Liyun Cao

Zhichun Yang
email: yangzc@nwpu.edu.cn

Yanlong Xu

Zhaolin Chen

Yifan Zhu

Shiwang Fan

Krupali Donda

Brice Vincent

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Pillared Elastic Metasurface with Constructive Interference for Flexural Wave Manipulation steering

Keywords: Elastic metasurfaces, Constructive interference, Flexural waves, Metamaterials, Elastic waves

A new conceptual design of pillared elastic metasurface (~0.495λ) with constructive interference (PEMC) is proposed to manipulate flexural waves.

• We systematically reveal the phase shift of transmitted waves across the improved subunits and establish an analytical model for multi-resonator subunits to accurately predict the phase shift and amplitude of the transmitted wave.

• The intrinsic physical of the constructive and destructive interferences through outof-plane vibration of the plate for the two adjacent subunits are analyzed.

• Finally, we experimentally verify deflecting and focusing functionality of the proposed PEMC without any drilling or grooving the original plate surface.

Introduction

Metasurface is one kind of ultrathin metamaterials, with a more compact and easily fabricated form. It has been recently proposed to realize extreme features of bulky metamaterials [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Zhu | Negative refraction of elastic waves at the deepsubwavelength scale in a single-phase metamaterial[END_REF][START_REF] Hu | On the modelling of membrane-coupled Helmholtz resonator and its application in acoustic metamaterial system[END_REF][START_REF] Zhu | Experimental study of an adaptive elastic metamaterial controlled by electric circuits[END_REF][START_REF] Zhu | Kirigami-based Elastic Metamaterials with Anisotropic Mass Density for Subwavelength Flexural wave Control[END_REF][START_REF] Lea Sirota | Tunable and reconfigurable mechanical transmissionline metamaterials via direct active feedback control[END_REF][START_REF] Zhou | Numerical and experimental investigation on broadband wave propagation features in perforated plates[END_REF][START_REF] Li | The SH0 wave manipulation in graded stubbed plates and its application to wave focusing and frequency separation[END_REF]. The studies on the metasurfaces with phase gradients, based on the generalized Snell's law (GSL) [START_REF] Yu | Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction[END_REF], have made a significant progress in the fields of optics [START_REF] Yu | Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction[END_REF][START_REF] Kildishev | Planar Photonics with Metasurfaces[END_REF][START_REF] Yu | Flat optics with designer metasurfaces[END_REF] and acoustics [START_REF] Assouar | Acoustic metasurfaces[END_REF][START_REF] Li | Tunable asymmetric transmission via Lossy acoustic metasurfaces[END_REF][START_REF] Zhu | Fine manipulation of sound via lossy metamaterials with independent and arbitrary reflection amplitude and phase[END_REF][START_REF] Fan | Tunable Broadband Reflective Acoustic Metasurface[END_REF]. Recently, the development of elastic metasurfaces, with complex inherent properties of different elastic wave types and mode conversions, has also attracted wide attentions of researchers.

The existing elastic metasurfaces can be classified into body wave metasurfaces [START_REF] Su | Elastic metasurfaces for splitting SV-and P-waves in elastic solids[END_REF][START_REF] Cao | Steering elastic SH waves in an anomalous way by metasurface[END_REF][START_REF] Xu | Steering of SH wave propagation in electrorheological elastomer with a structured meta-slab by tunable phase discontinuities[END_REF][START_REF] Longhai Zenga | Asymmetric transmission of elastic shear vertical waves in solids[END_REF][START_REF] Shen | Elastic wave manipulation by using a phasecontrolling meta-layer[END_REF], surface wave metasurfaces [START_REF] Xu | Deflecting Rayleigh surface acoustic wave by meta-ridge with gradient phase shift[END_REF], and plate wave metasurfaces [START_REF] Zhu | Anomalous Refraction of Acoustic Guided Waves in Solids with Geometrically Tapered Metasurfaces[END_REF][START_REF] Liu | Source Illusion Devices for Flexural Lamb Waves Using Elastic Metasurfaces[END_REF][START_REF] Lee | Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering[END_REF][START_REF] Zhang | Metasurface constituted by thin composite beams to steer flexural waves in thin plates[END_REF][START_REF] Tian | Elastic Phased Diffraction Gratings for Manipulation of Ultrasonic Guided Waves in Solids[END_REF][START_REF] Xu | Anomalous refraction control of mode-converted elastic wave using compact notch-structured metasurface[END_REF][START_REF] Xu | Deflecting incident flexural waves by nonresonant single-phase meta-slab with subunits of graded thicknesses[END_REF][START_REF] Lott | Effective impedance of a locally resonant metasurface[END_REF][START_REF] Qiu | Steering and focusing of fundamental shear horizontal guided waves in plates by using multiple-strip metasurfaces[END_REF][START_REF] Park | Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures[END_REF][START_REF] Cao | Disordered Elastic Metasurfaces[END_REF][START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF], according to the manipulated wave type. Among them, the plate wave metasurface is a hot one due to the universality of plate waves in engineering. For the first time, the concept of the optic metasurface or acoustic metasurface was extended to elastic wave metasurface in the field of solid mechanics by Zhu and Semperlotti [START_REF] Zhu | Anomalous Refraction of Acoustic Guided Waves in Solids with Geometrically Tapered Metasurfaces[END_REF]. They investigated the anomalous refraction of Lamb waves (plate waves in a relatively thick plate [START_REF] Giurgiutiu | Structural Health Monitoring with Piezoelectric Wafer Active Sensors[END_REF]) based on mode conversion by using geometric tapers. Meanwhile, Liu et al. [START_REF] Liu | Source Illusion Devices for Flexural Lamb Waves Using Elastic Metasurfaces[END_REF] designed a zigzag structure metasurface by grooving the plate to achieve a source illusion device for flexural waves (plate waves in a relatively thin plate [START_REF] Giurgiutiu | Structural Health Monitoring with Piezoelectric Wafer Active Sensors[END_REF]). In order to further improve transmittance, a high transmission metasurface for deflecting and focusing in-plane longitudinal waves in a plate [START_REF] Lee | Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering[END_REF] was proposed by breaking the intrinsic proportional density-stiffness relation. Furthermore, in one [START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF] of our previous works, phase accumulation of the multi-resonators was used to experimentally design and demonstrate the pillared metasurface, which is capable of deflecting vertically and obliquely incident flexural waves with high transmissions.

On the other hand, some tunable plate wave metasurfaces [START_REF] Li | Tunable modulation of refracted lamb wave front facilitated by adaptive elastic metasurfaces[END_REF][START_REF] Chen | A programmable metasurface for real time control of broadband elastic rays[END_REF][START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF][START_REF] Xia | In situ steering of shear horizontal waves in a plate by a tunable electromechanical resonant elastic metasurface[END_REF] were also proposed. An adaptive metasurface [START_REF] Li | Tunable modulation of refracted lamb wave front facilitated by adaptive elastic metasurfaces[END_REF] capable of modulating A0 Lamb waves in a plate were designed through the piezoelectric units. Furthermore, a programmable plate wave metasurface [START_REF] Chen | A programmable metasurface for real time control of broadband elastic rays[END_REF] with sensing-and-actuating units was introduced to adapt and reprogram flexural wave control functionalities in real time. Recently, a switchable multifunctional fish-bone elastic metasurface [START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF] is proposed to manipulate the transmitted plate waves. More recently, plate wave metasurfaces had been applied to total mode conversions from in-plane longitudinal waves to shear waves [START_REF] Kim | Transmodal elastic metasurface for broad angle total mode conversion[END_REF], splitting flexural waves [START_REF] Xu | Beam splitting of flexural waves with a coding meta-slab[END_REF], asymmetrical flexural wave transmission [START_REF] Cao | Asymmetric flexural wave transmission based on duallayer elastic gradient[END_REF] and isolating vibration [START_REF] Zhu | Total-internal-reflection elastic metasurfaces: design and application to structural vibration isolation[END_REF].

Almost all of the above mentioned elastic metasurfaces need to be grooved on the complete plate surface, which avoids the coupling interference (CI) between two adjacent subunits to design them independently. The CI is similar to the Fano interference [START_REF] Jin | Tunable Fano resonances of Lamb modes in a pillared metasurface[END_REF][START_REF] Fano | Effects of configuration interaction on intensities and phase shifts[END_REF][START_REF] Goffaux | Evidence of Fano-Like interference phenomena in locally resonant materials[END_REF][START_REF] Kosevich | Fano-like resonance phenomena by flexural shell modes in sound transmission through two-dimensional periodic arrays of thin-walled hollow cylinders[END_REF], which results from the coupling between individual unit-scatters in the continuum background, i.e., the host plate. However, the CI in our work pays more attention to the coupling of the adjacent subunits deviating from the resonance frequency.

In the present research, we improve the model reported in our previous paper [START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF] by the iron pillared-resonator subunits. The lowest dispersion band and vibration mode analyses are used to explain the working mechanism of the phase shift of transmitted waves. Furthermore, we indeed propose an analytical model for multi-pillared subunit to accurately predict the amplitude and phase shift of the transmitted wave. Based on the improved subunit model, the physical mechanisms of CI between the adjacent subunits are revealed, which can be classified into the constructive and destructive interferences. Further, by considering the CI, we propose the pillared elastic metasurface with constructive interference (PEMC) to manipulate flexural waves propagating in a plate. Finally, experimental validations are carried out to demonstrate the deflecting and focusing capabilities of the designed PEMCs. without slots. To clarify the difference between the current work and our previous one [START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF], the sketches of the previous elastic metasurface with slots and the present one without slots in a front view (xy-plane) are shown in Figs. 1(d) and 1(e), respectively.

