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# Theory of metascreen-based acoustic passive phased array

Yong Li, Shuibao Qi, B. Assouar

► To cite this version:

Yong Li, Shuibao Qi, B. Assouar. Theory of metascreen-based acoustic passive phased array. New Journal of Physics, 2016, 18 (4), pp.043024. 10.1088/1367-2630/18/4/043024. hal-04953678

HAL Id: hal-04953678

https://hal.science/hal-04953678v1

Submitted on 18 Feb 2025

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# Theory of metascreen-based acoustic passive phased array

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Yong Li<fnref n="1" />,<fnref n="2" />, Shuibao Qi<fnref n="1" />,<fnref n="2" />, and M. Badreddine Assouar<fnref n="1" />,<fnref n="2" />

E-mail: Yong.Li@univ-lorraine.fr, Badreddine.Assouar@univ-lorraine.fr

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**Abstract.** Metascreen-based acoustic passive phased array provides new degree of freedom to manipulate acoustic wave due to the fascinating capabilities of fully shifting phase, keeping impedance matching, and holding subwavelength spatial resolution. We develop acoustic theories to analyze the transmission/reflection spectra and the refracted pressure fields of a metascreen composed of elements with four Helmholtz resonators (HRs) in series and a straight pipe. We find that these capabilities are also valid under oblique incidence with large angle, with the underlying physics stemming from the hybrid resonances between the HRs and the straight pipe. By imposing desired phase profiles, the refracted fields can be tailored in anomalous yet controllable manner. Especially, two kinds of negative refractions are evidenced based on two distinct mechanisms: one is formed from the classical diffraction theory and the other is dominated by the periodicity of the metascreen. Positive (normal) and negative refractions can be converted by simply changing the incident angle, with the coexistence of two kinds of refractions in a certain range of incident angle.

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**Keywords:** Acoustic metascreen, Acoustic metasurface, Passive phased array, Wave manipulation

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<footnote n="1">¹CNRS, Institut Jean Lamour, Vandœuvre-lès-Nancy F-54506, France</footnote>

<footnote n="2">²Université de Lorraine, Institut Jean Lamour, Boulevard des Aiguillettes, BP: 70239, Vandœuvre-lès-Nancy 54506, France</footnote>
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# 1. Introduction

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Modulating wave propagation in desired way has always been of great interest to physicists and engineers. Optical lenses, as the most classic example, have been used to focus optical wave since the dawn of civilization. With the development of knowledge, there are amount of emergent requirements to manipulate wave field with more complex yet fascinating patterns, such as the superlenses beating the diffraction limit [1–3], non-diffracting self-accelerating beams in paraxial and non-paraxial region [4–7], twisting/vortex beams [8–10], and so on. Wave field can be regarded as the spatial interference of each elementary wave with desired amplitude and phase profile. This notion has stimulated the concept of *active* phased array in electromagnetic [11] and acoustic waves [12–14], which can be utilized to reshape the radiated wave field by providing a desired amplitude and phase shift as a function of position. However, these physically-distributed sources in the *active* way require to be driven individually with electric techniques, inevitably leading to the major drawbacks of high cost and complexity.

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In a different context, the emergence of engineered artificial metamaterials, which can provide unavailable properties in natural materials, has significantly broaden the horizon for acoustic wave manipulation [15–23]. Acoustic metascreen/metasurface, a family of novel metamaterials, has attracted increasing attention recently due to the advantageous features of the planar profile and subwavelength thickness comparing to bulky meta-structures [24–33]. Considerable effort has been dedicated to explore the possibilities to effectively mold the emerging wave field. Recent works have demonstrated both numerically and experimentally that metascreen can provide great flexibility to reshape the transmitted wavefront in a *passive* manner, by providing an additional momentum in the transverse direction. To form a desirable wave field with excellent performance, especially in the non-paraxial region, the *passive* structures should possess the abilities of transmitting sound effectively, shifting phase with $2\pi$ range, and holding a subwavelength spatial resolution to avoid the spatial aliasing effect [34]. A metascreen, composed of several hybrid elements with four Helmholtz resonators (HRs) and a straight pipe, has been proposed to achieve these tough goals [31]. Unfortunately, almost all of the previous designs have demonstrated the ability of wave shaping based on numerical simulations at normal incidence, without revealing the underlying physics properly. The case of oblique incidence, containing more rich physics and potentials, is only considered in limited literature yet without giving the applicable scope [28].

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In this work, a complete theory is developed for the metascreen to reveal the underlying mechanism at both normal and oblique incidence. It is found that the aforementioned goals can be realized within a certain range of incident angle. The theoretical pressure fields emerging from the metascreen are also derived based on the transmission spectrum of each element. With these formulas, four distinct wave manipulation properties are demonstrated: anomalous refraction, conversion from a propagating wave to an evanescent wave, negative refraction, and focusing from a cylinder wave. Especially, two types of negative refractions are observed based on different orders of diffraction. **Additionally, the negative and positive (normal) refractions can be inter-converted by simply changing the incident angle, with the**
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) Schematic diagram of an individual element of the acoustic metascreen (width *w* and height *h*) consisting of four Helmholtz resonators (HRs) in series connection and a straight pipe (width *w*₁ and height *h*). The width and height of the neck and cavity of the HRs are *w*₂, *h*₂ and *w*₃, *h*₃, respectively. The solid material forming the HRs has an identical height, *h*₁ = *w*₂. The symbols *p*ᵢ, *p*ᵣ, and *p*ᵡ represent the obliquely incident, reflected and transmitted sound waves. (b) A single HR is illustrated for the purpose of the corrected acoustic impedance of the cavity. (c) A single HR is located in front of a hard boundary (dashed line) to obtain the corrected acoustic radiation impedance at the conjunction between the neck and the straight pipe.</figcaption></figure>
coexistence of two kinds of refraction in a certain range of incident angle.

## 2. Analytical Model of Acoustic Metascreen

Imagine that an incident wave impinges obliquely on an individual element of the acoustic metascreen which is periodic in $x$ direction [cf. figure 1]. Four HRs are connected in series to construct one such element with a width of $w$ and a height of $h$, further forming a straight pipe with the linear combination of these elements. The geometrical parameters $w_2$, $h_2$, $w_3$ and $h_3$ denote the width and height of the neck and cavity of the HRs, respectively. The HRs are formed by solid materials with identical height, $h_1$, and the width of the straight pipe formed between adjacent elements is $w_1$. To obtain the theoretical properties of the element simply, these HRs are usually treated as lumped elements with an effective acoustic impedance boundaries and only the plane wave component is considered to mimic the impedance boundary. This treatment inevitably leads to large deviations for the resonant states due to the fact that the geometric size of the cavity is comparable to the working wavelength in our design [31]. The effect of higher modes in the cavity needs to be considered for a corrected acoustic impedance [cf. figure 1(b)]. Furthermore, the effect of the radiation impedance between the neck and the straight pipe should also be counted [cf. figure 1(c)]. After obatining these corrected impedance boundaries, the transmission and reflection spectra of the whole system can be presented under both normal and oblique incidence [cf. figure. 1(a)].

