Autler-Townes splitting and acoustically induced transparency based on Love waves interacting with pillared meta-surface

TL;DR

This paper investigates the interaction of Love waves with pillared meta-surfaces, demonstrating acoustic analogues of Autler-Townes splitting and Electromagnetically Induced Transparency through numerical simulations and model fitting.

Contribution

It introduces the first numerical demonstration of ATS and AIT phenomena in acoustic Love wave systems using pillared meta-surfaces, with model validation via AIC.

Findings

Demonstration of acoustic ATS and AIT phenomena.

Good agreement between models and numerical results.

Methodological approach for distinguishing ATS and AIT.

Abstract

Autler-Townes Splitting (ATS) and Electromagnetically Induced Transparency (EIT) are similar phenomena but distinct in nature. They have been widely discussed and distinguished by employing the Akaike information criterion (AIC). However, such work is lacking in acoustic system. In this work, the interaction of Love waves with two-line pillared meta-surface is numerically investigated by Finite Element Method. Acoustic analogue of ATS, Fabry-Perot resonance and cavity modes are first demonstrated in two lines of identical pillars by varying the distance between the pillar lines. By detuning the radius of one line of pillars, Fabry-Perot resonance along with two different pillar resonances give rise to the acoustic analogue of EIT (AIT) when the distance between the pillar lines is a multiple of half wavelength. ATS and AIT formula models are used to fit the transmission spectra, showing good agreements with numerical results. The quality of the fit models is quantitatively evaluated by resorting to the AIC. We show that theoretical and analytical discrimination between ATS and AIT are methodologically complementary. These results should have important consequences for potential acoustic applications such as wave control, designing of meta-materials and bio-sensors.

Equations

$$\text{NED}=\frac{{\iiint }_{\mathcal{D}}\frac{1}{2}{T}_{ij}{S}_{ij}^{\ast }(-z)dxdydz}{\lambda {\iiint }_{\mathcal{D}}\frac{1}{2}{T}_{ij}{S}_{ij}^{\ast }dxdydz},$$

$$\text{SH ratio}=\frac{{\iiint }_{\mathcal{D}}{u}_{SH}{u}_{SH}^{\ast }dxdydz}{{\iiint }_{\mathcal{D}}(u_x{u}_x^{\ast }+u_y{u}_y^{\ast }+u_z{u}_z^{\ast })dxdydz},$$

$$T_{ATS}=1-\frac{C_1({\Gamma }_1/2{)}^2}{(f-{\delta }_1{)}^2+({\Gamma }_1/2{)}^2}-\frac{C_2({\Gamma }_2/2{)}^2}{(f-{\delta }_2{)}^2+({\Gamma }_2/2{)}^2},$$

$$T_{AIT}=1-\frac{C_{+}({\Gamma }_{+}/2{)}^2}{(f-{\delta }_c-ϵ{)}^2+({\Gamma }_{+}/2{)}^2}+\frac{C_{-}({\Gamma }_{-}/2{)}^2}{(f-{\delta }_c{)}^2+({\Gamma }_{-}/2{)}^2},$$

$$\begin{array}{r}{T}_{ATS/AIT}=1-\frac{C_a({\Gamma }_a/2{)}^2}{(f-{\delta }_1{)}^2+({\Gamma }_a/2{)}^2}-\frac{C_b({\Gamma }_b/2{)}^2}{(f-{\delta }_2{)}^2+({\Gamma }_b/2{)}^2}\\ -\frac{(f-{\delta }_1)C_d({\Gamma }_d/2{)}^2}{(f-{\delta }_1{)}^2+({\Gamma }_d/2{)}^2}+\frac{(f-{\delta }_2)C_e({\Gamma }_e/2{)}^2}{(f-{\delta }_2{)}^2+({\Gamma }_e/2{)}^2},\end{array}$$

