⇦Chinese Physics Letters, 2024, Vol. 41, No. 1, Article code 014301 Acoustic Bilayer Gradient Metasurfaces for Perfect and Asymmetric Beam Splitting Jiaqi Quan (权家琪)1, Baoyin Sun (孙宝印)1, Yangyang Fu (伏洋洋)2*, Lei Gao (高雷)1,3, and Yadong Xu (徐亚东)1* Affiliations 1Institute of Theoretical and Applied Physics, School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2College of Physics, Nanjing University of Aeronautics and Astronautics & Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing 211106, China 3School of Optical and Electronic Information, Suzhou City University, Suzhou 215104, China Received 1 November 2023; accepted manuscript online 14 December 2023; published online 7 January 2024 *Corresponding authors. Email: yyfu@nuaa.edu.cn; ydxu@suda.edu.cn Citation Text: Quan J Q, Sun B Y, Fu Y Y et al. 2024 Chin. Phys. Lett. 41 014301    Abstract We experimentally and theoretically present a paradigm for the accurate bilayer design of gradient metasurfaces for wave beam manipulation, producing an extremely asymmetric splitting effect by simply tailoring the interlayer size. This concept arises from anomalous diffraction in phase gradient metasurfaces and the precise combination of the phase gradient in bilayer metasurfaces. Ensured by different diffraction routes in momentum space for incident beams from opposite directions, extremely asymmetric acoustic beam splitting can be generated in a robust way, as demonstrated in experiments through a designed bilayer system. Our work provides a novel approach and feasible platform for designing tunable devices to control wave propagation.
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 DOI:10.1088/0256-307X/41/1/014301 © 2024 Chinese Physics Society Article Text Ability to freely control acoustic wave propagation is significant in many applications, ranging from acoustic communication to biological imaging.[1,2] As the counterpart of electromagnetic metasurfaces,[3-5] acoustic metasurfaces have provided a novel approach for designing flat and compact devices that can manipulate sound waves in a two-dimensional (2D) planar situation.[6-8] In particular, acoustic gradient metasurfaces (AGMs) usually consist of periodic arrays of structurally graded subwavelength blocks that are designed as Helmholtz resonators[9,10] or coiled-up space structures.[11,12] AGMs introduce an in-plane phase gradient, which is equivalent to an additional wave vector, thus altering the fundamental law of reflection and refraction of acoustic waves at the interface.[13-15] Many promising applications have been proposed based on the concept of AGMs, such as asymmetric transmission,[16,17] perfect retroreflection,[18,19] acoustic holograms,[20] and orbital angular momentum generation.[21] The phase gradient has been demonstrated to have a degree of freedom for controlling sound wave propagation. Inspired by exciting advances in Moiré superlattices of 2D materials (e.g., graphene),[22-24] great efforts have recently been devoted to bilayer artificial 2D materials, i.e., bilayer metasurfaces,[25-28] in which many remarkable wave phenomena have been demonstrated, such as magic-angle lasers[29] and Moiré quasi-bound states.[30] Although great progress has been made in bilayer metasurfaces, most of them are simply cascaded by two metasurfaces without phase gradient modulation, and the core mechanism of controlling wave propagation relies on the concept of Moiré superlattices tuned by a twisting operation. Taking advantage of the new degree of freedom of the phase gradient, some studies have been performed to manipulate wave transmission using bilayer gradient metasurfaces.[31-33] However, most of the results are just simple superpositions of individual functions in each layer. Some recent works[34-36] have shown the existence of robust and exceptional diffraction rules in gradient metasurfaces, and this offers more opportunities beyond the generalized Snell's law for the control of wave propagation.