Reconfigurable and Phase‐Engineered Acoustic Metasurfaces for Broadband Wavefront Manipulation

A novel type of phase‐engineered acoustic metasurfaces with reconfigurable properties is reported, enabling the flexible broadband manipulation of reflected wavefronts. The participants of the metasurface are elements conceived to possess even‐distributed reflected phases covering 2π span with linearity and small acoustic energy loss. The reconfigurable property of the metasurface is implemented by rearranging the fixed meta‐elements based on the phase profile, which is related to the characteristic of a specific wavefront shape. The metasurface's capability is successfully demonstrated to achieve acoustic focusing and bending within the frequency range of 2300–2800 Hz, showcasing its feasibility and adaptability. To enhance its practical applications, porous materials are incorporated, leveraging the robustness of phase differences among the meta‐elements to achieve high acoustic energy cancellation. Effective sound attenuation occurs within the frequency range of 1300–3100 Hz, even under wide‐angle incidences ranging from −80° to 80°. The work paves the way for further research on reconfigurable acoustic metasurfaces in broad frequency regions and exerts favorable implications for generally applicable structures applied in multi‐fields including biomedical acoustics, noise control, and so on.


Introduction
[18] By engineering the acoustic wavefront elaborately, the application fields of acoustic metasurfaces are broadened into acoustic communication, [19][20][21][22] noise attenuation, [23][24][25][26][27][28][29][30][31][32] medical therapy, [33][34][35] and so on thus far.The design of the metasurface relies on the meta-atom, which functions as a catalyst for constructing the desired phase profile.44][45][46][47] Nevertheless, an undeniable fact is that the gradation of operating frequency range influences the presented performance, [41,[48][49][50] that is, efficiency and practicability in engineering acoustic wave, indicating the restriction in practical application ascribed to the narrow working frequency range.On the other hand, in numerous circumstances, metasurfaces work in a fixed pattern once manufactured.A diverse functionality calls for the reconfiguration and refabrication of the initial structure, while the structural dimensions need to be altered occasionally.To address this issue, acoustic metasurfaces with reconfigurable capability are proposed, [10,11,41] where meta-atoms needless to be modulated are inserted into an established frame so that the arrangement modes remain flexibly controllable.Reconfigurable metasurfaces can be designed on demand and are expected to be adopted in complex application scenarios.However, the acoustic metasurface behaving both broadband and reconfigurable performance is in the developing phase.
Moreover, acoustic wavefront conversion [51] takes a significant place in acoustic wave shaping, which is also achieved by introducing abrupt phase shifts along the metasurface based on the GSL.In this circumstance, the minimization of reflected wave energy is approachable, contributing to the potentially realizable functionality of highly efficient sound attenuation.Along this path, a metasurface with applicable multifunction deserves to be discussed.
In this work, we put forward a novel type of reconfigurable acoustic metasurface capable of manipulating wavefront within a broad frequency range.The meta-elements are designed as nonuniform coiled-up channels evolving from cascaded cavity series, the reflected phase covering the span of 2 presents linearity varying with frequency and equal phase difference within 2300 and 2800 Hz.Arranging the designed meta-elements in diverse modes corresponding to the specific wavefront shapes, the effective reconfigurability of the proposed metasurface is demonstrated through the typical functionalities of wavefront manipulation, acoustic focusing, and acoustic bending.Another functionality for broadband and quasi-omnidirectional sound energy cancellation is achieved by means of converting the propagating wave to an evanescent surface wave, which benefits from the robustness of phase differences remaining in the designed elements to a great degree.The high-efficiency sound attenuation performance is presented in the range of (1300,3100) Hz under a wide incidence of −80 and 80°, making it highly feasible for practical applications.