$$\sum (FE{M}_i-F_i{)}^2,i\in [1,N]$$

$${\overline{w}}_{ATS}=\frac{e^{-I_{ATS}/2n}}{e^{-I_{ATS}/2n}+e^{-I_{AIT}/2n}}$$

Full text

PACS numbers

68.60.Bs

I Introduction

Electromagnetically induced transparency (EIT) is a well-known physical effect in atomic systems that arise because of quantum destructive interferences between two excitation pathways to an upper atomic level Fleischhauer et al. (2005). Steep dispersion and low absorption take place in a sharp transparency window, which makes it very attractive for a plenty of potential applications in slowing light, enhancing optical nonlinearity and data storage Hau et al. (1999); Alotaibi and Sanders (2016); Heinze et al. (2013). Autler–Townes splitting Autler and Townes (1955) (ATS) which is the field-induced splitting of the optical response, is not associated with interference effects and has been described as an incoherent sum of two Lorentzians Abi-Salloum (2010). EIT and ATS may phenomenologically look similar, but they are different in nature, one being a quantum interference and the other a linear alternating current (AC) Stark effect. EIT and ATS were first observed in quantum/atom systemsBoller et al. (1991); Autler and Townes (1955). In recent years, classic analogues of EIT and ATS have attracted increasing interest in platforms such as photonics Peng et al. (2014); Liu et al. (2017); Wei and Jian (2017), optomechanics Dong et al. (2012); Safavi-Naeini et al. (2011); Weis et al. (2010), plasmonics Zhang et al. (2008); Liu et al. (2009); Han and Bozhevolnyi (2011) and metamaterials Liu et al. (2010a); Gu et al. (2012); Papasimakis et al. (2008). Many discussions have been devoted to their easily confused absorption or transmission spectra.Anisimov et al. (2011); Peng et al. (2014); Sun et al. (2014); Liu et al. (2016) Besides their differences in the physical mechanisms, the Akaike Information Criterion (AIC) has been proposed to quantitatively discern EIT from ATS,Anisimov et al. (2011) and the transition from ATS to EIT is thereby carried out.Giner et al. (2013) A crossover from EIT to ATS has been shown to exist in hot moleculesZhu et al. (2013), and in open ladder systemsTan and Huang (2014). In acoustic, the analogue of EIT, also referred to as AIT, has been investigated in different structuresRadeonychev et al. (2006); Liu et al. (2010b); Santillan and Bozhevolnyi (2011); Amin et al. (2015); Quotane et al. (2018), but the analogue of ATS and its comparison with AIT was only recently reportedJin et al. (2018). Additionally, the distinction and transition between acoustic analogue of ATS and AIT has not been quantitatively investigated before.

In the last two decades, phononic crystals have received increasing attention. Martínez-Sala et al. (1995); Liu et al. (2000); Khelif et al. (2003); Pennec et al. (2010); Zhu et al. (2011); Yang et al. (2014); Xu and Yuan (2018) They exhibit Bragg and/or hybridization band gaps to achieve control of elastic waves propagation, and apply to various aspects such as ratio frequency (RF) communications Khelif et al. (2003); Wu et al. (2009); Liang et al. (2010), acoustic isolators Meseguer et al. (1999); Olsson III and El-Kady (2009); Binci et al. (2016), sensors Talbi et al. (2006); Ke et al. (2011); Salman et al. (2015); Xu and Yuan (2018), thermoelectric materials Yu et al. (2010); Yang et al. (2014); Zen et al. (2014) and meta-materials Zhang and Liu (2004); Profunser et al. (2009); Zhu et al. (2011); Narayana and Sato (2012); Li et al. (2015); Guo et al. (2017). Phononic pillared meta-surface is a recently proposed structure stemming from the pillared phononic crystals. It consists of a single or a line of pillars on top of a slab, with which one can thoroughly investigate the pillar resonant properties. The pillar size, periodicity and the slab thickness are all sub-wavelength-scaledLi et al. (2011); Yu and Capasso (2014). Several studies have been devoted to the interaction of pillared meta-surface with Rayleigh waves Oudich et al. (2018) and Lamb waves Jin et al. (2017a, b), but no work has been done on Love waves, which are shear horizontal (SH) polarized surface acoustic waves (SAW). Love waves propagate in a thin guiding layer deposited on the surface of a semi-infinite substrate, and is therefore considered a compromise between Rayleigh waves and Lamb waves: Compared with Rayleigh waves, Love waves are well confined to the surface because of the thin guiding layer in which they propagate. In contrast to Lamb waves, Love waves exhibit device toughness since the guiding layer is deposited on the semi-infinite substrate. Love-wave-based pillared meta-surface is therefore suitable for use in wave control, sensors and design of metamaterials. Moreover, Love waves can excite pillar torsional mode Liu et al. (2019) which is impossible to be excited by Rayleigh waves and Lamb waves (S0 and A0) Jin et al. (2017a). In addition to the above qualities, their compatibility with liquid environment Lucklum and Li (2009); Ke et al. (2011) makes Love wave an ideal candidate for bio-sensor applications.

In this work, the interaction of Love waves with a two-line Ni pillar based meta-surface is investigated on a SiO2/ST quartz structure. Firstly, torsional mode in one line of cylindrical Ni pillars is demonstrated to be well excited by Love waves and gives rise to a sharp transmission dip due to a destructive interference. Secondly, acoustic analogue of ATS and cavity modes for Love waves are first demonstrated in two lines of identical pillars by varying the distance between the pillar lines. ATS appears when the distance is smaller than the half wavelength and a strong coupling is aroused between the pillar lines. Fabry-Perot resonance exists at the positions where the distance between the pillar lines is a multiple of half wavelength. The proximity of Fabry-Perot resonance with pillar torsional mode gives rise to the cavity modes with transmission enhancement at the edges of the dip induced by the pillars. Thirdly, the radius of one line of pillar is modified to detune the pillar resonant frequency. In the pillar coupling region, the coupling effect decreases with the increase of radius mismatch. When the distance between the pillar lines is a multiple of half wavelength, Fabry-Perot resonance along with the two different pillar resonances give rise to the AIT resonance. Then, ATS and AIT phenomena are first fitted respectively with corresponding formula models, showing good agreements. The Akaike Information Criterion (AIC) is then used to quantitatively evaluate the quality of the fit models, showing consistent results for the ATS and AIT cases, and the transition from ATS to AIT by increasing the distance between the pillars is illustrated. The AIT phenomenon is shown to be periodic due to the periodicity of Fabry-Perot resonance. The band structures and the transmission spectra are calculated with the finite element method(FEM, COMSOL Multiphysics*®*).