Theoretical design of the subunit of PEMC

The most obvious difference is that CI will exist at the junction of the adjacent subunits for the metasurface without slots (marked in Fig. 1(e)). The interference manifests itself as the coupling through out-of-plane vibration of the plate for the two adjacent subunits.

In addition, the previous subunits are improved by the iron pillared-resonator subunits.

Considering the CI, the present elastic metasurface is composed of the improved threedimensional (3D) subunits, periodically arrayed on the host plate, as shown in Fig. 1

(b).

The subunit design is one of the core design components. For simplicity, we firstly design the corresponding two-dimensional (2D) subunits, which is the cross-section view, as shown in Fig. 1(c). Then, it will be expanded to the 3D one.

Mechanism of phase shift in the multi-resonator subunit

The 2D subunit is based on the pillared resonators periodically arrayed and bonded on the surface of a thin aluminum alloy plate, as shown in Fig. 1 f k p = with lower velocity will have a larger phase shift. Except for the above same characteristics, the difference between Figs. 2(a) and 2(b) is that the wavenumber of the flexural wave in the plate with periodic iron pillared resonators is larger than that in the plate with periodic aluminum alloy pillared resonators at the same frequency. It indicates that periodic iron resonators are easier to make the flexural waves in the plate slow down, i.e., phase shift. Therefore, the iron pillared resonator is adopted to improve the subunits in our previous paper [START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF]. The targeted frequency in our design is 1 kHz, which is one of the typical low frequencies. In order to further reveal the interaction between the pillared resonators and flexural waves in the plate, the corresponding vibration modes of the iron resonator are analyzed. Figs. 2(c) and 2(d) show the z direction displacements in the pillared resonators associated with longitudinal vibration modes (longitudinal waves) and the x direction displacements in the pillared resonators associated with bending vibration modes (flexural waves), respectively. It can be found that when the pillar height is short (less than 6d), the longitudinal vibration of the pillared resonator (the longitudinal wave) is dominant, while when the pillar height is high (more than 14d), the bending vibration (the flexural wave) is dominant. From this propagation form in the resonators, it makes one easy to understand the dispersion bands in Figs. 2(a) and 2(b). When the propagating flexural waves in the plate encounter a resonator, parts of them are converted into longitudinal waves or flexural waves in the resonator, which will propagate in the resonator and then to the plate. This transmission process leads to slow flexural waves down to have a big phase shift. In this way, the wave velocity of the longitudinal wave or the flexural wave in the iron pillared resonator is lower than the one in the aluminum alloy resonator. In other words, the transmission speed in the iron resonator is slower than the one in the aluminum alloy resonator. The transmission process with the slower speed will lead to a larger phase shit. Then, we can easily understand that the improved subunits based on periodic iron resonators are easier to make the flexural waves phase shit.

Analytical transmission and phase spectra of the 2D subunit

In the current study, the number of pillared resonators is set as 25. It makes multiresonators structure approximate the above mentioned infinite periodic subunits to compose the subunits, as shown in Fig. 3(a), which will cause the phase shifts of transmitted flexural waves in a plate. The working mechanism of the phase shifts of transmitted flexural waves across the subunits has been revealed by the above lowest dispersion band and vibration mode analysis. The subunits with different pillar heights make it possible to meet the desired phase profile for designing a gradient metasurface.

In addition, the evenly distributed amplitude of transmitted waves across the subunits is also crucial in designing a metasurface without any undesired distortion for transmitted pattern [START_REF] Lee | Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering[END_REF][START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF]. Therefore, both amplitudes and phase shifts of transmitted waves across the subunits need to be accurately solve. To solve amplitudes and phase shifts of the transmitted waves, we establish the analytical model and divide the host plate into several regions, which are marked as the 1 st , 2 nd , …, 26 th regions, respectively, in Fig. 3(b). First, wave transmission in the (j-1) th region, j th region, and j th pillared resonator are studied, as shown in Fig. 3(c). From the above analysis of Figs. 2(c) and 2(d), we know that both flexural waves (bending vibrations) and longitudinal waves (longitudinal vibrations) will couple and exist in the pillared resonator. Therefore, for accuracy, they must be considered in the pillared resonator and host plate at the same time. It should be pointed out that flexural and longitudinal waves are equivalent to A0 and S0 lamb waves, respectively, in the low designed frequency of 1 kHz [START_REF] Giurgiutiu | Structural Health Monitoring with Piezoelectric Wafer Active Sensors[END_REF]. The governing equations for flexural waves and longitudinal waves in the host plate can be expressed in the following forms, respectively:

,

where , , and are the Young's modulus, density, and section moment of inertia of the plate, respectively. , in which and are the thickness of the plate and Poisson's ratio of the material, respectively. The general solutions of the displacements for the flexural wave equations in the (j-1) th and j th regions are:

( ) ( ) ( ) ( ) 4 2 1 1 1 1 4 2 2 2 1 1 2 2 2 1 , , 0
, , 0 -

w x t w x t E I d x t u x t u x t E x t r r n ¶ ¶ + = ¶ ¶ ¶ ¶ - = ¶ ¶ (1 ) 1 E 1 r 1 I 3 1 1 2 1 12(1 ) I d n - = 1 d 1 n , (2)
where the superscripts (j-1) and (j) indicate the (j-1) th and j th regions, respectively. The flexural wavenumber is , in which is the circular

frequency. , , , , , , , and are complex coefficients
where the longitudinal wavenumber is , , , , and are complex coefficients. and correspond to the positive-going propagating longitudinal waves, whereas and correspond to the negative-going propagating longitudinal waves.

The governing equations for the flexural wave and longitudinal wave in the pillared resonator j can be expressed in the following forms, respectively:

( ) ( ) ( ) ( )

1 1 1 1 1 1 1 1 - - ( 1) ( 1) ( 1) ( 1) ( 1) - - ( ) ( ) ( ) ( ) ( ) , , b b b b b b b b ik x ik x k x k x j j j j j i t ik x ik x k x k x j j j j j i t w x t A e B e C

e D e e w x t A e B e C e D e e

w w -----

= + + + = + + + 1/4 2 1 1 1 1 1 b d k E I r w ae ö = ç ÷ è ø 2 f w p = ( -1) j A ( -1) j B ( -1) j C ( -1) j D ( ) j A ( ) j B ( ) j C ( ) j D 1 ( -1) b ik x j A e - 1 ( ) b ik x j A e - 1 ( -1) b ik x j B e 1 ( ) b ik x j B e 1 ( -1) b k x j C e - 1 ( ) b k x j C e - 1 ( -1) b k x j D e 1 ( ) b k x j D e ( ) ( )
( ) ( ) 1 1 1 1 - ( 1) ( -1) ( -1) - ( ) ( ) ( ) , , l l l l ik x ik x j j j i t ik x ik x j j j i
= + = + 1/2 2 2 1 1 1 1 (1 ) l k E r w n é ù - = ê ú ë û ( -1) j P ( -1) j Q ( ) j P ( ) j Q 1 ( -1) l ik x j P e - 1 ( ) l ik x j P e - 1 ( -1) l ik x j Q e 1 ( ) l ik x j Q e , (4)
where , , and are the Young's modulus, density, and section moment of inertia of the pillared resonator, respectively. , in which and are the thickness of the resonator and Poisson's ratio of the material, respectively. The general solutions of the displacements for flexural wave and longitudinal wave equations in the j th pillared resonator are ,

where the superscript indicates the j th pillared resonator.

( ) ( ) ( ) ( ) 4 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 , , 0 , , 0 - w z t w z t E I d z t u z t u z t E z t r r n ¶ ¶ + = ¶ ¶ ¶ ¶ - = ¶ ¶ (1 ) 2 E 2 r 2 I 3 2 2 2 2
12( 1)

I d n - = 2 d 2 n ( ) ( ) ( ) ( ) 2 2 2 2 2 2 (P ) (P ) (P ) (P ) (P ) - - (P ) (P ) (P ) - , , j j j j j b b b b j j j l l ik z ik z k z k z i t ik z ik z i t w z t A e B e
= + + + = + (P ) j 1/4 2 2 2 2 2 2 b d k E I r w ae ö = ç ÷ è ø 1/2 2 2 2 2 2 2
(1 )

l k E r w n é ù - = ê ú ë û (P ) j A (P ) j B (P ) j C (P ) j D (P ) j P (P ) j Q 2 (P ) j b ik z A e -
j l ik z Q e ( ) j w ( ) j u ( ) j j ( ) j F ( ) j V ( ) j M ( ) j A ( ) j B ( ) j C ( ) j D ( ) j P ( ) j Q . ( 6
)
where , , , and are slope, axial force, shear force and bending moment, respectively. For simplicity, the origins of coordinates x and z are set at the interface among the (j-1) th region, j th region, and j th pillared resonator. Substituting Eqs. ( 2) and

(3) into Eq. (A13), we obtain

, , (7)
where is the transformation matrix between the state vector and the coefficient vector. It is shown in Appendix A. Substituting Eqs. ( 2), ( 3), [START_REF] Zhu | Kirigami-based Elastic Metamaterials with Anisotropic Mass Density for Subwavelength Flexural wave Control[END_REF], and (A13) into the boundary conditions among the (j-1) th region, j th region, and j th pillared resonator (Eqs.

