1P3-10
Topological phononic waveguide design on surface of 3D
object and its application to ball SAW sensor
Yuta Kono1‡, Motoki Kataoka1, Yusuke Hata1, and Kenji Tsuruta1* (1Okayama Univ.)
1. Introduction
SAW (surface acoustic wave) filters are now
widely used as frequency filters for mobile
communications such as smartphones. SAW sensors
have also been developed as a gas sensing device
which measures wave attenuation changes due to gas
adsorption by depositing a sensitive film on the SAW
propagation path. However, the device performanceis strongly degraded due to diffraction andattenuation when the waves propagate beyond acertain distance. Recently, it has been discovered thatwhen SAW is generated under certain conditions on
a spherical surface, the SAW propagates around theequator many times without attenuation anddiffraction[1]-[3]. This phenomenon has been shown todramatically improve the sensitivity of SAW sensors.
In the previous studies, the propagation of surface
acoustic waves on a sphere was investigated by using
a vibration source on a circular arc as shown in Fig.
1[1][2], where the aperture angle 2ߠ௔ defined as,2ߠܽ = ටߨ
ߣ
=ඨ݇ܽ
2ܽ

(1)

where ݇ and ܽ are the wave number and the radiusof the sphere, respectively. When equation (1) issatisfied, the SAW energy is confined to theequatorial part of the sphere.

Fig. 1 Coordinate system of a sphere

The propagation of SAWs confined in a narrow
orbit is not disturbed much even if the sphere is
distorted or inhomogeneous, and thus the lowest
attenuation and the highest number of circuits areexpected. When attempting to achieve higher
frequencies, the anisotropy and homogeneity of thesphere may affect diffraction, and high Q-value
propagation may not be obtained even whenequation (1) is satisfied. In this study, we examine
the possibility of robust propagation of surface
E-mail: ‡ pxjz2v8r@s.okayama-u.ac.jp*
tsuruta@okayama-u.ac.jp

acoustic waves on a sphere using topologicalphononics, which has been the focus of muchattention in recent years.
2. Topological phononic crystal
A phononic crystal is an artificial material consistingof periodic structures with different elastic properties.
By periodically arranging materials with different
elastic properties based on the order of the
wavelength of phonons propagating in a solid, it ispossible to realize phononic band gaps in which
acoustic/elastic waves at a specific frequency areprohibited to propagate and phononic waveguides
that guide the waves in a specific direction. Elastic
waveguides based on phononic crystals are in the
spotlight because of the possibility of controllingwave propagation along a desired path and frequency
by using band engineering methods similar to thoseused in semiconductor electronic design. However,the practical application of elastic waveguides basedon phononic crystals has been hampered by soundwave scattering at defects and transmission losses
due to waveguide bending. By adding the concept of
topology to the phononic band design, highly
efficient elastic wave propagation can be achieved
only at surfaces and interfaces where the propagating
modes at the interfaces are protected by the topologyof the band (“edge band”) against the disturbance.

Fig. 2 Phononic crystal on a thin substrate.

Figure 2 shows the phononic structure used in thisstudy. This phononic crystal consists of a hexagonallattice of three-dimensional unit cells withsnowflake-shaped holes. Snowflake-like unit cells
have been used in various studies on phononic
crystals as a very controllable structure for designingphonon band structures[5][6].
The hexagonal lattice has lattice constant a mm and thickness t  mm. The snowflake-likecrystalline scatterer has six legs of width w mm.
The mirror symmetry of the unit cell can be broken
by changing Δr, the difference between r1 and r2,while keeping the sum of r1 and r2 constant.
A further continuous decrease of Δr causes thegap to reopen. In addition, both K+ and K- pseudospin modes appear above and below the gap between
the K and K' points in the Brillouin zone,
respectively. The eigenmodes of the unit cell have
opposite energy flows. This indicates a topologicalphase transition. Thus, the combination of the twostructures results in the appearance of edge states in
the phonon bands, allowing elastic wave propagation
with robustness at the boundary.

interface is small due to the protected topologicaledge state. We confirmed that elastic waves
propagate robustly on curved surfaces, such as thesurface of a sphere, even in the edge state.

Fig. 5 Displacement field in a topological waveguide on
the equator of the sphere for elastic wave propagations at
85.4kHz.

Fig. 3 (a) Structure of phononic waveguide, and (b) its
phonon band diagram.

Figure 3(a) shows the supercell structure of thethin phononic crystal used in this study, and (b)shows the phonon band diagram of (a) obtainedusing eigen frequency analysis. The horizontal axisin Fig. 3 denotes the wave number and the verticalaxis is the frequency. Figure 3(b) shows a band gap
from 76 kHz to 83 kHz, where phonon propagation
is prohibited. The red and blue lines in Fig. 3(b) from83kHz to 88kHz indicate the edge states appear at theinterface corresponding to the respective colors in
Fig. 3(a). Here robust elastic wave propagation ispossible at the boundary surface.
We also designed a Z-shaped waveguide with asnowflake-shaped topological phononic structure.

Fig. 4 Out-of-plane displacement in Z-shaped waveguideat two interfaces. (a) 86.2kHz and (b) 87.7kHz.

The finite element method was used to solve thedifferential equations and simulate elastic wave
propagation. Figure 4 shows the out-of-plane
displacement distribution of a wave incident on theZ-shaped waveguide. The Z-shaped waveguide has
two obstructions, but the transmission loss at thisApplying the scheme of topological waveguidedesign described above. we develop a sphericalelastic waveguide based on phononic crystals and aspherical elastic waveguide. Figure 5 shows apreliminary result of an edge-mode propagation
along the spherical waveguide. It shows the SAW at
the edge mode (85.4kHz) is confined effectively in
the interface. Thus, the present scheme for
topological phononic crystal is shown to be
applicable to the surface of 3D-spherical object,which proves a promise for applying it effectively toball SAW device. The details of the properties of this
topological waveguide on a sphere will be discussed
in the presentation.
Acknowledgment
Part of this research was supported by a Grant-in-Aidfor Scientific Research (JP21H05020).
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