### 2.1. Corrected impedance of cavity

The corresponding model of a single HR to consider the corrected acoustic impedance of the cavity is shown in figure 1(b). The geometrical parameters keep the same to these shown in figure 1(a). The pressure field, $p(x, z)$, in the cavity ($w_2 \le x \le w_2 + w_3$) can be expanded in terms of the normal modes [35]:

$$p(x, z) = \sum_{n} \phi_{n}(z) \left[ A_{n} e^{-j k_{xn} (x - w_{2})} + B_{n} e^{j k_{xn} (x - w_{2} - w_{3})} \right], \quad (1)$$

where $\phi_n(z)$ is the $n$-th eigenmode in $z$ direction, $\phi_n(z)=\sqrt{2-\delta_{0n}}\cos[k_{zn}(z-h_3/2)]$, and satisfies the orthogonality, $\int_{\sigma} \phi_n(z)\phi_m(z)dz = \sigma\delta_{mn}$, with $\delta_{mn}$ representing the Kronecker Delta and $\sigma$ being the cross sections of pipes and apertures. $A_n$ and $B_n$ refer to the propagating coefficients in $+x$ and $-x$ direction. The wavenumbers in $z$ and $x$ direction are $k_{zn} = n\pi/h_3$ and $k_{xn} = \sqrt{k^2 - k_{zn}^2}$ with $k$ being the wavenumber $k = 2\pi/\lambda$. The time factor $e^{j\omega t}$ with $\omega = c_0k$ is always omitted as understood. The velocity, $v$, and the pressure, $p$, is connected with the momentum conservation equation,

$$v = \frac{j}{k\rho_0 c_0} \nabla p, \quad (2)$$

with $\rho_0 c_0$ referring to the characteristic acoustic impedance of the fluid medium. By substituting equation (2) into equation (1), we obtain the $x$ component of $v$, $u(x, z)$, in the cavity

$$u(x, z) = \frac{1}{\rho_0 c_0} \sum_n \frac{k_{xn}}{k} \phi_n(z) \left[ A_n e^{-j k_{xn} (x - w_2)} - B_n e^{j k_{xn} (x - w_2 - w_3)} \right]. \quad (3)$$
Considering the fact that the normal velocity should be zero, $u = 0$, at the hard boundary ($x = w_2 + w_3$), the first relation between the coefficients $A_n$ and $B_n$ can be obtained as below,

$$A_n e^{-jk_{xn}w_3} - B_n = 0. \qquad (4)$$

At the junction between the neck and the cavity, $x = w_2$, the continuity of velocity requires

$$u(x)|_{x=w_2} = U(x)|_{x=w_2}/h_2. \qquad (5)$$

Here $U(x)$ is the volume velocity in the neck, defined as $U = \int_{\sigma} u(x,z)d\sigma$. The velocity component along $z$ direction is treated to be identical due to the deeply subwavelength size of the neck. Replacing $u(x)$ with equation (3), multiplying $\phi_m(z)$, and integrating along the boundaries in $z$ direction, we can obtain

$$\int_{-h_2/2}^{h_2/2} \frac{U(x)|_{x=w_2}}{h_2} \phi_m(z) dz \\ = \frac{1}{\rho_0 c_0} \int_{-h_3/2}^{h_3/2} \sum_n \frac{k_{xn}}{k} \phi_n(z) \phi_m(z) [A_n - B_n e^{-jk_{xn}w_3}] dz. \qquad (6)$$

After utilizing the orthogonality of $\phi_m(z)$, the second relation between $A_n$ and $B_n$ yields

$$A_n - B_n e^{-jk_{xn}w_3} = \rho_0 c_0 \frac{kU(x)|_{x=w_2}}{k_{xn}h_3} \Phi_n, \qquad (7)$$

with $\Phi_n = \frac{1}{h_2} \int_{-h_2/2}^{h_2/2} \phi_n(z) dz$. Combining equation (4) and equation (7), these coefficients can be solved as

$$A_n = \rho_0 c_0 \frac{kU(x)|_{x=w_2}}{k_{xn}h_3(1 - e^{-2jk_{xn}w_3})} \Phi_n, \qquad (8a)$$

$$B_n = A_n e^{-jk_{xn}w_3}. \qquad (8b)$$

Substituting these coefficients into equation (1), the averaged pressure field, defined as $\bar{p}(x) = \frac{1}{h_2} \int_{-h_2/2}^{h_2/2} pdz$, at $x = w_2$ can be written as

$$\bar{p}(x)|_{x=w_2} = \sum_n \rho_0 c_0 \frac{kU(x)|_{x=w_2}(1 + e^{-2jk_{xn}w_3})\Phi_n^2}{k_{xn}h_3(1 - e^{-2jk_{xn}w_3})}. \qquad (9)$$

After obtaining the relationship between $\bar{p}(x)$ and $U(x)$ at $x = w_2$, the acoustic impedance, defined as $Z = \bar{p}/U$, of the cavity can be expressed as

$$Z_c = \frac{\bar{p}(x)|_{x=w_2}}{U(x)|_{x=w_2}} = \sum_n \rho_0 c_0 \frac{k(1 + e^{-2jk_{xn}w_3})\Phi_n^2}{k_{xn}h_3(1 - e^{-2jk_{xn}w_3})}. \qquad (10)$$

If the geometric size of the cavity is much smaller than the working wavelength, only the plane wave component, viz. $n = 0$, needs to be considered. Considering $k_{x0} = k$ and $\Phi_0 = 1$, then equation (10) is degraded to

$$Z_c = \frac{\rho_0 c_0 (1 + e^{-2jk_{x0}w_3})}{h_3 (1 - e^{-2jk_{x0}w_3})} = -j \frac{\rho_0 c_0}{h_3} \cot(kw_3), \qquad (11)$$

which is identical to the traditional form [36].
## 2.2. Corrected impedance of HR

In our previous design, the metascreen is composed of four HRs connected with a straight pipe. These HRs can radiate energy into the straight pipe through the neck, where the radiation impedance needs to be considered. The corresponding model is shown in figure (1)(c) and the dashed line referring to the boundary of the adjacent element is treated as a hard boundary. To describe the pressure/velocity field in the straight pipe, the Green function theory is employed [35]:

$$ \nabla^2 G(x, z) + k^2 G(x, z) = -\delta(x - x_0)\delta(z - z_0). \qquad (12) $$

The solution of equation (12) can be expressed as,

$$ G(x, z; x_0, z_0) = \sum_{n}^{\infty} \frac{\varphi_n(x) \varphi_n(x_0)}{2 j k'_{zn} w_1} e^{-j k'_{zn} |z - z_0|}, \qquad (13) $$

where $\varphi_n(x) = \sqrt{2 - \delta_{0n}} \cos[k'_{xn}(x + w_1/2)]$ is the n-th eigenmodes with $k'_{xn} = n\pi/w_1$ and $k'_{zn} = \sqrt{k^2 - k'_{xn}$ being the wavenumbers in x and z directions. The eigenmodes of the $\varphi_n(x)$ satisfy the orthogonality, $\int_{\sigma} \varphi_n(x) \varphi_m(x) dx = \sigma \delta_{nm}$. According to the Green theory, the pressure distribution in the straight pipe can be written as,

$$ p(x, z) = p_i(x, z) + \int_{-h_2/2}^{h_2/2} G(x, z; \frac{w_1}{2}, z_0) \left. \frac{\partial p(x_0, z_0)}{\partial x_0} \right|_{x_0=w_1/2} dz_0, \qquad (14) $$

with $p_i(x, z)$ representing the incident pressure. Substituting equation (2) into equation (14) yields

$$ p(x, z) = p_i(x, z) - \frac{j k \rho_0 c_0}{h_2} \int_{-h_2/2}^{h_2/2} G(x, z; \frac{w_1}{2}, z_0) dz_0 U(x_0) \bigg|_{x_0=w_1/2} . \qquad (15) $$

