Elastic bound states in the continuum with multi-polarization hybridization

Abstract

Bound states in the continuum (BICs) have garnered increasing interest for their outstanding ability to enhance wave-matter interactions and provide an infinite quality (Q) factor. However, elastic BICs have not been fully explored due to the complex polarization modes that differ from those of electromagnetic and acoustic waves. In this paper, we theoretically, numerically, and experimentally investigate elastic BICs with multi-polarization hybridization between flexural and longitudinal waves. Based on the hybridization, Friedrich-Wintgen (FW) BICs can be realized by degenerating two different-eigenmode resonances, unlike conventional realizations of two identical ones. Further, we demonstrate that the FW and accidental quasi-BICs can coexist in the multi-polarization elastic system by detuning two flexural resonances and simultaneously modulating the destructive interference. Our study reveals the rich properties of elastic BICs with multi-polarization hybridization and offers a method for achieving perfect mode conversion between flexural and longitudinal waves with a tunable Q factor.

Introduction

Bound states in the continuum (BICs), also regarded as trapped modes, are non-radiating localized states that coexist within the continuous spectrum of radiating waves¹. Initially proposed in quantum mechanics², BICs have since been identified in classical wave systems, including electromagnetic³, airborne/waterborne acoustic⁴, and elastic waves^5,6. BICs with extreme-high Q factors have paved the way for various applications in optical and photonics, including sensors^7,8, lasers^9,10, filters^11,12, nonlinearity generators^13,14, etc. Recently, the unique benefits of acoustic and elastic BICs have attracted increasing attention for applications in logic¹⁵, filters¹⁶, and communications¹⁷. Various types of acoustic BICs have been investigated, such as symmetry-protected^18,19,20, Friedrich-Wintgen (FW)^21,22,23, Fabry-Pérot^24,25, and accidental BICs^26,27,28,29, as well as collective BICs arising from coupled resonances³⁰. Comprehensive theoretical and experimental studies have been conducted to understand corresponding physical formation mechanisms and explore potential applications, such as sound absorption²² and acoustic sensor³¹.

Compared to optical and acoustic BICs, elastic BICs exhibit distinct characteristics due to the multiple polarization states among different wave modes³². Utilizing the polarization mismatch, genuine elastic BICs completely trapped within the resonator have been observed in solid resonators placed in nonviscous fluids³³. Recently, the genuine BICs, specifically polarization-protected BICs, have been both theoretically and experimentally demonstrated in a compact mechanical system made of cylindrical granular particles³⁴. Besides, elastic BICs have been explored in clusters of scatterers embedded in a background^35,36, layered systems composed of different materials without periodicity^37,38, multi-physics fields with interaction^39,40, and topological elastic lattices with non-Hermiticity⁴¹ or disclination⁴². For specific applications, perfect absorption of flexural waves is realized based on elastic quasi-BICs constructed by three parallel sub-beams⁴³. These elastic BICs can be summarized as symmetrical-protected, Fabry-Pérot, and FW BICs. Specifically, elastic FW BICs are typically formed by two identical resonators with the same eigenmode, while those constructed by any two degenerate resonances but different eigenmodes remain unexplored. Moreover, most previous studies on elastic BICs have focused solely on a single mode of elastic waves. Recently, elastic BICs for both asymmetric and symmetric Lamb wave modes have been achieved, utilizing the Fabry-Pérot resonance of the corresponding wave mode⁵. However, the system supports elastic BICs for only one wave mode at a time. Elastic BICs with multi-polarization hybridization (coupling between different modes), which are fundamentally different from their optical and acoustic counterparts, remain an open area of investigation.

In fact, multi-polarization hybridization is the key to realizing mode conversion between two different-polarization elastic waves. The ongoing development of elastic metamaterials^44,45 and metasurfaces^46,47 has offered new solutions for high-efficiency mode conversion. For instance, the conversion between longitudinal and transverse waves is achieved using anisotropic elastic metamaterials or metasurfaces designed based on impedance matching or generalized Snell’s law^48,49,50,51. The conversion from surface (edge) waves to bulk waves is achieved by pillared metamaterials. However, achieving perfect mode conversion (unitary efficiency) remains a challenge when utilizing a compact configuration. Especially, the conversion between flexural and longitudinal waves in two perpendicular polarization planes presents a significant difficulty due to the extreme impedance mismatch between these two wave modes⁵². To date, we have only noticed that one previous work⁶ achieved perfect mode conversion from flexural to longitudinal waves, induced by a kind of accidental BICs. In that study, the quasi-BICs enabling perfect mode conversion with extremely high Q factors are realized based on simultaneous destructive interference and Fabry-Pérot resonance. Due to configuration limitations, the system supports accidental BICs, which restricts perfect mode conversion to a single frequency.

Differently, in this study, we establish a general framework for simultaneous elastic FW and accidental BICs with multi-polarization hybridization between flexural and longitudinal waves. The elastic BICs are supported by a metamaterial with bilateral pillars connected to a waveguide at one end. The elastic FW BICs are achieved by bilateral pillars with degenerate Fano resonances, whether the eigenmodes are the same or not. The accidental BICs can be simultaneously supported by tuning the waveguide to modulate the destructive interference. A proper resonance detuning brings a balance of radiative decay rates between incident and converted other modes. As a result, the dual-frequency perfect mode conversion is achieved thanks to the coexistence of FW and accidental quasi-BICs. In addition to tuning the critical frequency via the incident angle, we present another approach for continuously adjusting the Q factors: modulating the radiative decay rates through adjusting the detuning degree of bilateral pillars, together with a precisely engineered waveguide that ensures destructive interference. We prove FW and accidental BICs with perfect mode conversion by theoretical and numerical analyses, complemented by experimental verifications.

Results

Platform for elastic BICs with multi-polarization hybridization

To clearly state elastic BICs with multi-polarization hybridization, a metamaterial plate with a simple configuration yet exhibiting typical resonance properties is studied. As shown in Fig. 1a, the metamaterial plate contains periodically arranged unit cells along an edge, with a periodic width denoted by ${w}_{{{\rm{s}}}}$. The unit cell is composed of bilateral pillars ${{{\rm{p}}}}_{1}$ and ${{{\rm{p}}}}_{2}$ connected to a waveguide ${{{\rm{p}}}}_{3}$. The bilateral pillars and waveguide have the same width, denoted by ${w}_{{{\rm{p}}}}$. The heights of bilateral pillars, and the length of the waveguide are denoted by ${l}_{1}$, ${l}_{2}$ and ${l}_{3}$, respectively. The bilateral pillars will hybridize the flexural and longitudinal waves in two perpendicular planes within the host plate. They have the same thickness ${h}_{{{\rm{b}}}}$ (equal to 2 mm throughout this paper) with the host plate and waveguide, which is in deep-subwavelength scale compared to the minimum flexural wavelength considered in this paper. To investigate the mechanism underlying FW BICs, a two-dimensional model that neglects the waveguide ${{{\rm{p}}}}_{3}$ (as shown in Fig. 1b, c) is employed. A model that incorporates the waveguide (see Fig. 1d) is used to elucidate the mechanism responsible for simultaneous FW and accidental BICs. Figure 1e, f presents sketches based on the temporal coupled-mode theory for FW BICs and the coexistence of FW and accidental BICs, respectively, as will be explained in detail in subsequent sections. To simplify the analysis and better reveal the mechanism of elastic BICs with multi-polarization hybridization, material damping is not considered in any simulations unless otherwise specified.

