulations. Moreover, this spatial-spectral filtering of acoustic waves is realized in a structure with a footprint up to 70 times smaller [29], and with the channels as many as 1.3 to 2.5 times [20,49] of that of previous designs operated in similar frequency ranges. Further optimization of the current design is expected to lead to even higher channel densities for acoustic rainbow trapping.

Experimental Results
The experimental results are shown in Figure 6. They are very similar to those predicted in Figure 4, except that shorter peaks are observed for f 2 = 2 kHz (Figure 6e,f) and f 3 = 3 kHz (Figure 6f). This is likely a combined result of attenuation along the spiral tube and background noise. The background noise is due to the broadband white noise exists everywhere and this experiment is performed without echoless chamber. Nevertheless, Figure 6 clearly indicates that the spiral structure with the Helmholtz resonators does indeed possess rainbow trapping capabilities, in accordance to the results obtained from numerical simulations. Moreover, this spatial-spectral filtering of acoustic waves is realized in a structure with a footprint up to 70 times smaller [29], and with the channels as many as 1.3 to 2.5 times [20,49] of that of previous designs operated in similar frequency ranges. Further optimization of the current design is expected to lead to even higher channel densities for acoustic rainbow trapping.

Conclusion
We have numerically and experimentally demonstrated that broadband acoustic waves can be filtered spectrally at different spatial locations when propagating along a spiral array of the Helmholtz resonators, which have subwavelength periods and a graded height. The linear variation in height led to a systematic modulation of the acoustic bandgap along the spiral, resulting in the rainbow trapping effect. This was verified through numerical simulations and experiments. The results showed that a total length of l = 0.3 m of the graded frequency selective structure with 40 spatial-spectral channels can be fabricated within an area of la × wa = 0.1 m × 0.1 m, which was up to 70 times smaller and had up to 2.5 times as many channels as that of previous structures operating in similar bandwidths.

Conclusions
We have numerically and experimentally demonstrated that broadband acoustic waves can be filtered spectrally at different spatial locations when propagating along a spiral array of the Helmholtz resonators, which have subwavelength periods and a graded height. The linear variation in height led to a systematic modulation of the acoustic bandgap along the spiral, resulting in the rainbow trapping effect. This was verified through numerical simulations and experiments. The results showed that a total length of l = 0.3 m of the graded frequency selective structure with 40 spatial-spectral channels can be fabricated within an area of l a × w a = 0.1 m × 0.1 m, which was up to 70 times smaller and had up to 2.5 times as many channels as that of previous structures operating in similar bandwidths.