Acoustic analogue of electromagnetically induced transparency and Autler–Townes splitting in pillared metasurfaces

Electromagnetically induced transparency (EIT) and Autler–Townes splitting (ATS) originating from multilevel atomic systems have similar transparency windows in transmission spectra which causes confusion when discriminating between them, despite the difference in their physical mechanisms. Indeed, Fano interference is involved in EIT but not in ATS. There has been significant interest in the classic analogues of EIT and ATS in recent years, such as in photonics, plasmonics, optomechanics; however, the acoustic analogue of ATS has been rarely studied. In this work, we propose to investigate these phenomena in a pillared metasurface consisting of two lines of pillars on top of a thin plate. The existence of Fabry–Pérot resonance and the intrinsic resonances of the two lines of pillars act as a three-level atomic system that gives rise to the acoustic analogue of EIT and ATS. Since the frequency of Fabry–Pérot resonance can be tuned by controlling the distance between the two lines, the underlying physics, whether Fano interference is involved or not, is quite clear in order to discriminate between them. The realizations of EIT and ATS are put forward to control elastic waves for potential applications such as sensing, imaging, filtering.

Among various pillared phononic crystals and metamaterials, a new type of structure that consists of a single or a line of pillars on top of a thin plate, also referred to as a pillared metasurface [39,40], has been recently proposed, where the intrinsic properties of the local resonances of a pillar are thoroughly investigated. In fact, a pillar can exhibit a monopolar compressional mode and a dipolar bending mode whose frequencies can be tuned by the geometric parameters of the pillar. The resonant modes can be coupled to the Lamb waves in the plate that result in a transmission dip due to the destructive interference between the incident wave and scattered wave by the resonant pillar. Fano resonance is found when two dissimilar pillars are introduced in each unit cell. The peak in the asymmetric transmission is always maintained whether the coupling of the two pillars is strong or weak [41].
Fano resonances were used to describe asymmetries in the autoionization spectra in atoms, named after the physicist Ugo Fano who first explained it in theory as the interference between individual resonances in the continuum [42]. They are also related to electromagnetically induced transparency (EIT) [43,44] that needs a discrete transition coupled to a continuum, giving rise to a transparency window in the absorption/transmission spectrum. Similar transparency windows can also be found in the Autler-Townes splitting (ATS) effect [45] but with essentially a different mechanism [46,47]: EIT is a result of Fano interference among several trans ition pathways, while ATS requires field-induced splitting of energy that does not require any Fano interference effects. EIT and ATS have been experimentally observed in quantum [48]/atom systems [49,50] as well as some classic systems such as in photonic crystals [51][52][53], metamaterials [54], plasmonics [55], optomechanics [56,57], and micro-resonators [46,[58][59][60][61], which are widely applied for controlling light at room temperature. Since the transparency profiles in the absorption/transmission spectrum of EIT and ATS closely resemble each other, they can be easily confused. Several methods [62][63][64][65] are proposed to discriminate between them based on observed absorption or transmission spectra, but not on the physical mechanism behind them (such as whether Fano resonance is involved or not). In another classic aspect, an acoustic system, Fano resonance and EIT are also investigated in several different structures [66][67][68][69][70]; however, the acoustic analogue of ATS has been rarely reported.
In this work, we propose pillared metasurfaces-two lines of pillars on top of a thin plate-and find that Fano interference originates from the Fabry-Pérot resonance between the two lines of pillars. By tuning the distance L between the two lines, the frequency of Fabry-Pérot resonance can be easily controlled as the wavelength at Fabry-Pérot resonance is two times the distance L. When the frequency of Fabry-Pérot resonance and the resonant frequency of two identical pillars are the same, Fabry-Pérot resonance becomes invisible as a bound state in the continuum. Detuning the resonant frequency of the two pillars, Fabry-Pérot resonance becomes stronger and the EIT effect is observed due to the destructive interference; on the other hand, when Fabry-Pérot resonance is far beyond the considered resonance frequency domain, the ATS effect is observed when the two pillars are strongly coupled. We realize EIT and ATS in these pillared metasurfaces by directly discriminating between the essential physical mechanisms to avoid any confusion, rather than employing some fitting-parameter methods from the obtained transmission spectra. Like the wide applications in optics or other systems, the realization of EIT and ATS in this work puts forward the control of elastic waves for potential applications such as sensing, imaging, filtering, among others, in micro or nano scale.

