Abstract
The geometricphase concept has farreaching implications in many branches of physics^{1,2,3,4,5,6,7,8,9,10,11,12,13,14}. The geometric phase that specifically characterizes the topological property of bulk bands in onedimensional periodic systems is known as the Zak phase^{15,16}. Recently, it has been found that topological notions can also characterize the topological phase of mechanical isostatic lattices^{13}. Here, we present a theoretical framework and two experimental methods to determine the Zak phase in a periodic acoustic system. We constructed a phononic crystal with a topological transition point in the acoustic band structure where the band inverts and the Zak phase in the bulk band changes following a shift in system parameters. As a consequence, the topological characteristics of the bandgap change and interface states form at the boundary separating two phononic crystals having different bandgap topological characteristics. Such acoustic interface states with large sound intensity enhancement are observed at the phononic crystal interfaces.
Main
We use a simple photonic crystal (PC) system to demonstrate geometric phase (GP) effects and the existence of topological transition points in acoustic systems. The experimental setup is shown in Fig. 1a. The PC is a cylindrical waveguide with periodically alternating crosssectional areas. Each unit cell has two wider tubes (tubeA) of length (1/2)d_{A} and inner radius r_{A} = 2.4 cm, sandwiching a narrower tubeB of length d_{B} and inner radius r_{B} = 1.5 cm. The tubes are filled with air (mass density ρ = 1.3 kg m^{−3}, speed of sound v = 343 m s^{−1}) and made of hard plastics to ensure that the inner surfaces meet the sound hardboundary condition. An example of a unit cell, with d_{A} = 3.0 cm and d_{B} = 5.5 cm (we refer to this configuration as ‘S1’), is shown in Fig. 1b–d, together with simulated pressure eigenfunctions of the lowest three eigenmodes at k = 0. Their corresponding eigenfrequencies are marked by red dots in Fig. 1e, which shows the acoustic band structure^{17,18,19}. It is clear that the lower two modes have their pressure gradient along the propagation direction k (Fig. 1b, c) and negligible pressure variation along the crosssectional directions. These two modes represent the two bandedge states of the second bandgap between the second and third bands, as shown by black curves in Fig. 1e. The mode shown in Fig. 1d has a clear pressure variation in the crosssectional planes. The dispersion of this mode is marked by the green line in Fig. 1e. For symmetry reasons, this mode cannot be excited with a plane wave, and is therefore not considered in the following discussions.
The two longitudinal modes can be further labelled as even (Fig. 1b) and odd (Fig. 1c) with respect to the centre of tubeA at k = 0. The eigenfrequencies of these two bandedge states can be tuned by varying Δd ≡ (d_{A} − d_{B})/2 (while keeping r_{A}, r_{B} and the total length a = d_{A} + d_{B} constant). Their eigenfrequencies as a function of Δd are plotted in Fig. 2. The results show that it is possible to tune the eigenfrequencies of the two bandedge states to cross each other, with accidental degeneracy occurring at Δd = 0.49 cm. The gap closing and reopening and the corresponding switching of the two bandedge states are analogous to the band inversion process in electronic systems^{20,21,22}, with Δd = 0.49 cm corresponding to a topological transition point in our acoustic system.
The parallel can be further seen by calculating the Zak phases of our PC systems. Originally defined for electronic systems, the Zak phase is a special type of Berry phase in onedimensional (1D) periodic systems^{15,23}. We extend the concept of the Zak phase to acoustics as follows. The GP for the nth isolated band is given by
where x is the axial coordinate, r represents positions in a crosssectional plane, ρ is the density of air, v is the speed of sound in air and u_{n, k}(x, r) is the periodic incell part of the normalized Bloch pressure eigenfunction of a state in the nth band with wavevector k—that is, . The factor 1/(2ρv^{2}) is the weight function of an acoustic system and is a constant in our system. On the other hand, with the pressure field being uniform in the lateral dimension, our system can be regarded as a quasi1D system^{17,18,19}. The Zak phase of an isolated band in such systems can also be obtained from the symmetry properties of bandedge states^{15,23}. As the unit cell has mirror symmetry with respect to its central crosssectional plane, the Zak phase can have only two values: 0 or π (see Supplementary Information I). If two states at the centre and edge of the Brillouin zone belonging to the same band have the same symmetry—that is, both even or odd with respect to the central plane of tubeA—then the Zak phase of this band is 0. Otherwise, the Zak phase is π.
