Fig. 1: Acoustic and optic modes in the chiral PnC and its comparison with classical diatomic chains.
From: Origin and tuning of bandgap in chiral phononic crystals
a Schematics of the conventional chiral subunit cell and its macroscopic polarizations under the longitudinal input \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}\). The green arrow denotes the longitudinal input mode \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}\); the blue arrows denote the polarizations determined by the bending deformation of the ligaments; the red arrows denote the polarizations determined by the stretch deformation of the ligaments. Therein, \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}\) is the longitudinal polarization determined by bending deformation; \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rb}}}}}\) is the rotational polarization determined by bending deformation; \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\) is the longitudinal polarization determined by stretch deformation; \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\) is the longitudinal polarization determined by stretch deformation. b The difference between the static mass and the dynamical inertia in the chiral subunit cell. The gray shadow refers to the extra inertia induced by the polarization coupling, such as \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}\) + \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rb}}}}}\) as well as \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\) + \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\). c Polarizations of the acoustic mode determined by the bending deformation of the ligaments. The inset is the zoom of the directions of the coupled polarization (\({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}\) + \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rb}}}}}\)). d Polarizations of the optic mode determined by the stretch deformation. The inset is the zoom of the directions of the coupled polarization (\({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\) + \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\)). e The relation between the static mass and the dynamical inertia in the classical diatomic chain. \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{l}}}}}\) means there is only longitudinal polarization in the classical diatomic chain. f Schematic of the classical diatomic chain. Polarization schematics of the acoustic mode (g) and optic mode (h) in the classical diatomic unit cell. i Dispersion-spectrum schematic of the classical diatomic unit cell. Therein, the shaded area indicates the bandgap.
