Unidirectional Wave Vector Manipulation in Two-Dimensional Space with an All Passive Acoustic Parity-Time-Symmetric Metamaterials Crystal

Exploring the concept of non-Hermitian Hamiltonians respecting parity-time symmetry with classical wave systems is of great interest as it enables the experimental investigation of parity-time-symmetric systems through the quantum-classical analogue. Here, we demonstrate unidirectional wave vector manipulation in two-dimensional space, with an all passive acoustic parity-time-symmetric metamaterials crystal. The metamaterials crystal is constructed through interleaving groove- and holey-structured acoustic metamaterials to provide an intrinsic parity-time-symmetric potential that is two-dimensionally extended and curved, which allows the flexible manipulation of unpaired wave vectors. At the transition point from the unbroken to broken parity-time symmetry phase, the unidirectional sound focusing effect (along with reflectionless acoustic transparency in the opposite direction) is experimentally realized over the spectrum. This demonstration confirms the capability of passive acoustic systems to carry the experimental studies on general parity-time symmetry physics and further reveals the unique functionalities enabled by the judiciously tailored unidirectional wave vectors in space.

Figure 1

All passive acoustic $PT$-symmetric metamaterials crystal. (a) Unidirectional focusing based on directional wave-vector matching. The upper-left inset is an illustration of the specular reflection induced by wave-vector matching ${\mathit{k}}_r=\mathit{q}+{\mathit{k}}_i$ ($|{\mathit{k}}_i|=|{\mathit{k}}_r|$) in 2D space. For different frequencies $k_0(\text{red})<k_0^{\prime }(\text{green})<k_0^{\prime \prime }(\text{blue})$, such reflections exist at different areas along the arc, so that the reflected waves are focused within a small zone. $\mathit{q}$ does not provide any wave-vector matching for backward incidence, thus leading to no reflection. (b) The fabricated passive acoustic $PT$-symmetric metamaterials crystal. The circumferential opening angle 60° in the $x$–$y$ plane is divided into six segments of equal opening angle 9.7°. (c) The real and imaginary part modulations. Each individual modulation is $T/2=30\mathrm{mm}$ in length within a planar waveguide of height $H=20\mathrm{mm}$ [39]. The real part modulation: 3D-printed groove-structured metamaterials of opening width $w=1.2\mathrm{mm}$, depth $h=3\mathrm{mm}$, and interval $p=2\mathrm{mm}$ (lower-right inset: enlarged photo). The imaginary part modulation: holey-structured metamaterials, viz., the $60\text{−}\mu \mathrm{m}$-thick mesh fabrics (upper-right inset: scanning electron microscope image). The average pore size and open area are $7\mu \mathrm{m}$ and 2%. Its acoustic impedance is about $4000\mathrm{Pa}\mathrm{s}/\mathrm{m}$.

Figure 2

Exact and passive acoustic $PT$-symmetric potentials. (a) Evolution of the acoustic $PT$-symmetric potential through refractive index modulations: I. Exact $PT$-symmetric potential $(\text{complex}\text{exponential}\text{modulation})\to \mathrm{II}$. Exact $PT$-symmetric potential $(\text{complex}\text{square}\text{−}\text{wave}\text{modulation})\to \mathrm{III}$. Passive $PT$-symmetric potential $(\text{truncated}\text{complex}\text{square}\text{-wave}\text{modulation})\to \mathrm{IV}$. Passive $PT$-symmetric potential ($3\pi /2$ in phase shift of the real part modulation). The red (blue) curves denote the real (imaginary) part modulations. (b) Arrangement of the real part and imaginary part refractive index modulators. The modulation period is $T_m=2T$. (c) Absolute value of the eigenvalues and (d) the second term of the eigenvectors of the scattering matrix, where lines and circles correspond to exact and passive $PT$-symmetric potentials [II and IV in (a)]. During the calculations, the modulation amplitude is set as $n^{\prime }=0.001n_0$ and the total length $L=100T$, with operating frequency being the Bragg frequency.

Figure 3

Unidirectional sound reflection and focusing. (a) Simulated (left) and measured (right) acoustic energy density fields for the forward (upper) and backward (lower) incidences at 3000 Hz. The white boxes mark the measurement areas. The arrows denote incidence directions. (b) Absolute acoustic pressure distributions for forward (upper) and backward (lower) incidences, along the white dashed lines marked in (a), from 2500 to 3200 Hz, with 50 Hz stepping. The dashed rectangles indicate the areas where the standing-wave ratios $G_f$ and $G_b$ are extracted ($x=-0.55\mathrm{t}\mathrm{o}-0.63\mathrm{m}$ and $x=0.94–1.02\mathrm{m}$). (c) Simulated and measured standing-wave patterns at 3000 Hz for the forward incidence. The red (black) arrow marks the peak and valley that we employ to calculate $G_f=p_{\text{peak}}/p_{\text{valley}}$ in experiment (simulation). The extraction of $G_b$ follows the same process. (d) Extracted contrast ratio $G_f/G_b$ versus frequency. The error bars are generated from four repeated measurements.

Figure 4

Unidirectional sound focusing over spectrum. (a) Simulated scattered energy density fields at 2500, 2850, 3050, and 3150 Hz. For forward incidences at different frequencies, strong reflections occur at different locations, marked by the white asterisks, to form focal areas. (b) Measured total energy density fields for the forward incidence at frequencies shown in (a). The interference pattern varies with frequency due to the change of reflected wave front normal, indicated by the white arrows. (c),(d) Simulated scattered energy density distributions along the dashed lines $x=-350\mathrm{mm}$ and $y=0\mathrm{mm}$ in (a). All the curves are normalized per the maximum. The shadow area in (d) indicates the spatial range of focal zones at multiple frequencies.