[37-42] Nevertheless, the underlying physics of manipulating wave propagation via bilayer metasurfaces with a phase gradient has not yet been fully explored. In particular, how the interlayer gap and the phase gradient contribute to diffraction effects is still unclear. In this Letter, we present a paradigm for accurate bilayer design of AGMs for one-way beam manipulation. The precise combination of different phase gradients in the bilayer metasurface results in an incident beam from the opposite direction following different diffraction routes in momentum space as it passes through the bilayer system. This process can be tuned by simply tailoring the size of the interlayer gap, producing an extremely asymmetric beam splitting effect that is not owned by each AGM. We know that acoustic beam splitters, which convert an incident beam into two or more beams,[43] play key roles in various applications such as rectifiers.[44] However, traditional methods to achieve beam splitting are bulky and of wavelength scale, which is not conducive to device miniaturization and limits applications. Our results provide a compact and flat route for designing acoustic beam splitters. Specifically, as shown in Fig. 1, as the interlayer gap increases from zero, a transition from a symmetric beam splitting effect to an asymmetric one is observed. For the lower incidence (i.e., $+z$ direction incidence), a perfect beam splitting effect occurs independent of the interlayer gap, originating from the diffracted wave of the dominant order that exists in the form of a propagating wave (PW). For the upper incidence (i.e., $-z$ direction incidence), an evanescent wave (EW) of certain special diffraction orders can emerge and the wave tunneling effect occurs in the interlayer region. As a result, the beam splitting effect is largely dependent on the gap size. When the interlayer gap reaches half the wavelength, the beam splitting effect disappears completely for $-z$ direction incidence, while it still holds perfectly for $+z$ direction incidence. The underlying physics is attributed to the broken symmetry of the PW in momentum space and the anomalous diffraction rule in AGMs. These findings are verified by both full-wave simulations and experimental measurements. Although this work focuses on acoustic waves, the revealed concept and physics can also be applied to other dynamic waves, such as electromagnetic waves, and in other disciplines. \it Model and Results.} To illustrate our idea, Fig. 1 shows a schematic of the proposed bilayer AGMs consisting of two gradient metasurfaces (i.e., AGM-1 and AGM-2) separated by an air gap. The two AGMs have the same thickness, and they are equipped with different phase gradients $\xi_{1}$ and $\xi_{2}$ along the interfaces, which meets the condition of $\xi_{1} >k_{0} >\xi_{2}$, where $k_{0}$ is the wave vector in air. This condition is the key to manipulating the diffraction wave in momentum space. As a concrete example, in this study we consider $\xi_{1} =\sqrt 2 k_{0}$ and $\xi_{2} ={\sqrt 2 k_{0} } / 2$ for bilayer AGMs. Due to the introduction of the phase gradient, both AGMs have anomalous diffraction properties that are governed by the parity-dependent diffraction rule,[35] i.e., $k_{x}^{\rm r/t} =k_{x}^{\rm in} +n\xi_{1/2}$, where $k_{x}^{\rm in}$ is the $x$-component wave vector of the incident wave, $k_{x}^{\rm r/t}$ is the $x$-component wave vector of the reflection/refraction wave, and $n$ is an integer indicating the diffraction order. For AGM-1 alone, perfect reflection happens for the incident beam with an angle of $\theta_{\rm in} \in ({-\theta_{\rm c},\, \theta_{\rm c}})$. Here $\theta_{\rm c}=\sin^{-1}({\gamma_{1} -1})\approx 24.5^{\circ}$ is the critical angle and $\gamma_{1} =\xi_{1} /k_{0}$. For AGM-2 alone, a perfect beam splitting effect occurs for the incident beam with $\theta_{\rm in} \in ({-\theta_{\rm c},\, \theta_{\rm c}})$. More details are provided in Note 1 of the Supplementary Information.
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Fig. 1. Concept of acoustic bilayer metasurfaces for extremely asymmetric beam splitting. (a) Schematic illustration of extreme beam splitting via bilayer gradient metasurfaces for positive incidence; the beam splitting effect accompanied with perfect transmission is independent of the interlayer air gap size of $d$. (b) Partial beam splitting via bilayer gradient metasurfaces for negative incidence, where the tunneling effect has a $d$-dependent splitting effect. BS$_{-1}$ and BS$_{1}$ are used to mark the diffraction beams which contribute from $n=-1$ and $n=1$ channels, respectively. The corresponding diffraction evolution in momentum space is shown by the equifrequency contours for (c) positive and (d) negative incidence.
