3Pb1-1
Design of Topological Phononic Structure and Application toThin Plate Elastic wave
Motoki Kataoka‡, Yuito Ohashi, Masaaki Misawa, and Kenji Tsuruta*(Grad. School Nat. Sci. Eng., Okayama Univ., Japan)

1.Introduction
As an analogy of topological insulators andsuperconductors, “topological acoustics”, whichapplies the concept of band topology to acoustic
dispersion, has attracted increasing attention inrecent years. By applying this concept to phononic
crystals, acoustic waveguides are expected to bemuch more robust against bending or defects at theedge-state frequencies than the conventional
phononic
waveguides
thatsuffers
frombackscattering at corners/defects. In the present
study, we design topological phononic crystals on
thin plastic plates, aiming at the development ofefficient elastic waveguides for next-generation
information devices. We focus on a snowflake-likestructure in the unit cell, which has been extensively
adopted due to its controllability of the dispersion inrecent years. Topological waveguides are designed
from the phonon dispersion analyses by finding edge
modes appearing at interfaces between crystals withdifferent band topologies. The transmission
efficiency of the waveguides is evaluatednumerically along with its experimental validation.
This study proves that the robust propagation ofelastic waves in thin plates is realized by topologicalphononic structures, and it paves the way to thedevelopment of new high-efficiency acousticdevices.
2. Numerical Design of 3-dimensional Topological
Waveguide with Snowflake Phononic Structure
In this study, valley-topological band structure of
the phononic crystal in plastic thin plates [1-5] wasdesigned by optimizing shape of the snowflake-likeholes in a hexagonal unit-cell. It was confirmed thatthe phase transition of the phonon band topology
occurred when the spatial symmetry of the unit cell
was destroyed by changing the length of the six
branches of the snowflake. Figure 1 shows thedesigned waveguide with two oppositely oriented
snowflake unit cells, band diagram of the supercell,and the edge mode appeared at the frequency
depicted in the red curve in the band diagram. Basedon the 3-dimensional analysis, we designed a Z*Corresponding author:tsuruta@okayama-u.ac.jp

shaped topological waveguide by arranging twotypes of unit cells one above the other with adjusted
lengths of six branches of snowflakes. The incident
wave at 85 kHz propagates efficiently along theinterface via the topologically protected mode.

Fig. 1 Designed supercell, band structure, and out-ofplane displacement of the edge mode.

3.Experiments on Elastic-wave Propagation inThin Plate Topological Waveguides
Next, we examined a Z-shaped waveguide based on
the topological phononic crystals fabricated on aplastic (polypropylene) plate, as depicted in Figure
2. The fabrication has been done using a 3D printerthat utilizes an optical stereolithography technology.
The number of unit cells in the fabricated waveguideis 161 and the dimension of the waveguide is 14 cm(W) x 5.8cm (D) x 3.8mm (H). In addition, absorbers
are attached to all the sides of the waveguide in orderto remove excess propagation modes.

Fig. 2Fabricated Z-shaped topological waveguide.

An elastic (Lamb) wave was excited at the left endof the waveguide by a piezoelectric transducer
attached to one side of the plate with an electricalsignal from a function generator amplified by anamplifier. Also, in order to measure wave
propagation accurately, reflective tape is attached tothe measurement points. The wave propagation
along the Z-shaped interface was confirmed andvisualized by measuring an out-of-plane
displacement with a laser Doppler vibrometer.
Figure 3 shows spacial distribution of thedisplacement (normalized by its maximum value) in
the waveguide for the incident wave at the frequencyof 84.2 kHz, which was predicted to be an edge modeby the supercell analysis. The figure clearly indicatesthat the wave excited at the input port was
propagating to the output port with littleattenuation/loss. Similar result was obtained byreversing the arrangement of topological phononic
crystals where another type of edge mode has been
predicted to be excited at the interface.

Fig. 3 Spacial distribution of (normalized) out-of-plane
displacement obtained with laser Doppler measurement.

4.Conclusion
In this study, we designed and fabricated a threedimensional phononic structure and an efficient
elastic-wave propagation via the topologicallyprotected edge mode was confirmed experimentallyby using a laser Doppler measurement. It was shownthat a topological phase transition occurs in eachphononic crystal with snowflake unit cell bychanging structural unit, and a topological edge state
appears at the boundary between the two types ofunit cells. The robustness of the waveguide designed
based on the band-structure analyses was confirmed
numerically and experimentally by observing wave
transmission via the topologically protected edge
states excited at frequency predicted from the bandstructure. Detailed analyses will be reported in thepresentation.
Acknowledgement
This work was supported in part by the JSPS
KAKENHI
Grant
Numbers 21H05020 and21K18877.
References
1. K. Okuno and K. Tsuruta, Jpn. J. Appl. Phys. 59,SKKA05 (2020).
2. A. Khelif, A. Choujaa, S. Benchabane, B. DjafariRouhani, and V. Laude, Appl. Phys. Lett. 84,4400 (2004).
3. Y. Pennec, B. Djafari-Rouhani, J. O. Vasseur, A.
Khelif, and P. A. Deymier,Phys. Rev. E. 69,046608 (2004).
4. J.-H. Sun and T. -T. Wu, , Phys. Rev. B. 71, 174303(2005).
5. X. Li and Z. Liu, , Phys. Lett. A. 338, 413 (2005).
6. D. Hatanaka and H. Yamaguchi, Phys. Rev. Appl.
13,024005,(2020)