(A1)-(A12)), we obtain

, ( 8
)
where and are the and matrices, respectively. They are given in Appendix A. From Eq.( 8), we obtain

. ( 9
)
From Eq. ( 9), we get , ,

where and indicate the coefficient vectors in the left interface of the j th region and the right interface of the (j-1) th region, respectively. N5 is obtained by selecting the first to sixth rows of the matrix N4 and has included the contribution of the

{ } { } T ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , , ,
j j j j j j j j j j j T j j j A B C D P w u F V M Q j = = v k j ( ) j F ( ) j V ( ) j M ( ) ( ) 1 = j j N × v k 1, 2 26 j = ! 1 N (P ) (P ) (P ) (P ) (P ) (P ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 3 , , , , , , , , , , , , , , , ,
j j j j j j T j j j j j j T j j j j j j N A B C D P Q A B C D P Q N A B C D P Q - - - - - - é ù ë û é ù = ë û 2 N 3 N 12 12 ´12 6 (P ) (P ) (P ) (P ) (P ) (P ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) 4 , , , , , , , , , , , = , , , , ,
j j j j j j T j j j j j j T j j j j j j A B C D P Q A B C D P Q N A B C D P Q - - - - - - é ù ë û é ù ë û 5 ( ) ( 1) j j L R N - = k k 2,3 26 j = ! ( ) j L k ( 1) j R - k pillared resonator.
For the j th region with a length , the relationship between the state vector and coefficients vector can be expressed as

, , (11)
where is the transfer matrix for waves propagating from the left to the right interface of the j th region and it is shown in Appendix A. From Eqs. ( 7), ( 10) and ( 11), for waves propagating from the right interface of the (j-1) th region to the right interface of the j th region, the transfer matrix can be expressed as

, . (12)
From Eq. ( 12), for waves propagating from the right interface of the 1 st region to the right interface of the 25 th region, the transfer matrix can be expressed as

. ( 13
)
Furthermore, for waves propagating from the right interface of the 1 st region to the left interface of the 26 th region, we get

. ( 14
)
From this equation, we can solve the amplitude and phase shift for the flexural wave propagating through these 25 pillared resonators. Eq. ( 14) can be rewrittend as It should be pointed out that the total thickness of the subunit L=49 mm is about half of the incident wavelength λ =98.9 mm [START_REF] Achenbach | Wave Propagation in Elastic Solids[END_REF], that is, these 25 pillared resonators constitute the sub-wavelength subunit.

. ( 15
6 ( ) ( ) j j R L N = × v k 2,3 25 j = ! 6 N ( ) 1 ( ) 1 ( ) 1 ( 1) 1 1 6 1 6 5 j j j j R R L R N N N N N N - - - - = = = k v k k 2,3 25 j = ! (25) 1 (24) 1 24 (1) 1 6 5 1 6 5 ( ) R R R N N N N N N - - = = k k k (26) (25) 1 24 (1) 5 5 1 6 5 = ( ) L R R N N N N N - = k k k 1 2 4 5 1 out i 6 5 n ( ) N N N N - = k k in = 1, ,0, ,0,

The design of 3D subunits with the CII

The above-designed subunit is embodied in its 2D model, as shown in Fig. 3(a).

The corresponding 3D one is shown in Fig. 4(a). The periodic boundary conditions need to be applied on the two long boundaries in the 3D strip-like subunit model. In order to display intuitively the 3D subunit applied by the periodic boundaries, another form is shown in Fig. 4(b). In our previous paper [START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF], slots are made at the subunit region (marked with the red dotted box in Fig. 4 The spacing size is determined by the parameters of the subunit width w1 and the filling ratio η, which is the ratio of the spacing width to the subunit width in one subunit.

We need to analyze the influence of the CII, which is related to the subunit width w1

and the filling ratio η, on the consistency of the single 3D subunit and the corresponding 2D subunit. The consistency of the single 3D subunit and the corresponding 2D subunit can be reflected in the differences between their phase shift and transmission amplitude for the different height h. Therefore, in the fixed design frequency, the phase shift ϕ1

and transmission amplitude |t1| for the 3D subunits can be expressed as ϕ1=Gϕ1(w1, η, h) and |t1|= G|t1|(w1, η, h), respectively, while that for the 2D subunits can be expressed as ϕ0=Gϕ0(h) and |t0|= G|t0|( h), respectively. In order to quantitatively evaluate the influence of the CII, related to the subunit width w1 and the ratio η, on the consistency of the 3D and 2D subunits, we define the influence coefficient as: ,

where n=13, which is the ratio of the maximum pillar height (considered in Fig. 3d

( ) ( ) ( ) ( ) ( )
1 0 1 0 1 1 1 1 1 , , , () 1 ,
,,

n d h d n d t t t h d w G w h G h n w G w h G h n f f f h h h h × = × = é ù L = × - ë û é ù L = × - ë û å å f L t L 4

The constructive and destructive interferences between the adjacent different subunits

The above twelve 3D subunits in the order shown in Fig. 4 The subunit consists of the square pillars and the plate under them, shown in Fig. 4(a), which are named as subunit pillars and subunit plate, respectively, for easy description. In order to reveal the intrinsic physics, the out-of-plane wave fields (z components of the displacements) on the mid-planes of subunit plates (OWMS), corresponding to the twelve black squares (the 3D subunits) in Fig. 4(e), are extracted, as shown in Fig. 5(c). It can be observed for one supercell that the difference of the extracted OWMSs between adjacent subunits from the # 1 subunit to the # 12 subunit is small. After the two supercells are arranged to design the metasurface, Fig. 5(c) shows the # 1 subunit in the lower supercell and the # 12 subunit in the upper supercell become adjacent two subunits, marked with black dotted box in Fig. 5(c). It can be seen that the difference of OWMSs between the two subunits is large, especially in terms of wave intensity and wavelength size. For example, the wave intensity for the # 1 subunit is about 2.5 times of that for the # To further quantitatively analyze the influence of the CI on the designed subunits, we select all adjacent two subunits, i.e., subunit and subunit, from the two supercell in Fig. 5(c). The values of i and are set to make the two subunits as the adjacent ones. The two adjacent subunits make up the two types of analytical models, the one has the slots and the other has no slots, as shown in the upper and lower subplots of Fig. 5(f), respectively. For the two types of analytical models, we calculate the transmitted fields and obtain the phase shift in the far field, respectively. For the first type of model with slots, the obtained phase shifts are accurate values without the influence of the CI on the OWMSs. For the second type of model without slots, there will be some deviation in the obtained phase shifts due to the influence of CI on the

# i # j 1 j i = ±
OWMSs. The difference values between the phase shifts for the two types of models are shown in Fig. 5(g). It can quantitatively analyze the influence of CI on OWMSs, which will determine the manipulation effect of the transmitted field. In order to make a comprehensive comparison, we also add the phase shift differences for the models composed of two identical subunits ( ) to Fig. 5(g), which are related to the 3D subunits applied by the periodic boundaries.

It can be clearly observed that the phase shift difference for the model composed of the # 1 subunit and the # 12 subunit reaches a large value of 2 rad, while the remaining differences are all less than 0.4 rad. The big difference value of 2 rad indicates that the transmitted wave field has been completely destroyed by the CI, the small difference value less than 0.4 rad indicates that the CI results in constructive interference. We can know from the above analysis of Fig. 5(c) that only the OWMSs between the subunit 1 and the subunit 12 in all adjacent subunits have big difference while the OWMSs for all other adjacent subunits have a small difference. It indicates that the CI between two adjacent subunits with very different OWMSs will lead to destructive interference while the CI between two subunits with small different OWMSs will lead to constructive interference. Therefore, we can take the advantage of the constructive interference to avoid having slots in the subunits.

Manipulating flexural waves by the PEMC

To design the PEMC, we need to take the advantage of the constructive interference and avoid the destructive one. For the deflecting metasurface, continuous phase shift within one supercell and an abrupt phase shift of 2π between two adjacent i j = supercells will make wave front linear change in the transmitted filed due to the wave periodicity [START_REF] Assouar | Acoustic metasurfaces[END_REF]. The OWMSs for the two adjacent subunits at the junction between the two adjacent supercells exhibit a larger difference and that for other two adjacent ones have a small difference. It will lead to the constructive interference exist all two adjacent subunits, except for that at the junction between the adjacent supercells.

Therefore, in the present study, taking the advantage of the constructive interference and avoiding the destructive one, we can just use one supercell without any slots to build the deflecting PEMC. It should be pointed out that this has one limitation, i.e., it is only suitable for incident Gauss beam whose width does not exceed the width of one supercell. However, we can simply eliminate this limitation by introducing only one slot at the junction of adjacent two supercells, which still greatly reduces the damage to the plate. However, for the focusing PEMC, there is no such limitation. The reason is that the phase shift for all two adjacent subunits is continuous without the abrupt phase shift. It enables our new conceptual design to be of great value for the focusing metasurface.