II Unit cell model

The acoustic meta-surface is constituted of two pillar lines deposited on a bi-layered matrix, of which the unit cell is shown in Fig 1. The matrix is a $2.4\mu\textrm{m}$-height silica guiding layer ($\rho=2200$kg/m3, $E=70\textrm{GPa},\nu=0.17$) that covers a $40\mu\textrm{m}$-height 90ST-cut quartz substrate (Euler angles=(0°, 47.25°, 90°), LH 1978 IEEE) which has been rotated 90 degrees around the $z$-axis from the ST-cut quartz, to be able to generate fast SH waves ($c_{SH,sub}$= 5000m/s) by the electric field. The shear wave velocity in the silica film is 3438m/s, less than that in the 90ST-cut quartz substrate, indicating the existence of Love waves. This structure is chosen since it has been experimentally validated for Love waves Liu et al. (2014a). The guiding layer thickness is sub-wavelength since only the fundamental Love mode exists to avoid physical complexity. The period along the $y$-axis is $a=2\mu\textrm{m}$. Two cylindrical Ni pillars on the silica film have a radius of $r_{1}=r_{2}=0.2a$ and a height of $h=0.6a$. The distance between the centers of the two pillars is denoted by $d$. The inclusions are chosen because of their strong contrast in density and elastic constants with regard to the matrix. However, similar results can also be obtained by using other materials such as gold or diamond. The length of the unit cell $L$ is $20a$ to insure a decoupling between the unit cells. Floquet periodic boundary conditions are applied along the $x$ and $y$ directions. The bottom of the substrate is assumed fixed. Love waves propagate along the $x$-axis (i.e. the $y$-axis of the ST-cut quartz), where Rayleigh waves cannot be generated Liu et al. (2014b) due to a zero electromechanical coupling factor in the substrate. The surfaces of the pillars coincide with the plane $z=0$. The wavelength normalized energy depth (NED)Liu et al. (2018) is calculated to select the surface modes:

[TABLE]

where $T_{ij}$ is the stress and $S_{ij}$ the strain. The star symbol (*) means the complex conjugate. $-z$ denotes that we integrate in the whole domain of the unit cell $\mathcal{D}$. $\lambda$ is the wavelength. This formula works well for a relatively large $k$ where the wave speed is less than the SH waves velocity of the substrate. For a relatively small $k$ (with wave speed greater than the SH waves velocity of the substrate), which corresponds to a large $\lambda$, we fix $\lambda$ to $2a$. Surface modes have a NED $<2$. The NED can filter out the bulk modes as well as the plate modes that appear in our finite-depth substrate which is supposed to be semi-infinite for Love waves.

SAW include SH type SAW and Rayleigh type SAW. The SH ratio is calculated to distinguish the displacement components:

[TABLE]

where $u_{x},u_{y}$ and $u_{z}$ are respectively the displacement along the $x,y,z$ directions. $u_{SH}$ is the SH displacement component that can be expressed as $u_{x}\cos\theta-u_{y}\sin\theta$, which is perpendicular to the wave vector $\bm{k}$. $\theta$ is the angle between $\bm{k}$ and the $y$-axis such that $\tan\theta=\frac{k_{x}}{k_{y}}$. In our case, $\bm{k}=\bm{k_{x}}$ and $u_{SH}=u_{y}$.

III resonant properties of single pillar line

Firstly, we study the resonant properties of the unit cell with only one pillar, which corresponds to a meta-surface with a single pillar line. The dispersion curves or band structure of SH modes (SH ratio$>$0.5) in the $x$ direction is shown in Fig 2(b). The X point is the irreducible Brillouin zone limit of the unit cell in the $x$ direction. The modes colors are determined by their NED values. The modes in red are well confined to the surface, and can therefore be excited by Love waves. Certain modes become pink as they are less confined to the surface. Two hybridization band gaps are observed respectively in the frequency range [177.3, 183.1] MHz and [501.8, 503.3] MHz, indicated in blue, which originates from the coupling of local resonant pillar modes and the SH SAW. Note that in the frequency range [100, 600] MHz, the wavelength ranges from 6.8 to 48.8$\mu m$. Therefore, our structure is indeed a meta-surface with sub-wavelength-scaled pillar size, periodicity and guiding layer thickness.