Fig. 1: Mechanisms of elastic BICs with multi-polarization hybridization.

a Metamaterial model with periodically arranged bilateral pillars ${{{\rm{p}}}}_{1}$ and ${{{\rm{p}}}}_{2}$ connected to a waveguide ${{{\rm{p}}}}_{3}$ along an edge, for perfect mode conversion between flexural (F) and longitudinal (L) waves. b, c Mechanism diagrams for FW BIC induced by degenerated first-first and first-second order resonances, respectively. $m$ represents the resonance order. d Mechanism diagram for coexistence of FW and accidental BICs arose from degenerate resonances and destructive interference of channels 1 and 2. e, f Sketches of temporal coupled-mode theory for FW BIC and coexistence of FW and accidental BICs by regarding bilateral pillars as a unified resonator $\bar{{{\rm{p}}}}$. ${S}_{{{\rm{in}}}}^{{{\rm{F}}}}$ represents the incident flexural waves, ${\gamma }_{1(2)}^{{{\rm{F}}}({{\rm{L}}})}$ represents the radiative decay rate of flexural (longitudinal) waves for pillar ${{{\rm{p}}}}_{1(2)}$, $\kappa$ denotes the near-filed coupling, $F{(L)}_{1(2)\pm }$ represents the amplitudes of flexural (longitudinal) waves from channel 1(2), where the symbols $+$ and $-$ denote incoming and outcoming waves, respectively.

Although the unit cells are periodically arranged along an edge (y-direction) of the plate, the dispersion relations with a typical Bloch periodicity $P$ along x-direction (inserts in Fig. 2a–c) are firstly analyzed (see Supplementary Note 1 for methods). The periodicity $P$ equals to a quarter of the wavelength at the second-order Fano resonance frequency, which is much less than the wavelengths considered in this paper. Consequently, the dispersion relations in this locally resonant system are not dictated by the small Bloch periodicity. From the dispersion relations, we can intuitively observe the multi-polarization hybridization between flexural and longitudinal waves induced by detuned bilateral pillars. The dispersion relations for bilateral pillars exhibiting degenerate and detuned first-order Fano resonances (abbreviated as first-first order resonances, Fig. 1b), and for those exhibiting degenerate resonances between the first-order Fano resonance of ${{{\rm{p}}}}_{1}$ and the second-order Fano resonance of ${{{\rm{p}}}}_{2}$ (abbreviated as first-second order resonance, Fig. 1c), are comprehensively analyzed by analytical solutions and simulations, as shown in Fig. 2a–c, respectively. The dispersion curves of flexural (orange dashed line) and longitudinal (grey dotted line) waves in the host without pillars are also plotted in Fig. 2a–c.

Fig. 2: Dispersion relations and eigenmodes of bilateral pillars.

a-c Dispersion relations of bilateral pillars exhibiting degenerate and detuned first-order resonances, as well as degenerate first-second resonance, obtained by analytical solutions and simulations. d–f Eigenmode profiles corresponding to frequencies marked at a–c. f also shows the eigenmode profile of bilateral pillars with detuned first-second resonances. The transverse (u) and vertical (w) displacements for the host are also enlarged to state the hybridization of bilateral pillars.

The bilateral pillars with degenerate first-first order resonance are completely symmetric. In such a case, the first two branches at Fig. 2a associated with the asymmetric and symmetric flexural resonances of bilateral pillars, respectively. The corresponding eigenmodes at ${f}_{{{\rm{A1}}}}$ and ${f}_{{{\rm{S1}}}}$ are shown in Fig. 2d, which are orthogonal. The enlarged transverse (u) and vertical (w) displacements of the host at these two frequencies are also shown in Fig. 2d. At ${f}_{{{\rm{A1}}}}$, the longitudinal displacement in the host is symmetric and will be cancelled out. The out-of-phase resonance of bilateral pillars induces the flexural deformation of the host. In contrast, at ${f}_{{{\rm{S1}}}}$, the symmetric flexural eigenmode leads to a cancellation of moments at the host interface. As a result, the flexural energy is trapped within the bilateral pillars without leakage. Therefore, ${f}_{{{\rm{S1}}}}$ meets the so-called symmetric-protected BIC, which will be demonstrated in the subsequent sections. Thus, the flexural resonance frequency ${f}_{{{\rm{S1}}}}$ hardly affected by the host. It is consistent with the first-order Fano resonance frequency ${f}_{0}$ (pink dash-dotted line) of corresponding single cantilever pillar, which satisfies the following transcendental equation⁵³:

$$\cos (l\rho {h}_{{{\rm{b}}}}/{D}_{{{\rm{F}}}}\sqrt{2{{\rm{\pi }}}{f}_{0}})\cdot \,\cosh (l\rho {h}_{{{\rm{b}}}}/{D}_{{{\rm{F}}}}\sqrt{2{{\rm{\pi }}}{f}_{0}})=-1$$

(1)

where $l$ is the pillar height in one side, ${D}_{{{\rm{F}}}}=E{h}_{{{\rm{b}}}}^{3}/12(1-{\upsilon }^{2})$ is the flexural stiffness, $\rho$, $E$ and $\upsilon$ are the density, Young’s modulus and Poisson’s ratio, respectively.

For bilateral pillars with detuned first-first Fano resonance frequencies (asymmetrical configuration), as shown in Fig. 2b, the first two branches correspond to the flexural resonances of upper and lower pillars, respectively. Differently, the dispersion relations exhibit the hybridization with flexural waves (first branch) and multi-polarization hybridization with both flexural and longitudinal waves (second branch) in the host. Due to the hybridization, the two flexural resonance frequencies ${f}_{{{\rm{A2}}}}$ and ${f}_{{{\rm{A3}}}}$, respectively corresponding to the upper and lower pillars, are lower than the corresponding Fano resonance frequencies (pink dash-dotted lines) predicted by Eq. (1). The eigenmode profiles at ${f}_{{{\rm{A2}}}}$ and ${f}_{{{\rm{A3}}}}$ are shown in Fig. 2e. The transverse (u) and vertical (w) displacements of the host are also described to further illustrate the multi-polarization hybridization. The deformations of the host at ${f}_{{{\rm{A2}}}}$ are similar with those at ${f}_{{{\rm{A1}}}}$ since the same kind of hybridization, whereas resonance at ${f}_{{{\rm{A3}}}}$ induces obvious deformations on both transverse and vertical (dominant) components, demonstrating the multi-polarization hybridization. The multi-polarization hybridization between flexural and longitudinal waves induced by detuned bilateral pillars plays a key role in constructing quasi-BICs with perfect mode conversion.