Meta-Element Design
The generalized Snell's law (GSL), which is derived from Fermat's principle of stationary phase, can be expressed as [4,5] sin where  re and  in separately refer to the reflected and incident angles of the acoustic wave,  is the wavelength of the acoustic wave in the air, Φ(x) is the reflected phase along x-axis direction, and x is the spatial location.Generally, the GSL is harnessed to form the multivarious acoustic wavefront via introducing the extra phase shift at the interface where acoustic wave impinges on the structure. [6]To form a meta-atom array with an appropriately additional phase, a highly efficient method is to disperse the 2 span into a certain number of uniform steps to substitute the continuous phase shift, for which the step of /4 corresponding to eight diverse elements is adequate. [5,10,41]In other words, the phase difference ΔΦ of adjacent elements ought to be /4.
To fulfill the requirement of wave manipulation within a relatively wide frequency region, the phase shift distribution of proposed elements is designed to behave gradient phases with constant phase difference within a target frequency range, as shown in Figure 1d, in which f l and f h represent the lower-and upperfrequency limits.
In previous studies, numerous metasurfaces designed for acoustic wave manipulation are constructed by the elements with a representative configuration, that is, the cavity-like structure, [52,53] for instance, the uniform cavities varying with heights, [54,55] and the uniform coiled-up channels with diverse internal dimensions. [5,7]The latter can also be regarded as the uniform cavities holding different cross-sectional areas (s 1 ) and heights (l 1 ) by the conversion of incident area (s 0 ), as shown in Figure 1a.Accordingly, the broadband characteristics of the uniform cavity are first studied to search for a desired configuration.To simplify the derivation process, the thermal viscous loss generated in the cavity is not considered.The deviations of phase shift are presented in Supporting Information.The relationship among phase shift  1,i , the cavity dimensions and the frequency f can be expressed as where c 0 is the sound velocity of air,  s1,i denotes the ratio of the sectional area of the incident wave (s 0 ) and the effective sectional area of the cavity (s 1,i ), i represents the parameters of the ith element for the homologous layer (representing the same variable in the following).Owing to the complexity of the formula, the slope of the total phase shift  1,i is calculated to verify the linearity, which can be written as It can be seen that the slope df is a function of frequency f, that is, the slope varies with frequency despite the geometric parameters ( s1,i and l 1,i ) of the cavity are adjustable, as illustrated in Figure 1b.Therefore, this configuration is incapable of meeting the proposed requirement.
An additional cavity with  s2,i and l 2,i is then connected to the original structure for a larger design freedom, as shown in Figure 1c.The phase shift of the derived structure can be calculated from where ϑ 1,i = − 2fl 1,i /c 0 , and ϑ 2,i meets the relationship that tan(ϑ 2,i ) = − 2fl 2,i /c 0  s2,i , and  s2,i = s 1,i /s 2,i denotes the crosssectional area ratio of the first layer to the second layer.The phase shifts of elements constructed with more layers can be obtained, similarly.Once the phase shift of the first element Φ 1 (f) is determined, the required phase shifts of the other seven elements can be obtained by The desired phase distribution of the eight elements is plotted in Figure 1d.The dimensions of each layer can be reciprocally deduced by selecting the parameters in turn from the first layer.

Theoretical Model and Designed Parameters
A conceptual diagram of the acoustic metasurface composed of meta-elements is illustrated in Figure 2a.The inset in the upper-right corner presents the deviations of the reflected acoustic waves of three representative functionalities in wavefront manipulations.According to the design process implemented above, the internal configuration for meta-elements of the metasurface is designed as a cascaded cavity series with multiple layers, as displayed in Figure 2c.The structure can be equivalent to nonuniform coiled-up channels (NUCC) after being bent.The overall dimensions of the NUCC are height H = 37 mm, length a = 25 mm and width L = 10 mm, within which each layer possesses different lengths and heights, except for the widths.The length and height of the ith cavity are denoted as w i and l i , respectively.The thickness of each wall of the proposed NUCC is t = 1 mm.The structural dimensions of the NUCC for eight elements (referred to as elements #1, #2, #3, …, #8) are elaborately designed and selected.The geometric parameters and the 2D schematics of each element are separately shown in Table 1 and Figure 2b.The phase shifts of these elements calculated by theoretical and numerical methods (Figure 2d) cover the whole 2 span and hold the roughly same slopes with an identical phase difference (/4) varying with the frequency range from 2300 to 2800 Hz.The phase differences between element #1 and elements #2 to #8 are displayed in Figure S3 (Supporting Information).The analytical results coincide with the numerical ones, which verifies the effectiveness of the theoretical model.In addition, the numerically predicted reflectance of these eight completely exceeds 0.8, in which the reflectance of elements #1 and #8 are located in the vicinity of one, as illustrated in Figure 2e.

