Acoustic metasurfaces for scattering-free anomalous reflection and refraction
AAcoustic metasurfaces for scattering-free anomalous reflection and refraction
A. D´ıaz-Rubio and S. A. Tretyakov Department of Electronics and Nanoengineering,Aalto University, P. O. Box 15500, FI-00076 Aalto, Finland
Manipulation of acoustic wavefronts by thin and planar devices, known as metasurfaces, has beenextensively studied, in view of many important applications. Reflective and refractive metasurfacesare designed using the generalized reflection and Snell’s laws, which tell that local phase shifts atthe metasurface supply extra momentum to the wave, presumably allowing arbitrary control ofreflected or transmitted waves. However, as it has been recently shown for the electromagneticcounterpart, conventional metasurfaces based on the generalized laws of reflection and refractionhave important drawbacks in terms of power efficiency. This work presents a new synthesis methodof acoustic metasurfaces for anomalous reflection and transmission that overcomes the fundamentallimitations of conventional designs, allowing full control of acoustic energy flow. The results showthat different mechanisms are necessary in the reflection and transmission scenarios for ensuring per-fect performance. Metasurfaces for anomalous reflection require non-local response, which allowsenergy channeling along the metasurface. On other hand, for perfect manipulation of anomalouslytransmitted waves, local and non-symmetric response is required. These conclusions are interpretedthrough appropriate surface impedance models which are used to find possible physical implemen-tations of perfect metasurfaces in each scenario. We hope that this advance in the design of acousticmetasurfaces opens new avenues not only for perfect anomalous reflection and transmission but alsofor realizing more complex functionalities, such as focusing, self-bending or vortex generation.
I. INTRODUCTION
The interest in quasi two-dimensional devices capableof manipulating waves revived with the formulation of thegeneralized reflection and Snell’s laws [1], which shows apossibility of tailoring the direction of reflected and trans-mitted waves by introducing gradient phase shifts at theinterface between two media. The generalized laws ofreflection and refraction have been applied for control-ling the direction of transmitted and reflected waves inelectromagnetism [2] and acoustics [3–13]. By appropri-ately varying the phase shift introduced along the meta-surface between 0 and 2 π the propagation direction ofthe reflected/refracted wave can be controlled. Theseapproaches enable tailoring the energy propagation di-rection, but with important restrictions of the efficiency(the amount of energy that is sent into the desired direc-tion is smaller than the energy introduced in the system,even for lossless metasurfaces).Recently, it has been demonstrated that some addi-tional considerations about the power conservation canbe applied over the conventional generalized reflectionand Snell’s laws for ensuring full control of electromag-netic energy flow [14–17]. This advance has attractedmuch attention due to the possibility of dramatic im-provements of conventional solutions, especially for steepreflection or transmission angles. For electromagneticwaves, the basis of this “second generation” of gradientelectromagnetic metasurfaces has been established andnumerically verified, and for reflective metasurfaces thetheoretical findings have been already confirmed experi-mentally [18].However, it appears that the synthesis tools for acous-tic metasurfaces do not benefit from the new knowledge.In the acoustic reflection scenario, the generalized reflec- tion law has been experimental demonstrated [4–8] usinglabyrinthine unit cells which provide a phase-shift profilewith the 2 π span in the reflection coefficient phase. How-ever, in view of the results of [14–17], the performance ofthese designs