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 $4\times 4$ 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.

Figure 1

Scheme showing the multiple layers and the interfaces of a TEP, as well as the incoming and outgoing propagating and evanescent flexural waves. The gray arrows show the $x-y$ plane unit vectors.

Figure 2

(a) Proposed device when operating at the lasing point. The size of the arrows is proportional to the strength of the wave. (b) Amplitude of the (left/right) transmittance/reflectance of the structure, in logarithmic scale. (c) Phase of the (left/right) transmittance/reflectance of the same structure, in units of $\pi$. All three layers have the same width of 20 cm. The thickness of the TEPs is equal to 2 cm, its Poisson's ratio is equal to 0.334, its density to 2790 $\text{kg}/{\text{m}}^3$, and its Young's modulus is equal to 70 GPa. In the active/lossy layers, i.e., PZT, the density is 7500 $\text{kg}/{\text{m}}^3$, while Young's modulus and Poisson's ratio are taken from the effective parameters, of Eqs. (6) and (7), and its thickness is 0.5 cm. The dark stars indicate the results obtained by using 3D FEM comsol results (see Appendix pp3).

Figure 3

(a) Contour plot of the amplitude of the eigenvalues versus frequency and $RS$. The dark stars give the results obtained by using FEM full 3D elasticity. (b) Eigenvalues for (increasing with arrow's direction) loss/gain values, i.e., $RS=$ (50, 75, 85, 100, 105, 110, 125, and 150) $\mathrm{\Omega }{\text{m}}^2$. Thick lines correspond to $RS=100\mathrm{\Omega }{\text{m}}^2$. The inductance of the piezoelectric layer is $LS=0.1\mathrm{H}{\text{m}}^2$. The thickness of the layers is equal to 2 cm, the density and Poisson's ratio of the passive layers are $2790{\text{kg/m}}^3$ and 0.334, respectively, while the density of the gain/loss layers is $7500{\text{kg/m}}^3$ and their thickness is 0.5 cm.

Figure 4

(a) Amplitude of the eigenvalues in the frequency domain where CPAL takes place for $LS=1.08$, 1.1, and 1.12 $\mathrm{H}{\text{m}}^2$ (red, black, and blue, respectively). For the higher frequency regime, see SM [8]. Points A and B indicate lasing and perfect absorption operation, respectively. (b) Transmittance and reflectance from the CPAL structure. The inset plots the output coefficient $\mathrm{\Psi }$ given in Eq. (9). The dark star curves are FEM-based results (obtained using the 3D full elasticity module of comsol [38]). (c) Numerically computed (3D FEM) snapshots of flexural energy ${\left|W\right|}^2$ for the FLASER in (a) at operating frequency 196.6 Hz (Point A) when the incident wave impinges from left (top) and right (bottom). (d) Same as in (c) but for the CPA mode (Point B) at the same frequency with ${\mathrm{\Omega }}_R^2=M_{21}^s{\mathrm{\Omega }}_L^1$. The colorbars are normalized by the amplitude of the incident waves.

Figure 5

Contour plot of $\mathrm{\Psi }$ versus frequency and $LS$ in logarithmic scale, i.e., $10{log}_{10}\left(\mathrm{\Psi }\right)$. Dark red regions correspond to the lasing regime ($\mathrm{\Delta }{M}^s\approx 0$). The inset shows a magnified view marked by the white dashed square.