Sample fabrication and experimental setup

To accurately fabricate the iron pillared resonator, the Wire Electrical Discharge Machining (WEDM) with a manufacturing precision of 0.01 mm is adopted. The manufactured iron pillared resonators are bonded to the middle of the aluminum alloy plate (1000 mm × 2000 mm × 1 mm) by a special adhesive. The enlarged view of the deflecting PEMC sample is shown in the top of Fig. 1(a). An array of four piezoelectric patches PZT-5A (20 mm × 20 mm × 0.3 mm) is bonded on the surface of the plate acting as actuators. The spacing of adjacent piezoelectric patches is set as the width of the piezoelectric patch according to the phased array theory [START_REF] Zhu | Negative refraction of elastic waves at the deepsubwavelength scale in a single-phase metamaterial[END_REF]. By rotating the direction of the bonded piezoelectric patch array, we can get different incident angles.

The exciting frequency is 1 kHz and the value of f•d (the product of the frequency and the plate thickness) is far less than 0.4 MHz mm [START_REF] Degertekin | Single mode Lamb wave excitation in thin plates by Hertzian contacts[END_REF]. Normal traction excitation of the piezoelectric patches can be used to generate a flexural wave (single Lamb mode A0)

with the high signal-to-noise ratio.

With the controlling of the waveform editor (based on the Labview software), the signal generator (Agilent 33220A) generates a 5-cycle tone burst signal

, where is the central frequency.

The power amplifier (HVPA05) drives the piezoelectric patches in a manner that the amplified voltage for the two middle piezoelectric patches is two times of that for the ones on both sides. The bonded four piezoelectric patches can approximately excite the transient flexural wave Gaussian beam. A layer of blue-tack is glued on the boundary edges to minimize reflections from the outer boundary of the test piece [START_REF] Zhu | Anomalous Refraction of Acoustic Guided Waves in Solids with Geometrically Tapered Metasurfaces[END_REF][START_REF] Liu | Source Illusion Devices for Flexural Lamb Waves Using Elastic Metasurfaces[END_REF]. The wave fields in the incident and transmitted areas are measured by the PSV-400 scanning laser Doppler vibrometer. In the scanning areas, more than six points in one flexural wavelength need to be scanned to ensure fine resolution.

PEMC for deflecting flexural waves

The PEMC without slots is formed by one supercell, as shown in Fig. 1(a).

According to the GSL, the theoretical refraction angle of the incident wave can be calculated by ( ) ( )

0 1 cos 2 5 sin 2 t c c w A f t f t p p = - é ù ë û 1 kHz c f = t q , (17)
where is the incident angle. is the spatial phase shift gradient. Firstly, a vertically incident flexural wave is considered, i.e., . In the supercell, the phase shift of subunits is set to increase along the opposite direction of the y-axis. The theoretical refraction angle of the transmitted wave is calculated to be 24.3 o according to Eq. ( 17). For obliquely incident flexural wave, when the incident angle meets with the relationship of , and the phase shift of subunits is set to increase along the y-axis, negative refraction will occur. As a typical example, the incident angle is set as , the corresponding theoretical refraction angle is calculated to be -8.

PEMC for focusing flexural waves

The focusing PEMC without slots is illustrated in the upper right corner of Fig.

2 2 ˆ( ) ( ) y F y F p f l = + - F ˆ=3 F l
is not a very different OWMS between adjacent subunits. Therefore, for all adjacent subunits, there are constructive interferences.

Conclusions

For the improved subunits (~0.495λ), the lowest dispersion band and vibration mode analysis reveal the mechanism of phase shift, that is, arrayed pillared resonators decrease the wavenumber of the flexural wave in the subunit due to slowing down. The corresponding analytical model is established and both of the amplitude and the phase shift of transmitted waves across the subunits are analytically solved. In addition, we

k E H k E H iE I k iE I k E I k E I k E I k E I k E I k E ik ik k I k N i k i é ù ê ú ê ú ê ú - - ê ú - ê ú ê ú ê ú ê ú ë û - - - - - 2 1 1 1 / (1 ) H d v = - 2 N 1 15 2 1 1 1 1 / 2 l E T k I i h = - 1 16 1 1 1 1 / 2 l E T k I i h = 3 17 2 2 2 b T iE I k = 3 18 2 2 2 b T iE I k = - 3 19 2 2 2 b T E I k = - 3 110 2 2 2 b T E I k = 3 21 1 1 1 b T iE I k = - 3 22 1 1 1 b T iE I k = 3 23 1 1 1 b T E I k = T EIk = - 2 2 2 2 2 211 - / l k E I T h i = 2 2 2 2 2 2 2 1 / l k E I T h i = 2 31 1 1 1 b T EI k = - 2 32 1 1 1 b T EIk = - 2 33 1 1 1 b T E I k = 2 34 1 1 1 b T E I k = 2 37 2 2 2 b T E I k = - 2 38 2 2 2 b T E I k = - 2 39 2 2 2 b T E I k = 2 310 2 2 2 b T E I k = 45 1 T = - 46 1 T = - 57 1 T = 58 1 T = 59 1 T = 510 1 T = 61 1 T = 62 1 T = 63 1 T = 64 1 T = 711 1 T = 712 1 T = 81 1 -b T ik = 82 1 b T ik = 83 1 b T k = - 84 1 b T k = 97 2 b T ik = - 98 2 b T ik = 99 2 b T k = - 910 2 b T k = 2 - 3 107 2 2

Fig. 1 (

 1 Fig. 1(a) depicts the PEMC structure (enlarged one is shown in the top of the figure)

Fig. 1 .

 1 Fig. 1. (a) Sketch of the PEMC capable of deflecting an incident flexural wave, with the enlarged

  (c). Here, two types of pillared resonators are analyzed, which are made of aluminum alloy (with the elastic modulus of , Poisson's ratio of , and the density of ) and iron (with the elastic modulus of , Poisson's ratio of , and the density of ). The elementary unit cells are illustrated in Figs. 2(a) and 2(b), where the black and grey areas represent the iron and aluminum alloy, respectively. Both of the plate and pillar thicknesses are d=1 mm. D denotes the constant lattice with 2 mm, and h is the pillar height. The periodicity is along the x-axis. Figs. 2(a) and 2(b) show the lowest dispersion bands (corresponding to flexural waves in the plates) for different heights of the pillared resonators. They are obtained by using the eigen frequency analysis of COMSOL MULTIPHYSICS 5.4 software. It can be observed from Figs. 2(a) and 2(b) that the wavenumbers k corresponding to the color lines (representing the plate with pillared resonators) are bigger than that corresponding to the black lines (representing the plate without resonators) at the same frequency. Furthermore, the wavenumber in the plate increases with the pillar height.According to , the wavenumber is inversely proportional to the corresponding wave velocity. Therefore, the wave velocity of the flexural wave propagating in the plate with the periodic structures decreases when the height of pillared resonator increase. After propagating the same distance, the flexural wave

Fig. 2 .

 2 Fig. 2. (a) and (b) show the lowest dispersion bands for different heights of the aluminum alloy and

Fig. 3 .

 3 Fig. 3. (a) Sketch of the 2D subunit with 25 pillared resonators. The host plate is divided into several

  . and correspond to the positive-going propagating flexural waves, whereas and correspond to the negative-going propagating flexural waves. and correspond to the positivegoing evanescent flexural waves, whereas and correspond to the negative-going evanescent flexural waves. The general solutions of the displacements for longitudinal wave equations in the (j-1) th and j th regions are: ,

  the reflected propagating flexural wave in the 1 st regions, respectively, , and are the transmitted propagating flexural wave, transmitted evanescent flexural wave and transmitted longitudinal wave in the 26 th

  s

Fig. 3 (

 3 Fig. 3(d) shows the analytical amplitude (red line) and phase shift (blue line) of

  (a)) to avoid the CI. In the present work, we in the 3D subunits instead of the slot. In this way, the CI between identical subunits (CII) will exist by the spacing, marked with red arrows in Fig.4(b).

Fig. 4 .

 4 Fig. 4. (a) the 3D subunit. (b) Another form of the subunit applied by the periodic

  ) to the thickness of the plate. The dependence of the influence coefficients and on w1 and η are shown numerically in Figs. 4(c) and 4(d). It can be clearly seen that these two influence coefficients (determine the influence of CII) are small when the values of w1 and η (determine the spacing size) are small. According to Figs. 4(c) and

  Fig. 4(e). Compared with Fig. 3(d), the average error of the phase shifts is very small

  (e) are integrated into a supercell as shown in Fig. 5(c). Two identical supercells, providing a stair-step (linearly gradient) discrete phase profile, compose the elastic metasurfaces in Figs. 5(a) and 5(b). For these two metasurfaces, the only difference between them is that one has no CI (with a slot) at the junction of the supercells and the other has CI (without a slot), as shown in the upper right corners of Figs. 5(a) and 5(b). The simulations of deflected flexural wave fields by the two metasurfaces are shown in Figs. 5(a) and 5(b). It can be clearly seen in Fig. 5(a) that the incident wave is deflected without any undesired distortion in the transmitted field. However, in Fig. 5(b), the transmitted field near the junction of two supercells is destroyed. The upper part of the transmitted field is hardly deflected. By comparing these two simulation results, we know that the destruction of the transmitted field is mainly due to the CI at the junction of the two supercells.