Pillared metasurfaces
We present in figure   Periodic conditions are applied to the two sides along the y axis and perfectly matched layers (grey) are applied to the two edges along the x axis. The period is a along the y axis and the distance between the two lines of pillars along the x axis is L. The fundamental anti-symmetric (A 0 ) Lamb mode wave is excited to propagate towards the x direction and is scattered by the two lines of pillars before exiting. 'M' is the middle point between the two lines on the surface. away from pillar 2) on the surface of the plate after the two lines of pillars is selected to detect u z , which is further used to calculate the transmission curves by normalizing to the u z on the same point without the pillars.

Acoustic analogue of EIT
EIT refers to the quantum interference of excitation with a three-level atomic system where a narrow transparency windows appears in an opaque region. In this pillared metasurface, the resonance of two pillars together with the Fabry-Pérot resonance act as a 'three-level atomic system' in acoustics. By detuning the height of the two pillars in order to separate the corre sponding dips in the transmission and by adjusting the length L such that the Fabry-Pérot resonance falls exactly between the two dips, one obtains the spectra shown in figure 2(a) for different values of the detuning. In this figure, the distance L is fixed to 230 µm, the radius of the two pillars is fixed to 25 µm and the heights of the pillars are symmetrically detuned around 245 µm. When h1 = h2 = 245 µm, the Fabry-Pérot resonance coincides with the compressional resonance frequencies of both pillars which means the round-trip phase shift of the wave between the two pillars adds up to 2π; then a Fabry-Pérot bound state in the continuum (BIC) is formed [71].
When the pillars are in resonant states, the displacement field in the plate can be regarded as the sum of the incident and scattered waves. As a consequence, the scattered wave is obtained as the subtraction between the full transmitted wave and the incident wave. In figures 2(b)-(d), we show the Nyquist plots of the scattered waves normalized to the incident wave at the far field for three examples of figure 2(a), namely curves in green (h1 = 240 µm, h2 = 250 µm), cyan (h1 = 242 µm, h2 = 248 µm) and blue (h1 = h2 = 245 µm). In this Nyquist plot, if a point locates at the +x, +y, −x or −y axes, it means that the phase of the scattered wave with respect to the incident wave at the corresponding frequency is 0, π/2, π/− π, −π/2, respectively. When the heights of the two pillars are different from 245 µm while keeping L = 230 µm, (in other words, the compressional resonance frequencies of the two pillars are different from that of the Fabry-Pérot resonance), the Fabry-Pérot resonance becomes visible in the transmission curve. The inner green and cyan ellipses in figures 2(b) and (c) show that Fabry-Pérot resonance becomes stronger with an increase in height difference as the green inner ellipse is larger, hence the peak p corresponds to a higher transmission (or weaker scattered field). The inner ellipse cuts the −x axis at p and d2, and is out of phase with respect to the incident wave, which gives rise to the peak p and dip d2 in the transmission in figure 2(a). The frequency of peak p corresponds to the Fabry-Pérot resonance and remains unchanged as the spacing distance L is fixed at 230 µm. For the green case, the position p of the inner ellipse is closer to the origin, so that the transmission peak is more towards 1, being an acoustic analogue of EIT. The transmission dips d1 and d2 follow the individual intrinsic resonant frequencies (circles in figure 2(a)), which is also supported from the vibrating states in the inserts. As a result, the acoustic analogue of EIT involves Fabry-Pérot resonance that contributes to the peak and pillar's intrinsic resonances that contribute to the two dips.