The transmission spectra is calculated by simulating the same dispersive SAW device introduced in our preceding works Liu et al. (2018); Yankin et al. (2014), which has been validated by experiment. The model consists of two parts of aluminum inter-digital transducers (IDTs) with the unit cell located between IDTs. This device model is periodic along the $y$ direction. Other lateral sides and the bottom are assumed fixed. The input IDT is given a $V_{0}=1V$ harmonic voltage signal. The output is measured by averaging the voltage difference between odd and even electrodes. Dissipation is neglected in the results shown in the manuscript. However, its effect is discussed in Appendix A supporting the validity of the presented features. The width of the electrodes of IDTs $L_{IDT}$ is updated for every frequency in the spectrum according to the relation $L_{IDT}=\frac{\lambda}{4}=\frac{v}{4f}$. $v$ is the velocity of Love waves for $H=2.4\mu m$, resulting from the basic dispersion relation of Love waves. That is, each frequency corresponds to a single wavelength. The frequency responses are then normalized by that of the matrix (without pillar), referred to as relative transmissions $\Delta S_{21}$. Fig 2(a) shows the transmission spectrum of the single pillar line. It can be seen that the transmission spectrum corroborates well with the band structure prediction. The displacement fields and the deformations at the two dips are shown in Fig 2(c). Due to their large SH component ratio, as well as the exclusive generation of SH waves by the electric field, we only show the transverse component $u_{y}$. The amplitude in pillar is normalized to that in the matrix and is indicated beside the pillar. The two dips correspond to the pillar’s intrinsic bending and torsional modes, respectively. The torsional mode induced transmission dip is better attenuated, since the excited torsional motion leads to, on the side opposite to the incident wave, a wave of identical amplitude and opposite phase, which is responsible for a destructive interference. To further confirm this mechanism, the emitted wave $Em$ is calculated by subtracting the incident waves $I$ (transmitted waves on the bare matrix without pillar) from the totally transmitted waves $T$: $Em=T-I$. $Em$ is then normalized by $I$. Fig 3(a) is the complex plot of the normalized emitted wave in the frequency range [500, 505] MHz around the torsional mode. The rose dot corresponds to the dip at resonant frequency of $502.1$ MHz. This mode falls at point (-1,0), which refers to the same amplitude with a phase shift of 180°with respect to the incident waves. This results in a destructive interference and a strong attenuation in transmission. The phase shift of full transmitted waves is shown in Fig 3(b). Red dotted lines represent the incident waves. The dip corresponds to a $\pi$ change in phase with respect to the incident waves. The phase shift is [math] ($2\pi$) before (after) the resonant frequency, meaning that the transmitted waves are in phase with the incident waves when they deviate from the resonant frequency. The corresponding phase derivative or group delay time $\tau_{g}$ of the transmitted waves is shown in Fig 3(c), with $\tau_{g}=\frac{d\phi}{d\omega}$, where $\phi$ denotes the phase and $\omega=2\pi f$ is the angular frequency. The waves are delayed at the resonant frequency. It can be seen that the resonance is characterized by a rapid variation of phase and amplitude.

By scaling the model in the same proportion, we can obtain the same dimensionless frequency $\omega a/c_{SH,sub}$ of a mode. Since the pillar intrinsic mode is almost independent of the periodicity, one can also tune the pillar mode frequency only by modifying the pillar size. Note that this frequency needs to be within the allowable operating range of the materials.

IV Two lines of identical pillars: Autler-Townes Splitting & cavity mode

Transmission spectra are then calculated around the torsional mode frequency for two lines of identical pillars. In addition to our intuitively predicted transmission dip, different phenomena appear when we gradually increase the distance $d$ between the pillar lines. When $d<1.4a$ (Fig 4(a)), a coupling effect arises between the pillar lines, causing a lifting of frequency degeneracy of the pillar torsional mode, and the original transmission dip splits into two dips with a transparency window in the middle, referred to as acoustic analogue of Autler-Townes Splitting (ATS). The coupling becomes stronger when $d$ gets smaller, as shown in Fig 4(a). Note that the pillar coupling also depends on the pillar mass. Therefore, the distance limit $1.4a$ can be detuned by changing the pillar size (radius or height). The displacement fields $u_{y}$ at the dips and peak for $d=0.5a$ are shown aside in Fig 4(d). It is found that the largest amplitude is located in the pillars for dip1 at 499.8 MHz. The two pillars are in opposite phase at dip1 frequency and in-phase at dip2 frequency, which is a feature of the ATS resonance that can be confirmed by calculating the phase difference between the two pillars. Since the pillars are in torsional mode in the range of measurement, all the points on the side of the pillar that faces the incident waves are in phase. The phases of $u_{y}$ on the wave-facing sides of the two pillars are probed. The cases of $d=0.5a$ and $d=a$ are shown as examples in Fig 5(a) and (b). Rose and green dots correspond to transmission dips and peaks, respectively. It can be seen that the two pillars have a phase difference of $\pi$ at the first dip, meaning that they are 180°out-of-phase. The phase difference is $2\pi$ at the second dip, indicating that they are in-phase. This reveals that the pillar vibrations are symmetrical at the dip1 frequency and asymmetrical at the dip2 frequency. We think this behavior leads to different quality factors of the two dips.