Fig. 5 .

 5 Fig. 5. (a) and (b) illustrate the two metasurfaces consist of two supercells. Enlarged views of the

  Figs. 5(a) and 5(b), are extracted, respectively, as shown in Figs. 5(d) and 5(e). Since

Fig. 6 .

 6 Fig. 6. The numerical simulation results for the vertically (a) and obliquely (c) incident waves across

7

  Fig. 3(d), the corresponding values of the pillar heights for all subunits with the

2

 2

Fig. 7 .

 7 Fig. 7. (a) shows the transmitted field of incident flexural waves propagating through the PEMC

2 t

 2 have revealed that the CI between the two subunits with larger different OWMSs lead to destructive interference, while the CI between the two subunits with smaller different OWMSs lead to the constructive interference. Further, combined with the constructive interference, pillared elastic metasurfaces have been analytically and numerically investigated to manipulate incident flexural waves in a plate. We find that the subwavelength focusing PEMC (~0.495λ) enables our new conceptual design to be of great value because has only the constructive interference due to the continuous phase shift for every two adjacent subunits. Finally, we experimentally have demonstrated the deflecting and focusing functionalities of the designed PEMC. This proposed conceptual design can ensure the integrity of the original plate without introducing any damage to its strength and stiffness. We expect the present work will open a new avenue for a variety of applications in the field of elastic metamaterials/ metasurfaces with the constructive interference.

2 -

 2 is investigated by the designed errors of the pillar heights. To this end, we will carry out numerical studies on the focusing performance [the focal distance of with f = 1 kHz] to show the sensitivity to designed errors. The values of the pillar heights for two lower precisions (Pr = 0.5 and 1.0 mm) are presented in Table B1 by rounding each value of Pr = 0.1 mm up to the nearest value. By employing these pillar heights separately, the PEMCs are designed and the transmitted fields are simulated. The normalized transverse and longitudinal intensities at the focal spot along two orthogonal directions [white dashed lines in Fig. 7(a)] are shown in Fig. B1. It can be seen that the designed pillar heights with precisions of Pr = 0.5 and 1.0 mm have a satisfactory focusing effect like that of Pr = 0.1 mm. This indicates that the focusing PEMC has good robustness. It should be pointed out that when the total width of the metasurface is constant, the focal distance can be arbitrarily designed in the range of more than half wavelength. The smaller the designed focal distance is, the smaller the error between simulated focal distance and theoretical one is, and vice versa [52, 53].

Fig. B1 .

 B1 Fig. B1. The focusing results at the frequency of 1 kHz with a focal distance of with different

  3l

  3l

Acknowledgment

This project is supported by the National Natural Science Foundation of China (Grant No.11972296), the 111 Project (No. BP0719007), the Institute CARNOT (ICEEL), La Région Grand Est, the Fundamental Research Funds for the Central Universities (No. 310201901A005). The first author is grateful to the support of Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant No. CX201936) and China Scholarship Council (CSC No. 201806290176).

Appendix A: Boundary conditions and the transfer matrixes

In Fig. 3(c), the positive directions of axial force , shear force V and bending moment M are marked at the interfaces among the (j-1) th region, j th region, and j th pillared resonator [START_REF] Graff | Wave motion in elastic solids[END_REF][START_REF] Cho | Energy flow analysis of coupled structures[END_REF]. The force and moment must be balanced at the connection position.

According to Fig. 3(c), the following boundary conditions must be satisfied.

Force balance in x direction:

Force balance in z direction:

(A2)

Moment balance:

At the connection position, the displacement and rotation angle are continuous, so the following boundary conditions must be satisfied.

Displacement continuity in x direction:

Displacement continuity in z direction:

Slope continuity:

At the top of the j th pillar resonator ( ), there is no constraint, hence the force and moment are 0.

The relationships between the slope and the displacement w, between axial force F and the displacement u, between shear force V and the displacement w, between bending moment M and the displacement w are:

respectively, where the superscript indicates space differentiation with respect to the corresponding coordinate.

The transformation matrix between the state vector and the coefficient vector is:

where .

The transfer matrix has the following components:

The transfer matrix can be expressed as:

. (A15)

For the j th region with a length , waves propagate from the left to the right. From Eqs. ( 2), ( 3) and (A13), the relationship between the physical quantities , , , , , in the right interface of the j th region and the complex coefficients , , , , , in the left interface of the j th region can be expressed as follows:

. (A16)

Here, , Eq. (A16) can be rewritten in matrix form as follows:

where is the transfer matrix for waves propagating from the left to the right interface of the j th region and it is , (A18) where and . For accurately solving the phase shift, the modifying thickness of the pillared resonator must be added to the length of the j th region, which can compensate the ignored thickness of the pillared resonator in boundary conditions between the (j-1) th and j th regions.

The wave fields for the 1 st and 26 th regions can be described as follows: The phase of can be calculated by using the following formula:

. (A20)

Appendix B: The sensitivity and limitation of the focusing PEMC For the focusing PEMC in Fig. 7(a), the values of the pillar heights in the subunits have a precision of 0.1 mm, as shown in the first row of Table B1. Further, the sensitivity of


Yifan Zhu
email: yifan.zhu@univ-lorraine.fr

Liyun Cao

Aurélien Merkel

Shi-Wang Fan

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Bifunctional superlens for simultaneous flexural and acoustic wave superfocusing

Keywords: Superlensing, Flexural waves, Acoustic waves, Phononic crystal, Multi-functionality

Superfocusing of acoustic and elastic waves is generally achieved by the combination of negative refraction and the enhancement of the evanescent waves. Here, we numerically and experimentally demonstrate the bifunctionality of a superlens that is able to simultaneously focus acoustic and flexural waves beyond the diffraction limit. The designed structure is composed of a two-dimensional arrangement of pillars that act as rigid scatterers for the sound waves and as resonant scatterers for the flexural waves. The band structure presents modes with negative dispersion bands allowing negative refraction for both types of waves within the frequency range 6.9-7.4kHz, which is induced by Bragg scattering effect. Edge modes that enhance the evanescent waves through resonant coupling appear around 7.2kHz for the flexural and sound wave. The simultaneous superlensing is then observed at this frequency. Our finding will enlighten multi-physical and multi-functional wave manipulations, and could have pragmatic applications involving multi-wave devices.

The concept of superlensing can be traced back to the Pendry's perfect lens in optics [1], which requires both abilities of the convergence of propagating waves and of the participation of non-propagating evanescent waves in the focal spot. The convergence of the waves emerges from the negative refraction of the waves and the evanescent waves are enhanced by surface resonant states that exist at the interface between the lens and the propagating medium. By analogy with the optical devices, acoustic superlensing and superfocusing have been a subject of intense research in the past decade. Indeed, acoustic superlens has been demonstrated using phononic crystals [2][3][4][5][6], acoustic metamaterials [7][8][9][10] and acoustic metasurfaces [11][12] for bulk waves, as well as for surface and spoof surface acoustic waves [13][14]. These findings have significant potential applications for wave imaging, due to the fact that super-resolutions are highly desired in imaging system and focused ultrasound is widely used in medical diagnosis.

Although the superlens for one kind of wave has been well demonstrated [1][2][3][4][5][6][7][8][9][10][11][12][13][14], the multiphysics manipulations [15][16][17] are much less explored and remains a challenging issue to be investigated and elucidated. Here, we propose a concept for a bifunctional superlens with simultaneous controls of flexural and acoustic waves. This concept, illustrated in Fig. 1(a), is based on a two-dimensional (2D) pillar-based phononic crystal attached to a solid plate. The host solid plate can only support A0 mode Lamb wave, viz., flexural wave. The incident wave and the output wave are flexural waves in the solid plate. The air medium on the top of the plate can support acoustic wave incidence and transmission. The bifunctionality derives from the design of the phononic crystal, where the pillars can act as rigid scatterers for the airborne waves and, at the same time, as resonant scatterers for the elastic waves. When flexural waves propagate into the phononic crystal, high-order guided elastic mode with negative refractive index will be excited. For acoustic waves, the acoustic impedance of the pillars is much larger than the acoustic impedance for air, so that scatterers are regarded as rigid ones. The obtained negative phase velocity mode is related to Bragg scattering effects due to the rigid feature of the scatterers.

As a result, our design is able to simultaneously achieve superlensing beyond the diffraction limit for these two kinds of waves. The multiwave focusing feature of the proposed phononic crystal will have promising applications in energy harvesting, since the multiwave focusing can improve the efficiency of energy harvesting at the elasto-acoustic interface. Our proposed strategy will also enlighten the design of multi-physics and multi-waves device. For example, in previous 2D acoustic wave experiments with parallel waveguide systems [18][19], we can predict that acoustic wave distributions in the air may also influence the elastic wave distributions in the solid waveguide plate. Therefore, it will be interesting to study the elastic (acoustic) wave phenomenon accompanied by acoustic (elastic) wave phenomenon. In addition to the bifunctionality feature, it is worth mentioning that this work has explored the experimental demonstration of flexural wave superlensing. Recently, wave manipulations in thin plates have been actively studied. Indeed, the design of flexural wave lens [20], flexural wave metasurfaces for wavefront manipulations [21][22][23][24], and the demonstration of negative refraction [25] or imaging [26] for Lamb waves have been demonstrated. Elastic waves in thin plates have more complex dispersion relation and mode conversion phenomenon in comparison to acoustic waves. Thus, the realization of flexural wave superlens will be of significant interest for wave physics, and have various practical applications, such as nondestructive testing or subwavelength imaging in the solids.

