Experimental realization of a pillared metasurface for flexural wave focusing

A metasurface consists of an array of subwavelength thickness units exhibiting inhomogeneous or modulated phase response that can arbitrarily manipulate refracted or reflected wavefronts. This field was rapidly developed in the domains of optics,^1–5 microwaves,^6–8 and acoustics.^9–18 Recently, it has been extended to elastic waves,^19–21 especially for Lamb waves in plates with diverse potential applications at different scales. The key feature to design an elastic metasurface is to realize a 2π phase span response by the constituting units while keeping a relatively high level of wave amplitude. As known by the classical wave motions, the propagating speeds of the Lamb waves depend on the elastic properties of the host medium and the plate’s thickness (for the flexural mode).^22,23 A large number of works focus on tailoring the plates to design composite in-plane geometries in order to fulfill the 2π phase shift span requirement.^24–34 Although in-plane geometries provide a wide platform to manipulate wave responses with different mechanisms, they will significantly reduce the stiffness of plates, which is very important as concerns the mechanical property of the structure.

An alternative solution is to design added out-of-plane units on the plate, for instance, by adding a thin patch with a size of the order of the wavelength for reflected wave response³⁵ or a set of slender pillars with a subwavelength total length for transmitted wave response.^36,37 Given that a pillar is able to exhibit rich resonant properties,^38–43 one can take advantage of the phase shift around the resonance to build the units of the array in the metasurface. For example, a set of graded pillars can produce different phase shifts in their transmission coefficient. However, in general, the variation of the phase around a resonance spans a region of π instead of 2π. In Ref. 38, some of the authors showed that the phase shift can cover a range of 2π if we superpose at the same frequency the fundamental compressional mode of the pillar with one of its bending modes. Based on this result, Ref. 44 proposed theoretically the design of a metasurface, constituted by a set of pillars with graded heights, to achieve subwavelength focusing and imaging of flexural Lamb waves. The purpose of this paper is first experimental realization of such a design.

In this work, we experimentally realize a pillared metasurface consisting of a transverse line of graded height resonant pillars with identical subwavelength diameter that is able to design various wavefront functions. Each pillar of the metasurface acts as a secondary excitation source with different transmitted phases through the interference between the incident wave and the re-emitted wave. It is found that the transmitted phase response covers a 2π shift, while the amplitude remains at a sufficiently high level. In Sec. II, we present the design process of the metasurface unit for phase and amplitude manipulation, which is further adopted to design a plane wave focusing effect. In Sec. III, we fabricate the pillared metasurface and experimentally demonstrate the focusing effect in good comparison with the numerically predicted results. Finally, we provide a summary of this work in Sec. IV.

We first consider the transmission through a line of identical pillars arranged on the plate in the frequency domain, as shown in Fig. 1(a). Periodic conditions are applied on the two sides of the unit cell along the x direction to simulate the infinite line of uniform pillars. Perfect matching layers (PMLs) are also applied to the two ends of the unit cell along the y direction to avoid wave reflections from the edges. An incident plane flexural wave (the dominant component is the out-of-plane displacement) source is excited by an out-of-plane face force, and the transmitted wave is detected at a point on the plate’s surface after the pillar. By analyzing the wavenumber of the transmitted wave with the Fourier transformation method, it is found that the transmitted wave is still dominated by the flexural wave (see the  Appendix for details). The transmission coefficient is defined as

where w₀ and w_ref are the out-of-plane displacement of the detection point with and without the pillar, respectively. Generally, two types of resonant modes, namely, bending and compressional modes, can be excited by the incident plane wave; then, the pillars play as a secondary source and emit the out-of-phase scattering wave. The transmitted wave results from the interference between the incident and re-emitted waves.³⁸ For a specific case, the bending and compressional modes can occur at the same frequency to enhance the re-emitting effect, making the amplitude of the scattering wave about 1.55 times that of the incident wave. After the destructive interference of the scattering and incident waves, the transmitted wave is dominated by the scattering wave, with an out-of-phase transmission coefficient of amplitude 0.55.

We set the pillar’s diameter d = 3.6 mm, pillar’s height h = 6.6 mm, plate thickness e = 6.0 mm, and unit width w = 4.8 mm. The pillar and plate are made of 3D-printing material polylactic acid (PLA) with Young’s modulus E as 3.5 GPa, Poisson’s ratio as 0.36, and density as 1240 kg/m³ (provided from the manufacturer). The above parameters are chosen appropriately such that the superposition effect of the bending mode and compressional mode can be achieved at 59.2 kHz corresponding to the simulated wavelength λ = 13.3 mm. The diameter of the pillar is only 0.27λ, being deep subwavelength. It should be mentioned that the size of the pillars can be scaled up and down to drive the working frequency to a lower or higher range, respectively. For instance, by changing the lengths, which are expressed in mm to micrometer, the resonance frequency will fall at 59.2 MHz instead of kHz.