For a bilayer AGM system, we first consider an acoustic beam normally incident on it from the positive direction, as shown in Fig. 1(a). It is first diffracted by AGM-2, and the diffraction mainly occurs in the transmission channel,[35] with the corresponding refraction angle given by $\sin \theta_{\rm 2t} =\sin \theta_{\rm in} +n\gamma_{2}$, where $\gamma_{2} =\xi_{2} /k_{0}$ and $n=\pm 1$ due to $\xi_{2} < k_{0}$ and $k_{x}^{\rm in} =0$. This means that a beam splitting effect occurs in the air gap with a splitting angle of $\theta_{\rm 2s} =\theta_{\rm 2t} =\pm 45^{\circ}$. As two beams travel and arrive at AGM-1, they will be diffracted by AGM-1. Due to $\theta_{\rm in} =\theta_{\rm 2s} >\theta_{\rm c}$, the diffraction in the transmission channel is dominant,[35] i.e., $\sin \theta_{\rm 1t} =\sin \theta_{\rm in} +n\gamma_{1}$. Due to $\theta_{\rm in} =45^{\circ }/-45^{\circ}$, then $n=-1/1$ is taken to achieve $\theta_{\rm 1s} =\theta_{\rm 1t} =-45^{\circ }/45^{\circ}$. As a result, the transmission wave that eventually passes through the whole bilayer system has a beam splitting effect, i.e., BS$_{-1}$ and BS$_{1}$, and the splitting beam obeys the following relationship: \begin{align} \sin \theta_{\rm s} =\sin \theta_{\rm in}+n({\gamma_{2}-\gamma_{1}}), \tag {1} \end{align} where $n=\pm 1$ and the range of the working incident angle is $\theta_{\rm in} \in (-\theta_{\rm c},\, \theta_{\rm c})$. Because the entire diffraction process in these two AGMs only involves the PW, such a splitting effect always holds for an arbitrary air gap size of $d$. For an incident beam from the opposite direction, the diffraction rule in bilayer AGMs enables a different process and beam splitting effect. For normal incidence, it is first diffracted by AGM-1, governed by $\sin \theta =\sin \theta_{\rm in} +n\gamma_{1}$. Due to $\gamma_{1} >1$, diffraction occurs simultaneously in the reflection and transmission channels,[35] among which reflection mainly takes the $n=0$ order, leading to total reflection (i.e. $\sin \theta_{\rm 1r} =\sin \theta_{\rm in}$), while transmission takes the $n=\pm 1$ order, resulting in a strong EW bounded at the transmitted side of AGM-1, with its parallel component wave vector of $k_{x}^{{\rm eva}} =\pm \xi_{1}$. Such an EW yields a tunneling effect through AGM-2, and the transmission efficiency of the whole bilayer system largely depends on the air gap size $d.$ For a small gap (e.g. $d < \lambda$ /2), such a tunneling effect is strong, but interestingly, the tunneling process through AGM-2 is still modulated by the phase gradient of $\xi_{2}$, producing anomalous diffraction of the EW. Due to $\xi_{2} < k_{0}$, tunneling of the EW through AGM-1 can occur via the diffraction channel of order $n=\pm 1$, which means the incident EW is efficiently converted into a PW. Eventually, the transmission wave is given by \begin{align} \sin \theta_{\rm s} =\sin \theta_{\rm in}+n({\gamma_{1} -\gamma_{2} \sigma}), \tag {2} \end{align} where $n=\pm 1$ and a coefficient of $\sigma \in[{0,\, 1}]$ depending on the interlayer gap is introduced to depict the contribution of the tunneling effect. When $d=0$, $\sigma =1$; when $d\to \infty$, $\sigma \to 0$. Similarly, the beam splitting effect (i.e., BS$_{-1}$ and BS$_{1}$) occurs, but it is dependent on $d$. Although both Eqs. (1) and (2) are similar in their formulation, i.e., only $\gamma_{1}$ and $\gamma_{2}$ are exchanged particularly for $\sigma \to 1$, they are ruled by different physical processes. The evolution of wave diffraction in momentum space for positive and negative incidences is shown by the equifrequency contours in Figs. 1(c) and 1(d), respectively. In the plot, the dashed circle, given by $k_{x}^{2} +k_{z}^{2} =k_{0}^{2}$, indicates the boundary between the PW region and the EW region. For positive incidence, as shown in Fig. 1(c), as it is modulated by $\xi_{2}$ and $\xi_{1}$ in succession, both the middle state and the final state are always located in the circle, implying that the entire diffraction process only involves the PW, leading to robust generation of beam splitting for arbitrary $d$. For negative incidence, as shown in Fig. 1(d), the normally incident waves are diffracted first by $\xi_{1} >k_{0}$ and then by $\xi_{2}$. The large vector of $\xi_{1} >k_{0}$ results in the middle state (see the yellow circles) of diffraction order $n=\pm 1$ located in the EW area, which is related to the tunneling effect depending on the interlayer gap. Thus, the precise combination of different phase gradients in the bilayer metasurface can lead to broken symmetry of the anomalous diffraction process in momentum space for two opposite incidences. However, we note that the asymmetric transmission associated with beam splitting demonstrated here is different from the asymmetric transmission arising from non-reciprocity, because our studied bilayer system is reciprocal without breaking the time reversal symmetry.