Integrating equation (15) along the boundary in z direction from $-h_2/2$ to $h_2/2$ and applying the orthogonality of $\varphi_n(x)$, we can obtain the averaged pressure at $x = w_1/2$ as shown below.

$$ \bar{p}(x) \bigg|_{x=w_1/2} = \frac{1}{h_2} \int_{-h_2/2}^{h_2/2} p_i(x, z) dz - \frac{j k \rho_0 c_0}{h_2^2} \int_{-h_2/2}^{h_2/2} \int_{-h_2/2}^{h_2/2} G(\frac{w_1}{2}, z; \frac{w_1}{2}, z_0) dz_0 dz U(x) \bigg|_{x=w_1/2}. \qquad (16) $$

Rearranging equation (16) yields

$$ \bar{p}_i(x) \bigg|_{x=w_1/2} = [Z' \bigg|_{x=w_1/2} + Z_d] U(x) \bigg|_{x=w_1/2}, \qquad (17) $$

where $Z' \bigg|_{x=w_1/2} = \bar{p}(x) \bigg|_{x=w_1/2} / U(x) \bigg|_{x=w_1/2}$ means the impedance at $x = w_1/2$ and $Z_d$ refers to the corrected radiation impedance of the neck

$$ Z_d = \frac{j k \rho_0 c_0}{h_2^2} \int_{-h_2/2}^{h_2/2} \int_{-h_2/2}^{h_2/2} G(\frac{w_1}{2}, z; \frac{w_1}{2}, z_0) dz_0 dz. \qquad (18) $$

After some derivations, the resulting $Z_d$ becomes

$$ Z_d = \frac{\rho_0 c_0}{w_1 h_2^2} \left[ \frac{1 - e^{-jkh_2} - jkh_2}{k^2} + \sum_{n=1} \frac{2k(1 - e^{-jk'_{zn}h_2} - jk'_{zn}h_2)}{k'_{zn}^3} \right]. \qquad (19) $$
Only the plane wave component ($n = 0$) contributes to the real part of $Z_d$ and the imaginary part stems from the influence of higher order modes, which can be regarded as a pure additional radiation reactance [35].

In order to obtain the whole acoustic impedance of the HRs, $Z_h$, in the straight pipe, three contributions need to be taken into account: the corrected impedance of the cavity, $Z_c$, the impedance of the neck, $Z_n = \rho_0 c_0 / h_2$, and the radiation impedance between the neck and the straight pipe, $\text{Im}(Z_d)$. According to the impedance transfer method [36], the acoustic effective impedance of the HRs at $x = w_1/2$ can be expressed as

$$Z|_{x=w_1/2} = Z_n \frac{Z_c + jZ_n \tan(kw_2)}{Z_n + jZ_c \tan(kw_2)}, \qquad (20)$$

By adding the corrected impedance stemmed from varied cross sections between the neck and the straight pipe, hence the whole acoustic impedance of the HRs, $Z_h$, can be expressed as

$$Z_h = Z|_{x=w_1/2} + j\text{Im}(Z_d). \qquad (21)$$

It is noteworthy to emphasize that only the imaginary part of $Z_d$ is necessary to be included for the corrected radiation impedance [35]. The real part of $Z_d$ is corrected with the impedance of HRs at $x = w_1/2$ [cf. equation (20)].

## 2.3. Transmission and reflection spectra

Considering a plane wave obliquely impinges on an acoustic metascreen composed of periodic element [cf. figure 1(a)]. Without loss of generality, we assume a unit incident plane wave, $p_i(x,y) = e^{-j(k \sin \theta_i x + k \cos \theta_i z)}$, with $\theta_i$ being the angle of incidence with respect to the normal. By employing the Bloch-Floquet periodic boundary condition, the reflected (region $z < 0$) and transmitted wave (region $z > h$) can be expressed as[37; 38]

$$p_r = \sum_n r_n e^{-jk\beta_n x + k\alpha_n z} \quad (z < 0), \qquad (22a)$$

$$p_t = \sum_n t_n e^{-jk\beta_n x - k\alpha_n(z-h)} \quad (z > h), \qquad (22b)$$

where the symbols $r_n$ and $t_n$ represent the reflection and transmission coefficients of the n-th modes with the horizontal wavenumber, $k\beta_n = k \sin \theta_i + 2\pi n / w$, and the vertical wavenumber, $-jk\alpha_n = k\sqrt{1 - \beta_n^2}$. At $z = 0$, the continuity of the velocity requires

$$u_i + u_r = \begin{cases} u_a & 0 \le x \le w_1 \\ 0 & \text{others} \end{cases}, \qquad (23)$$

where $u_a$, $u_i$ and $u_r$ are the velocity in the straight pipe, the velocity of incident and reflected waves at $z = 0$, respectively. Considering the pressure expression in equation (22a) and the conservation of momentum equation in equation (2), equation (23) becomes

$$-\alpha_0 e^{-jk\beta_0 x} + k\alpha_n \sum_n r_n e^{-jk\beta_n x} = -j\rho_0 c_0 \begin{cases} u_a & 0 \le x \le w_1 \\ 0 & \text{others} \end{cases}, \qquad (24)$$
179 Multiplying $e^{jk\beta_m x}$ and integrating along $x$ direction, we can obtain

$$ \int_{0}^{w} -\alpha_{0}e^{-jk\beta_{0}x}e^{jk\beta_{m}x}dx + \int_{0}^{w} \sum_{n} r_{n}\alpha_{n}e^{-jk\beta_{n}x}e^{jk\beta_{m}x}dx \\ = -j\rho_{0}c_{0}\int_{0}^{w_{1}} u_{a}e^{jk\beta_{m}x}dx. \qquad (25) $$

180 Utilizing the orthogonality of the exponential function, equation (25) can be simplified and the reflection coefficient can be expressed as shown below.

$$ r_n = \delta_{0n} - (-j\alpha_n)^{-1} R_w U_a(0) \Theta_n(\theta_i), \qquad (26) $$