Reconfigurable Metasurface for Broadband Wavefront Manipulation
To demonstrate the broadband wavefront manipulation capability of the proposed metasurface and its convenience for the reconfigurable structure, the arrangement modes of the designed NUCC are constructed to realize multifunction, acoustic focusing, and acoustic bending, which are enumerated as two typical functionalities of the wavefront manipulation.
Herein, the incident waves are all set as normal incidence.Thus, the GSL expressed in Equation (1) can be rewritten as In terms of acoustic focusing, the coordinate of the focal point is set as (0, y 0 ).sin  re in Equation ( 6) according to the desired wavefront shape can be described as [5] sin Adv. Physics Res.2024, 3, 2300128  By substituting Equation ( 7) into Equation ( 6) and introducing the initial condition Φ(x = 0) = 0, the phase profile along the x-axis ought to meet where the  here is chosen from the frequency of 2550 Hz, and y 0 is set as 0.4 m.The element number at each position along the x-axis is determined by opting the element with the nearest phase shift in the profile curve.The phase profiles required by the proposed wavefront performance and provided by the designed metasurface are illustrated in Figure 3a, represented by the black solid line and discrete red points, respectively.The 2D schematic diagram of the metasurface is depicted below.The reflected acoustic pressure fields at three typical frequencies of 2300, 2550, and 2800 Hz are attained by numerical simulations, as shown in Figure 3c-e.A focal spot is distinctly visible directly above the field's zero point at each appointed frequency, demonstrating the excellent broadband acoustic manipulation property of the metasurface.The coordinates (x 0 ,y 0 ) of the focal points for these three frequencies are severally (0, 0.30), (0, 0.40), and (0, 0.46).In addition, the normalized acoustic intensities along the x direction (y = y 0 ) and y direction (x = x 0 ) are calculated and plotted in Figure 3f-h For another functionality of acoustic bending, the relationship of the acoustic wave behaving arbitrary trajectory can be derived from Equation ( 6) as [14,41] dΦ (x) where x = x(y) is the desired trajectory, and x′(y) is its slope.In pursuit of a semicircular trajectory with the radius r and center coordinates (0, r), whose expression is x = √ r 2 − (y − r) 2 , the phase profile along x-axis can be obtained by substituting it into Equation ( 9) herein, r is set as 0.35 m.The desired phase profile and arrangement mode of the elements pertaining to the reconfigurable metasurface are reconstructed, which are separately expressed by the black solid line and blue scatter, as plotted in Figure 3b.Note that the dimensions of the elements are fixed, the multi-functionality is realizable simply by the variation of the arrangement.And the 2D schematics of the metasurface are displayed under it.More detailed diagrams are illustrated in Figure S1 (Supporting Information).The extra phase shift provided by the metasurface enables the acoustic beam to propagate in a semicircle line within a broadband frequency range from 2300 to 2800 Hz, as sketched in Figure 3i-k, which is identical as expected.As such, the multifunctional capability of the metasurface achieved by altering the arrangement of fixed elements is proved.It is noted that only two functionalities of the wavefront manipulation are presented here, whilst multivarious functions can be achieved by strategically managing the phase shift along the interface.