We next investigate four different PC configurations, with each having the same unit cell length a = 8.5 cm but a different Δd. These PC configurations are denoted as S1, S2, S3 and S4, and their corresponding Δd values are indicated by vertical dashed lines in Fig. 2. Specifically, Δd = −1.25 cm for S1, 2.25 cm for S2, −0.25 cm for S3, and 0.49 cm for S4. Figure 3 shows the calculated band structures of these four PCs. The Zak phases of each PC are calculated numerically using equation (1) and are labelled in red in Fig. 3. θ_{2}^{Zak} = π for the second band of both S1 and S3, whereas θ_{2}^{Zak} = 0 for the second band of S2. The second bandgap of S4 closes at the zone centre where the two modes become degenerate at k = 0 (Fig. 3c). This marks a topological transition point. This is also shown in Fig. 2, where a gap closing and reopening process can be seen if the parameters of the system are tuned continuously from those of S1 or S3 to those of S2, passing through a transition point at Δd = 0.49 cm. Furthermore, the system’s mirror symmetry is preserved when we vary Δd, and none of the first bandgaps of the four PCs closes during this process. It follows that the first bandgaps of these four systems remain topologically identical. Note that the topological property of a bandgap is determined by the summation of the Zak phases of all the bands below this gap, but has no dependence on the properties of the higher bands^{24,25}. For S1 and S3, θ_{2}^{Zak} = π, and so the topological characters of their first and second bandgaps are identical. In contrast, θ_{2}^{Zak} = 0 for S2, indicating that the second bandgap of S2 is topologically different from that of S1 or S3.
The determination of the GP has been theoretically proposed and experimentally done in cold atom^{16,26} and photonic systems^{27,28}. In our acoustic system, we follow a different scheme to determine the GP experimentally. The bulk band GP can be determined by measuring the reflection phase ϕ at the boundary of the PC. In a quasi1D system, ϕ of the reflected pressure field of a PC for frequencies inside the bandgap is a manifestation of its Zak phase^{24,25}. The reflection phase satisfies ϕ ∈ {−π, 0} or ϕ ∈ {0, π} (mod 2π) inside a bandgap, depending on the topological character of that gap^{25}. The topological character of a gap is related to the symmetry types (even or odd) of the bandedge states below or above this gap, while the Zak phase yields the relationship between the two bandedge states of a band. Thus, the Zak phase of a bulk band and the topological characters of the two bandgaps sandwiching this band can be related to each other through the symmetry types of the bandedge states. Owing to the inversion symmetry that is inherent in the system under consideration, the geometric Zak phase can take only two values: either 0 or π (see Supplementary Information I). It has been shown rigorously that the relationship between sgn(ϕ) of the first and second bandgaps and the Zak phase of the second band is given by^{24}.
where the subscripts indicate the number of bandgaps. In other words, measuring the signs of the reflection phases of the two bandgaps sandwiching an isolated bulk band provide sufficient information to determine the Zak phase of that band. If the signs are the same, the Zak phase of that band is π. Otherwise, it is zero. This is the first method we use to measure the Zak phase of the bulk bands. For the measurement, we add a homogeneous waveguide with an identical inner radius to that of tubeA between the loudspeaker and the PC. Figure 4a shows a schematic drawing of the setup used to measure the reflection phase. The measured fields in the second common bandgaps of S1, S2 and S3 are shown in Fig. 4b, together with that of a flat sound hard surface (steel plate) for reference. The reflection fields of S1 and S3 seem to be ‘advanced’ with respect to the steel plate, whereas the reflection field of S2 seems to be ‘delayed’. Thus ϕ_{2} values for the two topologically identical PCs—that is, S1 and S3—have the same sign, whereas ϕ_{2} for S2 has the opposite sign. In comparison, the measured reflection phases of the first common bandgaps, ϕ_{1}, of all three PCs have the same sign (Fig. 4c). Excellent agreement with the theoretical prediction is seen.