182 where $\Theta_n(\theta_i) = \frac{1}{w_1} \int_0^{w_1} e^{jk\beta_n x} dx$, $R_w = \rho_0 c_0 / w$ and $U_a(0)$ is the volume velocity in the straight pipe at $z=0$. Using the same procedure at $z=h$, the transmission coefficient can be expressed as

$$ t_n = (-j\alpha_n)^{-1} \Theta_n(\theta_i) R_w U_a(l). \qquad (27) $$

187 Substituting equation (26) into equation (22a), the total pressure at $z=0$ yields

$$ \begin{aligned} p(x, 0) &= p_i(x, 0) + p_r(x, 0) \\ &= 2e^{-jk\sin\theta_i x} - \sum_n (-j\alpha_n)^{-1} R_w U_a(0) \Theta_n(\theta_i) e^{-jk\beta_n x}. \end{aligned} \qquad (28) $$

188 Integrating equation (28) along $x$ direction from $-w_1/2$ to $w_1/2$, we can obtain

$$ \begin{aligned} \bar{p}_a(x, 0) = \frac{1}{w_1} \int_{-w_1/2}^{w_1/2} 2e^{-jk\sin\theta_i x} dx \\ - \frac{1}{w_1} \int_{-w_1/2}^{w_1/2} \sum_n (-j\alpha_n)^{-1} R_w U_a(0) \Theta_n(\theta_i) e^{-jk\beta_n x} dx, \end{aligned} \qquad (29) $$

189 with $\bar{p}_a$ being the averaged pressure in the straight pipe. It can be further simplified as

$$ 2\Theta_0^*(\theta_i) - \bar{p}_a(0) = Z_a U_a(0), \qquad (30) $$

190 with $Z_a = \sum_n (-j\alpha_n)^{-1} R_w |\Theta_n(\theta_i)|^2$ and the asterisk * on a variable denoting the conjugate complex. Similarly, the relation between the averaged pressure and volume velocity at $z=h$ can be written as shown below.

$$ \bar{p}_a(h) = Z_a U_a(h). \qquad (31) $$

195 Considering the fact that the width of the straight pipe, $w_1$, is quite smaller than the working wavelength, only plane wave component of the pressure and the volume velocity in the straight pipe need to be considered. Then the pressure and volume velocity in the region from the inlet to the crotch of the first HR (viz., $0 \le z \le a_1$) can be expressed as

$$ \begin{aligned} p_A(z) &= A^+ e^{-jkz} + A^- e^{jkz}, \\ U_A(z) &= \frac{A^+ e^{-jkz} - A^- e^{jkz}}{R_{w1}}, \end{aligned} \qquad (32) $$

200 where $A^+$ and $A^-$ are the propagating coefficients in $+z$ and $-z$ directions and $R_{w1} = \rho_0 c_0 / w_1$ refers to the acoustic impedance of the straight pipe. At $z=0$, equation (32) yields

$$ \begin{aligned} p_A(0) &= A^+ + A^-, \\ U_A(0) &= \frac{A^+ - A^-}{R_{w1}}. \end{aligned} \qquad (33) $$
Equation (33) can be revised in a matrix form as

$$ \begin{pmatrix} A^+ \\ A^- \end{pmatrix} = \mathbf{M}_1 \begin{pmatrix} \bar{p}_a(0) \\ U_a(0) \end{pmatrix}, \qquad (34) $$

where $\mathbf{M}_1$ is the transfer matrix,

$$ \mathbf{M}_1 = \begin{pmatrix} \frac{1}{2} & \frac{R_{w1}}{2} \\ \frac{1}{2} & -\frac{R_{w1}}{2} \end{pmatrix}. \qquad (35) $$

The expressions of the pressure and the volume velocity in the region between the first HR and the second HR (viz., $a_1 \le z \le a_1 + a_2$) are

$$ \begin{aligned} p_B(z) &= B^+ e^{-jk(z-a_1)} + B^- e^{jk(z-a_1)}, \\ U_B(z) &= \frac{B^+ e^{-jk(z-a_1)} - B^- e^{jk(z-a_1)}}{R_{w1}}. \end{aligned} \qquad (36) $$

At the position of the first HR (viz., $z = a_1$), the continuity of the pressure and the volume velocity require

$$ \begin{aligned} p_A(a_1) &= p_B(a_1), \\ U_A(a_1) &= U_B(a_1) + U_h(a_1), \end{aligned} \qquad (37) $$

where $U_h(a_1) = p_A(a_1)/Z_h$ is the velocity component of the HR at $z = a_1$. Combining equations (33) and (37), the transfer matrix between $A^+$, $A^-$ and $B^+$, $B^-$ is shown in equation (38).

$$ \begin{pmatrix} B^+ \\ B^- \end{pmatrix} = \mathbf{M}_2 \mathbf{N}_1 \begin{pmatrix} A^+ \\ A^- \end{pmatrix}. \qquad (38) $$

Here $\mathbf{M}_2\mathbf{N}_1$ is the transfer matrix,

$$ \mathbf{M}_2 = \begin{pmatrix} \frac{2-\alpha}{2} & -\frac{\alpha}{2} \\ \frac{\alpha}{2} & \frac{2+\alpha}{2} \end{pmatrix}, \qquad (39a) $$

$$ \mathbf{N}_1 = \begin{pmatrix} e^{-jka_1} & 0 \\ 0 & e^{jka_1} \end{pmatrix}, \qquad (39b) $$

with $\alpha = R_{w1}/Z_h$. Following the same procedure, the transfer matrices for other region of the element can be expressed as

$$ \begin{pmatrix} C^+ \\ C^- \end{pmatrix} = \mathbf{M}_2 \mathbf{N}_2 \begin{pmatrix} B^+ \\ B^- \end{pmatrix}, \qquad (40a) $$

$$ \begin{pmatrix} D^+ \\ D^- \end{pmatrix} = \mathbf{M}_2 \mathbf{N}_2 \begin{pmatrix} C^+ \\ C^- \end{pmatrix}, \qquad (40b) $$

$$ \begin{pmatrix} E^+ \\ E^- \end{pmatrix} = \mathbf{M}_2 \mathbf{N}_2 \begin{pmatrix} D^+ \\ D^- \end{pmatrix}, \qquad (40c) $$
where $C^{\pm}(D^{\pm})$ represent the coefficients in the region between second (third) HRs and third (fourth) HRs, $E^{\pm}$ refers to the region between fourth HRs and the outlet of the straight pipe, and $N2$ is

$$N_2 = \begin{pmatrix} e^{-jka_2} & 0 \\ 0 & e^{jka_2} \end{pmatrix}. \qquad (41)$$

At $z = h$, the relation between $E^{\pm}$ and the transmitted pressure/velocity, $\bar{p}_a(h)$ and $U_a(h)$, yields

$$\begin{pmatrix} \bar{p}_a(h) \\ U_a(h) \end{pmatrix} = \mathbf{M}_3 \mathbf{N}_1 \begin{pmatrix} E^+ \\ E^- \end{pmatrix}, \qquad (42)$$

with

$$\mathbf{M}_3 = \begin{pmatrix} 1 & 1 \\ \frac{1}{R_{w1}} & -\frac{1}{R_{w1}} \end{pmatrix}. \qquad (43)$$