High Acoustic Energy Cancellation with Robust Phase Difference
The functionalities described in Section 3.1 are achieved by leveraging the GSL in positive thinking, which requires maximizing the energy of the reflected wave to form a more effective functional wavefront.On the other hand, the GSL can be adopted for wave conversion to minimize the reflected wave in reverse thinking, thereby realizing high acoustic energy cancellation within a broad bandwidth.In this circumstance, the reflectance of the acoustic wave ought to be relatively low.Thus, the air domains of the original eight elements are replaced by the porous material, which ordinarily serves as an absorptive component.The polyurethane (PU) foam is employed here, and can be characterized as an equivalent medium by the Johnson-Champoux-Allard (JCA) model, [56,57] consisting of five transport parameters [ϕ,  ∞ , Λ, Λ′, ]: porosity, tortuosity, viscous and thermal characteristic lengths, and flow resistivity, respectively (Table 2).The deviations of two parameters, the equivalent wavenumber k eq and characteristic acoustic impedance Z eq , which describe the characteristics of the medium, are listed in Supporting Information.And the values of these two are obtained and plotted in Figure 4a.The phase shifts and reflectance of the proposed eight elements calculated theoretically and numerically are presented in Figure 4b-d.The phase shifts of the eight elements possess robust phase differences.Specifically, they exhibit linear variation over the phase range of 2 with basically consistent slopes and phase differences (/4) within the frequency range of (1800,2600) Hz, which benefits from the design of the structural configuration.Compared with the results in Figure 2e, the reflectance is effectively reduced by the porous medium, resulting from the internal energy dissipation.Moreover, the analytical and numerical results are in good agreement with each other, demonstrating the correctness of the analytical model.The GSL is then harnessed as a guide of the element arrangement with periodicity, which can be rewritten considering the diffracted wave as [53][54][55]58] sin where m denotes the diffracted order stemming from the periodicity of the structure.Supposing that the metasurface in a period is constructed by arranging the elements in the sequence, that is, elements #1, #2, #3, …, #8, then the factor dx can be simplified as 2/D owing to the linear variation of Φ(x), in which D is the periodic length of the metasurface.As such, Equation ( 11) can be expressed as In this arrangement, higher-order waves with anomalous reflected angles are proven to carry the most energy, [58] thereby should be removed for effective wave absorption.It is stressed that the specular reflection (m = 0) always exists.To address this issue, the method of transforming the higher-order propagation waves to evanescent surface waves along the metasurface is proposed.As its name suggests, the reflected angle  re of surface wave is ± 90°.To determine the value range of periodic length D, Equation ( 12) is solved by substituting in the minimum value of  within the frequency range of (1800,2600) Hz, that is, D ≤ /2 = 66 mm.To meet this requirement, the elements should be connected by the longer side, and four of the whole elements with the phase difference of /2 are selected to satisfy the linear characteristic of phase shifts.Herein, the arrangement of four elements #1, #3, #5, #7 (with the periodic length D 2 ) is adopted, and the arrangement of all eight elements (with the periodic length D 1 ) is also considered here for comparison, as demonstrated in Figure S2 (Supporting Information).
The relationships between the angles of reflected and incident waves for two conditions at the lower and upper frequencies of the target range (1800 and 2600 Hz) based on Equation ( 12) are illustrated in Figure 4e,f, respectively.It can be observed that the curves of m = ± n (n = 1, 2, …) are symmetric along the curve of m = 0 owing to the odd property of the sine function.For the metasurface with eight elements, the reflected waves with orders over ± 1 are all transformed into surface waves.The critical conditions indicate the cases that reflected angles  re = ± 90°, which are expressed by the variables with subscript c.Critical angles  ±1 c = ∓40.5 • of ± 1 order reflected wave marked by red arrows are obtained from Equation (12), that is, the + 1 and − 1 order reflected waves exist severally with the incident angle of (− 90, − 40.5)°and (40.5,90)°, which leads to the increase in reflectance.It is noteworthy that the +1 order reflected wave also exists at a frequency lower than 2600 Hz, specifically, from the critical frequency f c = 2144 Hz derived from  c = 2D, indicating that the increase in reflectance occurs during a broadband frequency domain.Otherwise, for the metasurface with four elements, the reflected waves with orders over 0 are all converted into surface waves (Figure 4f), thereby facilitating the decrease in reflectance with wide-angle incident waves in the whole target frequency range.
The reflected acoustic pressure fields are simulated by numerical methods with the typical incident angles at the lower fre-quency 1800 Hz and upper frequency 2600 Hz of the target frequency range, as displayed in Figure 5. Figure 5a shows the reflected pressure field with  in = − 60°at 1800 Hz, the metasurface here consists of eight elements.Only the m = 0 wave apparently exists, which matches well with the predicted curve in Figure 4e.The majority of the acoustic energy is transformed into the surface wave without propagating into the scattered field.Similarly, Figure 5b shows the reflected pressure field with  in = − 60°at 2600 Hz for the metasurface with four elements.The reflected wave propagates as the form of specular reflection with a relatively lower pressure amplitude, which corresponds to a lower reflectance.Thus, it is proved that the GSL remains effective in this case when the phase interval turns into /2.
Figure 5c,d present the reflected pressure fields of the metasurface with eight elements under  in = − 60°and 60°at 2600 Hz, respectively.In both circumstances, the incident angles  in locate out of the domain between two critical angles (− 40.5°and 40.5°), suggesting the presence of reflected waves with orders over 0. Comparing Figure 5d,b, m = + 1 wave propagates into the scattered field with a reflected angle of 51.5°, which agrees well with the reflected angle labeled by the red star in Figure 4e.More acoustic energy is scattered into the reflected field, leading to a higher reflectance.Additionally, the comparison of these two indicates that m = + 1 wave carries more energy than m = 0 and m = − 1 waves.The reflected performance for the metasurface with eight elements under critical incidence is displayed in Figure S5 (Supporting Information) to show a more evident conversion to surface wave, which suggests that the reconfigurable metasurface of four elements is a more appropriate candidate during the target frequency range under wide-angle incidence.