To further corroborate our conclusion, we use a second method to determine the Zak phase: by directly measuring the symmetry of the bandedge states^{23,25}. Direct measurement of bulk states is relatively straightforward in an acoustic wave system. We made pointbypoint measurements of the pressure distribution inside a unit cell in the bulk PC; the experimental results are shown in detail in the Supplementary Information II. Simply put, we found that for S1 and S3, the acoustic modes at k = 0 and k = π/a have different symmetries in the second band, meaning that the Zak phase must be π (ref. 24). For S2, the acoustic modes at k = 0 and k = π/a have the same symmetry in the second band, meaning that the Zak phase must be 0 (ref. 24). These results agree with the prediction of equation (2) using the reflection phase measurements. To the best of our knowledge, this is the first experimental determination of GP in the band structure of an acoustic system.
Although Zak phases are geometric properties of the bulk bands, they have important consequences at interfaces. With the bulk Zak phases of the second band determined, we can predict the presence or absence of interface modes at the boundary of different PC configurations^{24}. In our system, there must be an interface mode with a frequency inside the second common bandgap at the boundaries separating S1 and S2, and S3 and S2, which differ in their topological properties. In contrast, no such mode can exist in S1 and S3. A simulated distribution of the pressure at the PC interface between S1 and S2 is shown in Fig. 5a, where the green arrow indicates an interface mode predicted from bulk band GP considerations. In Fig. 5b–d, we show respectively the experimentally measured pressure response spectra at the interfaces of configurations S1 + S2, S3 + S2 and S1 + S3, together with the corresponding numerical simulations. Sharp transmission peaks are clearly observed for S1 + S2 and S3 + S2 at 3,836 Hz and 3,814 Hz respectively (Fig. 5b, c). These frequencies fall inside their second common bandgap. Such a peak is absent for S1 + S3 (Fig. 5d). These observations are consistent with predictions. For S1 + S2, we further measured the spatial distribution of the pressure field at the transmission peak frequency of 3,836 Hz. The result is shown in Fig. 5e. It can be seen that the pressure amplitude decays rapidly away from the PC interface (chosen to be x = 0)—a clear sign that the transmission peak appeared as a result of the interface state (details can be found in Supplementary Information III). Recently, some zerofrequency mechanical floppy modes similar to the interface states found here have been reported in certain isostatic lattices—that is, lattices comprising mass points connected by centralforce springs on the verge of mechanical instability. These floppy modes are localized at the domain wall between two isostatic lattices with distinct topological properties^{13}.
Knowledge of the bulk band GP in a PC allows us to create interface states which in turn have useful implications. For example, we can think of it as a recipe for acoustic field enhancement. Owing to the highly confined spatial characteristics, the sound intensity can build up at the interface states. As an example, we show the amplification of sound intensity using the interface of the S1+S2 system in Fig. 5f. The enhancement ratio τ = P_{c}^{2}/P_{f}^{2} exceeds 3,400, where P_{c} is the pressure at the PC interface and P_{f} is the pressure in free space. Note that, in this measurement, the PC system is excited directly at the interface (see Supplementary Information III). The enhancement ratio can be further increased if the loss and leakage of sound are reduced in our system. Acoustic field enhancement has obvious applications where strong sound intensities are required, such as sound detection^{29} and biomedical imaging^{30}: a PC system with such an interface state can be placed in close proximity to a sensor/receiver. The local sound intensity can be amplified at the interface, leading to increased sensitivity of the device at the targeted frequency.
We have shown that a periodic acoustic system is an excellent platform to realize advanced concepts such as band inversion and topological transition. The experimental samples can be made with minimal cost and the results are intuitively easy to understand. The acoustic system is rather unique in the sense that reflection phases at boundaries can be measured straightforwardly and, at the same time, the ‘field’ quantity (which is the sound intensity) can be measured pointbypoint inside the sample. These attributes made possible the straightforward determination of the geometric phase of a band. The present results pertain to systems that are periodic in one direction. As it is technically feasible to use 3D printing to fabricate periodic acoustic crystals that are periodic in two or three dimensions, the periodic acoustic system could be readily employed to explore and determine the relationship between the symmetry of the eigenmodes, geometric phases of the bands and the bulkinterface correspondence in higher dimensions^{31}.