For asymmetrical bilateral pillars, the degeneracy between the first-order Fano resonance of one-side pillar and higher-order Fano resonances of the other-side pillar can also induce a hybridization with a single longitudinal mode, similar to the symmetrical eigenmode at ${f}_{{{\rm{S1}}}}$, even though the eigenmodes are distinct. For example, Fig. 2c shows the dispersion relations for bilateral pillars with degenerate first-second Fano resonance, corresponding to the third branch (the first two branches correspond to the first-order flexural resonances of lower and upper pillars, and the eigenmodes at ${f}_{{{\rm{A4}}}}$ and ${f}_{{{\rm{A5}}}}$ are similar with those in Fig. 2e, thus are not shown here). This branch is similar with the dispersion curve associated with ${f}_{{{\rm{S1}}}}$, as further evidenced by the transverse and vertical deformations at ${f}_{{{\rm{A6}}}}$ shown in Fig. 2f. This behavior arises because the moments at the host interface are also cancelled by interference, due to the phase mismatch between bilateral pillars with distinct eigenmodes. This interference cancellation results in energy being trapped within the bilateral pillars. Subsequent sections will demonstrate that this kind of degeneracy with different-eigenmode resonances also supports FW BICs, unlike conventional realizations of two identical ones. For comparison, Fig. 2f also presents the eigenmode profile and enlarged displacements for bilateral pillars with detuned frequency between first-second order Fano resonances, denoted by detuned ${f}_{{{\rm{A6}}}}$, which is achieved by slightly reducing the length of pillar ${{{\rm{p}}}}_{2}$. The detuned resonances with multi-polarization hybridization between different orders allow multi-branch FW quasi-BICs with perfect mode conversion.

Construction of Friedrich-Wintgen BICs

To state the construction of FW BICs, we extend the temporal coupled-mode theory⁵⁴ in optics, which is applicable to general frameworks for resonators coupled with multiple input and output ports. Temporarily ignoring the waveguide ${{{\rm{p}}}}_{3}$, the metamaterial in Fig. 1a simplifies to a model featuring periodically bilateral pillars at its edge. The pillar thickness is in deep-subwavelength scale for all interested frequency, so that only fundamental mode is considered. Then, for incident flexural waves, when ignoring the structural loss, each-side pillar can be described by the temporal coupled-mode equation (sketch shown in Fig. 1e) for the lowest mode with the amplitude ${\tilde{a}}_{1(2)}={a}_{1(2)}{{{\rm{e}}}}^{{{\rm{i}}}\omega t}$ as¹

$$\frac{d}{dt}{\tilde{a}}_{1(2)}= \, \left({{\rm{i}}}{\omega }_{1(2)}-{\gamma }_{1(2)}^{{{\rm{F}}}}-{\gamma }_{1(2)}^{{{\rm{L}}}}\right){\tilde{a}}_{1(2)}+{{\rm{i}}}\kappa {\tilde{a}}_{2(1)}\\ +{{\rm{i}}}\sqrt{{\gamma }_{1(2)}^{{{\rm{F}}}}}\left({S}_{{{\rm{in}}}}^{{{\rm{F}}}}+{{\rm{i}}}\sqrt{{\gamma }_{1(2)}^{{{\rm{L}}}}}{\tilde{a}}_{1(2)}+{{\rm{i}}}\left(\sqrt{{\gamma }_{2(1)}^{{{\rm{F}}}}}+\sqrt{{\gamma }_{2(1)}^{{{\rm{L}}}}}\right){\tilde{a}}_{2(1)}\right)$$

(2)

where ${S}_{{{\rm{in}}}}^{{{\rm{F}}}}$ denotes the incident flexural waves, ${\gamma }_{1(2)}^{{{\rm{F}}}({{\rm{L}}})}$ is the radiative decay rate of flexural (longitudinal) waves for pillar ${{{\rm{p}}}}_{1(2)}$. The term ${{\rm{i}}}\kappa {\tilde{a}}_{1(2)}$ describes the near-field coupling between bilateral pillars induced by decaying flexural waves. The term ${{\rm{i}}}\sqrt{{\gamma }_{1(2)}^{{{\rm{L}}}}}{\tilde{a}}_{1(2)}$ denotes the multi-polarization hybridization induced by the resonance ${\tilde{a}}_{1(2)}$. Terms ${{\rm{i}}}\sqrt{{\gamma }_{2(1)}^{{{\rm{F}}}}}{\tilde{a}}_{2(1)}$ and ${{\rm{i}}}\sqrt{{\gamma }_{2(1)}^{{{\rm{L}}}}}{\tilde{a}}_{2(1)}$ describes the reradiation flexural and longitudinal fields arose from the resonance ${\tilde{a}}_{2(1)}$, respectively.

In the absence of incident flexural waves and converted radiative longitudinal waves ${\gamma }_{1(2)}^{{{\rm{L}}}}$, Eq. (2) can be expressed as $-{{\rm{i}}}\partial {{\bf{a}}}/\partial t={{\bf{H}}}{{\bf{a}}}$, where ${{\bf{a}}}={[{\tilde{a}}_{1},{\tilde{a}}_{2}]}^{{{\rm{T}}}}$ is the amplitude vector, ${{\bf{H}}}$ is the Hamiltonian matrix, which can be expressed as

$${{\bf{H}}}=\left[\begin{array}{cc}{\omega }_{1} & 0\\ 0 & {\omega }_{2}\end{array}\right]+{{\rm{i}}}\left[\begin{array}{cc}{\gamma }_{1}^{{{\rm{F}}}} & \sqrt{{\gamma }_{1}^{{{\rm{F}}}}{\gamma }_{2}^{{{\rm{F}}}}}-{{\rm{i}}}\kappa \\ \sqrt{{\gamma }_{1}^{{{\rm{F}}}}{\gamma }_{2}^{{{\rm{F}}}}}-{{\rm{i}}}\kappa & {\gamma }_{2}^{{{\rm{F}}}}\end{array}\right]$$

(3)

The eigenvalues of ${{\bf{H}}}$ can be expressed as

$${\omega }_{\pm }=\frac{{\omega }_{1}+{\omega }_{2}+{{\rm{i}}}({\gamma }_{1}^{{{\rm{F}}}}+{\gamma }_{2}^{{{\rm{F}}}})}{2}\pm \frac{\sqrt{{[{{\rm{i}}}({\gamma }_{1}^{{{\rm{F}}}}-{\gamma }_{2}^{{{\rm{F}}}})+{\omega }_{1}-{\omega }_{2}]}^{2}-{[2(\sqrt{{\gamma }_{1}^{{{\rm{F}}}}{\gamma }_{2}^{{{\rm{F}}}}}-{{\rm{i}}}\kappa )]}^{2}}}{2}$$

(4)

According to Eq. (4), if the following condition is satisfied¹

$$\kappa ({\gamma }_{1}^{{{\rm{F}}}}-{\gamma }_{2}^{{{\rm{F}}}})=\sqrt{{\gamma }_{1}^{{{\rm{F}}}}{\gamma }_{2}^{{{\rm{F}}}}}({\omega }_{1}-{\omega }_{2})$$