According to these transfer matrices, we can finally obtain the relationship for the pressure and volume velocity between $z = 0$ and $z = h$,

$$\begin{pmatrix} \bar{p}_a(h) \\ U_a(h) \end{pmatrix} = \mathbf{M} \begin{pmatrix} \bar{p}_a(0) \\ U_a(0) \end{pmatrix}, \qquad (44)$$

with the whole transfer matrix $\mathbf{M} = \mathbf{M}_3\mathbf{N}_1(\mathbf{M}_2\mathbf{N}_2)^3\mathbf{M}_2\mathbf{N}_1\mathbf{M}_1$. Combining equations (30), (31) and (44), we can obtain $U_a(0)$ and $U_a(h)$,

$$U_a(0) = \frac{2(m_{21}Z_a - m_{11})\Theta_0^*(\theta_i)}{m_{21}Z_a^2 - (m_{11} + m_{22})Z_a + m_{12}}, \qquad (45a)$$

$$U_a(h) = \frac{2(m_{11}m_{12} - m_{12}m_{21})\Theta_0^*(\theta_i)}{m_{21}Z_a^2 - (m_{11} + m_{22})Z_a + m_{12}}, \qquad (45b)$$

with $m_{ij}$ representing the element in $\mathbf{M}$. Substituting equations (45a) and (45b) into equations (26) and (27) yields the coefficients of the reflection and transmission as shown below.

$$r_0 = 1 - \frac{R_w |\Theta_0(\theta_i)|^2}{\cos \theta_i} \frac{2(m_{21}Z_a - m_{11})}{m_{21}Z_a^2 - (m_{11} + m_{22})Z_a + m_{12}}, \qquad (46a)$$

$$t_0 = \frac{R_w |\Theta_0(\theta_i)|^2}{\cos \theta_i} \frac{2(m_{11}m_{12} - m_{12}m_{21})}{m_{21}Z_a^2 - (m_{11} + m_{22})Z_a + m_{12}}. \qquad (46b)$$

The theoretical transmission and reflection spectra (amplitude and phase shift) predicted by equations (46a) and (46b) are shown in the figures 2(a) and 2(b). To evaluate the accuracy of the theoretical derivations, simulated results based on a model with same geometrical parameters are also illustrated with dots. A commercial software based on finite element method, COMSOL Multiphysics, is employed for the simulations. It can be found that the theoretical results agree excellently with the simulated ones, providing solid proof of the validity of the theoretical approach. One interesting phenomenon observed in figure 2 is that there are four transmission peaks within the domain of $0.28 < h/\lambda < 0.61$ where the local phase shift of the transmitted wave can cover $2\pi$ range.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) Amplitude (red line) and local phase shift (blue line) of the transmitted waves of the acoustic metascreen derived from the theoretical formulas. (b) The theoretical amplitude (red line) and the local phase shift (blue line) of the reflected waves of the acoustic metascreen. The corresponding dots refer to the simulated results with same geometrical parameters, showing the excellent agreement with the theoretical results. The geometric parameters for the derivations and simulations are $w = 0.2h$, $w_1 = 0.3w$, $h_1 = 0.02h$, $w_2 = 0.1w$, $h2 = 0.03h$, $w_3 = 0.5w$, $h_3 = 0.225h$, $a_1 = 0.1325h$, and $a_2 = 0.245h$.</figcaption></figure>

## 2.4. Physical interpretation

In order to understand the transmission peaks, we further explore the underlying physics of the acoustic metascreen. Equation (31) indicates the transmitted wave is formed by the radiation effect of the outlets, and $Z_a$ can be regarded as the output impedance. According to the theory of transmission line, the impedance of the pipe, $Z_p$, at $z_4 = a_1 + 3a_2$ can be obtained

$$Z_p|_{z=z_4} = R_{w1} \frac{Z_a + jR_{w1} \tan ka_1}{R_{w1} + jZ_a \tan ka_1} \quad (47)$$
Considering the fact that $Z_p|_{z=z_4}$ and $Z_h$ are in parallel connection, we can obtain the total impedance at $z = z_4$,

$$Z|_{z=z_4} = \frac{Z_p|_{z=z_4} + Z_h}{Z_p|_{z=z_4} Z_h}. \qquad (48)$$

Similarly, the impedance at $z = z_3 = a_1 + 2a_2, z = z_2 = a_1 + a_2$, and $z = z_1 = a_1$ can be expressed as

$$Z|_{z=z_i} = \frac{Z_p|_{z=z_i} + Z_h}{Z_p|_{z=z_i} Z_h}, \qquad (49)$$

with

$$Z_p|_{z=z_i} = R_{w1} \frac{Z|_{z=z_{i+1}} + jR_{w1} \tan ka_2}{R_{w1} + jZ|_{z=z_{i+1}} \tan ka_2}. \qquad (50)$$

The transfer impedance of the output load at $z = 0, Z_e$, can be finally obtained

$$Z_e = R_{w1} \frac{Z|_{z=z_1} + jR_{w1} \tan ka_1}{R_{w1} + jZ|_{z=z_1} \tan ka_1}. \qquad (51)$$

The averaged pressure, $\bar{p}_a(0)$ and volume velocity, $U_a(0)$, at $z = 0$ in the straight pipe can be correlated as

$$\bar{p}_a(0) = Z_e U_a(0). \qquad (52)$$

Combining equations (52) and (30), we can obtain

$$\bar{p}_a(0) = \frac{2\Phi^*_0(\theta_i)Z_e}{Z_a + Z_e}, \qquad (53a)$$

$$U_a(0) = \frac{2\Phi^*_0(\theta_i)}{Z_a + Z_e}, \qquad (53b)$$

where $Z_e + Z_a$ refers to the total impedance of the metascreen. Then the power transmitted into the metascreen at $z = 0$ is

$$W_m = \frac{1}{2} \mathrm{Re} [\bar{p}_a^*(0) U_a(0)] = \frac{1}{2} \left| \frac{2\Phi^*_0(\theta_i)}{Z_a + Z_e} \right|^2 \mathrm{Re}(Z_e). \qquad (54)$$

The incident sound power is

$$W_i = \frac{1}{2} \mathrm{Re} [\bar{p}_i^*(0) U_i(0)] = \frac{\cos \theta_i}{2R_w}. \qquad (55)$$

Then the power transmission of the metascreen is $t_I = W_m/W_i$. From equation (54), the resonant condition is

$$\mathrm{Im}(Z_a + Z_e) = 0. \qquad (56)$$

Meanwhile, it can be verified that

$$\mathrm{Re}(Z_e) = \mathrm{Re}(Z_a). \qquad (57)$$

Combining equations (56) and (57) yields

$$Z_a = Z_e^*. \qquad (58)$$
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Real part of the acoustic impedance of the effective load (red line), Re($Z_e$)/R<sub>w</sub>, and imaginary part of the total acoustic impedance of the metascreen (blue dotted line), Re($Z_e + Z_a$)/R<sub>w</sub>, at normal incidence, showing the four resonant states (black arrows) with $h/\lambda = (0.2793, 0.4892, 0.5740, 0.6163)$.</figcaption></figure>

The conjugation matching of $Z_a$ and $Z_e$ indicates that the incident energy can penetrate the metascreen at the resonant states, resulting in the total transmission, $t_I = 1$.