Broadband and Quasi-Omnidirectional Sound Absorption Performance
The sound absorption coefficient of metasurfaces with four elements and eight elements, and the uniform PU foam possessing the same height under normal incidence are predicted and illustrated in Figure 6a.Indeed, the absorptances of metasurfaces with four and eight elements are shown to be consistent across the frequency range of (1300,3100) Hz, which extends out the range of 500 Hz based on the target frequency domain.The absorptances of these two are significantly improved compared with the uniform porous material.The color lumps of Figure 6a,b covers the range from 0.7 to 1.0 of the reflectance to embody the superiority of the absorptive performance of the proposed metasurface more intuitively.To consider a more general application scenario, the absorptance of the diffusion field is defined as The results sketched in Figure 6b reflect the consistent tendency with Figure 6a.To quantitatively compare the performance of the metasurface with uniform PU foam, the average absorptances of these two within the range of (1300,3100) Hz are plotted in Figure 6c.It is observed that the average absorptance of metasurface with four elements (0.955) is 38.4% higher than that of the ordinary PU foam (0.690).
Furthermore, Figure 6d,e separately shows the absorptances of the metasurfaces with four and eight elements under the wide-angle incidence of (−80,80)°in the frequency range of (1300,3100) Hz.It is seen in Figure 6d that the absorptance exceeds 0.9 under a wide range of incidence within the broad frequency domain.The absorptance of the former one varies smoothly as the frequency changes, and holds effective absorptive performance under the quasi-omnidirectional incidence, while the absorptance of the latter one begins to drop sharply from the critical frequency f c , as mentioned in Section 3.2.The absorptance at f h = 2600 Hz appears a relatively obvious decrease corresponding to the incident angle of ≈−40°, as labeled by the silver star, which is in coincidence with the critical reflected characteristics demonstrated in Figure 4e.
Replacing the aforementioned PU foam with other porous materials, the acoustic performance of the individual elements and whole metasurface are shown in Figure S6 (Supporting Information).The phase shift distributions of eight elements maintain uniform linearity with a phase difference of ≈/4 within different frequency ranges, indicating the wave manipulation capability during a tunable frequency band.Moreover, the highly efficient absorptive band of the reconfigurable metasurfaces also changes accordingly, and the absorptance within an ultrabroadband (1000,8000) Hz remains relatively high.