By sweeping the pillar’s height at 59.2 kHz while maintaining other parameters as mentioned above, Fig. 1(b) shows the transmitted amplitude and phase responses of the displacement u_z. It is found that when the pillar’s height is swept from 4 to 11.4 mm, the transmitted phase can fully cover the range from −π to π, as shown with the black curve, and the transmitted amplitude keeps a relatively high level, as shown with the color map. Then, it is possible to manipulate the transmitted wavefront based on the generalized Snell’s law for various advanced functions, such as beam deflection, focusing, source illusion, and wave suppression among others. Focusing is one of the widest effects studied in wave physics, which is often used in non-destructive testing or signal reception as for sensing application.⁴⁵ Therefore, we select it to demonstrate the functionality of the proposed pillared metasurface. To design a plane wave focusing effect, it requires a gradient phase response along the transverse x axis in the metasurface. The continuous phase response profile $φx$ can be given as

where F and x are the focal length and x-coordinate position along the metasurface, respectively, and λ is the working wavelength. Since the real metasurface is made up of individual pillars and cannot represent a continuous phase shift, the continuous phase profile needs to be discretized into the required phase points according to pillar’s positions along the metasurface. In Fig. 2(a), we calculate the continuous phase profile (blue curve) by Eq. (1) for the plane wave focusing with a focal length F = λ and discretize it into 31 phase points (orange dots) for a pillared metasurface composed of 31 pillars. The number of pillars is limited by the maximum size capacity of 3D printer. The phase of the central unit is set as −π, corresponding to the strongest resonant status of the pillar. However, it should be noted that the choice of the central phase is not the key factor for the focusing effect, and other phases can be set as well. Once the discrete phases of metasurface units are obtained, the corresponding pillar’s heights can be easily retrieved from the black curve in Fig. 1(b). The final designed metasurface composed of 31 gradient pillars is shown in Fig. 2(b).

A focusing metasurface containing 31 pillars with the focal length as F = λ is consequently designed as shown in Fig. 3(a). To launch the incident wave, we followed a process close to the experiment. Namely, three piezoelectric ceramic transducer PZT patches with the dimension of 2.58 mm thickness, 50 mm length, and 10 mm width, shown as the dark green area in Fig. 3(a), are arranged on the upper surface of the plate to excite flexural waves by applying an electric field of amplitude of 10 V between the electrodes. The PZT patch made of lead zirconate titanate has material properties with piezoelectric constants e₃₁ = e₃₂ = −5.2 C m⁻² and dielectric permittivity ε₃₃ = 663.2 ε₀, with ε₀ being the vacuum permittivity. We fabricate the sample with the PLA material for experiment by using 3D printing technology, and its picture is presented in Fig. 3(b). The three identical PZT patches are glued to the upper surface of the plate for exciting flexural waves, as shown with the yellow stripe in Fig. 3(b); the patches are driven with a signal at the frequency of 59.2 kHz and an electric amplitude of 10 V. The out-of-plane displacement is measured with a Doppler laser vibrometer MSA-500 by Polytec supplied by analog displacement decoder model DD-300 with a sensitivity of 50 nm V⁻¹. The interested area of the focusing field is chosen as a square purple zone, L₁ = L₂ = 30 mm, and further divided in rectangular subsections, which is as large as the field of view of the objective with size 3.5 by 4.5 mm. The displacement is scanned along the x–y plane with a step of 335 µm in each subsection. Absorbing materials are applied to the surrounding of the sample during the experimental measurement in order to reduce the boundary reflecting effect as much as possible.

We present the simulated (E = 3.5 GPa) and experimental results of the focusing effect in Figs. 4(a) and 4(c). A clear focusing spot is observed with both approaches, supporting the focusing functionality of the pillared metasurface. The simulated focal length is found to be 19.9 mm corresponding to 1.50λ since the wavelength of the flexural Lamb wave is λ = 13.3 mm as mentioned in Sec. II. In the experiment, the wavelength of the Lamb wave is found to be 15.2 mm and the measured focal length is 22.6 mm corresponding to1.48λ. Therefore, the two focal lengths are very close to each other when expressed in units of the wavelength, while they show 15% deviation in their absolute values. We infer this to a deviation of the Young’s modulus E′ of the actual 3D-printed sample as compared to the value E provided by the manufacturer. To make a more quantitative comparison with experiment, we simulated the behavior of the fabricated sample at the same frequency of 59.2 kHz with a Young’s modulus of E′ = 5 GPa that reproduces the experimental wavelength. The result is shown in Fig. 4(b) where one can note a focal length of 22.0 mm (corresponding to 1.45λ) in good agreement with experiment. Additionally, the new simulation reproduces well the two experimental hot spots in the close vicinity of the metasurface at the level of the two pillars on both sides of the central pillar. However, caution should be taken that with the new value E′ of the Young’s modulus, the pillars in the fabricated sample no longer satisfy the exact phase shift conditions chosen from Fig. 2 during the initial design process. Another difference between the experimental results and the simulations is about the asymmetry of the measured pattern. We think that this can be attributed to some fabrication and experimental imperfections that will be mentioned below.

