Multi-dimensional wave steering with higher-order topological phononic crystal

The recent discovery and realizations of higher-order topological insulators enrich the fundamental studies on topological phases. Here, we report three-dimensional (3D) wave-steering capabilities enabled by topological boundary states at three different orders in a 3D phononic crystal with nontrivial bulk topology originated from the synergy of mirror symmetry of the unit cell and a non-symmorphic glide symmetry of the lattice. The multitude of topological states brings diverse possibility of wave manipulations. Through judicious engineering of the boundary modes, we experimentally demonstrate two functionalities at different dimensions: 2D negative refraction of sound wave enabled by a first-order topological surface state with negative dispersion, and a 3D acoustic interferometer leveraging on second-order topological hinge states. Our work showcases that topological modes at different orders promise diverse wave steering applications across different dimensions.

While many topological phases originate from the electronic systems 1-3 , they quickly gained the attention from other realms, spanning from optics, photonics 26,27 , electromagnetism [4][5][6][15][16][17] , to acoustics and phononics [18][19][20][21][22][23][24][25] . Due to the versatility offered by these classical systems, they rapidly become platforms to realize novel topological phases and to investigate the physics therein. However, relatively few efforts were devoted to the exploration of novel topological states for wave manipulation applications. This work is devoted to applying topological states to achieve novel wave steering at different dimensions. It is based on a simple realization of a 3D higher-order topological phononic crystal (PC) possessing a large bandgap that can be characterized by the nontrivial quantized bulk polarization. Differs from the acoustic analog of Su-Schrieffer-Heeger (SSH) model 10 or Kagome model 8 , our PC maintains nontrivial topology after a reversion of the center and corner of the unit cell, which means our PC does not have a topologically trivial counterpart. As a result of a ternary layer of topological protection, a hierarchy of 2D topological surface states (TSSs), 1D topological hinge states (THSs), and 0D topological corner states (TCSs) are observed. We then present the dispersion of the TSSs and show that it can lead to negative refraction at PC-air interfaces. In addition, we exploit the THSs as tailorable transport channels to realize a 3D acoustic interferometer. Our work showcases that topological states can be tailored for diverse and versatile wave steering applications across multiple dimensions.