Figure 3 illustrates the real part of the acoustic impedance of effective load, Re($Z_e$)/R<sub>w</sub>, and the imaginary part of the total acoustic impedance of the system, Re($Z_e + Z_a$)/R<sub>w</sub>. It can be observed that the resonant conditions and the impedance matching can be simultaneously achieved at four specific positions (black arrows), resulting in the total transmission. The positions agree excellently with the four peaks in transmission amplitude [cf. figure 2(a)]. By employing these hybrid resonances, induced by the HRs and the straight pipe, the individual element can achieve a high transmission and fully controlled phase shifts. With these properties, the elements can be employed to construct an acoustic metascreen. Although only the frequency response is investigated in figure 2, it is expected to achieve similar behaviors by tuning some geometrical parameters, such as $w_1$, at a fixed working frequency.

### 3. Elements of Acoustic Metascreen

The amplitude and phase shift of the transmitted coefficient varying with the width of the straight pipe, $w_1/w$, and the incident angle, $\theta_i$, is shown in figure 4. The black contour line demonstrates the region with transmission amplitude higher than 0.9 [cf. figure 4(a)]. Fully controlled phase shift can be achieved in the same region [cf. figure 4(b)]. By tuning $w_1/w$, viz. $w_3$, the reactance provided by the HRs is effectively tailored, resulting in the phase shift covering $2\pi$ span. Eight elements with steps of $\pi/4$ can be readily selected to implement a
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Maps of (a) transmitted amplitude, $|p_t/p_i|$, and (b) phase shift, $\phi_t/(2\pi)$, of the acoustic metascreen as functions of the width of the straight pipe, $w_1/w$ and the incident angle, $\theta_i$. The working wavelength is fixed with $\lambda = 0.1$ m. The solid and dashed lines enclose the region where amplitude larger than 90% and -3 dB, respectively. White circles show eight elements selected to cover $2\pi$ of phase shift with steps of $\pi/4$ at 30°.</figcaption></figure>

desired phase profile, still keeping the impedance matching. Interestingly, the fully controlled phase shift and high transmission can be achieved even for oblique incidence with incident angle smaller than 56° if the threshold of the transmission amplitude of each element is set to 0.9. The angle increases to 75° if we set the transmission amplitude of each element larger than -3dB, which means that 50% of incident power penetrates the element. Furthermore, with a fixed geometries, the phase shift is almost independent of the incident angle. These tremendous capabilities of fully tuned phase, high transmission and the wide-angle property, guarantee the excellent performance of the metascreen composed of these elements even for oblique incidence. For instance, in figure 4, eight elements with different geometries are identified for the case with incident angle of 30° (white circles). Full phase coverage can be realized by these elements with steps of $\pi/4$, meanwhile holding high transmission amplitudes (> 0.93). After discretizing the desired profile with finite steps and carefully

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<figure><img src="image_7.png" /><figcaption>**Figure 5.** Simulated pressure fields passed through the eight elements of the metascreen at normal incidence. Each case is calculated independently by impinging a plane wave with a wavelength ($\lambda$) of $2h$, clear showing that the transmitted phase can be shifted from 0 to $2\pi$. The parameters of $w_1/w$ are 0.299, 0.349, 0.428, 0.562, 0.189, 0.206, 0.226, 0.258, respectively.</figcaption></figure>

choosing corresponding geometries along the transverse direction (x), acoustic metascreen is constructed to tailor the phase in the desired manner. With the selected geometries, the phase shifts caused by the elements are almost independent of the incident angle. It should be emphasized that the spatial resolution of the formed array is as small as $\lambda/10$, allowing a sufficient control of the sound field even beyond the paraxial region.

To verify the phase shift and the amplitude predicted by the theoretical derivations, figure 5 shows the simulated pressure fields of the eight elements with the geometries identified from figure 4 at normal incidence. A plane wave with unit amplitude impinges on each structure. The incident wave penetrates the elements with little reflection and the phase of the transmitted wave can be effectively tuned to cover $2\pi$ span.

Figure 6 illustrates the effective acoustic impedance of the metascreen normalized to $R'_w = R_w/\cos\theta_i$ with the incident angles of 0° and 80°. The real part of the impedance of the effective load, $\text{Re}(Z_e)/R'_w$, varied around 1 from $w_1/w > 0.165$, indicating that the impedance is well matched, resulting in the high transmission [cf. figure 4]. With
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Real part of the acoustic impedance of the effective load (red line), $Re(Z_e)/R'_w$, and imaginary part of the total acoustic impedance of the whole system (blue dotted line), $Im(Z_e+Z_a)/R'_w$, with an incident angle of (a) 0° and (b) 80°. The resonant states is illustrated (a) with black arrows with $w_1/w = (0.171, 0.205, 0.310)$ and (b) with $w_1/w = (0.182, 0.283)$.</figcaption></figure>

the increase of the incident angle, the impedance, mainly $Re(Z_e)/R'_w$, becomes mismatch, leading to the low transmission with large incident angles. Interestingly, hybrid resonances, $Im(Z_e+Z_a)/R'_w = 0$, are also supported under oblique incidence [39].

## 4. Shaped beams with Acoustic Metascreen

The great abilities of the high power transmission, fully controlled phase shift, and deeply subwavenumber spatial resolution, $w = \lambda/10$, guarantee that the acoustic metascreen-based passive phased array can be employed to realize a wide range of wave manipulations. In this section, we theoretically and numerically investigated several interesting wave manipulations
using proposed acoustic metascreen, such as anomalous refraction, negative refraction, and focusing.

### 4.1. Theoretical pressure fields

The total transmitted pressure field emerging from the metascreen can be regarded as the synthesis of radiated wave from the elements. The radiated fields can be obtained with the Green function theory by treating the outlets of the elements as line sources with uniform volume velocities. The Green function of a line source located on a hard boundary can be written as [35]

$$G(x, y; x', y') = -\frac{j}{2}H_0^{(1)}(kR), \qquad (59)$$

where $R$ refers to the distance between the measured point $(x, y)$ and the source $(x', y')$, $R = \sqrt{(x-x')^2+(y-y')^2}$, and $H_0^{(1)}(*)$ represents the first kind of Hankel function of the zero order. The total pressure field is the integration of the Green function over the boundaries of metascreen ($\Sigma$)

$$p(x, z) = \int_{\Sigma} \left[ G(x, z, x', h) \frac{\partial p(x, z)}{\partial z} + p \frac{\partial G(x, x', z, z')}{\partial z} \right] d\sigma. \qquad (60)$$

Considering the fact that $\partial G/\partial z = 0$ and equation (2), equation (60) becomes

$$p(x, z) = -\frac{j\rho_0 c_0 k}{w_1} \sum_{n=1}^{\infty} \int_{(n-1)w}^{(n-1)w+w_1} G(x, x', z, h) U_a(x', h) dx', \qquad (61)$$

with $n$ referring to the $n$-th individual element.

### 4.2. Anomalous refractions at normal incidence

Under the guidance of the generalized Snells law [40; 41], the relation between the incident angle, $\theta_i$, and refracted angle, $\theta_t$, should be revised as

$$k \sin \theta_t = k \sin \theta_i + \xi(x), \qquad (62)$$

with $\xi(x) = d\phi_t/dx$ refers to the phase gradient.

