Theory of metascreen-based acoustic passive phased array

The metascreen-based acoustic passive phased array provides a new degree of freedom for manipulating acoustic waves due to their fascinating properties, such as a 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 properties are also valid under oblique incidence with large angles, with the underlying physics stemming from the hybrid resonances between the HRs and the straight pipe. By imposing the desired phase profiles, the refracted fields can be tailored in an anomalous yet controllable manner. In particular, two types of negative refraction are exhibited, based on two distinct mechanisms: one is formed from 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 types of refraction in a certain range of incident angles.


Introduction
Modulating wave propagation in a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 waves since the dawn of civilization. With the development of knowledge, a numberof methodsfor themanipulation ofwave fields with more complex yet fascinating patterns have emerged, such assuperlenses beating the diffraction limit [1][2][3], nondiffracting self-accelerating beams in theparaxial and non-paraxial regions [4][5][6][7], twisting/vortex beams [8][9][10], and so on. Wave fields can be regarded as the spatial interference of each elementary wave with thedesired amplitude and phase profile. This notion has stimulated the concept of anactive phased array in electromagnetic [11] and acoustic waves [12][13][14], which can be utilized to reshape the radiated wave field by providing the desired amplitude and phase shift as a function of position. However, these physicallydistributed sources in the active way needto be driven individually usingelectrical techniques, inevitably leading tomajor drawbacks such ashigh cost and complexity.
In a different context, the emergence of engineered artificial metamaterials, which can provideproperties unavailablein natural materials, hassignificantly broadened the horizon for acoustic wave manipulation [15][16][17][18][19][20][21][22][23]. Acoustic metascreens/metasurfaces, a family of novel metamaterials, haveattracted increasing attention recently due to the advantageous features of aplanar profile and subwavelength thickness comparedto bulky meta-structures [24][25][26][27][28][29][30][31][32][33]. Considerable effort has been dedicated to exploringthe possibilities for effectively molding the emerging wave field. Recent works have demonstrated both numerically and experimentally that metascreens can provide great flexibility inreshapingthe transmitted wavefront in a passive manner, by providing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 abilitytotransmitsound effectively, shiftphases with a p 2 range, and hold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 previous designs have demonstrated the ability to shape wavefrontbased on numerical simulations at normal incidence, without revealing the underlying physics properly. The case of oblique incidence, containingricher physics and that haspotential, is only considered in limited literature andwithout giving the applicable scope [28].
In this work, a complete theory is developed for the metascreen to reveal itsunderlying mechanism at both normal and oblique incidence. It is found that the aforementioned goals can be realized within a certain range of incident angles.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. In particular, two types of negative refractionare observed based on different orders of diffractions. In addition, the negative and positive (normal) refractions are interconverted by simply changing the incident angle, with the coexistence of two typesof refractionin a certain range of incident angles.

Analytical model of theacoustic metascreen
Imagine that an incident wave impinges obliquely on an individual element of the acoustic metascreen, which is periodic in the x direction (see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 geometric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 in 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 ( figure 1(b)). Furthermore, the effect ofradiation impedance between the neck and the straight pipe should also be included(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 (figure1(a)).

Corrected impedance of thecavity
Amodel of a single HR, to showthe corrected acoustic impedance of the cavity, is shown in figure 1(b). The geometrical parameters staythe same asthose shown in figure 1(a). The pressure field, ( ) p x z , , in the cavity 3 ) can be expanded in terms of the normal modes [35]: . The time factor w e j t with w = c k 0 is always omitted as understood. The velocity, v, and the pressure, p, is connected tothe momentum conservation equation, with r c 0 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 Considering the fact that the normal velocity should be zero, u=0, at the hard boundary ( = + x w w 2 3 ), the first relation between the coefficients A n and B n can be obtained as below, At the junction between the neck and the cavity, = x w 2 , the continuity of velocity requires = Here ( ) U x is the volume velocity in the neck, defined as The velocity component along the z direction is treated asbeing identical due to the deeply subwavelength size of the neck. Replacing ( ) u x with equation (3), multiplying by f ( ) z m , and integrating along the boundaries in the z direction, we can obtain . Combining equation (4) and equation (7), these coefficients can be solved as Substituting these coefficients into equation (1), the averaged pressure field, defined as , at = x w 2 can be written as . 9 After obtaining the relationship between¯( ) p x and ( ) U x at = x w 2 , the acoustic impedance, defined as =Z p U, of the cavity can be expressed as 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 which is identical to the traditional form [36].