(5)

one eigenvalue ${\omega }_{+}=\bar{\omega }+\bar{\gamma }+{{\rm{i}}}({\gamma }_{1}^{{{\rm{F}}}}+{\gamma }_{2}^{{{\rm{F}}}})$ becomes more lossy, while the other eigenvalue ${\omega }_{-}=\bar{\omega }-\bar{\gamma }$ is a purely real value, where $\bar{\omega }=({\omega }_{1}+{\omega }_{2})/2$ and $\bar{\gamma }=\kappa ({\gamma }_{1}^{{{\rm{F}}}}+{\gamma }_{2}^{{{\rm{F}}}})/2\sqrt{{\gamma }_{1}^{{{\rm{F}}}}{\gamma }_{2}^{{{\rm{F}}}}}$. The real eigenvalue indicates the so-called FW BIC with an infinite Q. Specially, identical resonators have the same resonance frequency and radiative decay rate, and the near-filed interaction satisfies $\kappa =0$. Then, there are ${\omega }_{+}=\bar{\omega }+2{{\rm{i}}}{\gamma }_{1}^{{{\rm{F}}}}$ and ${\omega }_{-}=\bar{\omega }$. The symmetrically bilateral pillars discussed in Fig. 2a correspond to this case. They induce the FW BIC at ${f}_{{{\rm{S1}}}}$ that corresponds to the symmetric eigenmode, which also known as the symmetry-protected BIC, resulting in trapped waves within the pillars.

Construction of simultaneous Friedrich-Wintgen and accidental BICs

When considering the waveguide ${{{\rm{p}}}}_{3}$ with a length ${l}_{3}$, the interference relation between incident flexural waves and scattering field will be reshaped. In this scenario, FW BICs are induced by bilateral pillars, while accidental BICs arise from the destructive interference modulated by the waveguide ${{{\rm{p}}}}_{3}$. The construction of FW BICs is similar to that described in the above section. To clearly state the accidental BICs, the bilateral pillars are treated as a unified resonator $\bar{{{\rm{p}}}}$. As shown in Fig. 1f, the Fano resonance of $\bar{{{\rm{p}}}}$ leaks leftward scattering flexural waves to the host plate (channel one) and rightward scattering flexural waves to the waveguide. The latter will be entirely reflected by the right boundary of the waveguide, with a portion being transmitted through the resonator $\bar{{{\rm{p}}}}$ to the host plate (channel two). The destructive interference in these two channels can eliminate the radiation of flexural waves, resulting in the emergence of a trapped mode.

As shown in Fig. 1f, the resonator $\bar{{{\rm{p}}}}$ hybridizes the incoming flexural waves (${F}_{1+}$, ${F}_{2+}$), incoming longitudinal waves (${L}_{1+}$, ${L}_{2+}$), outcoming flexural waves (${F}_{1-}$, ${F}_{2-}$), and outcoming longitudinal waves (${L}_{1-}$, ${L}_{2-}$). According to the temporal coupled-mode theory, the incoming and outcoming waves satisfy⁵⁴

$${{{\bf{s}}}}_{-}={{\bf{C}}}\cdot {{{\bf{s}}}}_{+}+{{\bf{d}}}a$$

(6)

where ${{{\bf{s}}}}_{-}={[\begin{array}{cccc}{F}_{1-} & {F}_{2-} & {L}_{1-} & {L}_{2-}\end{array}]}^{{{\rm{T}}}}$ is the outcoming vector, ${{{\bf{s}}}}_{+}={[\begin{array}{cccc}{F}_{1+} & {F}_{2+} & {L}_{1+} & {L}_{2+}\end{array}]}^{{{\rm{T}}}}$ is the incoming vector, the term ${{\bf{d}}}a$ denotes the radiation from resonance, ${{\bf{d}}}={[\begin{array}{cccc}{d}_{1} & {d}_{2} & {d}_{3} & {d}_{4}\end{array}]}^{{{\rm{T}}}}$ is the coupling vector between resonance mode and outcoming waves, $a$ is the amplitude of the localized mode inside the resonator, satisfying $da/dt=(i{\omega }_{0}-{\gamma }_{0})a$ in the absence of input, ${\gamma }_{0}$ is the radiative decay. ${{\bf{C}}}$ denotes the direct coupling between incoming and outcoming waves without resonator (background scattering), which can be expressed as

$$C=\left[\begin{array}{cccc}0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right]$$

(7)

On the other hand, according to the scattering relation by the resonator $\bar{{{\rm{p}}}}$, the radiation at the left and right ports can be respectively expressed as⁶

$${d}_{1}a={t}_{{{\rm{F}}}}{F}_{2+}-{F}_{2+}+{t}_{{{\rm{LF}}}}{L}_{2+}$$

(8)

$${d}_{2}a={r}_{{{\rm{F}}}}{F}_{2+}+{r}_{{{\rm{LF}}}}{L}_{2+}$$

(9)

where ${t}_{{{\rm{F}}}}$ and ${r}_{{{\rm{F}}}}$ are the transmission and reflection coefficients of flexural waves by the resonator, excluding the effects of the left and right boundaries. ${t}_{{{\rm{LF}}}}$ and ${r}_{{{\rm{LF}}}}$ are the coefficients converted from longitudinal waves. The derivation of these coefficients is provided in the Supplementary Note 2.

According to Eqs. (6) to (9) and the propagation relation of ${F}_{2-}={{{\rm{e}}}}^{-{{\rm{i}}}\phi }{F}_{2+}$, the destructive interference relation can be obtained and is expressed as

$${{{\rm{e}}}}^{-{{\rm{i}}}\phi }={r}_{{{\rm{F}}}}+{t}_{{{\rm{F}}}}$$

(10)

where $\phi =2{k}_{{{\rm{F}}}}{l}_{3}-3{{\rm{\pi }}}/2$ is the phase shift of one round-trip of flexural waves in the waveguide, ${k}_{{{\rm{F}}}}$ is the wavenumber of flexural waves. The term $3{{\rm{\pi }}}/2$ is induced by the decaying mode of flexural waves. On the other hand, the amplitude of the total scattering field satisfies ${|{r}_{{{\rm{F}}}}+{t}_{{{\rm{F}}}}|}^{2}=1$⁶, which means that the amplitudes in the left and right sides of Eq. (10) are always satisfied. Then, Eq. (10) can be expressed by its phase relation

$$2{k}_{{{\rm{F}}}}{l}_{3}-\frac{3{{\rm{\pi }}}}{2}={{\mbox{arg}}}({r}_{{{\rm{F}}}}+{t}_{{{\rm{F}}}})+2(n-1){{\rm{\pi }}}$$

(11)

where $n$ is a positive integer. The accidental BICs can be formed under the destructive interference condition described by Eq. (11). We should point out that the multi-polarization hybridization is a crucial factor in inducing accidental BICs based on the waveguide coupled with a resonator. The reason is that in the absence of multi-polarization hybridization, the wavefields only consists of a single mode that will be inevitably leaky, without any localized state.

