Extremely-huge wave localization in coupled multilayer heterogeneous phononic crystal resonators

Phononic crystals composed of materials with different elastic/acoustic properties show the strong ability to hinder or control the propagation of acoustic/elastic waves inside phononic crystals.^1–2) A classical physics effect is that the propagation of waves with frequencies in the elastic/acoustic band gaps is forbidden inside phononic crystals for some or all wave vectors. Hence, the phononic crystals can be exploited to design acoustic filters, acoustic shields, or acoustic rectifiers.^3–10) Otherwise, a few of novel behaviors in the bands of the band structures for phononic crystals have attracted growing interest due to their possible applications (such as acoustic imaging, acoustic lenses, and wave guides).^11–14) Specially, the physical defect created in the phononic crystal causes the appearance of new bands (known as the defect bands) in the absolute band gaps, leading to the wave localization phenomenon.^15–21) The wave is spatially confined in/near the defect where the pressure is amplified or the stress enhanced at the frequency localized inside the defect band. Some strategies have been provided to enhance the wave localization effect, such as increasing the size of phononic crystal,¹⁸⁾ altering the location of the defect,¹⁹⁾ employing rotated square inclusions,²⁰⁾ and using the coupled structure of two phononic crystal resonators (PCRs).^21–22) However, the acoustic coupling effect of multiple PCRs has been rarely investigated in order to obtain huge wave localization in a limited size.

A coupled multilayer heterogeneous structure composed of multiple two-dimensional concentric PCRs [Fig. 1] is presented to obtain a huge wave localization effect. When a strong interaction occurs in transmitted bands of the individual PCRs, the acoustic coupling effect and extremely-huge wave localization phenomenon are obtained in the proposed heterogeneous structure. Consequently, the coupled heterogeneous structure owns a greater amplification of acoustic pressure than each individual PCR structure.

Zoom In Zoom Out Reset image size

The proposed five-layer coupled heterogeneous structure (CPCR-L5) consists of five two-dimensional concentric PCRs (PCR1, PCR2, PCR3, PCR4, PCR5), as shown in Fig. 1(d). These PCRs are all composed of several polyoxymethylene (POM) cylinders in the airbackground. The structure PCR1 (or PCR2 and PCR5) is created through removing one rod (or one rod and nine rods) from a perfect 3 × 3 (or 3 × 3 and 5 × 5) phononic crystal. Meanwhile, the structure PCR3 (or PCR4) is actually a 2 × 2 phononic crystal. As comparisons, the similar heterogeneous structures composed of two, three, and four concentric PCRs are formed and named with two-layer CPCR-L2 [Fig. 1(a)], three-layer CPCR-L3 [Fig. 1(b)], and four-layer CPCR-L4 [Fig. 1(c)], respectively. Specially, the CPCR-L2 structure is composed of PCR1 and concentric PCR2, and CPCR-L3 composed of CPCR-L2 and concentric PCR3. CPCR-L3 and concentric PCR4 constitute CPCR-L4. The material properties of POM and air and the structural properties of each PCRs are listed in Tables I and II, respectively.

The calculations of band structures for all PCRs and their coupled heterogeneous structure can be carried out by using the plane wave expansion method and the supercell calculation.^17–19) In these calculations, the PCRs or their coupled heterogeneous structure seem as the supercells (Figs. 2–5). Moreover, frequency response of acoustic pressure amplification and acoustic pressure amplitude distribution for all the acoustic structures are calculated and simulated through utilizing the Acoustic Module of COMSOL MULTIPHYSICS. The simulation settings are according to those in Ref. 21.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 2 shows the calculated band structures for PCR1, PCR2 and their coupled structure CPCR-L2. From Fig. 2(a), it can be seen that the band structure of the 3 × 3 supercell with a point defect for PCR1 exhibits a defect band with a center frequency of 16.5 kHz among the range of 14.7–19.0 kHz, which is introduced by the point defect inside PCR1 and related to an acoustic wave localization mode.¹⁶⁾ And the partial second lowest band among 4.0–6.0 kHz can inspire a subwavelength resonant mode where a super-subwavelength acoustic wave localization effect exists in the cavity center of PCR1.²¹⁾ The calculated band structures for PCR2 [Fig. 2(b)] has the same tendency with that for PCR1 due to the similar structure for PCR1 and PCR2. But the center frequency of the defect band in the band structures for PCR2 is about 4.0 kHz that is 0.24 times smaller than that for PCR1 because the lattice constant for PCR2 is 4.16 times greater than that for PCR1 [Table II]. When putting a PCR1 structure at the cavity center of a PCR2 structure, the band structure of the resulting coupled structure CPCR-L2 [Fig. 2(c)] shows a more flat defect band with a center frequency of 3.830 kHz, arising from the interaction between the defect band of PCR1 and the second lowest band of PCR2.