Since we work around 500MHz with Love waves velocity $v$ around 4200 m/s, the wavelength $\lambda=v/f$ is therefore around 8.4$\mu$m. $d\approx 2a$ ($a=2\mu m$) indicates a distance around $\lambda/2$ corresponding to the Fabry-Perot (FP) resonance. This resonance is almost invisible in the transmission spectrum when it matches the pillar torsion mode, since the waves are almost totally reflected. However, when we change the distance between pillars around $\lambda/2$, the proximity of FP resonance with the pillar mode gives rise to the cavity modes at the two edges of the dip, as shown in Fig 4(b) and (c). In these cases, the two pillars act like partial reflectors, and the normally incident waves are multiply reflected to produce multiple transmitted waves with path difference equal to $n\lambda$, where $n$ is an integer. In this way, constructive interference occurs, leading to a resonant enhancement. The two pillars along with the guiding layer between them become a cavity to confine the waves. Nevertheless, in the transmission spectra, the behaviors for $d<2a$ (Fig 4(b)) are less marked than that for $d>2a$ (Fig 4(c)), where peaks rise at the lower edge of the dip, and give rise to Fano-like resonance line-shapes. However, we have verified that it can not be fitted by a Fano type formula. The displacement field $u_{y}$ at the peak and dip for $d=2.4a$ are shown aside in Fig 4(e), since the transmission curve for $d=2.4a$ presents the most confined peak. At the peak frequency, large amplitudes are observed for the three parts of the cavity (two pillars and the guiding layer in between). This behavior is different from that of ATS, the largest amplitude in the pillars occurs at the peak where the waves in the guiding layer are highly confined in the cavity.

To show more clearly the transition of the cavity mode with respect to the dip when $d$ changes, we draw the complex plots of the normalized transmissions in Fig 6(a). The cavity mode is manifested as an additional perturbation on the original ellipse. Since the phase changes clockwise, it can be seen that the cavity mode passes the dip as $d$ increases. When the cavity mode approaches to the dip, the perturbation decreases. For $d=2a$, the cavity mode coincides with the FP resonance and becomes almost invisible. Additionally, we can see that the behaviors for $d<2a$ and for $d>2a$ are quite similar, with the perturbation frequency either larger or smaller than the dip frequency.

Fig 6(b) shows the complex plots of the normalized transmissions of Love waves for $d=2a$ when we change the radius of both pillars from $0.196a$ to $0.204a$, indicating a shift in pillar resonant frequency. It is found that the cavity mode remains almost invisible, i.e. still coincides with the FP resonance. That is because the distance between pillars corresponding to the FP resonance is almost unchanged in our range of measurement (from 498 to 506 MHz), i.e. $\lambda/2$ is always around $d=2a$. Therefore, when we change the pillar vibration frequency, we obtain the parallel cavity modes for the same value of $d$.

To give an overview of the resonance behaviors, the dip and peak frequencies for different $d$ are shown in Fig 7. Blue dotted line is the dip frequency of the single pillar line. It can be seen that the pillar coupling induced ATS is in the region $d<1.4a$, where the two dips are mismatched with the single pillar resonant frequency. This coupling disappears when $d$ exceeds $1.4a$, and only one dip (quasi-zero transmission) remains. This dip matches the single pillar resonant frequency except when the cavity modes appear below (upon), the dip frequency shifts slightly upward (downward). The interaction between the pillars is much stronger for $d/a$=2 than that for $d/a$=4. Therefore, the resonances around $d$=$\lambda$ are too weak to be observed. Note that the frequency of the cavity modes changes much slower than that of the FP resonance. The $1^{st}$ and $2^{nd}$ FP resonance exists only in the very closed regions around $d=2a$ and $4a$, respectively. In these two cases, FP resonances are particular cases of the cavity modes when the later coincide with the dip. In the other regions, one should avoid to mix up the cavity mode with the FP resonance.

V Two lines of dissimilar pillars: Autler-Townes Splitting, cavity & Acoustically Induced Transparency

Since an increase of pillar radius or height will induce a decrease in the torsional mode frequency, one can gradually tune the position of the dip by modifying the pillar size. Here we modify the pillar radius as example since the radius is easier to be tuned in experimental process.

In the case below, the second pillar radius $r_{2}$ is tuned from $0.195a$ to $0.205a$, while the first pillar radius $r_{1}$ being fixed to $0.2a$. Fig 8 shows the transmission spectra of Love waves propagating through the two dissimilar pillar lines for different $d$ that remains unchanged for each case. Let un notice that the curves for different $r_{2}$ are shifted vertically with respect to each other. Dotted blue and rose lines denote the dip positions for a single line of pillars with radius equals to $r_{1}$ and $r_{2}$, respectively. Fig 8(a) corresponds to the case $d=0.5a$, where the coupling between the pillars is so strong that the two dips stay all the way mismatched with their corresponding single pillar dip positions. When $d$ gets larger, the coupling becomes weaker: Fig 8(b) corresponds to the case $d=a$. It is found that this coupling decreases with the increase of radius mismatch. In the case of $r_{2}=0.195a$ and $0.205a$, each dip almost coincides with the corresponding resonant frequency of one single pillar. In order to show the anti-crossing lines of ATS for $d<1.4a$, we plotted in Fig 9 the dip frequencies for different $d$ as functions of $r_{2}$. It can be seen that with the increase of pillar distance, the anti-crossing lines get closer to the crossing line (for $d=1.4a$), and that each anti-crossing line will rejoin its individual pillar resonant frequency when the radius mismatch is sufficiently large. This means the pillar coupling effect decreases when their distance or/and their radius difference increases.