Demonstration of elastic FW BICs

To demonstrate the elastic BICs with multi-polarization hybridization, we analytically solve the reflected coefficient ${R}_{{{\rm{F}}}}$ of flexural waves and corresponding energy conversion ratio $\delta$ to longitudinal waves. Here, ${R}_{{{\rm{F}}}}$ is the reflection coefficient of the metamaterial plate shown in Fig. 1a. This differs from the aforementioned ${r}_{{{\rm{F}}}}$, which represents the reflection coefficient of a single pair of bilateral pillars based on a two-dimensional model that does not consider boundary reflections and waveguide ${{{\rm{p}}}}_{3}$. See details in Supplementary Note 3. By excluding the outermost waveguide ${{{\rm{p}}}}_{3}$, we examine elastic FW BICs by focusing solely on bilateral pillars ${{{\rm{p}}}}_{1}$ and ${{{\rm{p}}}}_{2}$ at the end. The analytical dimensionless reflected energy of normal incident flexural waves ${|{R}_{{{\rm{F}}}}|}^{2}$, varying with the height ${l}_{1}$ of pillar ${{{\rm{p}}}}_{1}$ and the frequency, is shown in Fig. 3a. The height ${l}_{2}$ of pillar ${{{\rm{p}}}}_{2}$ is fixed as ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}=8.65$. The simulated reflected flexural energy, shown in Fig. 3b, closely aligns with the analytical results shown in Fig. 3a, confirming the validity of our derivation. Besides, the unitary total reflected energy of flexural and longitudinal waves (see Supplementary Note 4) throughout the calculation range, indicating the energy conservation and no other mode conversion. In Fig. 3a, the white dash-dotted line represents the first-order Fano resonance of the fixed pillar ${{{\rm{p}}}}_{2}$, while the pink dotted lines represent the $m{{\rm{th}}}$-order Fano resonances of pillar ${{{\rm{p}}}}_{1}$ with varying heights, as determined by Eq. (1). The intersections of these lines, marked by white circles, correspond to the vanishing linewidths, indicating ideal FW BICs with infinite $Q$ factor. These multibranch ideal BICs are induced by degenerate resonances of the bilateral pillars according to Eq. (5). Figure 3a illustrates that bilateral pillars exhibiting degenerate resonances but different eigenmodes (i.e., $m=2$ or 3) also induce ideal FW BICs, unlike conventional realizations of two identical ones.

Fig. 3: Demonstration of FW BICs based on solely bilateral pillars.

a Analytical and b simulated dimensionless reflected energy of flexural waves varying with height ${l}_{1}$ and frequency. ${l}_{2}$ is fixed with ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}$. Points A, B and C marks the perfect mode conversion. c Analytical and simulated mode conversion spectra for bilateral pillars of ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{1}$ and ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}$ (corresponding to Point A), as well as for single-side pillar of ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{1}$ and ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{2}$. d Incident (${w}_{{{\rm{in}}}}$) and reflected (${w}_{{{\rm{r}}}}$ and ${u}_{{{\rm{r}}}}$) wavefields corresponding to point A. e Simulated phase (indicated by color) and x-direction displacement (represented by directions and sizes of arrows) of the bilateral pillars at 0.7 kHz, 1.23 kHz, and 1.7 kHz. f Representations of $\log |{R}_{{{\rm{F}}}}|$ in complex frequency plane for ${{{\rm{p}}}}_{2}$ with different heights.

To achieve perfect mode conversion from incident flexural waves to reflected longitudinal waves, the height ${l}_{1}$ should be adjusted to detune the Fano resonance frequency and introduce the multi-polarization hybridization, thereby constructing quasi-BICs. In such a case, the purely real eigenvalue turns to a complex one with a tiny imaginary component, which results in a small leakage of flexural waves characterized by the radiative decay rate ${\gamma }_{{{\rm{F}}}}$. The leaky energy of flexural waves will exactly balance that of longitudinal waves (with the radiative decay rate ${\gamma }_{{{\rm{L}}}}$) induced by the multi-polarization hybridization at Fano resonances, under the balance condition of ${\gamma }_{{{\rm{F}}}}={\gamma }_{{{\rm{L}}}}$. As a result, the incident flexural waves will be completely trapped within the pillars without any backscattering. Only the converted longitudinal waves are allowed to leak, thus achieving the perfect mode conversion at quasi-BICs.

As shown in Fig. 3a, at multiple quasi-BICs realized by detuned Fano resonances of different orders, as marked by A, B and C, the reflected energies of flexural waves all reach zero, indicating that the perfect mode conversion to longitudinal waves is realized. It is necessary to point out that although only the low frequency range near the first-order Fano resonance of pillar ${{{\rm{p}}}}_{2}$ is considered, near the second one, the perfect mode conversion can also be realized by similar quasi-BICs, which are not depicted here. The comparison between a single-side pillar (see Supplementary Note 4) provides further compelling evidence that the perfect mode conversion is induced by FW quasi-BICs. The energy conversion ratio $\delta$ for the configuration corresponding to point A (${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{1}=12.35$, ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}=8.65$) is presented as an example to demonstrate the perfect mode conversion, as shown in Fig. 3c. For comparison, $\delta$ for single-side pillars with the heights of ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{1}$ and ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{2}$ are also depicted. The analytical and simulated mode conversion spectra agree well, with the peak value for point A approaching one. Accordantly, at this peak frequency (1.23 kHz), the incident flexural waves are converted into reflected longitudinal one, with almost no reflected flexural waves present, as shown in Fig. 3d.

Further, we extract the phase (indicated by color) and x-direction displacement (represented by directions and sizes of arrows) of the bilateral pillars at 1.23 kHz, along with those at 0.7 kHz and 1.7 kHz for comparisons, as shown in Fig. 3e. At 0.7 kHz and 1.7 kHz, far from the peak, only one pillar couples with the host plate. In contrast, at 1.23 kHz (quasi-BIC), both bilateral pillars strongly couple with the host plate. Their normal deformations align in the same direction, which is responsible for the generation of hybridized longitudinal waves in the host plate, thereby achieving the perfect mode conversion from flexural to longitudinal waves. In fact, this kind of perfect mode conversion realized by the balance between ${\gamma }_{{{\rm{F}}}}$ and ${\gamma }_{{{\rm{L}}}}$ is similar with the so-called critical coupling condition⁵⁵, but here is fulfilled by different radiative modes. As evidence, the representations of $\log |{R}_{{{\rm{F}}}}|$ in a complex frequency plane (see Supplementary Note 5) for ${{{\rm{p}}}}_{2}$ with different heights are analyzed, as shown in Fig. 3f. The higher and lower heights of ${l}_{2}$, corresponding to zero points below and above the real frequency axis, are similar with under and over losses in conventional optical and acoustic systems. For Point A, the zero point exactly locates on the real frequency axis, indicating the fulfilled critical coupling condition.

Demonstration of simultaneous FW and accidental quasi-BICs

Based on above discussions, here, the heights of bilateral pillars are set as ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{1}$ and ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}$, respectively, the same as those corresponding to Point A in Fig. 3a. In this scenario, the analytical and simulated dimensionless reflected energy of flexural waves versus waveguide length ${l}_{3}$ and frequency are shown in Fig. 4a, b, respectively. The trends of analytical and simulated results are in good consistence, further confirming the validity of the analytical model. In Fig. 4a, the white dash-dotted lines denotes the Fano resonance frequencies ${f}_{1}$ and ${f}_{2}$ of the bilateral pillars, while the green solid lines correspond to destructive interference relations predicted by Eq. (11). The predicted destructive interference relations show slight deviations from actual relations at short lengths, which is due to the influence of decaying flexural waves. When the length ${l}_{3}$ exceeds half of the wavelength at the Fano resonance frequency, the intersections of destructive interference relations and Fano resonances can accurately predict ideal accidental BICs (see Supplementary Note 4).

