Elliptic pillars based metasurface for elastic waves focusing in a plate

Metasurfaces are metamaterials of sub-wavelength thickness made of a large number of discrete scatterers that locally and collectively modify the scattering properties of an incident wave. Initially developed in the domain of electromagnetic waves,¹ the working principle of metasurfaces has since been extended to many other fields of wave physics, such as optics,^2–4 acoustics,^5–9 and wave propagation in elastic media.^10–14 The specific design of the metasurfaces allows for unusual and particularly appealing properties if compared to those of ordinary homogeneous materials, such as the ability to deviate,^15–18 focus,^19,20 guide, or filter^21,22 waves. In particular, sub-wavelength filtering in precise frequency ranges^23–25 or controlling the transmission direction of a plane wave (either by deflecting it according to a desired refraction angle²⁶ or by focusing it at different targeted points) have been shown as very promising applications in terms of waves propagating within a half-space or plate-like structures. To achieve such functionalities, a recent study²⁷ proposed the design of a sub-wavelength metasurface consisting of a row of cylindrical pillars for robust deep sub-wavelength focusing and imaging of elastic waves on a plate of thickness e = 145 μm. The sub-wavelength pillar diameter was fixed to d = 120 μm, while their heights were varied around a specific value of h = 240 μm. Such parameters allowed for the superposition of the second bending and the first compressional resonances, which was indeed required to allow the transmission phase shift at resonance to reach 2π (which cannot be achieved when the two resonances are separated). A gradual change in the height of the pillars around this reference value allowed us to tune the phase shift over a 2π range while maintaining a high level of transmission. However, fabricating such a metasurface with variable pillar heights at the microscale, while feasible, requires a technological step for each height variation, regardless of the lithography process employed. Consequently, such fabrication is time-consuming and costly and could potentially impact the reproducibility of the metasurface. Consequently, we propose a different approach based on the design of pillars of elliptical shape characterized by a constant height. In this case, the ellipticity of the scatterers is the key parameter for the superposition of a bending mode and a compressional mode, i.e., the two involved resonances to achieve the required phase shift. The design of the proposed metasurface consists of an array of resonant pillars of different ellipticity parameters, allowing it to realize a gradient phase shift fully covering a 2π range.

The paper is organized as follows: Sec. II presents the different resonance modes of an individual pillar on a substrate based on the analysis of the dispersion curves and the deformation maps. The focus is on the comparison between the properties of cylindrical pillars and those with an elliptical geometry. It is shown that the breaking of rotational invariance considerably modifies the nature of some modes and their resonant frequencies. In Sec. III, the transmission curves of a row of identical pillars to determine the resonance modes that are most sensitive to the ellipticity of the pillars are investigated with the aim of determining the conditions under which the two resonant modes can be superimposed (thus producing the 2π phase shift required for the architecture of the metasurface). In this transmission study, the effects of tilting the major axis of the pillar by an angle φ with respect to the incident wave are also presented. This tilt reveals additional resonance modes compared to the case of pillars oriented along the direction of propagation of the incident wave, but without affecting the major goal of the metasurface. Finally, in Sec. IV, we will apply these results to the design of different metasurface geometries, which will enable an incident Lamb wave of type A₀ to be focused at different points on the plate.

