Self-dual singularity through lasing and antilasing in thin elastic plates

We show here that a coherent perfect absorber and laser (CPAL) enabled by parity-time ($\mathcal{PT}$)-symmetry breaking may be exploited to build monochromatic amplifying devices for flexural waves. The fourth-order partial differential equation governing the propagation of flexural waves in thin-elastic piezoelectric plates leads to $4\ifmmode\times\else\texttimes\fi{}4$ transfer matrices, and captures the interplay between propagating and evanescent waves that is translated into $\mathcal{PT}$-symmetry properties specific to elastic plate systems. We thus demonstrate the possibility of using CPAL for such systems and we argue the possibility of using this concept to detect extremely small-scale vibration perturbation with important outcomes in surface science (imaging of nanometer vibration) and geophysics (improving seismic sensors like velocimeters). The device can also generate finite signals using very low exciting intensities and/or alternatively can be used as a perfect absorber for flexural energy, by tailoring the left and right incident wave amplitude and phase, with evident energy harvesting applications.