Fig. 4: Demonstration of coexistence of FW and accidental quasi-BICs.

a Analytical and b simulated dimensionless reflected energy of flexural waves varying with frequency and waveguide length ${l}_{3}$ in the scenario of ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{1}$ and ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}$. c Reflected flexural energy considering only singe-side pillar ${{{\rm{p}}}}_{1}$ with the height of ${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{2}$. d Mode conversion spectra corresponding to lengths ${L}_{1}$ and ${L}_{2}$ in a. e Simulated x- and z-direction phases (indicated by color) and displacements (represented by directions and sizes of arrows) corresponding to points ${{{\rm{P}}}}_{1}$ and ${{{\rm{P}}}}_{2}$.

Despite slight deviations at shorter lengths of ${l}_{3}$ considered here, it is evident that the ideal accidental BICs with vanishing linewidth, marked by white circles, are induced due to the crossings of destructive interference and Fano resonance. Similarly, the waveguide length ${l}_{3}$ should be tuned to achieve quasi-BICs with perfect mode conversion. As shown in Fig. 4a, the quasi-BICs near the two flexural resonance frequencies ${f}_{1}$ and ${f}_{2}$ differ significantly in linewidth and sensitivity to ${l}_{3}$. The wider linewidth and lower sensitivity to ${l}_{3}$ near ${f}_{2}$ is because the bilateral pillars induce FW quasi-BICs, thereby the radiative decay rate balance ${\gamma }_{{{\rm{F}}}}={\gamma }_{{{\rm{L}}}}$ is slightly influenced by the waveguide with a length mismatching the destructive interference relation. On the contrary, the BICs near ${f}_{1}$ is purely accidental BICs achieved by destructive interferences, with the corresponding quasi-BICs highly influenced by the waveguide length. For comparison, the reflected energy for the metamaterial with periodically single-side pillar (${l}_{1}/{h}_{{{\rm{b}}}}={\mu }_{2}$) versus waveguide length is shown in Fig. 4c. In this scenario, only accidental BICs induced by destructive interference sensitive to ${l}_{3}$ are realized, as marked by white circles.

Thanks to the coexistence of FW and accidental quasi-BICs, perfect mode conversion at dual frequencies can be realized at the same waveguide length ${l}_{3}$, such as frequencies according with points ${{{\rm{P}}}}_{1}$ and ${{{\rm{P}}}}_{2}$ (corresponding to length ${L}_{1}$), as well as ${{{\rm{P}}}}_{3}$ and ${{{\rm{P}}}}_{4}$ (corresponding to length ${L}_{2}$) in Fig. 4a. For these two lengths, Fig. 4d shows the mode conversion spectra, where both peak values approach one for each case. The linewidth for the second peak is wider than that for the first one, resulted from the relatively larger radiative decay rate of the system at FW quasi-BICs. To demonstrate this point, for the waveguide length ${L}_{1}$, we extract the x- and z-direction phases (indicated by color) and displacements (represented by directions and sizes of arrows) at frequencies corresponding to points ${{{\rm{P}}}}_{1}$ and ${{{\rm{P}}}}_{2}$, respectively, as shown in Fig. 4e. For points ${{{\rm{P}}}}_{1}$ corresponding to an accidental quasi-BIC, the upper pillar and the waveguide exhibit a strong coupling, inducing the destructive interference in the host plate, and consequently leading to trapped flexural modes with perfect mode conversion. In this case, the lower pillar contributes minimally. Differently, for points ${{{\rm{P}}}}_{2}$ corresponding to a FW quasi-BIC, the waveguide at the end is minimally involved, and the quasi-BIC arises from the coupling between bilateral pillars, similar to those shown in Fig. 3e. Besides, the amplitude of trapped flexural modes for ${{{\rm{P}}}}_{1}$ is much larger than that for ${{{\rm{P}}}}_{2}$, compared with the results shown in Fig. 4e, indicating greater energy leakage at FW quasi-BICs. Consequently, the linewidth of FW quasi-BICs is wider than that of accidental BICs.

Continuously tuned high Q factor

Above demonstrated FW and accidental BICs are analyzed for normal incident flexural waves. For oblique incident flexural waves, there is a critical frequency when the radiation angle ${\theta }_{{{\rm{L}}}}^{{{\rm{r}}}}$ of converted longitudinal waves equals to $90^\circ$. The radiation angle is determined by ${\theta }_{{{\rm{L}}}}^{{{\rm{r}}}}=\arcsin ({k}_{{{\rm{F}}}}\,\sin {\theta }_{{{\rm{in}}}}/{k}_{{{\rm{L}}}})$, from which, we can obtain the critical frequency ${f}_{{{\rm{c}}}}=6{\sin }^{2}{\theta }_{{{\rm{in}}}}/\pi \sqrt{\rho {h}_{{{\rm{b}}}}^{5}/{D}_{{{\rm{F}}}}}$ that is relevant to the incident angle ${\theta }_{{{\rm{in}}}}$. Below the critical frequency ${f}_{{{\rm{c}}}}$, the calculated ${\theta }_{{{\rm{L}}}}^{{{\rm{r}}}}$ is imaginary, and the impendence of longitudinal waves is infinite, which will suppress the radiation of longitudinal waves. Therefore, the mode conversion disappears below the critical frequency. Using dimensionless incident angles ${\theta }_{{{\rm{in}}}}\cdot {l}_{2}/{h}_{{{\rm{b}}}}=28$ as an example, Fig. 5a show the reflected flexural energy varying with the height ${l}_{1}/{h}_{{{\rm{b}}}}$ and frequency (${l}_{2}$ the same as that used in Fig. 3a). The green dashed and white dash-dotted lines correspond to the critical frequency ${f}_{{{\rm{c}}}}$ and Fano resonance frequency of ${{{\rm{p}}}}_{2}$, respectively. Below the critical frequency, the unitary dimensionless energy is achieved due to the disappeared mode conversion. When the critical frequency approaches the resonance one, the quasi-BICs with perfect mode conversion (marked by blue crosses) close to ideal BICs (marked by white circles), resulting in a higher $Q$ factor and narrower linewidth. It means that the $Q$ factor can be fine-tuned by adjusting the critical frequency relevant to the incident angle.