For $d=2a$ as shown in Fig 8(c), two pillars with different radius give rise to two dips with a transparency window in the middle. These peaks have a narrower line-shapes compared with the cases of ATS (see also Fig 11). Each dip is consistent with the corresponding dip frequency of one single pillar since no more coupling exists, which means each of them originates from individual pillar’s torsional mode. The displacement fields $u_{y}$ for $r_{2}=0.202a$ are presented in Fig 10. It can be seen that each dip corresponds to a large amplitude in a single pillar, indicating the attenuation of transmission at each pillar’s resonant frequency due to the destructive interferences. The peak correspond to a Fabry-Perot resonance since $d$ is close to $\lambda/2$. The two detuned pillar modes act as two partial reflectors and are hence able to support the constructive interference of the FP resonance, where large amplitudes are observed for the three parts of the cavity (two pillars and the guiding layer in between). This three-resonance system induced transparency window is referred to as the Acoustic analogue of Electromagnetically Induced Transparency, also called Acoustically Induced Transparency (AIT). The peak rises and gets wider with the increase of radius difference. As for $d=2.4a$ shown in Fig 8(d), where a cavity mode peak is observed for the two identical pillars, it is found that with the increase of pillar radius mismatch, the peak confinement decreases and the dip1 becomes evident for $r_{2}=0.198a$ and $0.202a$. Compared with the case of $d=2a$ (Fig 8(d)), the peak between two dips is less confined due to the red shift of cavity mode with respect to the pillar resonant frequency. Note that AIT requires a well excited resonance between the two dips. Therefore, in the case of $d=2.4a$, the transparency window is only two dips resulting from the different resonant frequencies of two pillars.

The transmission spectra of AIT are similar with those of ATS, however, they originate from different mechanisms. ATS appears only when the two pillars are coupled to each other, and exists even when the two pillars are identical. AIT appears when $d$ is out of the pillar coupling region and only when the two pillars are different. The pillars and the cavity interact at the peak. Moreover, AIT requires a clearly identified 3-level resonant system, which is not the case for ATS.

Besides the different mechanisms related to ATS and AIT as presented above, corresponding analytical formulas of ATS and AIT for transmission spectra can be used to fit the numerical data to better distinguish these different transparency windows. The transmission curves for ATS can be written as the sum of two separate inverse Lorentzian profiles representing the two dips, while the transmission for AIT is expressed as the difference of a broad Lorentzian profiles and a narrow one with a similar central frequency Peng et al. (2014); Liu et al. (2016):

[TABLE]

[TABLE]

where $C_{1}$, $C_{2}$, $C_{+}$, $C_{-}$ are the amplitudes of the Lorentzian profiles, $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{+}$, $\Gamma_{-}$ are their full width at half maximum (FWHM). $\delta_{1}$, $\delta_{2}$, $\delta_{c}$ are the central frequencies with $\epsilon$ denoting a possible slight shift on $\delta_{c}$. $\Gamma_{1}$, $\Gamma_{2}$, $\delta_{1}$, $\delta_{2}$ and $\delta_{c}$ can be directly taken from the transmission spectra.

In an intermediate state, the transmission spectra can be fitted by a transition formula that considers both the features of ATS and AIT:

[TABLE]

where $C_{a}$, $C_{b}$, $C_{d}$, $C_{e}$, $\Gamma_{a}$, $\Gamma_{b}$, $\Gamma_{d}$, $\Gamma_{e}$ are parameters to be determined. Note that this formula can also be used to fit the ATS and AIT cases.

Fig 11 shows the numerical data (black dots) together with the best fit functions $T_{ATS}$ (red lines) and $T_{AIT}$ (blue lines) for the four values of $d$ in Fig 8 which stand for strong coupling ATS, weak coupling ATS, AIT and intermediate states, respectively, in the case of $r_{2}=0.202a$ and $r_{1}$ fixed to $0.2a$. The best fit functions are determined by resorting to the least-squares method, that is, by searching the fit parameters that minimize the sum of squared errors between numerical data and fit function:

[TABLE]

where $N$ is the number of values calculated by FEM. $F_{i}$ is the value of $T_{ATS}$ or $T_{AIT}$ corresponding to each data frequency. It can be seen that, as expected, for $d=0.5a$ and $a$ where coupling exists between the pillars, $T_{ATS}$ fits the numerical data much better than $T_{AIT}$. Whereas for $d=2a$ , $T_{AIT}$ fits the numerical data better than $T_{ATS}$. For $d=2.4a$, both $T_{ATS}$ and $T_{AIT}$ do not fit the numerical data well. In this case, $T_{ATS/AIT}$ (green line) fits the data better.