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Fig. 2. (a) Bilayer gradient metasurfaces designed by filling the subwavelength slit array with different media. The supercell in AGM-1 contains two unit cells. AGM-2 is designed by simply placing each unit cell in AGM-1 twice over a period. (b) Calculated transmission versus the interlayer gap side for positive incidence. (c) The same as (b) but for negative incidence. Simulated field patterns for the incident beam from opposite directions striking the designed bilayer metasurface system when the interlayer size is $d=0$ (d) and $d=0.5\lambda$ (e). In the plots, the BS$_{-1}$ and BS$_{1}$ indicate the efficiency of each splitting beam and BS$_{-1}+$BS$_{1}$ indicates the total transmission.
The proposed concept can be realized by filling the subwavelength slit array with different media to construct bilayer AGMs, as shown in Fig. 2(a). To obtain the required phase shift along the interface that covers a complete 2$\pi$, the refractive index of the $j$th unit cell is $n_{j} =1+(j-1)\lambda /mh$, where $m$ is the number of unit cells in the supercell. In this work, the operating wavelength is $\lambda =10$ cm (i.e., 3430 Hz), and we consider the simplest case of $m=2$ for AGM-1, i.e., the supercell for AGM-1 only contains two unit cells. The period is $p={\sqrt 2 \lambda}/2$, the width of each unit cell is $a=0.5p$, the slit width is $w = 0.47p$, and the thickness is $h = 0.5\lambda$. In this design, the corresponding phase gradient is $\xi_{1} =\sqrt 2 k_{0}$. Based on the binary unit cells in AGM-1, we place them twice in a period to construct AGM-2, i.e., the supercell for AGM-2 also contains two unit cells, and the corresponding phase gradient is $\xi_{2} ={\sqrt 2 k_{0}}/2$. COMSOL Multiphysics is used here to verify the gap-dependent beam splitting effect. For positive incidence, Fig. 2(b) shows the numerically calculated transmission efficiency of two splitting beams (i.e., BS$_{-1}$ and BS$_{1}$) for different air gaps, both of which are around 50%, i.e., the efficiency of BS$_{-1}$ oscillates from 26.3% to 49.5% and the efficiency of BS$_{1}$ oscillates from 48.2% to 68.6%. Their total transmission is nearly unity and almost independent of $d$, and a small oscillation (i.e., from 93.5% to 99.8%) is caused by the interference of weak reflected waves in the gap region. Note that the unequal efficiency feature in both BS$_{-1}$ and BS$_{1}$ stems from the fact that the whole bilayer system in this design is symmetrically broken in the $x$ direction. It has been numerically proved (not shown here) that such an unequal feature will disappear in a mirror-symmetric design by shifting AGM-1, exhibiting almost the same splitting efficiency for both BS$_{1}$ and BS$_{-1}$. For negative incidence, as shown in Fig. 2(c), transmission efficiency gradually decreases as $d$ increases and the tunneling channel is closed. At $d=0.5\lambda$ the transmission efficiency is as low as 0.4%. Therefore, both symmetric beam splitting and asymmetric beam splitting can be realized in bilayer AGMs by altering the gap size. Furthermore, the simulated field patterns illustrate the performance of the symmetric and asymmetric beam splitting effect in a bilayer metasurface system. When $d=0$ [see Fig. 2(d)], asymmetric diffraction orders disappear and beam splitting occurs for both negative and positive incidences. When $d=0.5\lambda$, as shown in Fig. 2(e), for negative incidence a strong EW is bounded at the transmission side and cannot tunnel through the air gap, resulting in total reflection of the incident beam. For positive incidence, a perfect beam splitting effect is seen, as the incident beam is