Fig. 5: Demonstration of continuously tuned high $Q$ factor.

a Dimensionless reflected energy by periodically bilateral pillars varying with height ${l}_{1}$ and frequency for ${l}_{2}/{h}_{{{\rm{b}}}}={\mu }_{2}$, when the incident angle of flexural waves satisfies ${\theta }_{{{\rm{in}}}}\cdot {l}_{2}/{h}_{{{\rm{b}}}}=28$. b Mode conversion spectra at different incident angles. c $Q$ factors of quasi-BICs with perfect mode conversion tuned by incident angles. d, e $Q$ factors of quasi-BICs with perfect mode conversion tuned by height deviation $\Delta l$, under the detuning of first-first and first-second order resonances, respectively. For different deviation values, the quasi-BICs with perfect mode conversion is realized by modulating the waveguide length. f, g Dimensionless reflected energy of flexural waves versus frequency and waveguide length for $\Delta l/{h}_{{{\rm{b}}}}=0.05$ and $\Delta l/{h}_{{{\rm{b}}}}=13.75$.

For instance, considering the FW BIC induced by the degenerate first-first order Fano resonances of the bilateral pillars, the corresponding mode conversion spectra at different incident angles are depicted in Fig. 5b, with all peak values reaching one. Based on the mode conversion spectra, Fig. 5c shows the $Q$ factors calculated as $Q={\bar{f}}_{{{\rm{c}}}}/\Delta f$, where ${\bar{f}}_{{{\rm{c}}}}$ is the peak frequency, and $\Delta f$ is the full width at half maximum of the peak. At ${\theta }_{{{\rm{in}}}}\cdot {l}_{2}/{h}_{{{\rm{b}}}}=36$, the linewidth is less than 1 Hz, and the $Q$ factor exceeds 7000, indeed approaching a BIC with infinite $Q$. It should be noted out that the $Q$ factor tuned by adjusting the critical frequency is both suitable for FW and accidental BICs, although we take FW BICs as examples.

Besides, there is another method for achieving high $Q$ factor for quasi-BICs with perfect mode conversion, which involves detuned bilateral pillars along with a precisely modulated waveguide. Figure 3a has indicated that multibranch ideal BICs are achieved when the bilateral pillars with degenerate resonances (white circles). The corresponding pillar heights are referred to as ideal heights. If there is a slight deviation from the ideal height in one of the bilateral pillars, a small detuning in the Fano resonances between the bilateral pillars will be introduced. This detuning allows a minor energy leakage of flexural waves, which will balance the leaked energy of longitudinal waves by approaching the destructive interference relation, realized by precisely modulated waveguide length. The closer the pillar height is to the ideal one, the smaller the resonance detuning, and the less the energy leakage, resulting in a higher $Q$ factor for quasi-BICs with perfect mode conversion.

To demonstrate the high $Q$ factor realized by detuned bilateral pillars, we examine two scenarios: the detuning between the first-first order and first-second order Fano resonances of bilateral pillars, as indicated by the inserts in Fig. 5d and e, respectively. Figure 5d and e show the corresponding $Q$ factors varying with the height deviation $\Delta l/{h}_{{{\rm{b}}}}$, where $\Delta l={l}_{2}-{l}_{1}$. For the first kind of detune, a zero deviation $\Delta l=0$ indicates completely symmetric bilateral pillars. A slight deviation of $\Delta l=0.01$ leads to the $Q$ factor exceeding ${10}^{6}$, and with the increasing of deviation, $Q$ factor decreases gradually due to the increased energy leakage. Similarly, for the second kind of detune, the first-order Fano resonance frequency of ${{{\rm{p}}}}_{1}$ approaches the second-order one of ${{{\rm{p}}}}_{2}$ near the deviation of $\Delta l/{h}_{{{\rm{b}}}}=13.8$, leading to a high $Q$ factor about ${10}^{7}$, actually approaching an infinite $Q$ factor. Deviating $\Delta l/{h}_{{{\rm{b}}}}$ larger or smaller than 13.8 both results in a decreased $Q$ factor due to the increasing detuning. As examples, Fig. 5f and g show the dimensionless reflected energy of flexural waves for $\Delta l/{h}_{{{\rm{b}}}}=0.05$ and $\Delta l/{h}_{{{\rm{b}}}}=13.75$, respectively corresponding to the detuning of first- and second-order Fano resonances. The quasi-BICs with perfect mode conversion are also displayed, marked by white circles. For both cases, the positions of quasi-BICs almost overlap with those of corresponding ideal BICs. The linewidths are extremely narrow and are almost invisible in the scales of Fig. 5f and g, consistent with the high $Q$ factors. The $Q$ factor can be tuned continuously by the detuned bilaterial pillars connected to a meticulously modulated waveguide.

Experimental validations

In the experiment, it is challenging to generate ideal plane waves using an array of several piezoelectric patches. To facilitate ideal excitation, simplified strip-like beam models are considered, as illustrated in Fig. 6a, requiring only a single exciting piezoelectric patch. It has been demonstrated that for normal incidence, the results from the simplified model align well with those from the corresponding periodic unit cells separated by small slits⁶. As shown in Fig. 6a, we fabricated two specimens: specimen one contains detuned bilateral pillars at both ends of the beam, while specimen two contains detuned bilateral pillars connected to a waveguide at the right end. Specimen one is tested to verify the perfect mode conversion induced by the FW quasi-BIC, corresponding to point A in Fig. 3a. Specimen two is tested to demonstrate the dual-frequency perfect mode conversions induced by simultaneous FW and accidental quasi-BICs, corresponding to length ${L}_{1}$ in Fig. 4a. The parameters are provided in Supplementary Note 8. In experiments, the inherent damping of the 3D printed material should be considered. The influences of material damping on BIC behaviors are analyzed in Supplementary Note 6.

Fig. 6: Experimental demonstrations.

a Experimental setup. The two test specimens with the end featuring detuned bilateral pillars (specimen one) and detuned bilateral pillars connected to a waveguide (specimen two) are presented. Specimen one contains unit cells in both ends to intuitively observe mode conversion phenomenon. b and c Tested capture coefficient $\varsigma$ of flexural waves for the two test specimens. Analytical and simulated results are also displayed for comparison. d Snapshots of Supplementary Movie 1 and 2 for specimen one at frequencies of 1.23 kHz and 1.6 kHz, corresponding to traveling and standing waves, respectively. e Normalized measured velocity response versus time for specimen one subjected to a 5-peak tone-burst excitation. f Transient full wavefields at 14.6 ms and 16. 8 ms corresponding to times in e. g Normalized power spectra for incident and reflected responses marked in e.

We use the measured capture coefficient $\varsigma$ of flexural waves (see Methods section) to characteristic the mode conversion ratio $\delta$ to longitudinal waves since there are no other modes. As shown in Fig. 6b and c, the measured capture coefficients for specimens one and two are presented alongside corresponding analytical and simulated results for comparison. For specimen one, the measured capture coefficient $\varsigma$ reaches 0.95 at the peak frequency 1.23 kHz in Fig. 6b, with a corresponding Q factor of about 9. For specimens two, the measured capture coefficients reach 0.91 and 0.97 at the two peak frequencies (0.616 kHz and 1.288 kHz) in Fig. 6c, with the corresponding Q factors of 40 and 16.5, respectively. The alignment of the curves from these three methods, along with the measured high capture coefficients, confirms the perfect mode conversion by FW quasi-BIC and the dual-frequency perfect mode conversions enabled by the coexistence of FW and accidental quasi-BICs.