In this research, we consider a thin aluminum plate on which a 2D phononic crystal will be arranged. The flexural waves propagate in this aluminum plate, having a Young's modulus of Eal=70×10 9 Pa, a mass density of ρal=2700kg/m 3 and a Poisson's ratio of νal=0.33. The plate thickness is of t0=0.2cm on which is glued our phononic crystal as depicted in Figs. 1(a-b). On the other hand, the acoustic waves propagate in the air with a mass density of ρ0=1.21kg/m 3 and sound velocity c0=343m/s. The phononic crystal marked by the blue color in Fig. 1(a) is 3Dprinted from poly-lactic acid (PLA) having a Young's modulus of EPLA=3.2×10 9 Pa, a mass density of ρPLA=1250kg/m 3 , and Poisson's ratio of νPLA=0.35 [27][28]. The unit cell of the phononic crystal is shown in Fig. 1(c), corresponding to the red region marked in Fig. 1(a). It consists of two pillars, which are built on a PLA substrate having a thickness t1=0.2cm. The lattice constant is p=4.75cm. The pillars are composed of two superposed cylinders. The based cylinders of the Pillars 0 and 1 have a radius of R0=0.96cm and R1=1.44cm with a height of H0=H1=0.5cm, respectively. The top cylinders have radius of r0=0.8cm and r1=1cm with a height h0=h1=0.5cm, respectively. We properly choose the different dimensions of the Pillars 0 and 1 in order to achieve negative refraction in the desired frequency range. The optimization of the structural parameters, viz., the different radius of two stages pillars, help to locate the surface mode within the negative refraction bands, which eventually lead to the superlensing effect for flexural and acoustic waves. The first ingredient to realize the superlensing effect is to seek for negative dispersion bands in the band structure of the phononic crystal [5]. Using the unit cells exposed in Figs. 2(a-c), we numerically compute the band structures (red circles in Figs. 2(d-f)) by COMSOL Multiphysics 5.4a of the elastic waves propagating in the plate attached to the phononic crystals, the acoustic wave propagating in the air surrounding the pillar structure, and for both elastic waves and acoustic wave propagating in the composite structure. The waves travel through the phononic crystal along the x-direction which corresponds to the ΓM path of the first Brillouin zone. In Fig. 2(d), one can see the existence of a branch for elastic waves with negative slope within 4.8-9.1kHz range. The Γ and M points in this mode (k=0.02 and k=1) are marked in the figure by blue circles. It is noted that for the range of k=0.02-0.38 (viz., 7.4-9.1kHz), the curve is with positive slope, while for the range of k=0.4-1 (viz., 4.8-9.1kHz), the curve is with negative slope. Thus, the frequency range of 4.8-7.4kHz marked by blue shadow is a pure negative refraction region for this mode. The dispersion curve of the flexural wave propagating in the aluminum plate is plotted (black line) in Fig. 2(d). The crossing point between the negative refraction curve and the dispersion curve of aluminum plate (near point A) indicates that the flexural wave propagating in the aluminum plate will be coupled to the phononic crystal array and be converted to a high-order elastic phononic mode with negative refraction phenomenon. The dispersion curve of aluminum plate at the crossing point has been folded several times in the band structure of phononic crystal. This can be numerically explained by the fact that the designed lattice constant (p=4.75cm) of the phononic crystal part is larger than the wavelength of acoustic and flexural waves.

Similarly, In Fig. 2(e), the dispersion curve of the sound wave propagating in the air is plotted (blue line) in Fig. 2(d). A branch for acoustic waves with negative slope within 6.9-9.0kHz range is marked by blue shadow. Combining the band structures for elastic and acoustic waves in Figs. 2(d-e) implies that the joint negative refraction band is within the range of 6.9-7.4kHz. The working frequencies are selected in this range, and marked by points A (7.1kHz) and B (7.25kHz) for elastic wave and acoustic wave, respectively. The band structure for the composite structure for both elastic and acoustic waves are shown in Fig. 2(f). In this figure, the unbounded acoustic mode is marked by the yellow color. Such an unbounded acoustic mode is located in the frequency range (>6.4kHz) where the dispersion curve of the acoustic wave has higher k value than that of flexural wave. The chosen working points A and B are located in the unbounded zone, indicating that the elastic mode used in this study is within the acoustic cone and thus would radiate acoustic energy from plate to air. This allows the exchange between elastic energy and acoustic energy, and makes some coupled elasto-acoustic mode possible. The simulations of the negative refraction in the phononic crystal for flexural and acoustic wave with plane wave incidences are shown in Supplementary Note 1. The second ingredient to realize the superlensing effect is to seek for edge modes in the supercell to ensure the enhancement of evanescent waves at the interface between the finite phononic crystal and the surrounding medium. Figures 3(a-c) show the simulated eigenmodes of the supercells in the solid, air, and composite structure, respectively. In these numerical simulations, the periodic conditions are applied on the upper and bottom boundaries, as marked by the blue lines. Red arrows indicate the propagating direction (x-direction). Black arrows indicate the enhancement of evanescent waves.

For the simulations of the single wave in Figs. 3(a-b), the z-component displacement distribution and acoustic pressure distribution are displayed. The edge resonant modes are observed at the frequency of 7.1kHz and 7.25kHz for elastic and acoustic waves, indicating that the evanescent waves can be enhanced at the interface of the array marked by the black arrows.

The two working frequencies are within the joint negative band of 6.9-7.4kHz, and are located at the frequency of A and B (7.2kHz) in Figs. 2(d-f). This means that we have a good coupling between the incident wave, the negative refraction wave in the phononic crystal, and the edge modes that will enhance evanescent wave. This agreement is achieved by choosing the optimized structural parameters of the Pillars 0 and 1.

Figure 3(c) shows the simulations of simultaneous elastic and acoustic waves in the composite structure. We can see that for the mode at 7.1kHz (corresponding to mode A in Fig. 2), edge modes can be observed in both elastic domain and acoustic domain. However, at 7.25kHz, (corresponding to mode B in Fig. 2) edge modes can only be observed in acoustic domain, but disappears in elastic domain. We can conclude that mode A at 7.1kHz is the coupled mode that supports edge modes for both elastic and acoustic waves, but mode B at 7.25kHz is not coupled and only supports edge mode for acoustic wave. The field distribution in the lattices of the eigenmodes at the frequency 7.1kHz and 7.25kHz for elastic and acoustic wave are shown in Figs. 3(d-e), corresponding to the points A and B in Figs. 2(c-d) with k=0.66 and k=0.68, respectively. Horizontal error bars are provided, which is obtained from three sets of measured data. The error bars indicate the influence of the positional accuracy. The simulated spatial resolution of the superfocusing, defined as the full width at half maximum (FWHM), is 1.9cm (viz., 0.36λ1, λ1=5.21cm is the wavelength of flexural wave in the aluminum plate at 7.2kHz).. The realization of the superlensing emphasizes the role of the phononic crystal's edge modes in the amplification of the evanescent waves.

On the other hand, Fig. 4(c) presents the simulated and experimental acoustic intensity distributions of the superlens for the acoustic waves at 7.2kHz. The simulation details for acoustic domain are shown in Supplementary Note 2. The PZT source for flexural wave could radiate sound energy into air due to the location in the unbounded region in Fig. 2(f), but the radiation efficiency is limited. We therefore choose the individual source for acoustic wave to ensure enough acoustic power. In the experiment, the source for acoustic waves is a small loudspeaker placed on the plate. The acoustic wave field measurements are accomplished by 1/8inch-diameter Brüel&Kjaer type-4138-A-015 microphone and Brüel&Kjaer PULSE Type 3160.

The size of the analyzed region is 9cm×4cm (x=11-15cm, y=-4. 5-4.5cm). The measured acoustic field also agrees well with the simulated one, for which the subwavelength focusing spot is shown in Fig. 4(c). Figure 4(d) provides the simulated and experimental normalized focal intensity along y-direction (x=12cm, y=-4. 5-4.5cm). The numerical FWHM is 2.1cm (viz. 0.44λ2, λ2=4.76cm is the wavelength of acoustic wave at 7.2kHz). This size is still subwavelength, and beyond the diffraction limit of 0.5λ, showing fine superlensing effect in the air. Furthermore, via the simulations, we find that the bifunctional superlens has a good performance in a certain bandwidth, within about 7.0 to 7.3kHz. This bandwidth is within the joint negative refraction band (6.9-7.4kHz), also around the two eigenfrequencies of edge modes (7.1kHz and 7.25kHz).

In experiment, some factors can influence the performance of the superlens. For example, the material properties may be affected due to additively manufactured parts. In order to demonstrate the robustness of the proposed superlens, we provide numerical simulations with different parameters which consider the possible error in the 3D printing fabrication. In Figs. 4(ef), the elastic modulus of PLA and the size (radius) of the pillars, are varying from 90% to 110% of the original one [27][28]. The simulated FWHM for flexural wave in Figs. 4(e) is ranging from 0.36λ1 to 0.4λ1 with fluctuant parameters, and the simulated FWHM for acoustic wave in Figs.