By putting a PCR3 structure at the cavity center of the CPCR-L2 structure, a coupled structure CPCR-L3 is created and its band structure is shown in Fig. 3. From Fig. 3(a), we can see that the calculated band structure of the 2 × 2 supercell for PCR3 exhibits a narrow band gap with a higher center frequency of 20.35 kHz (related to that for PCR1 or PCR2) and a bandwidth of 0.9 kHz because of a smaller lattice constant of 0.010 m and a fairly low filling fraction (∼34.2%) of the cylinder in the unit cell for PCR3. Besides the band structure for PCR3 has a wide first band sweeping from 0 to 19.9 kHz, which reveals that acoustic wave can propagate through the PCR3 structure at a wide frequency range. From Fig. 3(b), one can see that the band structures for CPCR-L3 has the nearly same bands at 0–10 kHz with that for CPCR-L2 since the acoustic wave propagation is allowed in the PCR3 structure among 0–19.9 kHz. However, the center frequency (3.91 kHz) of the defect band for CPCR-L3 is higher than that (3.830 kHz) for CPCR-L2 originating from the wave interaction between PCR3 and CPCR-L2.

When refitting a CPCR-L3 structure by adding a concentric PCR4 structure, a coupled structure CPCR-L4 is formed and its calculated band structure is shown in Fig. 4. It can be obviously viewed that there are no any complete band gaps ranging from approximately 0 to 8.5 kHz in the band structures for PCR4, but only a very narrow partial band gap along the Γ–X direction (x-direction) over a range of 2.75–3.84 kHz [Fig. 4(a)]. This is because the PCR3 structure exhibits an extremely low filling fraction (15.2%) of cylinders. Additionally, Fig. 4(b) displays that the band structures for CPCR-L4 has remarkably five complete band gaps (marked with blue) between lots of plat bands (1.7–5.0 kHz) compared with one for PCR4. The defect band (a center frequency of 3.91 kHz) in the band gap for CPCR-L3 [Fig. 3(b)] translates several flat bands over approximately 4.0–4.6 kHz between the fourth and the fifth band gap for CPCR-L4 [Fig. 4(b)], due to the strong coupling of acoustic waves between low-frequency bands (0–5 kHz) for PCR4 and the defect band for CPCR-L3. Furthermore, these emerging flat bands can induce multiple wave localization states.

The coupled heterogeneous structure CPCR-L5 (composed of five two-dimensional concentric PCRs) is lastly created consisting of a PCR5 and CPCR-L4 structures. To explore deeply the performance of the PCR5 structure with multiple point defects, a conventional PCR with the same size, lattice constant and filling fraction of cylinders is chosen. The calculated band structures for PCR, PCR5 and CPCR-L5 structure are performed and shown in Fig. 5. The results of a supercell calculation for PCR [Fig. 5(a)] show that there is a flat defect band emerging in the band gap, which is centered a little above the midgap frequency at around 4.14 kHz. When the number of cylinders completely removed in PCR structure increases from one to nine, the PCR5 structure consequently created has interesting band structures [Fig. 5(b)] where there are more defect bands between several gaps compared with that for PCR. Hence, the defect bands occurring in gaps intensively depend on the structural point defect, demonstrating a potential application on adjustable frequency control for acoustic filters. Additionally, Fig. 5(c) shows that there exist more and wider gaps and flatter bands in the band structures for CPCR-L5 relative to that for PCR5 or CPCR-L4 over the range of 3.0–5.0 kHz, due to the cross interaction between the single low complete gap for PCR5 and the multiple complete gaps for CPCR-L4. The increased flat bands demonstrate that there will appear more wave localization states inside the CPCR-L5 structure.