Corrected impedance of theHRs
In our previous design, the metascreen is composed of four HRs connected to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]: The solution of equation (12) can be expressed as, nm . According to the Green theory, the pressure distribution in the straight pipe can be written as, Integrating equation (  Rearranging equation (16) yields means the impedance at = x w 2 1 and Z d refers to the corrected radiation impedance of the neck After some derivations, the resulting Z d becomes Only the plane wave component (n=0) contributes to the real part of Z d and the imaginary part mainly stems from the influence of higher orders of evanescent modes, which can be regarded as a pure additional radiation reactance [35].
In order to obtain the whole acoustic impedance of the HRs attached to the straight pipe, Z h , three factors need to be taken into account: the corrected impedance of the cavity, Z c , the impedance of the neck, r = Z c h n 0 0 2 , and the radiation impedance between the neck and the straight pipe, ( ) Z Im d . According to the impedance transfer method [36], the acoustic effective impedance of the HRs at = x w 2 1 can be expressed as By adding the corrected impedance stemmingfrom thevaried cross sections between the neck and the straight pipe, the whole acoustic impedance of the HRs, Z h , can be expressed as It is worth emphasizingthat only the imaginary part of Z d needs 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 2 1 (seeequation (20)).

Transmission and reflection spectra
We considera plane waveobliquely impingingon an acoustic metascreen composed of periodic element ( figure 1(a)). Without loss of generality, we assume an incident plane wave with unity ampliutude, = , with q 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] where the symbols r n and t n represent the reflection and transmission coefficients of the nth modes with the horizontal wavenumber, b , and the vertical wavenumber, At z=0, the continuity of the velocity requires where u a , u i and u r are the velocity in the straight pipe, andthe velocity of the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 å a a r - Utilizing the orthogonality of the exponential function, equation (25) can be simplified and the reflection coefficient can be expressed as shown below.
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 Substituting equation (26) into equation (22a), the total pressure at z=0 yields withp a being the averaged pressure in the straight pipe. It can be further simplified as j R a n n w n i 1 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.
Considering the fact that the width of the straight pipe, w 1 , is muchsmaller than the working wavelength, only the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.   z a 0 1 ) can be expressed as Equation (33) can be revised in a matrix form as where M 1 is the transfer matrix, The expressions of the pressure and the volume velocity in the region between the first HR and the second HR (viz., At the position of the first HR (viz., = z a 1 ), the continuity of the pressure and the volume velocity require 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).
Here M 2 and N 1 are the transfer matrices, a a a a = -- Following the same procedure, the transfer matrices for other region of the element can be expressed as According to these transfer matrices, we can finally obtain the relationship for the pressure and volume velocity between z=0 and z=h,   The theoretical transmission and reflection spectra (amplitude and phase shift) predicted by equations (46a) and (46b) are shown infigures 2(a) and 2(b). To evaluate the accuracy of the theoretical derivations, simulated results based on a model with thesame geometrical parameters are also illustrated with dots. Commercial software based on the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 l < < h 0.28 0.61 where the local phase shift of the transmitted wave can cover the p 2 range.

