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Acoustic topological one-way waveguides with tunable widths using spinning components

We propose the topological one-way waveguide for acoustic waves whose width can be flexibly adjusted. The waveguide is constructed by a heterostructure where an ordinary phononic crystal is sandwiched by two time-reversal-symmetry-broken (TRS-broken) phononic crystals with their cylinders spinning in an opposite manner. The waveguide mode is confined to the ordinary phononic crystal and exhibits the gap-less and asymmetric dispersion. Therefore, we can tune the width of the waveguide by adjusting the thickness of the ordinary phononic crystal, and the waveguide mode is one-way transport which is robust against various types of local disorders and arbitrary bends. Owing to these, this acoustic topological one-way waveguide can meet the requirements of more applications compared with conventional waveguides and conventional one-way waveguides based on chiral surface waves.

Although the one-way waveguide is a direct application of the chiral surface state, in most cases, since the domain wall that separates the two topologically distinct materials cannot be very thick, the chiral surface wave has to be localized inside a narrow channel. Therefore, the carried energy usually cannot be so high, otherwise the nonlinear effect is unavoidable. On the other hand, limited by the thin waveguide channel, the one-way waveguides based on chiral surface waves will face many limitations in potential device applications, for example interfacing with conventional waveguides of much wider widths. Recently, M Wang et al theoretically and experimentally demonstrated that the width of the one-way waveguide channel can be greatly enlarged if an ordinary photonic crystal carrying Dirac points is used as the domain wall to separate two magneto-optical photonic crystals with gapped Dirac cones [45]. Later, large-area one-way surface magnetoplasmons [46] and photonic topological valley-locked waveguides with tunable widths [47] were also demonstrated. We are also aware that J-Q Wang et al experimentally realized the extended topological valley-locked surface acoustic waves very recently [48]. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
In this work, we proposed the acoustic topological one-way waveguide with a tunable width which overcomes the shortcomings of the conventional one-way waveguides based on the chiral surface modes, such as the excessive concentration of energy and the lack of width degree of freedom. Instead of connecting two TRSbroken phononic crystals directly [41][42][43][44], the waveguide is formed by a heterostructure where an ordinary phononic crystal is sandwiched by two TRS-broken phononic crystals. The ordinary phononic crystal is composed of cylinders arranged into a triangular lattice and carries Dirac points, while the two TRS-broken phononic crystals are the same as the ordinary one except that their cylinders are spinning in an opposite manner which lifts the Dirac points to generate band gaps with opposite nonzero gap-Chern numbers. As the ordinary phononic crystal acts as the domain wall to separate the two topologically distinct TRS-broken phononic crystals, two waveguide modes with their group velocities in the same direction are supported. The dispersion of the waveguide mode is gapless and very similar to that of the chiral surface mode supported on the interface formed by the two TRS-broken phononic crystals directly, irrespective of the thickness of the ordinary phononic crystal. Therefore, the one-way transport waveguide mode is robust against almost all kinds of local disorders and arbitrary bends. Moreover, as the acoustic wave is extended inside the ordinary phononic crystal, the width of the waveguide channel can be flexibly tuned by adjusting the thickness of the ordinary phononic crystal.
Due to the flexible control of the width, this type of acoustic one-way waveguide has unique advantages in potential applications. For example, since the backward scattering is totally suppressed inside the one-way waveguide, when we abruptly reduce or increase the thickness of the ordinary phononic crystal, the guided wave will become more localized or more extended along the transverse direction due to energy conservation, thereby realizing the energy squeezing or decompressing. On the other hand, when interfacing with other waveguides of arbitrary widths, an optimized width can greatly reduce the insertion loss.