Results
Phononic crystal with higher-order topology. Our PC comprises a cubic array of orthogonally aligned aluminum rods (treated as sound-hard objects) along the x-, y-, z-directions, respectively, in an air background. All rods have a square cross-section with a side length = 1.8 cm, and their axes are separated by /2, where = 4 cm is the lattice constant. The PC belongs to a non-symmorphic space group no. 223 28 and has a glide symmetry !" = / #!" |  To verify the simulated results, we fabricated this 5×5×5 PC cube enclosed in aluminum plates, whose picture is shown in Fig. 3e without the top aluminum plate for viewing purposes. Arrays of holes are drilled on the aluminum plates. The holes are blocked when not in use. Sound sources can be placed at different positions and excite the PC cube through the holes. First, we insert a microphone well inside the PC through the holes to measure the bulk response. The normalized pressure field is plotted in the black curve in Fig. 3f, exhibiting a large bandgap in the frequency range 4.9 -7.0 kHz as predicted by the simulation. Then the hinge response is also observed, which exhibits a dominant peak near 5.14 kHz followed by a plateau extending to about 6.8 kHz, as shown in the green curve in Fig. 3f, implying the existence of localized states on the surfaces or at the hinge. The response at the corner is plotted in Fig. 3f in red curve. Only one dominant peak at 6.72 kHz is observed, agreeing with the numerical simulation. To further confirm the existence of THSs and TCSs, we mapped out the acoustic field distribution on the surfaces, hinges, and corners. The pressure field maps obtained at 5.20 kHz and 6.72 kHz are shown in Figs. 3g and 3h, and clearly demonstrate the localization of the sound wave on the hinges and at the corner, respectively, affirming the coexistence of second-order and third-order topological states in our PC. The stars in Figs. 3g and 3h represent the position of sound sources.
Wave transport leveraging THSs. Higher-order topological states give rise to new possibilities for wave manipulation. For example, THSs naturally offer tailorable wave transport channels that can be tailored to versatile shapes. As a proof-of-principle demonstration of potential practical applications of the higher-order topological states, an interferometer based on the PC cube are fabricated and characterized. As illustrated in Fig. 4a, three rectangular waveguides, labeled port 1, 2, and 3, are connected to three corners of the aforementioned 5×5×5 PC cube. The location of these waveguides ensures three-fold rotational symmetry about the diagonal axis of the PC cube. The cross-section of each waveguide is set as 1.6×0.1 cm. Figure 4b shows the simulated acoustic field distribution when a wave with frequency 5.16 kHz incidents from port 1. The THSs are excited and indeed functions as waveguiding channels, which can be exploited to construct an interferometer. Figure 4c plots the result of two in-phase waves incident to the cube from ports 1 and 2. Owing to the constructive interference, the amplitude of the outgoing sound at port 3 is doubled. In contrast, when two out-of-phase waves incident onto the same ports, destructive interference occurs, and consequently, the suppression of the outgoing sound at port 3 is observed, as shown in Fig. 4d. Figure 4e gives a picture of the experimental setup of such a THS interferometer. Two speakers are used as sound sources, placed at the end of two aluminum waveguides with a rectangular cross-section. The other ends of the waveguides are connected to the two corners of the PC cube. We measure the sound amplitude in the output waveguide at the upper right corner. The transmittance for 5.0 -5.5 kHz in Fig. 4f is peaked at 5.27 kHz when the two speakers are in-phase, while it is minimized at the same frequency when the two speakers are outof-phase. These results are strong evidence of the constructive and destructive interferences attributed to two separate but equivalent hinge paths.

Discussion and Conclusions
We present a simple design of an acoustic 3D high-order TI that can simultaneously support topological states at three different dimensional hierarchies. The negative dispersion of the TSS makes negative refraction easily attainable. Note that our negative refraction occurs at the PC-air interfaces instead of between two different PC boundaries, making it feasible for wave-steering applications. The presence of higher-order topological states, in particular, the THSs, brings even more intriguing effects.
First, THSs along different hinges can relay the transport of sound waves, guiding the propagation to bend around corners towards different directions. As a result, the inputs and output are not on the same plane. Similar effects can only be attained by using a 3D double-zero-index medium previously 31 , which relies on the stringent tuning of system parameters. In comparison, the topological protection of THSs endows additional robustness. Second, it is easy to see the design of the THS interferometer affords great versatility. For example, by choosing the position of input and output ports, the interferometer can easily guide waves towards different directions in the 3D space (as shown in Fig.   4b). Such functionality is unobtainable for any 2D TIs.
In summary, our work presents convincing cases that higher-order topological wave crystals can benefit wave-steering applications. As topological notions can be universally applied to other realms of physics, such as photonics and electromagnetism, we believe our work is an important step for a broad area of next-generation technology and devices.

Methods
Simulations. The band structures, Bloch wavefunctions, and acoustic field distributions were calculated using the acoustic module in COMSOL MULTIPHYSICS. The Wannier centers are obtained by computing the Wilson loop operator defined as: Experiments. All PC samples and their claddings were machined from aluminum alloy and then were assembled in the lab. The frequency responses were measured by frequency scans. A waveform generator (Keysight 33500B) was used to generate a monochromatic signal to drive a loudspeaker through an audio power amplifier. The signals were detected by a 1/4-inch microphone (PCB Piezotronics Model-378C10) and were then recorded by a digital oscilloscope (Keysight DSO2024A).
For the negative refraction experiments, the raster-scans of acoustic fields were performed using a house-built motorized translation stage. For the measurement of THSs and TCS, manual scans were performed by inserting the microphone to all small ports on the claddings (Fig. 3e).