Figure 1(a) reports a schematic representation of the unit cell geometry adopted for the calculation of the dispersion diagrams [Figs. 1(b) and 1(c)]. The dispersion diagrams of the elliptic shaped pillars [Fig. 1(c)] are compared with those of an equivalent cylindrical pillar structure [Fig. 1(b)]. The dispersion diagrams are calculated via the commercial finite element software Comsol Multiphysics^©. The solid mechanics module is exploited under the following boundary conditions: Floquet–Bloch periodicity is applied along the x- and y-directions [refer to Fig. 1(a)], whereas free surface conditions are set on the top and bottom surfaces of the plate and the boundaries of the pillars. In all subsequent simulations, the following constant geometrical parameters are adopted: a thickness of the plate e = 145 µm, a height of the pillars h = 240 µm, a small axis value of the elliptic pillars e₁ = 20 µm, and a lattice parameter a = 100 µm. The ellipticity parameter e₂ will be used as the adjustable parameter. All these geometrical parameters are recalled in Fig. 1(a). Silicon is chosen as the material for the entire structure. The crystallographic axes [100], [010], and [001] of the face-centered cubic unit cell are aligned to the (x, y, z) directions in Fig. 1(a). The components of the elastic tensor are taken as C₁₁ = 16.57 × 10¹⁰ Nm⁻², C₁₂ = 6.39 × 10¹⁰ Nm⁻², C₄₄ = 7.962 × 10¹⁰ Nm⁻², and the density of silicon is ρ = 2331 kg m⁻³. Figures 1(b) and 1(c) show the dispersion curves in the 0–12 MHz frequency range for cylindrical-shaped pillars (i.e., e₂ = e₁ = 20 µm) and for elliptical-shape pillars with an ellipticity parameter e₂ = 50 µm, respectively. Solid red curves correspond to the three first Lamb modes of the plate in the absence of pillars. To label the different modes of the pillars in the rest of the paper, we number the resonant modes of the cylindrical pillars in ascending order of frequency in Fig. 1(b). Then, the modes of the elliptic pillars presenting similar mode shapes can be equally labeled in Fig. 1(c). In the cylindrical case [Fig. 1(b)], the first two sets of doubly degenerated low frequency modes 1–2 and 3–4 are the first and second order bending modes of the pillar along the x and y directions, occurring at f = 0.65 MHz and f = 3.99 MHz, respectively. At higher frequencies (in order of increasing frequency), the first torsion mode (5), the compressional mode (6), and the degenerate third-order bending mode (7) appear. The degeneracy of the bending modes is lifted for the elliptic pillar due to the loss of radial symmetry. Since e₁ is kept constant, two bending modes with displacements along the y-direction are noticeable at the same frequencies as for the cylindrical pillar, and the other two bending modes, with displacements along the x-direction are found around 0.65 and 3.99 MHz, respectively. Let us note that due to the increase in e₂, these other two modes with displacements along the x-direction are upshifted in frequency. As expected, this shift gets more pronounced as the difference between e₁ and e₂ increases. This phenomenon is also observed for higher order bending modes (second order bending modes 3 and 4, as well as third order bending mode 7). The torsional mode (5) is now present at lower frequencies (f = 4.17 MHz) because the moment of inertia of the pillar is increased due to a higher mass that is also distributed further from the center of inertia, hence lowering the frequency. The shape of the pillar also likely modifies the effective rotational stiffness, making the evolution of the mode with respect to e₂ more complex. The frequency of the compressional mode (6) is hardly modified since the deformation of the pillar is only in the out-of-plane direction, thus unaffected by the in-plane profile of the pillar.

In this section, we investigate the effect of the resonant modes of the pillars on the transmission (amplitude and phase) of the A₀ Lamb mode propagating across a line of identical elliptic pillars, of height h, arranged on a plate of thickness e, with pillar centers positioned every a = 100 µm in the direction perpendicular to the impinging wave front. For these calculations, we have considered the effect of the two following geometrical parameters, namely, the ellipticity parameter e₂ and the orientation of the pillars with respect to the direction of an incident elastic wave (x-direction) indicated by angle φ (see the schematic representation of the structure in Fig. 2). Two different values of φ are investigated as follows: φ = 0° and φ = 45°. The method used to compute the resonances is as follows: an oscillating force is applied, normal to the surface of the plate, along a line (AB) parallel to the crystallographic direction (010) of the silicon (referred to as the y-axis in the following), and the resulting normal displacements are recorded as a function of frequency on a line (CD) located on the other hand of the pillar row. An infinite line of identical pillars is obtained by applying Floquet–Bloch Periodic Boundary Conditions (PBCs.) on the boundaries perpendicular to the y-axis, and Perfect Matched Layers (PMLs.) located at each end of the plate are used to avoid any reflection from the boundaries.

We first compute u_z on the line (CD) against the frequency for a row of elliptical pillars whose major axis, designed by e₂, is oriented in the direction of propagation of the incident wave, i.e., φ = 0°, as a function of e₂ ranging between 20 and 70 µm by steps of 10 µm. Since the A₀ mode is mostly polarized in the z-direction, we define the transmission as the displacement component u_z at the line (CD) when the pillars are modeled and normalized by u_z at the line (CD) when the pillars are removed in the simulation. The transmission is shown in Fig. 3 for different values of e₂. The presence of dips in the transmission curves reflects the excitation of pillar modes by the incident wave. In the explored frequency range (0–12 MHz), the first four resonance modes are identifiable through more or less sharp dips in the transmission curve and can be unambiguously ascribed to the three first bending modes (2), (4), and (7), and to the first compressional mode (6) (see Fig. 1). We can note that mode (7) is only observed for e₂ = 20 µm, as this mode shifts rapidly to frequencies outside of the range of interest for larger values of e₂. Second, no signature corresponding to the torsional mode (5) is observable. Actually, due to the symmetry of the incident wave, this mode cannot be excited by the incident Lamb wave. Modes (2) and (6) are rather insensitive to variations of the parameter e₂ defining the elliptic shape of the pillar, whereas the second order bending mode (4) rapidly shifts toward high frequencies as e₂ increases. Finally, modes (4) and (6) end up overlapping at the frequency of 7.48 MHz when the value of e₂ reaches 50 µm.