4(f) is ranging from 0.44λ1 to 0.46λ1. The results suggest that the potential errors of the parameters lead to a very small influence on the superlensing effect, which ensure the good agreement between numerical simulation and experimental measurement. To conclude, we have numerically and experimentally introduced a bifunctional superlens for simultaneous realization of flexural and acoustic wave superlensing by a single structure. The designed structure is a 3D-printed pillar-based phononic crystal array glued on a thin aluminum plate. Numerical studies have revealed that the designed phononic crystal can simultaneously produce negative refraction and edge modes to amplify the evanescent waves, which are the two necessary ingredients for superfocusing. By both numerical simulation and experimental measurements, the flexural and acoustic waves superfocusing are realized in the plate at 7.2kHz.

The FWHMs for flexural and acoustic waves focusing are 0.36λ1 and 0.44λ2, respectively. The proposed bifunctional superlens has a frequency bandwidth extending from 7.0kHz to 7.3kHz.

The multiwave focusing of the proposed phononic crystal has promissing applications in energy harvesting, since it can increase the total energy power at the interface between solid plate and the air medium. In addition, the proposed design strategy, viz., designing a structure that acts as rigid scatterers for the acoustic waves and as resonant scatterers for the flexural waves, could be extended to other bifunctional wave manipulations. Therefore, our findings may pave the wave to multiphysical and multifunctional devices [29][30][31][32].

See Supplemental Materials for the simulated negative refraction for flexural and acoustic waves, and the COMSOL simulation details.

Fig. 1 .

 1 Fig. 1. Design of the bifunctional superlens. (a) Schematic diagram of the superlens for flexural

Fig. 2 .

 2 Fig. 2. The band structures. The schematic diagram of the phononic crystal for (a) flexural wave

Fig. 3 .

 3 Fig. 3. Numerical simulation of the edge modes. (a) The eigenmode of the edge mode for the

Figure 4 (

 4 b) exhibits the simulated and experimental normalized focal intensity along y-direction (x=12.5cm, y=-4.5-4.5cm).

Fig. 4 .

 4 Fig. 4. Demonstration of bifunctional superlens. (a) The simulated and experimental acoustic

Acknowledgements

This work is supported by the Air Force Office of Scientific Research under award number

Data Availability Statement

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

The physical conditions to produce superlensing effect for both elastic and acoustic waves are now established by designing the suitable phononic structure, verified by the realization of both negative refraction and the enhancement of the evanescent waves responsible to carry the sub-wavelength features. We then will show the numerical and experimental demonstration of our bifunctional superlens. Since the eigenfrequencies of edge mode for elastic and acoustic waves are adjacent 7.1kHz and 7.25kHz, we choose 7.2kHz as the joint working frequency for the bifunctional superlens. Figure 4(a) shows the simulated and experimental intensity distributions (the square of z-component displacement, w 2 ) of the superlens for flexural waves at 7.2kHz, evidencing a focusing spot. In the experiment, the source for flexural wave is a piezoelectric transducer (PZT) with a radius of 1.3cm glued tightly on the plate as shown in Fig. 1(b), for achieving good couplings between the source and the plate. The PZT source is excited FA9550-18-1-7021, and by la Région Grand Est and Institut Carnot ICEEL.


Shi-Wang Fan

Yan-Feng Wang
email: yswang@tju.edu.cn

Liyun Cao

Yifan Zhu

A-Li Chen

Brice Vincent

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Yue-Sheng Wang

Acoustic Vortices with High-order Orbital Angular Momentum by a Continuously Tunable Metasurface

Keywords: acoustic vortices, broadband tunable metasurfaces, high-order orbital angular momentum

Metasurfaces-based acoustic vortex generators formed by fixed microstructures can only transfer the first-order orbital angular momentum (OAM) at a single frequency in a waveguide. Here, the multiple order acoustic vortices are realized through a helical matesurface design which is geometrically compact, broadband and tunable. The proposed metasurface consists of matched helixes rotating into the corresponding helical channels allowing to modulate the reflected phase shifts of the individual units. The units are arranged in a compact circular configuration, and then passively transforming an incident plane wavefront into the desired spiral wavefront with a controllable topological charge. The reported theoretical, numerical and experimental results evidence that multiple orders of OAM modes can be continuously generated by using only a single metasurface. The tunable vortices design is relevant to the pragmatic applications of metasurfaces in real-time acoustic communications, spanners and tweezers.

Acoustic metasurfaces (AMs) are constituted by rationally designed individual units with preconceived phase and amplitude responses to efficiently modulate wavefronts. 1,2 However, there still is one particularly technical difficulty to broaden and tune the operating frequencies. 3,4 Indeed, most of the existing AMs have fixed and rigid microstructures that can only operate at a single or a narrow frequency range. To break through this restriction, a few effective ways have been proposed to design integrally adjustable AM structures 5,6 or independently reconfigurable units of AMs. [7][START_REF] Ma | Proc. Natl. Acad. Sci[END_REF][START_REF] Tian | [END_REF][10][11][12][13][14][15] These works of tunable design have been explored to deal with multiple acoustic functions including asymmetric transmission, [5][6][7] sound shaping, [START_REF] Ma | Proc. Natl. Acad. Sci[END_REF][START_REF] Tian | [END_REF] acoustic focusing, 10,11 flat reflected manipulation, 12,13 curved cloaking and illusion, 14,15 and so on.

Among various functions of AMs, acoustic vortex (AV) is one of the most promising applications to transfer orbital angular momentum (OAM) for micro-fluidic mixing, 16 high-speed communication, 17 acoustic tweezers or spanners, 18,19 etc. The vortex beam is characterized by a spiral pressure wavefront exp(imθ), where the phase factor ϕ=mθ is linearly proportional to the azimuth angle θ. The integer m indicates the topological charge (TC), corresponding to the OAM order of the vortex carried, which determines the total phase accumulated in one full annular loop around the propagation axis. [20][21][22][23][24] Through the designed AMs, AVs can be produced in an infinite free space, 25,26 or in a waveguide system. [27][28][29][30] The vortices in free space can eliminate the boundary effect and maintain the OAM pattern in the near field. However, due to the divergence of the radiation cone angle, the pressure amplitude of the AV decreases and the pressure profile expands with the propagation distance increasing. Then, the vortex signal is severely limited by its transmitted distance in the infinite free space. Also, AVs generated in a waveguide system with a suitable wave number might be more beneficial to the remote signal propagation. Recently, the Helmholtz resonators were utilized to get AVs, and reduce the AMs thickness down to 1/2 wavelength. 27,28 A coiling-space AM was designed with 1/3 wavelength. 29 Due to the fixed microstructures, they were only used to generate first-order AVs at a single frequency.

In this letter, we introduce the matched helical mechanism [10][11][12] to design a continuously tunable acoustic metasurface (TAM). By changing a helical depth, we can dynamically manipulate the reflected phase shifts of the acoustic wave from the TAM unit. Then, the higher-order AVs can be achieved. The field amplitude and phase distributions of the higher-order AVs are demonstrated through both analysis and simulation. Experiments are further carried out to evidence the realization of the multiple OAM states by the reconfiguration of only one TAM. In addition, the helical design of the TAM can decrease its thickness down to less than 1/4 wavelength at a relatively low frequency. Figure 1(a) shows the case of an acoustic plane wave which is incident on an AM, and then reflected as a vortex beam from its surface. The AM consists of M rows in the radial direction, in which each row of the planar layer is divided into N sectors, as shown in Fig. 1(b). To achieve the AVs with a TC of the integer value m, the proposed circular configuration should cover the phase shift range of 2mπ with the same phase gradient. Then, the desired phase shift reflected from the j-th sector on each row is given by

ϕ 1,1 ϕ 1,2 ϕ 1,3 ϕ 2,1 ϕ 2,2 ϕ 2,3 ϕ M,1 ϕ M,2 ϕ M,3 ϕ M,N ϕ 2,N ϕ 1,N ϕ M,N-1 ϕ 2,N-1 ϕ 1, N-1 R T 1 T 2 T 3 H T 1 T 2 T 3 (
In this general configuration, N controls the resolution of the generated helical wavefront and determines the maximum TC that the AM can produce, while M controls the amount of the reflected power.

For theoretical interpretation, in order to generate AVs in a rigid waveguide tube with the radius of R, the reflected field can be represented as the superposition of cylindrical Bessel modes: ,

where the axial wave number is kz 2 =k 2 -km,n 2 ; k=2π/λ is the wave number, in which λ=c0/f is the wavelength with f being the incident frequency and c0 the sound velocity; km,n represents the radial wave number which is the nth positive root of the equation dJm(km,nR)/d(km,nR)=0; Jm(km,nR) is the m-order Bessel function; and Am,n is the modal amplitude. We construct the compact TAM as an assembly of eight fanlike sections (N=8) of helical channels over the whole azimuth, as shown in Fig. 1(c). This number of sections gives a reasonably good

, 2 ( 1)
, ( =1, 2, , ), >Max (2 +1, 4).

M N m j N j N N m f p = - ××× , , , exp( )
( ) exp[ ( )] m n m n m m n z m n p i t A J k R im kz w q = - +
åå resolution for generating vortices with TC of |m|≤3 according to Eq. (1). For generating more higher-order vortex beams or finer resolutions, the number of sections can be increased accordingly.