Results and discussions
2.1. The topological one-way waveguide mode The phononic crystal waveguide investigated in this work is schematically shown in figure 1, which contains three domains marked as A (red), B (gray) and C (blue), respectively. The phononic crystals are composed of identical cylinders with the radius r arranged into a triangular lattice with the lattice constant a embedded in water. In domain A (C), the cylinders are spinning in the clockwise (anticlockwise) direction with the spinning circular frequency Ω < 0 (Ω > 0), while in domain B, the cylinders are static. The operation frequency is within the band gaps of domains A and C (they share the same band gap), but on the passband of domain B. Therefore, the guided wave is almost uniformly extended inside domain B but decays into domains A and C. A thin, soundpermeable, and unspinning shell is coated on each cylinder to avoid direct contact between the cylinder and water, so that spinning will not drive water to move. The shell is so thin that the scattering of the shell can be safely ignored.
The bulk bands of domain B and domains A and C are plotted in figures 2(a) and (b), respectively. Throughout the paper, we employed the multiple scattering technique [49,50] to calculate all band dispersions and simulate the propagation of the waveguide mode. For the spinning cylinder, taking both the Doppler effect and Coriolis force into account, the nth-order Mie coefficient for a time-harmonic ( w e i t ) incident acoustic wave is given by [31,51,52] l r r l where J n and H n 1 ( ) are the Bessel function and Hankel function of the first kind, k 0 is the wavenumber in the background medium, r 0 and r denote the mass densities of the background and the cylinder, )being the frequency correction for the nth vector cylindrical wave due to the Doppler rotational effect [52] and c being the sound speed inside a static cylinder, and the auxiliary function R n is expressed as Equation (1) reduces to the expression for static cylinders when Ω = 0. When W ¹ 0, it is easily found that ¹ -D D n n for ¹ n 0, indicating that TRS of the system is broken. When the cylinders are static, there is a pair of Dirac points formed at the K and K′ points due to the lattice C 6v symmetry. While the cylinders are spinning in the clockwise (counterclockwise) direction, because of the breaking of TRS, the Dirac points are lifted to a form a band gap with gap-Chern number 1 (−1) [36], see figure 2(b). For the heterostructure as shown by figure 1, domain B which is topologically trivial forms as the domain wall separating two topologically distinct domains A and C. According to the principle of bulk-edge correspondence [16,17], there are two gap-less waveguide states with positive group velocities (propagating rightward) supported in the phononic waveguide (shown by figure 1) at the frequencies within the band gap as shown by the cyan region in figure 2(b). The material of the cylinder used here is close to the porous rubber. We chose such parameters of the material to achieve a large band gap for a small spinning speed. By doing this, the domains A and C can be thin in the simulations due to the short penetration depths, reducing the amount of calculation greatly. However, we should emphasize that in principle, any material of the cylinder can be used.
The surface band dispersions are calculated by using the supercell calculation method [53]. In the calculations, we used the supercell that contains only one layer along the x direction and 18 layers along the y direction, and periodic boundary conditions are applied to both directions. We note the supercell as AB n C, indicating that there are n layers of static cylinders in the middle while clockwise and counterclockwise spinning cylinders of equal numbers are distributed on the upper and lower, respectively. The insets in figure 3 depict the normalized scattering cross sections of the cylinders inside the supercell. For the surface state, only the cylinders near the interface or inside domain B are excited. Figure 3(a) shows the projection of bulk bands and the surface bands for the supercell AB 0 C (no static cylinder contained inside the supercell). It is shown that there are indeed two gap-less surface states with positive group velocities (marked by the black circles) on the interface formed by domain A on the upper and domain B on the lower. The other two surface bands which are not marked by circles are for the interface when domain B is above domain A. For the supercell AB 6 C (6 layers of static cylinders in the middle), the bands are shown in figure 3(b). There are also two gap-less waveguide states with positive group The radius and the mass density of the cylinder are = r a 0.12 and r = -800 kg m . 3 In the static, the sound speed of the cylinder is = c 400 m s . It is interesting that the dispersions of the surface band for AB 0 C and the waveguide band for AB 6 C are very similar. This similarity is due to the fact that the effective Hamiltonians of the three domains around the K and K' valleys process the same Fermi velocity [49]. According to the k.p method [48,54], near the K valley, the effective Hamiltonians are expressed as where s x y z , , are Pauli matrices, q q ,  (4) we can see that the wavefunction of the waveguide mode is just regarded as the combination of the chiral surface mode for AB 0 C and the bulk mode in domain B [45]. Therefore, the distributions of the scattering cross sections of cylinders in domains A and C for both supercells are almost the same, see the insets of figure 3. The waveguide shown by figure 1 is a one-way waveguide since no leftward propagating waveguide mode is supported. To achieve the waveguide that supports only the leftward propagating modes, we just need to swap domain A and domain C, see figure 5. We note that although only two typical cases (AB 0 C and AB 6 C) are shown, the conclusions are the same for AB n C of an arbitrary and non-negative integer n.