For a more quantitative comparison, we compare the size of the simulated and experimental focal spots in terms of their corresponding wavelengths, and Figs. 4(d) and 4(e) display the intensity profile along the x and y directions crossing the focal spot, respectively. For the simulated (E = 3.5 GPa) case, the full width at half maximum (FWHM) obtained from the intensity field along x and y axes is 0.46λ and 0.99λ, respectively, as shown with the brown curves. The FWHM along the x axis in simulation is subwavelength, originating from the interference and diffraction of the gradient secondary sources.^11,44,46 For the simulated (E′ = 5 GPa) case, the FWHMs along x and y axes become 0.90λ and 0.91λ, respectively, as shown with the green curves. One can note that we lose the subwavelength FWHM along x, which can be explained by the deviation of the actual pillars from an accurate design based on the correct value E′ of the Young’s modulus. For the experimental case, the FWHMs along x and y axes are 0.78λ and 0.39λ, respectively, as shown with the blue curves in Figs. 4(d) and 4(e). The FWHM along the x axis is also above λ/2 and qualitatively close to the second simulation. As concerns the experimental FWHM along the y axis, it has a complicated shape due to the asymmetry of the whole transmission pattern. This asymmetry may result from different imperfections in the sample or possibly in the experimental excitation of the incident wave, in particular, the lateral boundary conditions and the induced reflection in the experimental configuration, the fabrication imperfections and the gluing, the bandwidth of the transducers, and the proximity of the transducers to the sample, which can generate an imperfect plane wave.

From the above discussion, one can conclude that the focusing effect is observed in all three above-mentioned approaches and the relative focal lengths in terms of corresponding wavelength are very close to each other. This indicates a robust focusing property of the metasurface despite the actual differences between experiment and simulation. It is further expected that if the ratio between the focal length and the metasurface length decreases, one can achieve a deep sub-diffraction focusing effect with the transverse FWHM much smaller than half a wavelength. Moreover, in a recent theoretical analysis,⁴⁴ we have shown that the focusing effect is robustly conserved over a frequency range of about 6% on each side of the working frequency.

In summary, we numerically and experimentally demonstrated a pillared metasurface, consisting of a line of pillars with gradient heights, for focusing an incident plane flexural wave into a spot. We took advantage of the superposition of the bending and compressional modes of the pillar on a homogeneous plate that is able to enhance the out-of-plane scattering wave with amplitude larger than that of the incident wave. After the destructive interference between the scattering and incident waves, the transmitted wave is out of phase. The phase response of the pillared units spans a 2π shift range, and the amplitude response remains at a relatively high level. We designed and fabricated a focusing metasurface and compared the measured focusing spots quantitatively with simulations. The FWHM along the x axis for the simulated spot is subwavelength due to the interference and diffraction of the re-emitted waves in the near field. The experimental and simulated relative focusing lengths have a good agreement, showing the strong robustness of the focusing metasurface. This metasurface can also be extended to surface waves, such as Rayleigh waves, propagating on the surface of a substrate.⁴⁷ The simple, compact, and robust design of the proposed pillared metasurfaces without tailoring the plate open a new avenue for advanced elastic wave functions in great potential applications for MEMS, civil engineering, aerospace engineering, and marine engineering.

W.W. and J.I. contributed equally to this work.

This work was supported by the National Natural Science Foundation of China (Grant No. 11902223), the Shanghai Pujiang Program (Grant No. 19PJ1410100), the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the Fundamental Research Funds for the Central Universities, and the High-Level Foreign Expert Program. This work was also partially supported by the French EIPHI Graduate School (Contract No. “ANR-17-EURE-0002”).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

In Fig. 5(a), the out-of-plane displacements in the blue rectangular area are selected to perform the Fast Fourier Transformation (FFT) to analyze the wavenumber of transmitted waves. First, both flexural/antisymmetric (black) and symmetric (blue) Lamb mode incident waves are independently excited in the bare plate without the pillars. The positions of the black and blue intensity peaks correspond to the wavenumber values of the antisymmetric and symmetric Lamb modes, respectively. Then, we excite the antisymmetric Lamb wave in the plate with the pillars, and the intensity is plotted as the dotted red line. It is found that there are two peaks occurring at the wavenumbers of the antisymmetric and symmetric Lamb modes. The intensity of the antisymmetric Lamb mode is over one magnitude bigger than that of the symmetric Lamb mode, supporting that the transmitted waves after the pillars are dominated by the flexural/antisymmetric Lamb wave.