Fano resonance
In this section, we will discuss how EIT deviates to ATS in this pillared metasurface. Before considering the scattering by the two lines of pillars, we first study the case of a single line of pillars to give a basic view of the transmission properties. In this calculation, the radius of the pillars is r = 25 µm and the height is h = 245 µm. As shown by the red line in figure 3, a transmission dip appears in the frequency domain [6,8] MHz associated with the monopolar (compressional) resonance of this pillar that locates at f = 7.19 MHz (see the real part of u z ), well isolated from other intrinsic resonances in this frequency domain. The compressional mode of the pillar can be excited by the incident A 0 Lamb wave dominated by the displacement component u z and emits the same A 0 Lamb wave. When the incident and emitted A 0 Lamb waves are out of phase, destructive interference occurs, resulting in a transmission dip in the spectrum.
Considering two lines of such identical pillars separated by a distance L, the transmission properties versus distance L are plotted in figure 3 with L varying from 100 µm to 350 µm. One can observe that the transmission spectra are more complex than that of a single line case.  Fabry-Pérot interference occurs when the wavelength is two times the distance L. At L = 100 µm, Fabry-Pérot resonance is far above the frequency domain [6][7][8] MHz, so that it is the coupling of the two identical pillars that gives a splitting in the transmission with two dips. When L increases to 200 µm, Fabry-Pérot resonance red-shifts into the frequency domain [6][7][8] MHz and interacts with one of the resonance frequencies, hence producing an asymmetric Fano type resonance. Then at L = 230 µm, the Fabry-Pérot resonance coincides exactly with the zero of transmission of both pillars, and the transmission spectrum containing a single dip reveals the case of a BIC [71]. When L continues to increase, the Fabry-Pérot resonance moves to a much lower frequency range and the transmission spectrum displays a broad dip characteristic of a low coupling between the two pillars. The inserts in figure 3 also indicate that the two pillars couple each other when L < 230 µm, behave as out of phase resonant vibration when L = 230 µm, and have very weak coupling when L becomes much higher than 230 µm.
In order to better understand how the Fabry-Pérot resonance changes around L = 230 µm, we take a smaller step in L, as 225 µm, 228 µm, 229 µm, 230 µm, 232 µm, and 235 µm, and show the transmission spectra in a zoom-in frequency domain [7, 7.5] MHz in figure 4(a). When L = 225 µm, 232 µm and 235 µm, there is a slight peak and dip at the right side of the main dip; however, it is difficult to observe them for L = 228 µm, 229 µm, 230 µm. as the Fabry-Pérot resonance becomes a BIC. We further select a middle point M on the surface of the plate between the two lines and detect the amplitude of u z on this M point. Normalized to the u z of the same position M without pillars, the relative amplitude is plotted in figure 4(b), which clearly shows the evolution of an asymmetric profile. This profile first decreases to almost zero at L = 229 µm then increases again with an increase in L from 225 µm to 235 µm. In addition, the profile suffers a phase change when L transverses 229 µm manifested by the fact that the dip follows the peak or vice-versa.
In figure 5(a), Nyquist plots of the scattered wave normalized to the incident wave at point M are shown for L = 200 µm, 229 µm, and 250 µm. They exhibit a blue ellipse in the positive half y space, a dot at the origin, and a green ellipse in the negative half y space, respectively. The blue and green ellipses stand for scattered waves by Fabry-Pérot resonance and the pink dot at the origin shows that Fabry-Pérot  resonance is invisible at L = 229 µm. Then, in figure 5(b), we show the Nyquist plot of the scattered waves normalized to the incident wave at the far field point that is detected for the transmission calculation. Following the explanation of the Nyquist plot of the normalized scattered wave in section 3, for L = 229 µm, the scattered field at point b is out of phase with respect to the incident field and its amplitude is almost the same, slightly smaller; consequently, the transmitted field almost vanishes while it remains in phase with the incident field.
For L = 200 µm and 250 µm, the Nyquist plots of the scattered waves at the far field in figure 5(b) exhibit a protruding shape with respect to the pink ellipse, which is associated with visible Fabry-Pérot resonance. The blue and green protruding curves cut the −x-axis at point a (very close to x = −1) and point c (about x = −1.4), respectively. Point a corresponds to the 0 phase in figure 4(c) as is the case for point b. The x = −1.4 at point c means that the amplitude of the scattered wave is 1.4 times that of the incident wave while they are out of phase, so that a new transmission peak appears (as seen in figure 3) and the phase is π or −π. The real part of u z at points a, b, and c are shown in figure 5(d). The real part field at L = 229 µm behaves as a transition of those at L = 200 µm and 250 µm when destructive interference occurs.

