Unidirectional Zero Reflection and Perfect Absorption via Exceptional Points in Active Piezoelectric Willis Metamaterials

Abstract

Electro-momentum coupling in piezoelectric metamaterials with broken inversion symmetry enables asymmetric elastic wave transport by linking macroscopic electric fields to momentum, an effect analogous to Willis coupling in elastic media. A one-dimensional layered piezoelectric metamaterial integrated with shunt circuits, consisting of a resistor, inductor, and strain-proportional voltage feedback gain, is proposed to achieve dynamic control of frequency-dependent stiffness and damping through electromechanical interactions. Tuning the circuit parameters yields direction-dependent wave scattering at targeted frequencies. Dynamic homogenization reveals macroscopic constitutive relations exhibiting both Willis and electro-momentum couplings. Non-Hermitian exceptional points are identified, where scattering eigenmodes coalesce and produce extreme asymmetries in wave response. Near these points, the system realizes unidirectional zero reflection (UZR) and unidirectional perfect absorption (UPA), achieving complete absorption from one direction and total reflection from the opposite side. The findings demonstrate a compact and reconfigurable platform for tunable, directional elastic wave control using passive-active hybrid metamaterials, opening new avenues for programmable devices in acoustic isolation, wave-based computing, sensing, and energy manipulation in solid media.

Figures (8)

• Figure 1: Schematic and modeling framework for electro-momentum coupling in shunted piezoelectric metamaterials.a, One-dimensional waveguide comprising a layered piezoelectric composite embedded between two aluminum sections and terminated with perfectly matched layers (PML). Elastic waves incident from either direction exhibit asymmetric transmission ($t_f$, $t_b$) and reflection ($r_f$, $r_b$) due to the spatial asymmetry introduced by the shunted composite. The structure features a uniform poling direction, while the arrangement of heterogeneous piezoelectric layers breaks inversion symmetry to support directional wave propagation. b, Zoomed-in view of a representative unit cell consisting of three serially connected piezoelectric layers PZT-4, BaTiO$_3$, and PVDF each interfaced with a shunt circuit composed of a resistor ($R_i$), inductor ($L_i$), and strain-proportional voltage feedback source ($V_{0i} \varepsilon$). These circuits actively tailor the electromechanical response of each segment and modulate the macroscopic dynamic behavior through piezoelectric coupling. c, The shunted layer is modeled as an effective elastic layer with a modified electro-mechanical elastic constant $\check{C}_i$, incorporating both material properties and circuit-induced effects. This framework enables tunable control of wave propagation through circuit parameters, where the resistor introduces loss, the inductor defines resonance behavior, and the voltage feedback allows dynamic modulation of stiffness.
• Figure 2: Wave decomposition in a layered piezoelectric composite for forward and backward incidence.a, Schematic of elastic wave propagation from the left (forward incidence) through a layered structure composed of alternating piezoelectric segments with effective electro-mechanical elastic constants $\check{C}_1$, $\check{C}_2$, and $\check{C}_3$, bounded by homogeneous elastic layers $\check{C}_0$. Each segment is associated with local forward ($A_i$) and backward ($B_i$) wave components. The incident wave ($A_{0,\text{inc}}$) is partially reflected ($B_{0,\text{ref}}$) and transmitted ($A_{0,\text{tran}}$), with no incoming wave from the right ($B_0 = 0$). b, The same configuration under backward incidence, where the wave enters from the right. The incident wave ($B_{0,\text{inc}}$) results in partial transmission ($B_{0,\text{tran}}$) and reflection ($A_{0,\text{ref}}$), with no incoming wave from the left ($A_0 = 0$). This layer-wise decomposition into forward and backward components enables analytical evaluation of reflection and transmission coefficients under both excitation directions using the transfer matrix method.
• Figure 3: Effect of shunt circuit parameters on directional reflection behavior. Frequency-dependent reflection magnitude ($|r|$, top row) and phase ($\angle r$, bottom row) for forward (solid lines) and backward (dashed lines) wave incidence are shown under different shunting conditions. Identical circuit parameters are applied to all piezoelectric layers. a, In the open-circuit case (no shunt), the reflection magnitudes are symmetric, while phase asymmetry arises solely due to structural asymmetry. b, Introducing a resistor ($R = 50~\mathrm{k}\Omega$) and inductor ($L = 1~\mathrm{H}$) leads to asymmetry in both magnitude and phase of the reflection ratio, with losses induced by the resistor breaking amplitude symmetry. The LC resonance condition contributes to frequency-selective enhancement of this asymmetry. c, Adding voltage feedback ($V_0 = 10~\mathrm{MV}$) further modifies the reflection characteristics across the entire frequency spectrum, introducing tunability and stronger contrast between forward and backward responses. These results illustrate how circuit parameters enable directional control over elastic wave reflection in shunted piezoelectric composites.
• Figure 4: Parametric analysis of unidirectional zero reflection (UZR) and its relation to exceptional points (EPs).a, Colormap of the contrast ratio $\alpha = r_f / r_b$ between forward and backward reflection as a function of resistance ($R$) and voltage feedback amplitude ($V_0$), with inductance fixed at $L = 1~\mathrm{H}$. Identical circuit parameters are applied to all piezoelectric layers. A sharp peak in $\alpha$ identifies the condition for UZR at $0.15~\mathrm{MHz}$, corresponding to $R = 78.38~\mathrm{k}\Omega$ and $V_0 = 20.42~\mathrm{MV}$. b, Frequency-dependent reflection magnitude (top) and phase (bottom) for forward (solid) and backward (dashed) incidence, confirming complete suppression of forward reflection at the UZR frequency along with a $\pi$ phase difference. c, Absolute differences in reflection magnitude (top) and phase (bottom) between forward and backward incidence, showing a pronounced asymmetry at $0.15~\mathrm{MHz}$. d, Eigenvalue trajectories of the scattering matrix in the complex plane showing the coalescence of $\lambda_1$ and $\lambda_2$ at the EP. e–g, Frequency-dependent plots of the imaginary parts (e), real parts (f), and magnitudes (g) of the eigenvalues, confirming the occurrence of a second-order exceptional point at the UZR frequency. The $\pi$ phase jump observed in b and c is a spectral signature of the EP. These results confirm that UZR coincides with an exceptional point of the scattering matrix composed of reflection and transmission coefficients ($r_f$, $r_b$, $t_f$, $t_b$).
• Figure 5: Dynamic homogenization of a layered piezoelectric composite.a, Schematic of a one-dimensional periodic piezoelectric composite consisting of alternating heterogeneous piezoelectric layers embedded between aluminum waveguide sections and terminated with perfectly matched layers (PML). Due to the lack of inversion symmetry and spatial variation, elastic waves exhibit direction-dependent reflection ($r_f$, $r_b$). b, The composite is dynamically homogenized into an equivalent continuum domain characterized by effective macroscopic properties. These include the stiffness tensor $\tilde{\mathbf{C}}$, effective density $\tilde{\mathbf{\rho}}$, and piezoelectric coupling tensors $\tilde{\mathbf{A}}$ and $\tilde{\mathbf{B}}$. Importantly, the homogenized model also incorporates the Willis coupling tensors $\tilde{\mathbf{S}}$, $\tilde{\mathbf{S}}^\dagger$ (describing coupling between momentum and strain), and the electro-momentum coupling tensors $\tilde{\mathbf{W}}$, $\tilde{\mathbf{W}}^\dagger$ (capturing coupling between electric field and momentum). The homogenized medium retains the same poling direction as the original layered system and reproduces its macroscopic wave behavior. This framework enables effective modeling of asymmetric wave propagation in complex piezoelectric composites.