Methods
Experiments.
All PC samples were fabricated using 3D printing. Polylactic acid, a kind of hard plastic, was the material chosen to construct the unit cells. The tube walls were printed with sufficient thickness to ensure a suitable rigidity to reproduce a sound hard boundary. A series of small side ports were opened to house the microphone for the measurement of the spatial pressure distribution (Fig. 5e). The ports were sealed with airtight putty when not in use. The PCs were 3Dprinted one unit cell at a time, and were connected with super glue, with the joints further sealed with vacuum grease to minimize loss. S1, S2 and S3 each consisted of ten unit cells. For measurements of the pressure response spectra, a pulse with a central frequency of 3,840 Hz was generated with an arbitrary waveform generator (Agilent 33220A). The pulse was fed into a loudspeaker as the sound source. In this case the system was excited from its end. The pressure signals were picked up at the PC interface, using a 1/4inch microphone (PCB Piezotronics Model378C10), and then recorded with a digital oscilloscope (Agilent DSO6012A). The measurements were typically repeated up to 256 times with a 0.5 s repetition period, and then averaged to minimize random errors and obtain a highly stable readout. A Fourier transform was then performed to obtain the pressure response as a function of frequency. The relatively shallower depths of bandgaps in Fig. 5b–d are attributed to the noise level of our setup. However, this does not affect our arguments and conclusion.
For the measurement of the spatial pressure distribution and sound intensity amplification, a small port was opened at the interface and connected to the loudspeaker. The loudspeaker was driven by a lockin amplifier (Stanford Research SR850) to generate a monochromatic signal with the same frequency as the interface state (3,836 Hz). The pressure field was manually mapped by moving the microphone across each of the side ports on the PCs. The measured signal was also detected by the same lockin amplifier. With a sufficient time constant, the readout is highly stable. The sound intensity amplification was determined by removing the PC interface while keeping the sound source and the microphone in place. For the measurement of the reflection phase, each system was also excited with a monochromatic sound (3,836 Hz for the second bandgap, and 2,100 Hz for the first bandgap). The pressure field, together with the relative phase (with respect to a reference signal), was then manually mapped. In measurements of the spatial pressure distribution, the finite size of the microphone (the diaphragm diameter is ∼4 mm) introduces an uncertainty in position ≤2 mm, which then propagates into the measured pressure. Error bars in experimental data were calculated based on this uncertainty.
Simulations.
All simulations were performed using the ‘Acoustics module’ of COMSOL Multiphysics, a commercial finiteelement solver package. A full threedimensional geometry was used. Air (density ρ = 1.3 kg m^{−3}, speed of sound v = 343 m s^{−1}) is chosen as the fluid medium. Eigenmode calculations were carried out to find the pressure eigenfunctions of the PC unit cells as well as their band structures. Frequency domain calculations were performed to obtain the pressure response spectra. To calculate the Zak phase, we used a discretized form of equation (1) as follows:
where we have selected N points from k = π/a to k = −π/a. Equation (3) gives the Zak phase in the limit Δk_{i} = k_{i+1} − k_{i} → 0. The periodic gauge was applied in the calculation of the Zak phase, implying that for the two eigenmodes at the edges of the Brillouin zone. For each isolated band, we first calculated the pressure eigenfunction using the eigenmode analysis module of COMSOL, and then normalized the eigenfunction with the orthogonal relationship to obtain the periodicincell part of the eigenpressure u_{n, k}(x, r).
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Acknowledgements
G.M. thanks Ke Sun for his technical support with measurements and signal processing. M.X. thanks Mengyuan He for his help with the numerical calculation of the Zak phase. This work was supported by the Hong Kong Research Grants Council (Grant No. AoE/P02/12).
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M.X., Z.Q.Z. and C.T.C. provided the theoretical framework. M.X. carried out the numerical simulations. G.M. fabricated the samples and carried out experimental measurements. M.X., G.M., Z.Q.Z. and C.T.C. wrote the manuscript. All authors were involved in the analysis and discussion of the results.
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Xiao, M., Ma, G., Yang, Z. et al. Geometric phase and band inversion in periodic acoustic systems. Nature Phys 11, 240–244 (2015). https://doi.org/10.1038/nphys3228
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