To clearly demonstrate the perfect mode conversion, we also measure the transient full wavefields of the two specimens. The time-varying full wavefields for specimen one at the peak frequency of 1.23 kHz and a comparison frequency of 1.6 kHz are shown in Supplementary Movie 1 and 2, respectively. For specimen two, the time-varying full wavefields at the two peak frequencies (0.616 kHz and 1.288 kHz) are provided in Supplementary Movie 3 and 4, respectively, and the wavefield at a comparison frequency of 0.71 kHz is shown in Supplementary Movie 5. Results presented in Supplementary Movies show that for both specimens, the wavefields resemble traveling waves at peak frequencies, but exhibit standing wave patterns at other compared frequencies. It indicates that at the peak frequencies, most of flexural waves are trapped in the pillars or waveguide. The allowed small amount of leaked flexural waves are perfectly converted to longitudinal waves without reflection, consistent with the simulated findings—specifically, point A in Fig. 3a, as well as points ${{{\rm{P}}}}_{1}$ and ${{{\rm{P}}}}_{2}$ in Fig. 4a. To avoid repetition and provide a more detailed analysis, we focus on specimen one, which demonstrates perfect mode conversion at point A.

Figure 6d presents snapshots from Supplementary Movie 1 and 2 for specimen one, corresponding to the peak frequency of 1.23 kHz and a comparison frequency of 1.6 kHz, respectively. Note that the transient normal wavefields shown in Fig. 6d include the incident wave components. This makes them different from the result shown in Fig. 3d, where the normal displacement is zero due to the removal of incident waves. The absorption area (blue tack) in the middle region can completely absorb flexural waves, but an observable flexural wavefield remains in the left test area at 1.23 kHz. This occurs because the longitudinal waves converted by the right bilateral pillars cannot be fully absorbed by the absorption area, and are reconverted to flexural waves by the left bilateral pillars due to the reciprocity. In contrast, at 1.6 kHz far away from the peak, as seen in Supplementary Movie 2, the wavefield in the right test area forms a standing wave, and the normal velocity in the left test area is near zero, indicating the little mode conversion, as shown in Fig. 6d.

As further evidence, a 5-peak tone-burst signal with the central frequency of 1.23 kHz is excited to measure the transient full wavefields. The measured normalized out-of-plane velocity response of a probe over time is shown in Fig. 6e, which contains mainly two wave packets: the incident and reflected responses, highlighted in the blue and green regions. The transient full wavefields for incident (14.6 ms) and reflected (16.8 ms) waves are shown in Fig. 6f. Both the transient full wavefields and probe response indicate a significantly reduction in the overall amplitude of the reflected flexural waves. As illustrated by Fig. 6e, the amplitude at the center of the reflected wave packet (indicated by blue arrow), corresponding to the central frequency of 1.23 kHz, approaches zero. This demonstrates that the small amount of leaked flexural waves at quasi-BICs are perfectly converted to longitudinal waves. To further confirm this, we perform a fast Fourier transform on the incident and reflected responses to obtain their power spectra, as shown in Fig. 6g. Near-zero amplitude in the central frequency 1.23 kHz is found, thanks to the prefect mode conversion induced by the quasi-BIC. Overall, the experiments in both the frequency and time domain modes are consistent, demonstrating the perfect mode conversion enabled by quasi-BICs with multi-polarization hybridization.

Conclusions

In this study, we theoretically, numerically, and experimentally demonstrate elastic quasi-BICs with multi-polarization hybridization, enabling perfect mode conversion between flexural and longitudinal waves by bilateral pillars connected to a waveguide. We demonstrate that FW BICs can be induced by bilateral pillars with degenerate resonances, whether the eigenmode profiles are identical or distinct. Besides, the system supports the coexistence of FW and accidental quasi-BICs, with the later arose from the encounter of Fano resonance and destructive interference. The quasi-BICs with perfect mode conversion enables a continuous tunable of $Q$ factors, from single digits to millions. The high $Q$ factors can be achieved either by adjusting the critical frequency to approach the Fano resonance or slightly detuning the bilateral pillars to allow tiny energy leakage. This work provides insights into the physics of elastic BICs with multi-polarization hybridization. The quasi-BICs enabling perfect mode conversion from flexural to longitudinal waves offers an approach for structural vibration control, while those with high $Q$ factors have potential applications in high-resolution filtering and high-sensitivity sensing of elastic waves.

Methods

Numerical simulations

All simulations in this work are conducted in the Solid Mechanics modulus in the software of COMSOL Multiphysics. Results in Figs. 3, 4, 6 are obtained by conducting the Frequency Domian study. A perfectly matched layer with a length triples the wavelength is applied at the left end to avoid boundary reflections. Periodic boundary condition is applied at the two side boundaries. Results in Fig. 2 are obtained by conducting the Eigenfrequency study with the left and right boundary condition of the host set as Bloch boundaries. The material parameters pertain to a 3D-printed resin material, and are determined through experimental measurements (refer to Supplementary Note 8).

Experimental setup

The test specimens are fabricated using photosensitive resin through 3D printing. See Supplementary Note 7 for details. As shown in Fig. 6a, a PZT patch ($20\,{{\rm{mm}}}\times 20\,{{\rm{mm}}}\times 0.5\,{{\rm{mm}}}$), driven by the built-in generator in PSV-500, is bonded on the beam to excite the incident flexural waves. A blue-tack layer, covering more than two-wavelength region, is pasted on the middle region of specimen one and the left region of specimen two to simulate a non-reflective boundary for flexural waves. The out-of-plane velocities in the corresponding scanning areas are measured using a PSV-500 vibrometer. The results shown in Supplementary Movies are obtained in the time domain mode under the excitation of a single-frequency sinusoidal signal, while the results in Fig. 6e, f are measured using a 5-peak tone-burst signal with the amplitude satisfying $A(t)=[1-\,\cos (2\pi {f}_{{{\rm{c}}}}t/5)]\sin (2\pi {f}_{{{\rm{c}}}}t)$, where ${f}_{{{\rm{c}}}}=1.23$ kHz is the central frequency. For determining the capture coefficient $\varsigma$ of flexural waves, the out-of-plane complex velocities ${v}_{1}$ and ${v}_{2}$ of points 1 and 2 (see Supplementary Fig. S7) are measured in the frequency domain mode under a chirp signal. The distance ${s}_{1}$ between the two point is less than one eighth wavelength. The distance of point 1 from the PZT patch and that of point 2 from the bilateral pillars (${s}_{2}$) are larger than a wavelength to ensure the far-filed assumption. The transfer function between the two points can be expressed as ${H}_{12}={v}_{1}/{v}_{2}$. Then, the capture coefficient can be obtained by

$$\varsigma =1-\frac{{H}_{12}-{{{\rm{e}}}}^{-{{\rm{i}}}{k}_{{{\rm{F}}}}{s}_{1}}}{{{{\rm{e}}}}^{{{\rm{i}}}{k}_{{{\rm{F}}}}{s}_{1}}-{H}_{12}}{{{\rm{e}}}}^{-2{{\rm{i}}}{k}_{{{\rm{F}}}}({s}_{1}+{s}_{2})}$$

(12)