Conclusion
We have introduced a reconfigurable metasurface composed of specialized meta-elements featuring nonuniform coiled-up channels, enabling versatile manipulation of wavefronts across a broad spectrum.This innovative metasurface holds immense potential across various fields.The meticulously designed metaelements exhibit phase linearity within the (2300,2800) Hz range, maintaining identical phase differences and a high reflectance exceeding 0.8.These characteristics are essential for constructing a broadband metasurface.In the process of establishing this reconfigurable metasurface, fixed elements are meticulously arranged in a customizable configuration, conforming to specific wavefront shapes guided by the principles of generalized Snell's law.Broadband acoustic focusing and bending serve as illustrative examples, demonstrating the metasurface's feasibility and adaptability.The phase differences exhibit remarkable stability, and with the addition of porous materials, the reflectance decreases significantly.This innovation enables highly efficient sound absorption within the (1300,3100) Hz range, even under wide-angle incidences ranging from −80°to 80°, achieved through wavefront conversion.Our work presents a robust and practical solution for constructing broadband acoustic metasurfaces, offering flexibility for diverse applications across multiple scenarios.

Figure 1 .
Figure 1.Design procedure of metasurface.Schematics of a) uniform cavity and c) cascaded cavity series.b) Phase shifts of the uniform cavity during the target frequency band.d) Desired phase shift distributions of eight elements to meet the requirement of the GSL within the target frequency range.

Figure 2 .
Figure 2. Concept view of phase-engineered acoustic metasurface for broadband wavefront manipulation, and characteristics of the proposed nonuniform coiled-up channels (NUCC).a) The schematics of the meta-elements' arrangement for the acoustic metasurface.The upper-right inset displays the diagrams of reflected waves for three typical types of acoustic wavefront manipulation.b) 2D schematics of meta-elements #1 to #8 designed to be utilized in the metasurface.c) The evolution process of the designed configuration.d) Phase shifts and e) reflectance of elements #1 to #8 during the frequency range of (2300, 2800) Hz.

Figure 3 .
Figure 3.The realization of acoustic focusing and acoustic bending.The phase profiles (solid line) and designed arrangement modes (discrete points) for a) acoustic focusing and b) acoustic bending.The 2D diagrams of the metasurfaces are sketched beneath the corresponding profiles.The reflected acoustic pressure fields and normalized intensities along x direction (y = y 0 ) and y direction (x = x 0 ) for acoustic focusing at three specific frequencies: c,f) 2300 Hz; d,g) 2550 Hz; and e,h) 2800 Hz. i-k) The reflected acoustic pressure fields for acoustic bending at 2300, 2550, and 2800 Hz, respectively.
, based on which the amplitude positions of the focal points are interpretable.More detailed characteristics of acoustic focusing within the range of 2300-2800 Hz are displayed in Figure S4 (Supporting Information).

Figure 4 .
Figure 4. a) Characteristic parameters of the PU foam filled in the elements.Calculated and simulated b) phase shifts and c,d) reflectance of designed elements.The relationships between the reflected angles and incident angles for reconfigurable metasurface with e) eight elements and f) four elements.The red arrows in e) are the critical incident angles for the reflected angles of ± 90°.The red star represents the opted incident angles for exploring the reflected pressure field.

Figure 5 .
Figure 5. Reflected acoustic pressure fields of the reconfigurable metasurface a) with the incident angle of −60°at 1800 Hz; b,c) with the incident angle of 60°and −60°at 2600 Hz, respectively (all with four elements); b) with the incident angle of −60°at 2600 Hz (with eight elements).The black and blue arrows separately represent the directions of incident and reflected waves.

Figure 6 .
Figure 6.Sound absorption performance of the metasurface with four and eight elements.Comparison of  between metasurfaces with four and eight elements, and uniform porous material with the same height a) at normal incidence and b) in diffusion field.The color lumps in a,b) represent the absorptance exceeding 0.7.c) Comparison of the average  between the metasurface with four elements and uniform porous material.Comparison of  with a wide-angle incidence of (−80,80)°in the frequency range of (1300,3100) Hz between metasurfaces with d) four and e) eight elements.

Table 1 .
Geometric parameters of the designed elements.

Table 2 .
Values of five transport parameters for the used polyurethane foam in the JCA model.