In order to realize the anomalous refraction at normal incidence, $\theta_i = 0$, a linear phase profile should be yielded, indicating a constant value of $\xi(x)$. The relationship between the transmitted angle, $\theta_t$, and the constant phase gradient, $\xi$, under normal incidence is illustrated in the figure 7. The theoretical and simulated results, obtained at six wavelengths away from the center of the metascreen with a plane wave of $5\lambda$ width incidence, are also plotted for the comparison and an excellent agreement is observed.

After the desired profile, $\xi$, is selected, the metascreen can be constructed by suitably discretizing the profile in $\pi/4$ steps and arranging these elements in the transverse direction, $x$. Additional phase shift is provided by the metascreen so that the transmitted propagation is inevitably redirected to the desired angle due to the conservation of the transverse momentum. Figure 8 illustrates the theoretical and simulated pressure fields of a metascreen with $\xi/(2\pi) = 50/8$, clearly showing that the metascreen generates extra transverse momentum
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Relationship between the transmitted angle ($\theta_t$) and the phase gradient ($\xi/(2\pi)$) of the metascreen at normal incidence. Black line is the predicted curve from equation (62). Red dots and blue rectangles show the corresponding theoretical and simulated angles obtained at six wavelengths away from the center of the metascreen.</figcaption></figure>

on the normally incident wave, resulting in a bending transmitted wave propagating with predicted desired angles (black arrows in the output region).

The formula in equation (62) implies the fact that there is a critical value with $\xi/(2\pi) = 10$. No transmitted angle is allowed when $\xi/(2\pi) > 10$, indicating that the conversion from propagating wave to evanescent wave in $z$ direction with energy concentrated near the metascreen. Figure 9 illustrates the theoretical and simulated normalized sound pressure level distributions of an acoustic metascreen with $\xi/(2\pi) = 100/9$ at normal incidence with a width of $8\lambda$. The acoustic energy is confined near the metascreen and rapidly attenuated in $z$ direction. **The proposed metascreen can provide an extra momentum profile along $x$ direction for the conversion from the propagating wave to the evanescent wave with concentrated energy near the surface.** In the simulations, the existing interaction between the elements and other orders of diffraction enlarges the reflected energy, resulting in the slight difference between the theoretical and simulated results. The energy concentration near the surface may provide an alternative method to generate surface acoustic wave with some structures supporting its propagation, such as the corrugated structure [28; 42].

### 4.3. Negative refractions with oblique incidence

The individual element possesses the capability of shifting almost identical phase within a certain range of incident angle, while keeping a high transmission as shown in figure 4(b). This indicates that our proposed metascreen can effectively tune the wave propagation even for oblique incidence. **The scalar diffraction theory, however, is limited and higher orders of diffraction need to be counted when the incident angle becomes larger** [43]. Then the
<figure><img src="image_10.png" /><figcaption>**Figure 8.** (a) Theoretical and (b) simulated pressure fields (snapshot in time) of an acoustic metascreen with phase gradient $\xi/(2\pi) = 50/8$ and working wavelength $\lambda = 0.1$ m. The black arrows indicate the incident wave and the predicted redirected angle of the transmitted waves, $\theta_t = 38.7^\circ$.</figcaption></figure>

non-local effect from the periodicity will play a crucial role in the wave shaping when the wavelength is comparable to the period of the metascreen. The generalized Snell's law should be revised [28]

$$k \sin \theta_t = k \sin \theta_i + \xi(x) + n_G G. \qquad (63)$$

where $G = 2\pi/\Lambda$ is the amplitude of the reciprocal lattice vector with $\Lambda$ being the periodicity of the phase profile of the metascreen.

The formula of the theoretical pressure field [cf. equation (61)] needs to be revisited under oblique incidence

$$p(x,z) = -\frac{j\rho_0 c_0 k}{w_1} \sum_{n=1}^{\infty} \int_{(n-1)w}^{(n-1)w+w_1} G(x,x',z,h) e^{i\phi_c(x)} U_a(x',h) dx', \quad (64)$$

with $\phi_c(x) = kx \sin \theta_i + n_G x G$ revealing the incident phase difference at the boundary of the metascreen ($z = 0$). In the following calculations, the metascreen designed for the normal incidence is employed to challenge the performance for the oblique incidence.
<figure><img src="image_11.png" /><figcaption>**Figure 9.** (a) Theoretical and (b) simulated sound pressure level (normalized by the maximum value) distributions of an acoustic metascreen with phase gradient $\xi/(2\pi) = 50/8$ and working wavelength $\lambda = 0.1$m. The green arrows indicate the propagating direction of incident wave.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Relationship between the transmitted angle, $\theta_t$, and the incident angle, $\theta_i$. Black line is the predicted curve from equation (63). Red dots and blue rectangles show the corresponding theoretical and simulated $\theta_t$ obtained at six wavelengths away from the center of the metascreen.</figcaption></figure>
In order to investigate the behaviors at oblique incidence, $\xi$ is fixed to 50/8 to construct the metascreen, indicating that the bending angle at normal incidence should be 38.68°. The black solid line in figure 10 shows the predicted relationship between the transmitted angle and the incident angle. The theoretical and simulated ones are also plotted with red dots and blue rectangles for comparison and excellent agreement is obtained. For negative incident angle defined as the clockwise direction, $-50° \le \theta_i \le 0°$, classical diffraction plays dominant roles in the transmitted wave forming, indicating $n_G = 0 \ddagger$. For the positive incident angle defined as the anti-clockwise direction, $n_G = 0$ is also valid for the cases with small incident angle ($\theta_i < 22°$) and $n_G$ becomes no-zero values (in our case, $n_G = -3$) when $\theta_i > 14.5°$ governed by the surface momentum matching condition. As a result, the curve of $\theta_t$ versus $\theta_i$ exhibits four interesting features: (i) negative refraction with $n_G = 0$ (the region $-38.7° < \theta_t < 0°$), (ii) positive (normal) refraction (region $0° < \theta_t < 22°$, (iii) negative refraction with $n_G = -3$ (region $14.5° < \theta_t < 22°$), and (iv) coexistence of the normal and negative refraction (region $22° \le \theta_t \le 50°$). By modulating the spatial resolution, $w$, and the phase gradient, $\xi(x)$, the conversion from a propagating wave to an evanescent wave can also be achieved within a certain range of incident angle [28].

Theoretical and simulated pressure fields in the figure 11 clearly illustrate the negative refraction. The oblique incidence with small negative angle ($\theta_i > -38.7°$) can not provide enough momentum to cancel the additional transverse momentum generated by the metascreen so that the wave vector of the transmitted wave always lies on the same side of the surface normal as the incident wave vector, namely, negative refraction. Transmitted wave emerges from the metascreen with an angle of $21.5°$ when an incident plane wave with an angle of $-15°$ impinges on the structure [cf. figures 11(a) and 11(b)]. The plane wave has a unity amplitude and a width of $5λ$. For the incidence with positive angle of $35°$, the theoretical and simulated pressure fields demonstrate that the metascreen redirects the transmitted waves with an angle of $-42.6°$, situated at the same side of the incidence [cf. figures 11(c) and 11(d)]. The transmitted angles of the negative refraction agree well with the predicted ones (black arrows in the output region). Even though the negative refractions are realized with both positive and negative incident angles, the physical mechanisms are totally different. For the incidence with negative angle, the transmitted field is formed following classical diffraction theory, $n_G = 0$. However, for the incidence with positive angle, the periodicity of the metascreen plays a dominant role in the field shaping with $n_G = -3$. The asymmetric property for positive and negative incidence may be used to realize some novel functions, such as the asymmetric acoustic transmission [44–48].