diffracted by AGM-2 and AGM-1 in the form of a PW in both processes. Figure 3 displays a phase diagram of wave propagation in the designed bilayer AGMs with different phase gradients. To illustrate transmission feature for opposite incidences, an asymmetric parameter is defined as $\eta ={({T_{\rm negative} -T_{\rm positive}})}/{({T_{\rm negative}+T_{\rm positive}})}$, where $T_{\rm positive}$ and $T_{\rm negative}$ represent the transmission for positive and negative incidences, respectively. The phase diagram can be divided into four regions, with region I being a complex diffraction area due to two small phase gradients within $k_{0}$ and region III a tunneling area because of two phase gradients larger than $k_{0}$. The asymmetric transmission behaviors are located in regions II and IV, due to different wave vector evolution processes for both side incidences. In these regions, for one incidence the wave diffraction always follows a PW route, which is independent of the interlayer gap; while for the opposite incidence, the EW stemming high-ordered diffraction takes part in the diffraction process, resulting in a $d$-dependent diffraction transmission and associated splitting effect.
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Fig. 3. Phase diagram of wave propagation in the designed bilayer AGMs with different phase gradients for $d=0.5\lambda$, divided into four regions labeled I, II, III, and IV. The asterisk indicates the specific case of $\xi_{1} =\sqrt 2 k_{0}$ and $\xi_{2} ={\sqrt 2 k_{0}}/2$ discussed in this work.
The revealed asymmetric beam splitting effects are verified by acoustic experiments with the setup shown by Fig. 4(a). AGM-1 and AGM-2 are designed as an empty waveguide and a coiling-up structure to obtain two phase elements, 0 and $\pi$ respectively. The working frequency is 3430 Hz, and the wall thickness, height, and width of the coiling-up structure are $t=0.2$ cm, $h=0.5\lambda$, and $a={\sqrt 2 } / 4\lambda$. More details about the cell design can be found in Note 2 of the Supplementary Information. The designed AGMs with coiling-up structures have desired performances to individually realize total reflection (AGM-1) and beam splitting (AGM-2) effects, which are verified by the measurement results shown in Note 3 of the Supplementary Information. The designed bilayer AGMs are fabricated by three-dimensional printing with epoxy resin [see Fig. 4(b)]. In experiments, the incident beam is generated by a speaker array, and the acoustic field is scanned using a moving pressure probe (PCB 378C01) with a step of 1.0 cm. The experimental measurement is performed in an anechoic chamber to avoid undesired environmental noise. For $d=0$, Figs. 4(c) and 4(d) show the simulated and measured results for positive and negative incidences, and the scanning areas in experiments are locked at $z/p$ from 7 to 10 and $-7$ to $-10$. The simulated and measured results demonstrate consistently the symmetric beam splitting effect, which is further confirmed by the far-field results [see Fig. 4(e)]. For $d=0.5\lambda$, the pressure field distributions of the simulated and measured results are shown in Figs. 4(f) and 4(g), and they agree with each other. An extremely asymmetric beam splitting effect is observed and further confirmed by the corresponding far-field information in Fig. 4(h). In addition, the extremely asymmetric beam splitting effect has a certain bandwidth response from 3.24 to 3.54 kHz (see Note 4 of the Supplementary Information for details). The effect is quite robust as it is insensitive to some perturbations, such as horizontal slip of the bilayer system (see Note 5 of the Supplementary Information for details).