Waveguide of tunable width
Since the guided wave is almost confined to domain B, the thickness of domain B can be regarded as the equivalent width of the one-way waveguide. The thickness of domain B is not fixed but flexibly controlled. Here, we assume that the spinning velocity of each cylinder is controlled by each individual motor. When the cylinder is spinning clockwise, anticlockwise or static, it belongs to domain A, C or B, respectively. For instance, when the layer of cylinders in domain A or C adjacent to domain B become static, the cylinders fall into domain B, enlarging the thickness of domain B. to the contrary, when the layer of cylinders in domain B adjacent to domain A (C) start to spin clockwise (counter-clockwise), they fall into domain A (C), and the thickness of domain B is reduced. Therefore, the waveguide of tunable widths can be achieved by precisely controlling the spinning of Our work provides a new type of mechanical control method for active tunable phononic crystals and metamaterials [55,56]. Due to the flexibility of the waveguide channel width, we can adjust the widths of the left and right ends individually, and connect the two ends to other waveguides with different cross sections. Therefore, this topological one-way waveguide with tunable width can be used as a connector for waveguides with different cross sections. On the other hand, because of the backscattering immunity, the energy of the guided wave can be squeezed by abruptly reducing the width of the waveguide channel or decompressed by abruptly enlarging the width. Figure 4 shows that the acoustic wave propagates inside a topological one-way waveguide which has varying width along the propagation direction. The line source is located at x = 10a. Before x > 20a, the layer number of domain B is 4, and it is abruptly changed to 8 at x = 20a and then suddenly reduced to 0 at x = 38a. As we can see, the acoustic wave cannot propagate leftward and no backward scattering takes place even though there are large geometry mismatches. In figure 4(b), we showed the normalized energy fluxes of the leftward and rightward propagating waves as functions of the dimensionless frequency by the black and red symbol lines, respectively. It is clearly that within the band gap of domain A or C as shown by the shaded region, the leftward propagation is totally suppressed. We also depicted the pressure fields at the lines l 1 and l 2 (as marked by the white dashed lines in figure 4(a)). The pressure fields at the line l 1 are vanishing small, in stark contrast to the pressure fields at the line l , 2 which again confirms that there is no leftward propagating mode supported. The vanishing small pressure fields at the line l 1 is attributed to the evanescent waves.
When the acoustic wave propagates from a relatively narrower channel into a relatively broader channel, the field intensities are decreased. On the contrary, the field intensities are remarkably enhanced when the acoustic wave goes from a broad channel into a very narrow channel. This flexible control of the field intensities is beneficial to the adjustment of the interaction between acoustic wave and matter. When the domain A and domain C are swapped, as we have discussed previously, the acoustic wave can only propagate leftward instead, see figure 5.

Robustness against disorders and bends
Due to the topological protection, the one-way transport is robust against local disorders and in principle any kinds of bends. In figure 6(a), we showed the propagation of the acoustic wave in the waveguide with a 90°bend. Domain B of the waveguide has 4 layers. It is seen that the acoustic wave can pass through the bend without any reflection. We also showed that the acoustic wave propagates inside a waveguide with disorders in figure 6(b). Without disorders, domain B of the waveguide has 6 layers. Within the range 20a < x < 30a, we considered that each cylinder in domain B is displaced from its original position by d 1.5 along the x direction and / d 3 2 along the y direction, where d -0.5 0.5   is a number randomly chosen for each cylinder. And we also removed 6 cylinders during the range 40a < x < 45a. We can see that the acoustic wave is still focused in domain B and the backscattering is totally suppressed.

Conclusions
In summary, we have proposed the use of a heterostructure composed by an ordinary phononic crystal sandwiched between two TRS-broken phononic crystals to achieve the acoustic topological one-way waveguide with the width of the waveguide channel can be flexibly tuned by adjusting the thickness of the ordinary phononic crystal. The one-way transport of the guided acoustic wave is protected by the gap-Chern numbers of the two TRS-broken phononic crystals, and thus robust against various kinds of local disorders and arbitrary bends. The tunability of the width enables the topological one-way waveguide to find some unique applications, such as interfacing with waveguides of arbitrary widths and squeezing or decompressing the acoustic energy flexibly.

Acknowledgments
This work was supported by National Natural Science Foundation of China (NSFC) (Nos. 11904237 and 12174263).

Data availability statement
The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.