By fitting the numerical data to the model fits of $T_{ATS}$ and $T_{AIT}$ for different $r_{2}$ ($r_{1}$ fixed), the relation between the fits parameters and $r_{2}$ can be obtained as shown in Fig 12, where Fig 12(a) presents the fit parameters of the ATS model for $d=0.5a$ and Fig 12(b) are those of the AIT model for $d=2a$. The parameters of ATS model describing each Lorentzian curve are independent of each other. It can be seen that $\Gamma_{1}$ increases with the increase of radius mismatch while $\Gamma_{2}$ increases with the increase of $r_{2}$. We think this is resulting from the symmetrical/asymmetrical vibration of the two pillars at dip1/dip2 in the ATS cases. $C_{1}$ is relatively stable while $C_{2}$ presents a slight tendency to increase. The parameters of AIT model are related to each other. As can be seen in Fig 12(b), these four parameters all increase as the radius mismatch increases and are almost symmetrical in our range of measurement. Despite a larger values of $\Gamma_{+}$ compared with $\Gamma_{-}$, $C_{+}$ and $C_{-}$ are almost the same for different $r_{2}$ .

With the increase of distance between pillars, the transition from ATS to AIT can be quantitatively studied by evaluating the quality of these model fits. The Akaike information criterion (AIC)Burnham and Anderson (2002) is used to discern AIT from ATS, which provides a method to select the best model from a set of models. This criterion quantifies the amount of information lost, i.e. the degree of unfitness, and is given as $I_{j}=2k-2ln(L_{j})$, where $k=4$ is the number of unknown parameters and $L_{j}$ the maximum likelihood for the considered models, i.e. $j=$ATS or AIT. Since we already found the best fit functions of $T_{ATS}$ and $T_{AIT}$, it is sufficient to calculate the likelihood of these two functions. Then, the AIC weight $W_{j}=e^{-I_{j}/2}/\sum_{1}^{N}e^{-I_{j}/2}$ can give the relative likelihood of a candidate model. $N$ is the number of considered model. In our case, $N=2$ as only two models are involved. Since we have more than one calculated data for each model, we utilize the AIC mean per-point weightAnisimov et al. (2011) $\overline{w}_{j}=e^{-I_{j}/2n}/\sum_{1}^{N}e^{-I_{j}/2n}$ to calculate the statistically synthesized likelihood of the candidate model. $n$ is the calculated data number. The AIC mean weight can be rewritten as:

[TABLE]

with $\overline{w}_{ATS}+\overline{w}_{AIT}=1$.

Fig 13 shows the AIC mean weight of the ATS and AIT models as function of the distance between pillars, in the case of $r_{2}=0.202a$ and $r_{1}=0.2a$. It can be seen that, as expected, the AIC mean weight of ATS model is dominant for the small distance region, which means it is preferable to use the ATS model. When $d$ increases, the AIC mean weight of AIT model starts to increase as well, and becomes dominant for $d\in$[$1.8a$, $2.2a$]. Since cavity mode is not presented at the peak for $d=1.8a$, and starts to interact with the dip1 for $d=2.2a$, the AIT is hence in the distance range of $d\in$[$1.9a$, $2.1a$]. When $d$ continues to increase, the AIC mean weight of ATS becomes dominant again until $d$ increases to around $4a$, i.e. the position of the second FP resonance, where we found the second AIT position for $d\in$[$3.9a$, $4.1a$] . The periodicity of AIT, which is a behavior that hasn’t been addressed before in the existing literature, is due to the periodic apparition of the FP resonance. The AIC mean weight of the second AIT region is smaller than that of the first AIT region, since the interaction of the system decreases with the increase of distance between pillars $d$. Note that the ATS in our case exists only for $d<1.4a$, therefore the transition region $d\in$[$1.4a$, $1.8a$] as well as the larger distances other than the AIT regions do not represent ATS or AIT. We can see that the theoretical distinction between ATS and AIT helps the comprehension of the analytical results. On the other hand, this criterion is useful when we cannot theoretically rule out the AIT phenomenon. This happens for example when the radius mismatch is so large that the peak is much wider with respect to the two dips. In this case, although FP resonance is still present between two dips, the transmission spectra can not be well fitted by the AIT model and therefore cannot be ascribed to an AIT.

The above effects can also be obtained by detuning the pillar height. The heights of the two pillars are increased/reduced from the original height $h=0.6a$ by the same amount $\Delta h$. That is, $h_{1}=h+\Delta h$ and $h_{2}=h-\Delta h$. The radius of both the two pillars $r_{1}$ and $r_{2}$ are fixed at $0.2a$. Fig 14 shows the transmission spectra of Love waves through the two lines of pillars when $d=2a$, i.e. in the region of AIT. It can be seen that by decreasing $\Delta h$, the two dips approach to each other and the peak with almost unchanged frequency decreases and becomes invisible for identical pillars.