When the pillars are oriented with an angle φ = 45° with respect to the incident wave, all the modes observed on the dispersion curves inFig. 1(c) have their signature on the transmission curve. Compared to the φ = 0° case, in addition to the separation of bending modes (1, 2) and (3, 4) in the e₁ and e₂ directions, we now observe the occurrence of extremely sharp dips associated with torsional modes labeled 5 and 8. These torsional modes are only visible on the transmission curves for e₂ ≥ 60 µm. Indeed, below e₂ = 60 µm, the first order torsional mode, is overlapped with the second order bending mode (3), which is much broader in the transmission spectrum. The second order torsional mode referred to as (8), lies above the frequency window studied as far as e₂ < 60 µm since this mode shifts to higher frequencies as e₂ decreases. To better highlight the signature of the torsional modes, we report in Fig. 5 a zoom of the red (e = 60 μm) and blue (e = 70 μm) boxes depicted in Fig. 4. Figure 5(a) reports the two fundamental torsional modes, while Fig. 5(b) shows the third-order bending mode (7) and second-order torsional mode (8).

It is worth noting that the first-order torsional mode (5) evolves in the same way as both the bending modes along the e₂ axis and the compressional one, namely, they shift toward higher frequencies as e₂ increases. In contrast, an opposite behavior is observed for the second order torsional mode (8). Furthermore, these two torsional modes show up on the transmission curve as extremely sharp peaks, reflecting very high-quality factors. The resonant frequencies of the various modes observed between 0 and 12 MHz are plotted as a function of e₂ in Fig. 6 for the two angles φ = 0° [Fig. 6(a)] and φ = 45° [Fig. 6(b)].

Varying the orientation of the pillars with respect to the incident wave has almost no influence on the frequencies of modes (2), (4), and (6). It reveals, in addition to the torsional modes (5 and 8), bending ones along the y direction (modes 1, 3, and 7), whose resonant frequencies are independent of e₂. This is understandable since these additional modes only depend on the e₁ value, which remains constant here. We also note that the dips of transmission are deeper for φ = 0°. As shown in Fig. 6(a), bending modes (2) and (4) present a high sensitivity to the value of e₂, with a particularly rapid evolution of the second order mode (4). Therefore, this mode crosses the evolution of the compressional mode (6) for e₂ = 50 µm. This salient feature makes it possible to use elliptic pillars to create a gradient metasurface for plane-wave focusing. Indeed, it is well known that, whereas a variation of the phase around a resonance generally spans a range of π, a phase shift covering 2π can be obtained when the fundamental compressional mode of the pillar is superposed with one of the bending modes. Since each resonant pillar of the metasurface acts as a secondary emitter source of waves with a different phase shift, the metasurface can be designed with a suitable arrangement of pillars such that all the transmitted waves interfere constructively and create a focal point on the plate based on their relative path difference.

One can note that in Fig. 6(b), a second crossing occurs between modes 7 and 8, at e₂ = 68 μm, which could potentially be used to make the focusing gradient metasurface. However, branch 8, associated with the second-order torsional mode, has a very narrow signature in the transmission [Fig. 5(a)] that makes the phase variation too sharp on a small frequency range and may not be suitable for the design of the metasurface. Therefore, in the next section, the focusing functionality is demonstrated for φ = 0° around e₂ = 50 μm, for which the bending and compressional modes overlap in a large frequency range.

At frequency f = 7.48 MHz, where the first compressional mode and the second bending mode overlap, the wavelength of the incident A₀ Lamb wave is equal to λ = 445 µm (see red line in Fig. 1, where k_x.a = 2.11 for the A₀ mode at f = 7.48 MHz). The phase of the central pillar (y = 0) is set to −π, corresponding to the strongest resonant status. We first present the focusing effect at a point M whose coordinates are (x₀, y₀) = (λ, 0). For this focal point, the discrete theoretical phases Φ(y) of the 30 other pillars of the line are calculated from Eq. (2) as a function of their positions y [Fig. 7(a)] and compared to the numerical calculation of the phase as a function of e₂ [Fig. 7(b)]. In Fig. 7(b), the solid black line depicts the phase response of a pillar with varying ellipticity ranging from 15 to 200 μm at a frequency of 7.48 MHz. For each position y of the pillar on the metasurface, the corresponding phase value is determined along the red dashed lines shown in Fig. 7(a). For instance, at the position y = 5 μm, the phase is zero [Fig. 7(a)], which, according to Fig. 7(b), corresponds to a value of e₂ = 172 μm. The evolution of the ellipticity parameter e₂ is represented as a function of the pillars’ positions in the top panel in Fig. 7(a), covering the phase variation of 2π for the set of 31 pillars.