The predicted curve in figure 10 also implies that the normal refraction and negative refraction can be achieved simultaneously within a special range of incident angles. The theoretical pressure field of the metascreen with an incident angle of $21°$ is shown in figure 12. It can be observed that the transmitted wave is split into two main lobes with angle of $79.5°$ (normal refraction) and $-63°$ (negative refraction), showing the good agreement with the

<footnote>$\ddagger$ $n_G = 1$ is another permitted value, however, the amplitude of the transmitted wave of this branch is quite smaller than that of $n_G = 0$, therefore, this branch is neglected in the figure 10.</footnote>
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Negative refraction. (a) The theoretical and (b) the simulated normalized pressure fields (snapshot in time) of the metascreens with incident angle $\theta_i = -15^\circ$. (c) The theoretical and (d) the simulated normalized pressure fields (snapshot in time) of the metascreens with incident angle $\theta_i = 35^\circ$. The working wavelength is fixed as 0.1 m. The black arrows indicate the propagating direction of the incident waves and the redirected angles of the negative refractions ($\theta_t = 21.5^\circ$ and $\theta_t = -42.6^\circ$).</figcaption></figure>

436 predicted ones §.

4.4. Acoustic focusing from a cylinder wave

To further demonstrate the great capabilities of the proposed metascreen to control wave propagation flexibly, we also implement an acoustic focusing effect with desired quadratic phase profile. A line source located at $(x, z) = (x_s, z_s) = (6\lambda, -5\lambda)$ is employed to radiate an incident cylinder wave considering that each element with a fixed geometry generates almost identical phase shifts within a certain range of incident angle [cf. figure 4]. To form a focus spot with focal length of $f = 5\lambda$, the phase profile provided by the metascreen can be written as

$$ \phi(x) = k \left[ \sqrt{(x-x_s)^2 + f^2} - f \right] + \phi_c(x), \qquad (65) $$

where the first term is the desired profile to focus a plane wave, and the second term, $\phi_c = k \left[ \sqrt{(x-x_s)^2 + z_s^2} - z_s \right]$, compensates the arrival phase difference along the boundary

§ Theoretical and simulated pressure fields are not comparable because the role of $n_G$ in the theoretical derivations is implemented in the incident part.
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Positive and negative refractions. (a) The theoretical and (b) the simulated normalized pressure fields (snapshot in time) of the metascreens with incident angle $\theta_t = 21^\circ$ and working wavelength of 0.1 m. The black arrows indicate the propagating direction of the incident wave and the redirected angles of the normal refraction ($\theta_t = 79.5^\circ$) and negative refraction ($\theta_t = -63^\circ$).</figcaption></figure>

of the metascreen ($z = 0$).

The theoretical pressure field from a line source is

$$p(x, z) = -j\rho_0 c_0 k G'(x, y, x_s, z_s), \qquad (66)$$

with $G'(x, z, x_s, z_s)$ being the Green function of a line source in free space,

$$G'(x, z, x_s, z_s) = -\frac{j}{4} H_0^{(1)} k [(x - x_s)^2 + (z - z_s)^2]. \qquad (67)$$

Then the emergent pressure field from the metascreen can be expressed as

$$p(x, z) = -\frac{j\rho_0 c_0 k}{w_1} \sum_{n=1}^{\infty} \int_{(n-1)w}^{(n-1)w+1} A_c(x) G(x, x', z, h) e^{i\phi_c(x)} U_a(x', h) dx', \quad (68)$$

where $A_c(x) = -j\rho_0 c_0 k G'(x, 6\lambda, 0, -5\lambda)$ and $\phi_c$ refer to amplitude and phase difference at the boundary of the metascreen ($z = 0$) from the line source. In the derivations, the elements are selected according to equation (65) and then the amplitude and the phase difference are compensated in the incident part to ensure the identical geometries in theoretical and simulated cases.

The theoretical and simulated normalized pressure fields are illustrated in figure 13. The corresponding distributions clearly demonstrate the focusing effect at the designed focal plane (dashed lines). The excellent agreement between the theoretical/simulated results and the predicted ones witnesses the abilities of metascreen even for the cylinder wave from the line source. Due to the fully controllable phase pattern, acoustic lenses can be designed to focus energy effectively at an arbitrary position [26; 49; 50].
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Acoustic lens. (a) The theoretical and (b) the simulated normalized pressure fields (snapshot in time) of the metascreen providing a local phase profile shown in equation (65). A line source located at $(x, z) = (6\lambda, -5\lambda)$ is employed to radiate a cylinder wave in (b). The dashed lines refer to the focal planes at $z = 5.5\lambda$.</figcaption></figure>
5. Summary and Discussion

In summary, we develop a complete theory to analyze the transmission/reflection spectra of the individual element of the acoustic metascreen. The great capabilities of shifting phase with $2\pi$ span, keeping well matched impedance, and holding deeply subwavelength spatial resolution, are well achieved even for the oblique incidence within a certain range of incident angle. The underlying physics has been well revealed that the desired phase pattern can be fully tuned by changing the reactance provided by the HRs, while keeping impedance matching to the surrounding medium with hybrid resonances.

By carefully discretizing desired phase profile and arranging eight types of element, acoustic metascreen is constructed to yield additional transverse momentum. Anomalous refraction and conversion from propagating wave to evanescent wave have been demonstrated at normal incidence. Under oblique incidence, two kinds of negative refractions are observed stemming from distinct mechanisms. One is based on the classical diffraction theory, and the other is dominated by the periodicity of the metascreen. Significantly, the negative and positive (normal) refractions can be inter-converted through simply changing the incident angle, with the existence of two kinds of refractions in a certain range of incident angle. Acoustic lens is also demonstrated to focus acoustic energy with a line source rather than normal plane wave, validating the wide angle feature of the metascreen. Both the theoretical and simulated results agree excellently with the predicted ones, providing solid supports that the metascreen can control wave propagation flexibly and concisely.

Our planar *passive* phased array composed of acoustic metascreen, overcoming the drawbacks of *active* phased array, takes advantages of simple configuration and extreme acoustic performance and hence shows great potentials in diverse wave-shaping applications, such as the non-diffracting beams [31], vortex beams [9], acoustic focusing [51], acoustic holography [52], and so on. The concept of *passive* phased array can also be extended into under water [53], nondestructive evaluation [54], and micro-fluidic systems [55].

Acknowledgments

This work was supported by the FEDER "Fonds Européen de Développement Régional" (project "MASTER") and the "Région Lorraine". Y.L. wishes to thank Dr. Gaokun Yu and Dr. Xishan Yang for fruitful discussions.

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