Valley, as a new degree of freedom, raises the valleytronics in fundamental and applied science. The elastic analogs of valley states have been proposed by mimicking the symmetrical structure of either two-dimensional materials or photonic valley crystals. However, the asymmetrical valley construction remains unfulfilled. Here, we present the valley anisotropy by introducing asymmetrical design into elastic metamaterials. The elastic valley metamaterials are composed of bio-inspired hard spirals and soft materials. The anisotropic topological nature of valley is verified by asymmetrical distribution of the Berry curvature. We show the high tunability of the Berry curvature both in magnitude and sign enabled by our anisotropic valley metamaterials. Finally, we demonstrate the creation of valley topological insulators and show topologically protected propagation of transverse elastic waves relying on operating frequency. The proposed topological properties of elastic valley metamaterials pave the way to better understanding the valley topology and to creating a new type of topological insulators enabled by an additional valley degree of freedom. IntroductionElastic waves, possessing plenty of degree of freedoms (DOFs) including frequency, phase and polarization, have demonstrated tremendous promise in a variety of applications including target detection, information processing and biomedical imaging 1-4 . Recently, topology has been proposed as a new DOF in manipulating waves in both photonic and phononic systems, exhibiting remarkable impact not only on fundamental science such as condensed matter physics, but also on engineering applications such as low loss devices and waveguides 5-9 . In photonics, the photonic spin Hall effect has been achieved by taking advantage of spin DOF, which opens up an avenue of spin-dependent light transport and one-way spin transport 10-13 . In phononics, mechanical patterns and deformation have been employed as a new DOF to enable the elastic topological states 8,9,[14][15][16][17][18][19] .Recently, valley, the degenerate yet inequivalent energy extrema in momentum space, has emerged as a new dimension in manipulating waves in electronics, photonics and phononics 5,6,[20][21][22][23][24] . In graphene and transition metal dichalcogenides, valley-selective circular dichroism and valleyHall effect due to the long lifetime of valley polarization and non-zero Berry curvature have been studied for the promising applications in information carrier and storage 20,21,23,24 . As the concept of valley is introduced into the classic system, the photonic and phononic valley crystals are proposed, showing potential applications such as information processing via valley-dependent transportation 5,6,22 . However, existing designs of valley metamaterials are limited to the inherent spatial inversion symmetry of the physical system, where the typical Berry curvature distribution in the Brillouin zone follows Ω −# = Ω # . The valley metamaterials without spatial inversion symmetry have not yet bee...

We report the design of GaAs-based monolithic valley phononic crystals (VPnCs) with multiple complete phononic bandgaps, which support simultaneous valley-protected edge states with different symmetries above GHz. Rotation of triangular holes in the unit cells breaks the mirror symmetry, and this orientation degree of freedom enables the structures to exhibit different valley vortex chiralities. We numerically demonstrate the transport of multi-band valley-protected edge states with suppressed backscattering at the sharp corners of the interfaces between different VPnCs. Such monolithic semiconductor structures pave the way for ultra-high frequency topological nanophononic applications by using the lithographic technique.

We report the design of silica-based 1D phononic crystals (PnCs) with topologically distinct complete phononic bandgaps (PnBGs) and the observation of a topologically protected state of elastic waves at their interface. By choosing different structural parameters of unit cells, two PnCs can possess a common PnBG with different topological nature. At the interface between the two PnCs, a topological interface mode with a quality factor of ~5,650 is observed in the PnBG. Spatial confinement of the interface mode is also confirmed by using photoelastic imaging technique. Such topologically protected elastic states are potentially applicable for constructing novel phononic devices. 2 Topological phenomena in quantum Hall, quantum spin Hall systems and topological insulators have been extensively studied in condensed matter physics 1,2 . A hallmark of such phenomena is topologically protected edge states which are robust against defects and imperfection. The presence of the topological edge states is due to topological characters of bulk electronic bands, which is called the bulk-edge correspondence 3,4 . Recently, topological concepts have also been extended to bosonic systems including photonic and phononic structures which support the topologically protected states of light [5][6][7][8][9][10][11][12][13][14][15][16] , acoustic 17-24 and mechanical [25][26][27][28][29][30] waves for various applications. Most of experimental studies in mechanical systems have focused on discrete structures such as coupled pendula 27,28 or granular chains 30 .Although they are useful for describing topological concepts in mechanical systems, continuous solid structures supporting topological elastic waves are highly expected to realize practical high speed phononic applications. In contrast to the intensive theoretical studies of the topological elastic waves [31][32][33][34] , there is a lack of an experimental demonstration in the continuous structures. One of the main challenges is due to high modal densities of elastic waves in continuous-solid structures, preventing the formation of complete PnBGs with topologically distinct properties. Very recently, experimental demonstration of topological elastic waves using continuous structures has been reported 35,36 . However, the structures have only partial phononic bandgap (PnBG) which can cause loss of the topological elastic states by coupling with other propagation modes. Thus, it remains a challenge to realize topological elastic waves in continuous structures with complete PnBGs.To the best of our knowledge, experimental demonstration of topological elastic waves in continuous structures with complete PnBGs is very limited even in one-dimensional (1D) periodic system due to their high modal densities.In this report, we report the experimental realization of topological interface state in solidstructured quasi 1D phononic crystals (PnCs) 37 . In 1D periodic systems, topologically protected edge states are zero-dimensional (0D), localized at the interface between two PnCs...

We propose and design a phononic crystal (PnC) cavity for efficient photoelastic modulation. A strongly confined acoustic field in the cavity enhances light-sound interaction, which results in efficient phase modulation of light. As one of the possible configurations, an acoustic cavity formed in a quasi-one-dimensional (quasi-1D) PnC was investigated. By carefully tuning geometrical parameters, we successfully designed a high-Q cavity mode for a longitudinal wave within a complete phononic band gap. The acoustic Q was calculated to be as high as 9.5 × 104. This enables efficient optical modulation by a factor of 2.5 compared with a bar-type structure without PnCs.