To deeply study the dependence of the acoustic wave localization enhancement on the layer number (or the number of PCRs used to build the heterogeneous structure) of the coupled multilayer heterogeneous structure, we have simulated the acoustic pressure amplification versus frequency for CPCR-L2, CPCR-L3, CPCR-L4 and CPCR-L5 structures (the acoustic pressure amplification is defined as the ratio of the cavity center pressure to the incident pressure). The simulated results [Fig. 6] demonstrate that varying from 3.0 to 6.0 kHz, the peak values of pressure amplification vs. frequency curve for CPCR-L2, CPCR-L3, CPCR-L4 and CPCR-L5 structures are 8.2 at 3.830 kHz, 16 at 3.910 kHz, 48 at 4.460 kHz and 156 at 4.463 kHz, respectively. Hence, the resonance frequency and corresponding maximum pressure amplification (that is the peak value of pressure amplification) of the coupled multilayer heterogeneous structure increase with layer number (varying from 2 to 5). The resonance frequency enhancement phenomenon is induced due to the decrease of the space between cylinders inside the coupled structure when layer number increasing, while the increasement of maximum pressure amplification arises from the enhanced acoustic wave coupling between each PCRs. Additionally, the maximum pressure amplification of five-layer CPCR-L5 is 19, 9.8 and 3.0 times greater than those of two-layer CPCR-L2, three-layer CPCR-L3 and four-layer CPCR-L4, respectively. As a result, the average ratio of the maximum pressure amplification enhancement to layer number can reach 2.73, which reveals that both maximum pressure amplification and wave localization effect for the coupled multilayer heterogeneous structure enhance markedly with its layer number.

Zoom In Zoom Out Reset image size

In order to clearly observe the enhanced acoustic wave localization effect in the coupled multilayer heterogeneous structures, we perform simulations on the acoustic pressure fields in CPCR-L2, CPCR-L3, CPCR-L4, and CPCR-L5 structures at their respective resonant frequencies (related to Fig. 6). Figure 7 shows that the resonant acoustic waves are significantly confined in the cavity centers of CPCR-L2, CPCR-L3, CPCR-L4 and CPCR-L5 structures, respectively, and the enlarged cavity pressures are also achieved. As increasing the layer number (or the number of PCRs) of the coupled multilayer heterogeneous structures, the localization effect of acoustic waves and cavity pressure enhance obviously. Hence, the strongest acoustic resonant mode and the enhanced localization effect of acoustic waves are obtained in the CPCR-L5 structure when layer number varying from 2 to 5.

Zoom In Zoom Out Reset image size

In order to exhibit the performance superiority of the proposed coupled heterogeneous structure, the pressure amplification spectrum for the PCR structure [Fig. 5(a)] is calculated and shown in Fig. 8. From Fig. 8, it can be seen that the pressure amplification for PCR reaches a maximum value of 4.6 at the frequency of 4.14 kHz. Because the maximum pressure amplification of the five-layer CPCR-L5 structure can approach 156 [Fig. 6] which is about 33.9 times larger than that of PCR, the five-layer CPCR-L5 structure has extremely stronger wave localization compared with the single conventional PCR. This huge wave localization is achieved by the acoustic coupling between all the PCRs inside the five-layer CPCR-L5 structure. Hence, the proposed coupled multilayer heterogeneous structure is useful to improve acoustic sensor sensitivity, frequency resolution of sound signal, and acoustic energy harvesting efficiency.

Zoom In Zoom Out Reset image size

In summary, the coupled multilayer heterogeneous structure composed of multiple two-dimensional concentric PCRs has been presented to obtain extremely-huge localization effect of acoustic wave. Due to the acoustic wave interaction between PCRs inside the proposed coupled multilayer structure, the extremely-huge wave localization effect is achieved in coupled multilayer heterogeneous phononic crystals, and the cavity pressure obviously increases with layer number. The calculated results obtained by numerical analysis demonstrate that the proposed coupled five-layer heterogeneous structure CPCR-L5 (optimal choice) exhibits ∼33.9 times larger maximum pressure compared with the single conventional PCR with the same size. This proposed structure can be further useful in fields of acoustic energy harvesting, acoustic sensing, frequency selection and sound concentration.

Acknowledgments