What is and what is not an acoustic analogue of ATS?
ATS requires no Fano interference, which is still induced from the coupling of the resonators. Therefore, for two separated transmission dips, if there is no coupling between the two resonators (which means the dips are very close to those of the isolated resonators), it is not an ATS.
In section 4, we showed that for L = [100, 350] µm, the coupling effect between the two pillars occurs when L is smaller than 230 µm while very weak coupling occurs when L is larger than 230 µm. In figure 6, the Nyquist plot of a scattered wave at the far field point within [6,8] MHz for L = 100 µm and 350 µm are plotted as black and blue curves, respectively, where the two pillars are identical (no detuning frequency). Fabry-Pérot resonance is beyond the frequency domain [6,8] MHz for both cases. For L = 100 µm, two ellipses cross with an inner close-shape curve that cuts the −x-axis three times closer to x = −1, giving rise to two transmission dips as seen in figure 3, as an ATS. For L = 350 µm, the two ellipses merge as a heart shape without any interaction and cut the −x-axis only once at a point a little exceeding x = −1, so that there is only one transmission dip whose phase is π or −π.
For L = 100 µm, we keep the height of the first pillar fixed as h1 = 245 µm and make a sweep in the height of the second pillar from 220 µm to 270 µm. The frequencies of the two resonators are then detuned with respect to each other. The evolution of individual resonant frequency when the height changes from 220 µm to 270 µm is also plotted as a red circle-dotted line, whereas the resonant frequency for an individual pillar with a height at 245 µm is plotted as a blue circle-dotted line. From the transmission spectra in figure 7, the ATS exhibits an avoided crossing effect. When h1 = h2 = 245 µm, the two dips in the transmission deviate the most from the individual values, showing the strongest coupling between the two pillars. When h2 is larger or smaller than 245 µm, such deviation decreases, and the two dips move closer to the individual values, showing a weakening of the coupling effect. Therefore, when h1 = h2 = 245 µm or h2 is close to 245 µm, such as h2 = 240 µm/250 µm, it is ATS because there is no Fano interference effect involved and the coupling effect between the two pillars is strong. When h2 is far away from 245 µm, the transmission dips result from individual resonances without the coupling effect, so that it is not ATS.

Summary
In this work, we realized an acoustic analogue of EIT and ATS in pillared metasurfaces, especially since the acoustic analogue of ATS has rarely been reported in the literature [72]. We constructed a metasurface consisting of two lines of pillars separated by a distance L where a Fabry-Pérot resonance can appear between the two lines at a wavelength which is two times the distance L. At a specific case L = 230 µm, Fabry-Pérot resonance and the pillar's compressional mode have the same frequency, Fabry-Pérot resonance has zero width and becomes invisible as a BIC. At this L, an acoustic analogue of EIT was realized by making the heights of the two pillars slightly different. In that case, the Fabry-Pérot resonance becomes visible and stronger and appears as a transparency window between two dips in the transmission. In contrast, the Fabry-Pérot resonance shifts beyond the working band when L = 100 µm or L = 350 µm, so that no Fano interference occurs. ATS is induced by the strong coupling between two resonators. It is found that for L = 100 µm, only when the heights of the two pillars are the same or very close is the coupling effect strong and the two transmission dips are ATS; when the two heights are far away from each other, the transmission dips are very close to those resulting from individual resonances, so they are not ATS. For L = 350 µm, it is a similar case as no coupling effect is involved, and the two transmission dips are not ATS.
We realized and distinguished an acoustic analogue of EIT and ATS by the essential mechanism, whether Fano interference is involved. Moreover, we clarified what was ATS or not by discriminating whether a strong coupling effect occurs. The realization of EIT and ATS in acoustics can be applied to control elastic waves for potential applications such as sensing, imaging, and filtering.