Transmodal Fabry-Pérot Resonance: Theory and Realization with Elastic Metamaterials

We discovered a new transmodal Fabry-Pérot resonance where one elastic-wave mode is maximally transmitted to another. It occurs when the phase difference of two dissimilar modes through an anisotropic layer becomes odd multiples of $\pi$ under the reflection-free and weak mode-coupling assumptions. Unlike the well-established Fabry-Pérot resonance, the transmodal resonance must involve two coupled elastic waves between longitudinal and shear modes. The investigation into the origin of wiggly transmodal transmission spectra suggests that efficient broadband mode conversion can be achieved if the media satisfy the structural stability condition to some degree. The new resonance mechanism, also experimentally characterized, opens up new possibilities for manipulating elastic wave modes as an effective alternative to generating shear-mode ultrasound.

Figure 1

(a) An illustration of TFPR in a mode-coupled layer surrounded by an isotropic medium. At TFPRs, the incident $L$ wave can be maximally converted to the $S$ wave as it propagates through the mode-coupled layer, which can be achieved by EMMs. An EMM with slender, tilted microvoids inserted in an isotropic medium is also illustrated. (b) The $L$-to-$L$ (red) and $L$-to-$S$ (blue) transmission as a function of frequency $f$ ($d$: layer thickness) for different $C_{11}/C_{66}$ values of the mode-coupled layers. The stiffness coefficients ($C_{11}$,$C_{66}$,$C_{16}$) of the layers are (25.2, 12.9, 9), (18, 18, 9), and (12.9, 25.2, 9) in GPa units. The mass density of all layers is $2000\mathrm{kg}/{\mathrm{m}}^3$. The background medium is aluminum (${\rho }_0=2700\mathrm{kg}/{\mathrm{m}}^3$, $C_{11}^0=79.7\mathrm{GPa}$, $C_{66}^0=26.7\mathrm{GPa}$) under the plane stress condition. The transmission spectra were calculated by the transfer matrix method.

Figure 2

Transmission spectra as a function of $\mathrm{\Delta }\varphi =(k_{qs}-k_{ql})d$ for (a) the weakly coupled layer with $F_{\mathrm{SI}}=3.0$ and ${\widehat{C}}_{16}=0.35$ ($C_{11}=45\mathrm{GPa}$, $C_{66}=18\mathrm{GPa}$, $C_{16}=10\mathrm{GPa}$, $\rho =2400\mathrm{kg}/{\mathrm{m}}^3$) and (b) the strongly coupled layer with $F_{\mathrm{SI}}=7.5$ and ${\widehat{C}}_{16}=0.69$ ($C_{11}=30\mathrm{GPa}$, $C_{66}=18\mathrm{GPa}$, $C_{16}=16\mathrm{GPa}$, $\rho =2300\mathrm{kg}/{\mathrm{m}}^3$). For the unit layer thickness ($d=1\mathrm{mm}$), the first TFPR frequency $f_{\text{theory}}$ is predicted to be 2.7 and 1.5 MHz for the weakly coupled and strongly coupled media, respectively. The background medium used here is the same aluminum used in Fig. 1. An $L$ wave is incident in the $x$ direction. Blue and red curves represent $T_S$ and $T_L$, respectively. Solid lines represent the exact results while dotted lines represent the theoretical results in Eqs. (5)–(6). The insets show the particle velocity ($v_x$,$v_y$) distributions of the QL and QS modes at $\mathrm{\Delta }\varphi {|}_{\text{theory}}=\pi$ and $3\pi$.

Figure 3

Contours of the $L$-to-$S$ conversion rate $T_S$ on the $fd-F_{\mathrm{SI}}$ and $\mathrm{\Delta }\varphi -F_{\mathrm{SI}}$ planes for two different sets of anisotropic media. (a) High-$F_{\mathrm{SI}}^{\text{coup}}$ media with the instability primarily induced by strong mode coupling were considered. To vary $F_{\mathrm{SI}}$ continuously from 5 to 20, we changed ($C_{11}$,$C_{66}$,$C_{16}$) simultaneously from point $A$ to $A^{\prime }$ in Fig. S2 of Ref. [16]. (b) High-$F_{\mathrm{SI}}^{\text{back}}$ media with the instability induced by the stiffness mismatch were considered. Likewise, to vary $F_{\mathrm{SI}}$ from 5 to 50, we changed ($C_{11}$,$C_{66}$,$C_{16}$) from point $B$ to $B^{\prime }$ in Ref. [16]. The mass densities of all media are the same as that of the background medium, aluminum ($\rho =2700\mathrm{kg}/{\mathrm{m}}^3$).

Figure 4

(a) SEM images (with 2-mm scale bar) and mode shapes of $3\times 3{\mathrm{mm}}^2$ EMM unit cells of weakly (left) and strongly (right) mode-coupled EMMs at $f_{\max }\approx 100\mathrm{kHz}$. [Slit length, thickness, rotation angle ${\theta }_s$] $=$ [2.25 mm, 0.5 mm, 50°] (weakly coupled) and [2.9 mm, 0.5 mm, 65°] (strongly coupled). The arrows denote the displacement-field direction while the color levels correspond to the displacement magnitude. (b) The transmission spectra of $T_S$ (blue) and $T_L$ (red) near $f_{\text{theory}}=116\mathrm{kHz}$ for the weakly coupled and $f_{\text{theory}}=112\mathrm{kHz}$ for the strongly coupled. Solid lines are simulation results while circles are experimental results.