VI Conclusion

In this work, the interaction of Love waves with two lines of cylindrical Ni pillars are investigated on a silica film deposited on a 90ST quartz substrate. Firstly, pillar intrinsic torsional mode is demonstrated to be well excited by Love waves. One line of pillars can give rise to a sharp transmission dip due to a destructive interference. Secondly, acoustic analogue of Autler-Townes Splitting (ATS) and Fabry-Perot resonance of Love waves are demonstrated in two lines of identical pillars by varying the distance between the pillar lines. ATS appears when the distance is smaller than the half wavelength and a strong coupling is aroused between the pillar lines, causing the pillar mode induced transmission dip to split into two dips with a transparency window in the middle. This coupling decreases with the increase of pillar distance. We demonstrated the different pillar vibration symmetries at the two dip frequencies, which lead to different dip widths. Fabry-Perot resonance exists at the positions where the distance between the pillar lines is a multiple of half wavelength. The proximity of Fabry-Perot resonance with pillar intrinsic mode gives rise to the cavity modes with transmission enhancement on the two edges of the single dip. We avoided to mix up the FP resonances with the cavity modes by presenting the different frequency variation with respect to the distance between the pillars. Thirdly, the radius of one line of pillar is modified to detune the pillar resonant frequency. In the pillar coupling region, the coupling effect decreases with the increase of radius mismatch, and the two dips will rejoin their individual pillar mode frequencies. When the distance between the pillar lines is a multiple of half wavelength, Fabry-Perot resonance along with the two different pillars’ resonances give rise to the Acoustically Induced Transparency (AIT). Same phenomena can also be obtained by detuning the pillar height. Then, with similar transparency window in the transmission spectra, ATS and AIT phenomena are first fitted with the corresponding formula models, showing good agreements. The fit parameters are demonstrated as functions of the geometrical parameter. The Akaike information criterion (AIC) is then used to quantitatively evaluate the quality of the fit models, which illustrates the transition from ATS to AIT as well as the periodicity of AIT by increasing the distance between the pillar lines. The theoretical and analytical differentiation of ATS and AIT should be used together to discriminate the assignment of the observed spectrum to one or the other physical mechanism. The results presented in this study could be used to potential acoustic applications such as wave control, meta-materials and bio-sensors.

Appendix A Dissipation consideration

To consider the losses by dissipation, we recalculated the critical transmission curves by changing the Young’s modulus $E$ of the silica film and the pillars to complex numbers $E(1+j\epsilon)$, with $\epsilon$ equals to $5\times 10^{-4}$ and $1\times 10^{-3}$. Corresponding transmission spectra for different phenomena (dips, ATS, cavity modes and AIT) are shown in Fig 15. It can be seen that all the main resonances and dips are visible for $\epsilon$ below $1\times 10^{-3}$. With $\epsilon$=$1\times 10^{-3}$, we can estimate a propagation length around 2mm. In experiments, Love waves can propagate for a few mm. We have already realized 500 MHz devices with distances between transmitter and receiver of a few mmTalbi et al. (2015). This means the parameter $\epsilon$ is below $1\times 10^{-3}$, which indicates that our system is robust for practical applications.

Appendix B Cavity mode and FP resonance

B.1 two lines of identical pillars

In order to show the behavior of the cavity modes at the vicinity of the transmission dip, we plotted in Fig 16(a) the transmission spectra for $d=2.4a$ when all the pillars’ radius vary from 0.196$a$ to 0.204$a$. It can be seen that we obtain almost the same cavity mode besides the dip at different frequencies. Along with Fig 6(b) and Fig 16(a), we can present in Fig 16(b) the relation between cavity modes (black lines) and FP resonance (red line). When we change the pillar resonant frequency (blue lines), FP resonance remains at almost the same position. For each radius value, the FP resonance is a particular case of the cavity modes when the later coincide with the dip.

B.2 two lines of dissimilar pillars

Fig 17 gives the same results as in Fig 7, but with two lines of dissimilar pillars. Fig 17(a) is the case of $r_{1}=0.2a$ and $r_{2}=0.201a$. Dip1 (dip2) corresponds to the second (first) pillar. It can be seen that outside the ATS region, dip2 becomes invisible after $d$ exceeding 2.2$a$. This is because on the one hand, the interaction between the two pillars becomes too weak and on the other hand, the radius mismatch is too small. This dip become visible again when $d$ approaches to $4a$, since the interaction between the two pillars regains its strength in the AIT region. When we enlarge the radius difference (see Fig 17(b)), most of the dip1 in this region becomes visible. Moreover, we can see that the cavity modes cross the center of the two dips at $d$=$2a$ and $4a$, where we obtain the AIT resonance of FP type. This behavior is compatible with the cavity mode-FP resonance relation that we presented in Fig 16(b). FP resonances are particular cases of the cavity modes when the later fall between the two dips.