Semi-Dirac cone and singular features of two-dimensional three-component phononic crystals

Abstract

Due to accidental degeneracy, a semi-Dirac point is realized at the center of the Brillouin zone in a two-dimensional phononic crystal (PC) consisting of a square array of core-shell-structure elliptical cylinders in water. In the vicinity of the semi-Dirac point, the dispersion is linear along the X direction, but it is quadratic along the Y direction. The semi-Dirac point is formed by the degeneracy of dipole and quadrupole modes, through accurately adjusting the radius of the cores and shells, the two modes will coincide and the dispersion relation will become linear. It is worth to be emphasised that the frequency of the semi-Dirac point is very low in our designed PC, and this is exactly the special advantage of a three-component system. Since the dispersion relation is different in the vicinity of the semi-Dirac point, some new features may be seen. Firstly, the anisotropic transmission phenomenon is demonstrated. A PC slab is placed in a rectangular waveguide where the sound hard boundary conditions are used on the upper and lower walls; a plane wave impinges on the PC slab along the X direction at the semi-Dirac point frequency, and total transmission can be achieved, so that the sound energy transmissivity is also equal to one. In the meantime, the waves experience no spatial phase changes when they are transmitting through the PC slab; this behavior indicates that the PC can be equivalent to zero index medium along the X direction. However, when the plane wave is incident along the Y direction, the transmitted field is very weak, and the sound energy transmission is nearly zero. Secondly, the properties of the semi-Dirac point can be applied to design acoustic diode. The scatterers of the PC are arranged in triangular prism shapes and placed into a straight waveguide; when the wave is incident along the X direction, it can be transmitted through the PC slab and emerge in the right area, but when the waves is incident from the opposite direction, it will be totally reflected back. Therefore, the semi-Dirac point in PC provides a way to realize the acoustic diode. Finally, the unidirectional wave-front shape effect can also be observed in our considered system. We put a square sample with 16-by-16 coating rods into water medium. When a tightly focused Gaussian beam impinges on the PC sample along the X direction at the semi-Dirac point frequency, the outgoing wave will be modulated to a plan wave. Whereas, when the incident wave along the Y direction, the Gaussian beam will be totally reflected. In conclusion, the singular features of semi-Dirac point in PC will provides an advantageous means to manipulate acoustic waves and exploit new functional materials.