Figure 8 shows the out-of-plane displacement (u_z) field distribution and the transmitted amplitude around the focal point (i.e., along the white vertical line), which shows a significant focusing effect with a sub-wavelength full width at half maximum (FWHM) of λ/3. It should be noted that this sub-wavelength focusing can be further improved by increasing the number of pillars and, hence, the lateral length of the metasurface. The right panel in Fig. 8 shows the intensity of the field along the red line, normalized to the intensity of the field crossing the plate without the metasurface. One can see that the intensity at the focal point is 3.7 times higher than that of the incident field, presenting a compelling opportunity for energy harvesting applications. Moreover, we have checked that incorporating a disorder in the value of e₂ with a fluctuation of +/− 1 μm resulted in a 15% reduction of the intensity at the focal point. This provides an acceptable range of tolerance in the microfabrication process.

As a further example of the metasurface's ability to manipulate transmitted waves, we have simulated the focusing effect of a row of 31 pillars. The procedure to design the metasurface is the same as that described above: once the position y of each pillar has been fixed and the position of the focusing spot (x₀, y₀) chosen, the value of the phase Φ(y) is calculated from Eq. (2), and the value of the ellipticity parameter e₂ is thus determined from the calculated variation of e₂ vs Φ. The results obtained for two focusing spots with coordinates (2λ, 0) and (λ, λ) are reported in Fig. 9.

In this paper, the study of different elliptical pillar resonance modes has enabled us to propose a type of elastic metasurface consisting of a line of resonant pillars with constant height and variable ellipticity. The pillars have been arranged on a homogeneous plate. Our design can provide full phase range manipulation and high transmission. The 2π variation of the phase was made possible by superimposing, for a given value of ellipticity, a bending mode and a compressional mode. A particularly striking advantage of this structure is that, due to its constant height, it is likely that it can be relatively easily fabricated at the micro- and even nano-scale using current nanofabrication techniques, i.e., metal-assisted chemical etching (MACE)²⁸ or deep reactive ion etching (DRIE),²⁹ which allow for pillars with very high aspect ratios. Moreover, this structure can be adapted to the wavelength of the incident wave by varying the height of the pillars and determining a new value of e₂ related to the two modes to be superimposed at the frequency of the incident wave. We have demonstrated the possibility of precisely selecting the focus points by adjusting the ellipticity of the pillars. This capability not only enhances our control over the focus point but also facilitates a significant concentration of energy, presenting valuable applications in integrated devices, for instance in electro-mechanic designs used in optomechanical applications. To illustrate this, we refer to our recent paper,³⁰ where an acoustic wave was injected into an optomechanical cavity inside a nanobeam by means of surface acoustic waves generated in front of the nanobeam by a set of interdigital transducers (IDTs). Incorporating a metasurface in front of the IDTs offers the opportunity to focus the energy directly inside a connected, freestanding nanobeam. We have also emphasized the possibility of activating torsional modes by changing the orientation of the pillars with respect to the direction of propagation of the incident wave. These modes can be superimposed with compression and bending modes, but the phase shift analysis for the associated e₂ values shows phase variations of π only, so they cannot be used to create a wave-focusing metasurface. Nevertheless, these torsional modes are characterized by very high-quality factors that can make them valuable in high-sensitivity sensors or as very selective filters.

L.C., R.A.-G., C.C., M.M., Y.P., and B.D.-R. thank the MAGNIFIC, funded by the European Union’s Horizon Europe research and innovation program under grant agreement No. 101091968. A.G. and G.L. thank the Chist-ERA Call 2021, funded by national agencies AEI (Spain), ANR (France), and AKA (Finland) under Grant Agreement No. CHIST-ERA-20-NOEMS-003.

The authors have no conflicts to disclose.

L. Carpentier: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Gueddida: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). G. Lévêque: Formal analysis (equal); Methodology (equal); Supervision (equal); Validation (equal). R. Alcorta-Galvan: Validation (equal); Visualization (equal); Writing – review & editing (equal). C. Croënne: Validation (equal); Visualization (equal); Writing – review & editing (equal). M. Miniaci: Validation (equal); Visualization (equal); Writing – review & editing (equal). B. Djafari-Rouhani: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Y. Pennec: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

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