Physical interpretation
In order to understand the transmission peaks, we further explore the underlying physics of the acoustic metascreen. Equation (31) indicates that 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 thetransmission line, the impedance of the pipe, Z p , at = + z a a 3 Similarly, the impedance at = = + = = + z z a a z z a a 2 , The averaged pressure,¯( ) p 0 a and volume velocity, ( ) U 0 a , at z=0 in the straight pipe can be correlated as Combining equations (52) and (30), we can obtain where + Z Z e a refers to the total impedance of the metascreen. Then the power transmitted into the metascreen at z=0 is The incident sound power is 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 1 I . Figure 3 illustrates the real part of the acoustic impedance of effective load, ( ) Z R Re e w , and the imaginary part of the total acoustic impedance of the system, + ( ) Z Z R Im e a w . 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 the transmission amplitude ( 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.

Elements of theacoustic metascreen
The amplitude and phase shift of the transmitted coefficient varying with the width of the straight pipe, w w 1 , and the incident angle, q i , is shown in figure 4. The black contour line demonstrates the region with transmission amplitude higher than 0.9 ( figure 4(a)). Fully controlled phase shift can be achieved in the same region ( figure 4(b)). By tuning w w 1 , viz. w 3 , the reactance provided by the HRs is effectively tailored, resulting in the phase shift covering a p 2 span. Eight elements with steps of p 4 can be readily selected to implement the desired phase profile, whilekeeping the impedance matching. Interestingly,fully controlled phase shift and high transmission can be achieved even for oblique incidence with an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 −3dB, which means that 50% of the incident power penetrates the element. Furthermore, withfixed geometries, the phase shift is almost independent of the incident angle. These tremendous properties, i.e.fully tuned phase and high transmission within wide incident angles, guarantee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 p 4, meanwhile holding high transmission amplitudes (>0.93). After discretizing the desired profile with finite steps and carefully choosing the corresponding geometries along the transverse direction (x), the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 l 10, allowing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 unity 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 a p 2 span. Figure 6 illustrates the effective acoustic impedance of the metascreen normalized to q ¢ = R R cos w w i withincident angles of 0°and 80°. The real part of the impedance of the effective load,   , indicating that the impedance is well matched, and resulting inhigh transmission ( figure 4). With anincrease inthe incident angle, the impedance, mainly ¢ ( ) Z R Re e w , becomes mismatched, leading tolow transmission with large incident angles. Interestingly, hybrid resonances, , are also supported under oblique incidence [39].

Shaped beams with acoustic metascreens
The great propertiesof the high power transmission, fully controlled phase shift, and deeply subwavelength spatial resolution, l = w 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 investigateseveral interesting wave manipulations using theproposed acoustic metascreen, such as anomalous refraction, negative refraction, and focusing.

Theoretical pressure fields
The total transmitted pressure field emerging from the metascreen can be regarded as the synthesis of aradiated wave from the elements. The radiated fields can be obtained using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] Considering the fact that ¶ ¶ = G z 0 and equation (2), equation (60) becomes , , , ,d ,

Anomalous refractions at normal incidence
Under the guidance ofgeneralized Snell's law [40,41], the relation between the incident angle, q i , and refracted angle, q t , should be revised as referingto the phase gradient. In order to realize the anomalous refraction at normal incidence, q = 0 i , a linear phase profile should be yielded, indicating a constant value of x ( ) x . The relationship between the transmitted angle, q t , and the constant phase gradient, ξ, under normal incidence is illustrated infigure 7. The theoretical and simulated results, obtainedsix wavelengths away from the center of the metascreen with a plane wave of l 5 width incidence, are also plotted forcomparison and an excellent agreement is observed.
After the desired profile, ξ, is selected, the metascreen can be constructed by suitably discretizing the profile in p 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 x p = ( ) 2 50 8, clearly showing that the metascreen generates extra transverse momentum on the normally incident wave, resulting in a bending transmitted wave propagating with predicted desired angles (black arrows in the output region).
Equation (62) implies that there is a critical value with x p = ( ) 2 10. No transmitted angle is allowed when x p > ( ) 2 10, indicatingthe conversion from propagating wave to evanescent wave in the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 x p = ( ) 2 100 9 at normal incidence with a width of l 8 . The acoustic energy is confined near the metascreen and rapidly attenuated in the z direction. The proposed metascreen can provide an extra momentum profile along the x direction for the conversion frompropagating wave toevanescent wave with theenergy concentrated near the surface. In the simulations, the existing interaction between the elements and other orders of diffractions enlarges the reflected energy, resulting in aslight difference between the theoretical and simulated results. The energy concentration near the surface may provide an alternative method forgeneratingsurface acoustic waves with some structures supporting its propagation, such as the corrugated structure [28,42].