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Fig. 4. Experimentally measured results. (a) Experimental setup with fabricated samples of AGM-1 and AGM-2. (b) Schematic plot of the coiling-up structure and the fabricated supercells for two AGMs. [(c), (d)] Experimental results of field distributions for positive and negative incidences when $d=0$, respectively. (e) The corresponding far-field plot of the beam splitting effect, where the measured position is 10$\lambda$ away from the center of the exit end. [(f), (g), (h)] The same as (c), (d), and (e), respectively, but for $d=0.5\lambda$.
In conclusion, we have theoretically and experimentally explored the diffraction behaviors in bilayer gradient metasurfaces. It is found that the precise combination of different phase gradients in bilayer AGMs provides a route to manipulate wave propagation, in particular producing extremely asymmetric effects that are tuned by the interlayer gap. This approach is universal for all wave systems, such as electromagnetic or acoustic waves, and the fundamental physics is caused by anomalous diffraction rules enabled by the phase gradient in AGMs and the associated symmetry-broken evolution of wave diffraction in the bilayer system. Our work provides a new theory for the control of wave propagation and a promising platform for designing compact meta-devices that can have many applications in perfect beam splitters, one-way propagation and asymmetric diffraction elements. Acknowledgments. This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1404400), the National Natural Science Foundation of China (Grant Nos. 11974010, 12274225, and 12274313), and the Fundamental Research Funds for the Central Universities (Grant No. NE2022007). References Metamaterial-based real-time communication with high information density by multipath twisting of acoustic waveAcoustic reporter genes for noninvasive imaging of microorganisms in mammalian hostsElectromagnetic metasurfaces: physics and applicationsFlat optics with designer metasurfacesPlanar gradient metamaterialsAcoustic metasurfacesUnderwater acoustic metamaterialsDesign of Acoustic/Elastic Phase Gradient Metasurfaces: Principles, Functional Elements, Tunability, and CodingTheory of metascreen-based acoustic passive phased arrayAn underwater planar lens for broadband acoustic concentratorCreation of acoustic vortex knotsAcoustic metasurface by layered concentric structuresAmplitude-modulated binary acoustic metasurface for perfect anomalous refractionLight Propagation with Phase Discontinuities: Generalized Laws of Reflection and RefractionAsymmetric acoustic metagrating enabled by parity-time symmetryExtremely Asymmetrical Acoustic Metasurface Mirror at the Exceptional PointAsymmetric transmission of acoustic waves in a waveguide via gradient index metamaterialsMultifunctional reflection in acoustic metagratings with simplified designAcoustic planar surface retroreflectorFine manipulation of sound via lossy metamaterials with independent and arbitrary reflection amplitude and phaseAsymmetric Generation of Acoustic Vortex Using Dual-Layer MetasurfacesTunable correlated states and spin-polarized phases in twisted bilayer–bilayer grapheneGraphene bilayers with a twistTunable strongly coupled superconductivity in magic-angle twisted trilayer grapheneMoiré photonics and optoelectronicsEmpowered Layer Effects and Prominent Properties in Few‐Layer MetasurfacesPlanar metasurface retroreflectorMiniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrationsMagic-angle lasers in nanostructured moiré superlatticeMoiré Quasibound States in the ContinuumObservation of full-parameter Jones matrix in bilayer metasurfaceControllable asymmetric transmission via gap-tunable acoustic metasurfaceFull Complex‐Amplitude Engineering by Orientation‐Assisted Bilayer MetasurfacesSound vortex diffraction via topological charge in phase gradient metagratingsReversal of transmission and reflection based on acoustic metagratings with integer parity designParity-protected anomalous diffraction in optical phase gradient metasurfacesScattering of Light with Orbital Angular Momentum from a Metallic Meta-Cylinder with Engineered Topological ChargeMulti-functional high-efficiency light beam splitter based on metagratingAnomalous wavefront control of third-harmonic generation via graphene-based nonlinear metasurfaces in the terahertz regimeOptical beam splitting and asymmetric transmission in bi-layer metagratingsSwitchable bifunctional metasurfaces: nearly perfect retroreflection and absorption at the terahertz regimeMechanism Behind Angularly Asymmetric Diffraction in Phase-Gradient MetasurfacesSingle-sided acoustic beam splitting based on parity-time symmetrySound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator
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