is not optimal because significant energy isspread in unwanted directions. Theoretical studies basedof inhomogeneous impedance along the metasurface [5]show the coexistence of more than one reflected wave.For perfect control of anomalous reflection or, in otherwords, for allowing arbitrary changes of the direction ofreflected plane waves, we need to ensure perfect steeringof all the incident power into the desired direction avoid-ing generation of parasitic waves propagating in otherdirections or losses in the system.On the other hand, the same approach based on thegeneralized Snell’s law has been applied for the design ofrefractive acoustic metasurfaces [9–13]. The direction ofthe transmitted wave is controlled by linearly modulatingthe local phase shift in transmission through the meta-surface. For the design of unit cells, different topologieshave been used, including space-colling structures [9],slits filled with different density materials [10] or straightpipes with Helmholtz resonators in series [11, 12]. Oneproblem addressed in the design of refractive metasurfacewas the required impedance matching of each meta-atomin order to obtain total transmission. In this sense, sub-stantial improvements in the design of matched unit cellhave been achieved by using tapered labyrinthine units[13]. However, as it has been recently demonstrated forthe electromagnetic scenario [14, 15, 19], by ensuring per-fect matching in the microscopic design of the metasur-face (individual design of each meta-atom) we cannot ob-tain the proper macroscopic behavior of the metasurface.In this work we present the foundations for the syn-thesis of perfect acoustic metasurfaces, overcoming the a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug fundamental limitations of conventional designs. Thestudy covers two different scenarios: anomalous reflec-tion and transmission of acoustic plane waves. With thepurpose of simplifying the presentation and emphasizingthe novelty of this approach, in both cases the analysisstarts with a comprehensive overview of the known ap-proaches based on the generalized reflection and Snell’slaws. After identifying the weaknesses of current designs,we propose new methods which ensure perfect control ofacoustic energy in reflection and refraction. Finally, weinterpret the theoretical findings in terms of the physicalproperties of metasurface unit cells and give examples ofpossible realizations. II. ACOUSTIC METASURFACES FORREFLECTION
In this section, the reflected wavefront manipulationis studied. Particularly, we focus the study on anoma-lous reflection of acoustic plane waves. This fundamentalfunctionality is the base of many interesting applicationssuch reflection lenses, plane wave to surface wave conver-sion or acoustic retro-reflectors.
A. Design based on the generalized reflection law
In order to understand the current status of the syn-thesis methods, we start with the analysis of reflectivegradient metasurfaces based on the generalized reflec-tion law (we use the same short-hand notation, GSL, forboth generalized Snell’s law and the generalized reflectionlaw). If we consider the scenario illustrated in Fig. 1(a)where the incident and reflected waves propagate in ahomogeneous medium with density ρ and sound speed c ,assuming time-harmonic dependence e jωt , the incidentand reflected pressure fields can be written as p i ( x, y ) = p e − jk sin θ i x e jk cos θ i y , (1) p r ( x, y ) = Ap e − jk sin θ r x e − jk cos θ r y , (2)where p is the amplitude of the incident plane wave, k = ω/c is the wavenumber in the background mediumat the operation frequency, θ i and θ r are the incidenceand reflection angles, and A is a constant which relatesthe amplitudes of the incident and reflected waves. Thevelocity vectors associated with these pressure fields ( (cid:126)v = jωρ ∇ p ) read (cid:126)v i ( x, y ) = p i ( x, y ) Z (sin θ i ˆ x − cos θ i ˆ y ) , (3) (cid:126)v r ( x, y ) = p r ( x, y ) Z (sin θ r ˆ x + cos θ r ˆ y ) , (4)where Z = cρ is the characteristic impedance of thebackground medium.Assuming that the field beyond the metasurface is zero(an impenetrable metasurface), the system can be con-veniently modeled by the equivalent circuit shown in 𝑝 i (𝑥, 𝑦) 𝑣 i (𝑥, 𝑦) 𝑝 r (𝑥, 𝑦) 𝑣 r (𝑥, 𝑦) 𝑥 𝑦 𝑛 𝜃 𝑖 𝜃 𝑟 (a) − 𝑛 ∙ 𝑣 tot 𝑝 tot 𝑍 𝑠 (b) FIG. 1: (a) Schematic representations of the desiredmetasurface behavior for the anomalous reflectionscenario. (b) Equivalent circuit proposed for theanalysis of reflective metasurfaces.Fig. 1(b) where the impedance Z s models the specificimpedance of the metasurface. GSL designs are basedon the assumption that a linear gradient phase shift[ ∂ Φ x ∂x = k (sin θ r − sin θ i )] is introduced by the metasur-face. In other words, the metasurface is characterized bythe local reflection coefficient with the unit amplitude,which can be written asΓ( x ) = e − jk sin θ r x e − jk sin θ i x = e jk (sin θ i − sin θ r ) x = e j Φ x , (5)where the reflection phase is defined as Φ x = k (sin θ r − sin θ i ) x . The reflection coefficient is related with the sur-face impedance as Γ = Z s − Z i Z s + Z i , where Z i = Z / cos θ i represents the specific acoustic impedance of the inci-dent wave at the metasurface. From this expression, theimpedance which models the metasurface can be foundas Z s ( x ) = j Z cos θ i cot(Φ x / . (6)Figure 2 presents a numerical study of the conventionaldesigns based on the GSL. The surface impedance mod-eled by Eq. (6) is purely imaginary [see Fig. 2(c)], solossless implementations are possible for this kind of re-flective metasurfaces. Actual implementations of the de-sired impedance profiles can be obtained by using simplerigidly terminated waveguides with different lengths or,exploiting the longitudinal character of acoustics wavesand so the absence of cutoff frequency, “space-colling”particles with labyrinth channels [4–8]. In these cases,each meta-atom has to be carefully tailored for indi-vidually implementing the required surface impedanceprofile and producing the required local phase shift.For the purposes of this study we assume that the re-quired impedance profile has been realized and model themetasurface (using COMSOL software) as an impedanceboundary described by Eq. (6). Figures 2(a) and 2(b)show the results of numerical simulations when the meta-surface is illuminated normally ( θ i = 0 ◦ ) and the reflec-tion angles are θ r = 30 ◦ and θ r = 70 ◦ , respectively. Fromthe comparison of these two examples, it is easy to seetwo important issues: First, the “quality” of the reflectedwave decreases when the reflection angle increases, due -1 -0.5 0 0.5 100.511.522.53 -1.5-1-0.500.511.5 (a) -0.5 0 0.500.511.522.53 -1.5-1-0.500.511.5 (b) -0.5 0 0.5-50 (c) (d) FIG. 2: Real part of the scattered pressure field for ametasurface designed according to Eq. (6) when: (a) θ i = 0 ◦ and θ r = 30 ◦ ; (b) θ i = 0 ◦ and θ r = 70 ◦ . (c)Surface impedance described by Eq. (6) when θ i = 0 ◦ and θ r = 70 ◦ . (d) Efficiency of the GSL gradientmetasurfaces as a function of the reflection angle.to parasitic reflections in other directions; Second, theamplitude of the reflected wave changes with the reflec-tion angle, although this behavior it is not contemplatedin the design statement ( A = 1). Clearly, the simpledesign philosophy described by Eq. (6) does not ensurethe perfect conversion of energy between the incident andreflected plane waves and it cannot be considered as anaccurate method for the design of metasurfaces for largevalues of the reflection angle.The conclusions extracted from the analysis of the nu-merical simulations can be understood as an impedancemismatch problem. Although the metasurface providesthe desired phase response, the incident and reflectedwaves have different specific impedances, so part of theenergy cannot be redirected into the desired direction.Since the metasurface is assumed to be lossless, part ofthe incident energy