Each section is designed to be composed of three rows (M=3) of helical channels in the radius (more rows can be employed for a larger radius). The total diameter of the cylindrical TAM is R=7 cm. By considering Eq. ( 2), the corresponding propagable wave numbers and the limiting frequencies for the fundamental mode are listed in the Table 1. The other modes gradually fall in the evanescent state and are finally captured in the near field of the surface at low frequencies, while the vortex fields will be disturbed by the superposition of higher-order Bessel-like modes at high frequencies, see the details in supplementary material.

From Table 1, we can know that the first, second and third order vortices can be generated at the frequency ranges of [1.44, 4.16], [2.38, 5.23] and [3.28, 6.25]kHz, respectively. These three AVs can be produced in the intersection frequency ranges as [3.28, 4.16]kHz. However, there is no common frequency range between them when m≥4. Therefore, the continuously frequency range considered in the present work will be from 1.5 to 6.0kHz.

The designed TAM consists of matched helixes rotated into helical channels from bottom to top, as shown in Fig. 1(c). Based on the matched helical mechanism, [10][11][12] when the matched helix is tuned by rotating, the channel length characterized by the helical depth (h) will be changed. Each sector in each row can be regarded as an independent tunable unit. The reflected phase shift can be then varied by changing the helical depth to in real-time generate multiple order AVs in broadband frequency range. The total height of the TAM is H=55mm. The three rows have the radius of r1=10mm, r2=20mm and r3=30mm, respectively. The thread leads in three rows are T1=70mm, T2=130mm and T3=240mm, respectively. The unit wall and blade thickness are t=1mm. The thin total thickness of the TAM, the easy tunable operation of the unit, and the plane wavefront reflected from the helical channels can be simultaneously satisfied with tradeoffs between three parameters: the thread lead, the helical radius, and the central column, in the limit of approximate equality. The central column radius is r0=10mm. To ensure the computational accuracy of the reflected field, the center column will be considered in the first row in the following calculations. To illustrate the broadband tunability, we should first obtain the relation between the phase shift and the helical depth in a continuous frequency domain. We present the computing process for the first-row (T1) as an example. Then, the equivalent propagation path [Le(h)] of the theoretical model is developed in Fig. 2(a) to calculate the phase shifts by using the transfer matrix method. 14 Therefore, the normalized amplitude and phase shift of reflection can be calculated by

where the area ratio is S12=(S1-S2)/(S1+S2). It can be found from Eq. ( 3) that the reflected amplitude [A=abs(pr)] is always equal to 1 by MATLAB programming calculation. This means total reflection from the TAM without any energy loss. Variation of the phase shift [ϕ=arg(pr)] with the incident frequency and helical depth for the theoretical model is calculated and illustrated in Fig. 2(c). We refer to supplementary material for the theoretical derivations and simulated settings in detail.

Then, the finite element model for the helical units in the first-row is built in Fig. For quantitative comparison with considering the limited frequencies in Table 1, the cut lines of the phase shift at the frequencies of 1.6, 2. It is observed that the theoretical and numerical results for these three-rows are consistent with each other in terms of the number of full 2π ranges and the varying phase shift trend. Therefore, the reflected acoustic waves can be manipulated broadband by continuously adjusting the helical depth.

All the needed phase values by Eq. ( 1) and the corresponding helical depths at an operational

S i k L h p A i S ikL h f

+ - = = + -
frequency can be obtained directly from the phase shift responses. Note that the interactions between units are very weak and negligible, see supplementary material. For clarity, the tuned helical depths for multiple OAM modes at different frequencies are listed in supplementary material. First, the vortex transition from the near to the far field in the waveguide is concerned in the analysis and simulation, see supplementary material. In contrast, we discuss the limited propagation distance of the vortex in an infinite free space. It is found that the OAM pattern can propagate at x (cm)

x (cm) x (cm) x (cm) x (cm) y (cm) y (cm) (b) f=3.6 kHz 1 -1
most 2λ, where the pressure amplitude are reduced to indistinguishable, see supplementary material.

We also present the acoustic pressure fields including the amplitude and phase of the vortex using the thermoviscous acoustics model. The results indicate that the influence of the thermoviscous loss on the acoustic performances can be ignored (see supplementary material for details).

We then further systematically investigate the patterns of the multiple order AVs at different frequencies, as shown in Fig. 3. The reflected phase and amplitude distributions of the multiple order vortices with |m| =l (l=1, 2, 3) at frequencies of 1.6, 2.6 and 3.6 kHz are illustrated in Fig. 3(a), respectively. The observation transverse plane is located at z=50 cm. In the left side, it can be observed that the wavefront have changed 2π, 4π and 6π in an annular loop of the phase fields.

The corresponding pressure amplitude profile also has an approximate symmetric distribution with null pressure at the central region. Likewise, the phase variation of the multiple order vortices and the corresponding null amplitude in cores can be exhibited at a single frequency of 3.6 kHz in the left side of Fig. 3(b). This cross-section is located at z= 60 cm. In addition, although the helical channels of the TAM structure have a definite clockwise spiral direction, the reverse vortex fields with m=-l (l=1,2,3) can also be well formed in the three right-hand side panels of Fig. 3. These all prove the generations of the first three order AVs. Note that the OAM mode propagation in the waveguide is the superposition of multiple Bessel modes especially at high frequencies. 30 In addition, the sound waves reflected from the helical channels may not have a perfect plane wavefront and may be accompanied by the parasitic or high-order scattering. 31 Thus, the null cores in some cases are not strictly located in the geometrical center or have a small asymmetry in the amplitude profiles. profiles are observed between the simulations in Fig. 3 and measurements in Fig. 4. For quantitative comparison, the circular cut lines of the phase profiles at r=6 cm in the radial position (dashed lines with arrow) are plotted in Fig. 4(e). The phases have shifted from -π to π for one time and two times in an annular loop for the first-order and second-order TAM configurations, which clearly shows that the TCs of the corresponding generated AVs are m=1 and 2, respectively. In addition, Figure 4(f) shows that both the simulated and the experimental phases have shifted 2mπ for the multiple order OAM modes at the single frequency of 3.6kHz. This proves that the designed TAM possesses the characteristic of continuously tunable TC in the waveguide system.

To conclude, we have proposed a continuously tunable acoustic vortex generator based on a reconfigurable AM with theoretical analysis, numerical simulation, and experimental demonstration.

The designed TAM is formed by the matched helical channels to modulate the reflected phase shifts of the individual units. This adjustable design is capable of spatially spiraling the acoustic wave over

FIG. 1 .

 1 FIG. 1. (a) The schematic diagram of the realization of acoustic vortex in a cylindrical waveguide. Wave front

FIG. 2 .

 2 FIG. 2. (a) The equivalent theoretical model and (b) the numerical simulation model of the helical units for the

  Theoretical analysis (T 1 ) (d) Numerical simulation (T 1 )

  2(b) by using the commercial software COMSOL Multiphysics 5.4. The simulated phase shift responses for different frequencies and helical depths are shown in Fig. 2(d).

  6 and 3.6 kHz [dashed lines in Figs. 2(c) and 2(d)] are plotted in Fig. 2(e). Similarly, we could get the phase shifts for the other two-rows (T2 and T3). The corresponding theoretical and simulated results are shown in Figs. 2(f) and 2(g), respectively, see supplementary material.

FIG. 3 .

 3 FIG. 3. (a) The reflected phase and amplitude distributions of the multiple order AVs with |m|=l (l=1, 2, 3) at

  ) x (cm) x (cm) x (cm) x (cm) x (cm) y (cm) y (cm)

FIG. 4 .

 4 FIG. 4. (a) Schematic of the experimental setup. (b) Photographs of the metasurface sample. Experimental

  Fig. 4(b). The details of the devices and the measurement processes are found in supplementary

Table 1 .

 1 The

	km,1R (fm,1)	km,2R (fm,2)
	m=1 1.84 (1.44 kHz) 5.33 (4.16 kHz)
	m=2 3.05 (2.38 kHz) 6.71 (5.23 kHz)
	m=3 4.20 (3.28 kHz) 8.02 (6.25 kHz)
	m=4 5.32(4.15 kHz) 9.28(7.24 kHz)

propagable wave numbers (limiting frequencies fm,n) for the fundamental modes in the cylindrical waveguide with diameter R=7 cm.

Acknowledgments:

This work is supported la Région Grand Est, by the Institut CARNOT ICEEL, and by the National Natural Science Foundation of China (Nos. 11872101 and 11991031). The first author is grateful for the support of the National Natural Science Foundation of China (No. 11502123), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (No. 2015JQ01), and the China Scholarship Council (CSC No. 201807090043).

a wide frequency range and producing the twisted multiple order AVs in the waveguide system. Each vortex order at different frequencies and the multiple OAM models at a single frequency are realized by the reconfiguration of only one TAM. We expect that the TAM can yield AVs with remarkable flexibility and suggest exciting possibilities in a variety of applications. The latter include the real-time acoustic communication, micro-particles manipulation, and micro-fluidic fast mixing.

See the supplementary material for the propagable frequency range, the theoretically derivation, the interactions between units, the tuned helical depth, the vortex transition in the waveguide, the vortex in the free space, the influences of the thermoviscous loss effects and the experimental measurement in detail.