Negative refractions with oblique incidence
The individual element possesses the capability of shifting analmost identical phase within a certain range of incident angles, while maintaining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 diffractions need to be counted when the incident angle becomes larger [43]. Then the non-local effect from the periodicity will play a crucial role inwave shaping when the wavelength is comparable to the period of the metascreen. The generalized Snellʼs law should be revised as [28] is the amplitude of the reciprocal lattice vector with Λ being the periodicity of the phase profile of the metascreen.
The formula of the theoretical pressure field (equation (61))needs to be revisited under oblique incidence , , ,e ,d , 6 4 n n w n w w x a 0 0 revealing the incident phase difference at the boundary of the metascreen (z=0). In the following calculations, the metascreen designed fornormal incidence is employed to challenge the performance foroblique incidence.
In order to investigate the behaviors at oblique incidence, ξ 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 anegative incident angle defined in the clockwise direction,   . By modulating the spatial resolution, w, and the phase gradient, x ( ) x , the conversion from a propagating wave to an evanescent wave can also be achieved within a certain range of incident angles [28].
Theoretical and simulated pressure fields infigure 11 clearly illustrate the negative refraction. The oblique incidence with small negative angle (q > - 38.7 i ) cannot provide enough momentum to cancel the additional transverse momentum generated by the metascreen. Therefore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. The 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 (figures 11(a) and (b)). The plane wave hasunity amplitude and a width of l 5 . Forincidence with a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 onthe same side of the incidence (figures 11(c) and (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 0 G . However, for the incidence with positive angle, the periodicity of the metascreen plays a dominant role in the field shaping with =n 3 G . The asymmetric property for positive and negative incidence may be used to realize some novel functions, such asasymmetric acoustic transmission [44][45][46][47][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 anangle of  79.5 (normal refraction) and - 63 (negative refraction), showinggood agreement with the predicted ones 4 .

Acoustic focusing from a cylinder wave
To further demonstrate the great capabilityof the proposed metascreen to control wave propagation flexibly, we also implement an acoustic focusing effect with thedesired quadratic phase profile. A line source located at s s 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 angles (see figure 4). To form a focus spot with afocal length of l = f 5 , the phase profile provided by the metascreen can be written as where the first term is the desired profile to focus a plane wave, and the second term, , compensates the arrival phase difference along the boundary of the metascreen (z=0).
The theoretical pressure field from a line source is By carefully discretizing thedesired phase profile and arranging eight types of elements, theacoustic metascreen was constructed to yield additional transverse momentum. Anomalous refraction and conversion from propagating wave to evanescent wave were demonstrated at normal incidence. Under oblique incidence, two typesof negative refractionwere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, negative and positive (normal) refractions can be inter-converted through simply changing the incident angle, with the existence of two types of refraction in a certain range of incident angles. An acoustic lens is also demonstrated to focus the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 evidencethat the metascreen can control wave propagation flexibly and concisely.
Our planar passive phased array composed of anacoustic metascreen, overcoming the drawbacks of anactive phased array, takes advantageof thesimple configuration and extreme acoustic performance, and hence shows great potentialin diverse wave-shaping applications, such asnon-diffracting beams [31], vortex beams [9], acoustic focusing [51], acoustic holography [52], and so on. The concept of apassive phased array can also be extended into underwater [53], nondestructive evaluation [54], and micro-fluidic systems [55].