has to be reflected into other direc-tions (into 0 ◦ and − ◦ in the example of Fig. 2). Thereflections into parasitic directions can be estimated in-troducing reflection coefficient calculated in terms of the respective impedances: R = Z r − Z i Z r + Z i = cos θ i − cos θ r cos θ i + cos θ r . (7)Because the metasurface is an impenetrable boundary,the total pressure of the incident and reflected waves (1+ R ) is equal to the pressure of the wave redirected intothe desired direction ( A GSL ), in analogy to transmissionof electromagnetic waves through electric-current sheets[22]: A GSL = 1 + R = 2 cos θ i cos θ i + cos θ r . (8)We can now define the efficiency of the metasurfaceas the ratio between the incident power and the powerreflected in the desired direction. The acoustic power canbe expressed in terms of the intensity vector (cid:126)I = 12 Re[ p(cid:126)v ∗ ] , (9)where ’ ∗ ’ represents the complex conjugate. Due to theperiodicity of the system only the normal component ofthe intensity vector will take part in the power balance.The normal component of the incident power is P i = ˆ n · (cid:126)I i = − p Z cos θ i . (10)The normal component of the power carried in the de-sired reflection direction can be calculated as P r = ˆ n · (cid:126)I r = A p Z cos θ r , (11)and the efficiency reads η = | P r || P i | = A cos θ r cos θ i . (12)Substituting the wave amplitude A from Eq. (8) we canfinally estimate the efficiency of conventional metasur-faces as η GSL = (cid:18) θ i cos θ i + cos θ r (cid:19) cos θ r cos θ i . (13)Figure 2(d) represents the efficiency estimation givenby Eq. (13) and its comparison with the numerical re-sults. In order to find the power efficiency from numer-ical results we calculate the amplitude of the reflectedplane wave into θ r , A COMSOL . This amplitude can becalculated as A COMSOL = 1 D (cid:90) D p r · e jk sin θ r x dx, (14)where D is the metasurface period, and the efficiency isobtained using Eq. (12). It is possible to see how the effi-ciency of the generalized reflection law metasurfaces dra-matically decreases when the reflection angle increases. -1 -0.5 0 0.5 100.511.522.53 -1-0.8-0.6-0.4-0.200.20.40.60.81 (a) -0.5 0 0.500.511.522.53 -1-0.8-0.6-0.4-0.200.20.40.60.81 (b) -0.5 0 0.5-2024 (c) θ r E ffi c i e n c y ( η ) Theoretical predictionNumerical validation (d)
FIG. 3: Real part of the scattered pressure field for ametasurface designed according to Eq. (19) when: (a) θ i = 0 ◦ and θ r = 30 ◦ and (b) θ i = 0 ◦ and θ r = 70 ◦ . (c)Surface impedance described by Eq. (19) when θ i = 0 ◦ and θ r = 70 ◦ . (d) Efficiency of the gradientmetasurfaces for A = 1 as a function of the reflectionangle. B. Lossy metasurfaces for anomalous reflection
As we have seen, the amplitude of the reflected wavein the conventional design is not equal to the amplitudeof the incident plane wave ( A GSL (cid:54) = 1). If we design ametasurface which arbitrarily changes the direction of thereflected wave keeping the amplitude A = 1, the pressurefield at metasurface can be written as p tot ( x,
0) = p (1 + e j Φ x ) e − jk sin θ i x . (15)The corresponding total velocity at the metasurface reads (cid:126)v tot ( x,
0) = p Z (sin θ i + sin θ r e j Φ x ) e − jk sin θ i x ˆ x + (16) p Z ( − cos θ i + cos θ r e j Φ x ) e − jk sin θ i x ˆ y. (17)At this point, we have to satisfy the boundary condi-tion at the metasurface. We can do that by defining thesurface impedance which models this metasurface as Z s ( x ) = p tot ( x, − ˆ n · (cid:126)v tot ( x, . (18) Introducing the expressions for the desired total pressure(15) and velocity (17) into this equation, we find theimpedance which models such metasurfaces: Z s ( x ) = Z e j Φ x cos θ i − cos θ r e j Φ x . (19)Figure 3 presents the results of a numerical study ofacoustic metasurfaces based on Eq. (19). As in the pre-vious case, Figs. 3(a) and 3(b) show simulated results forthe metasurface illuminated normally and designed forthe reflection angles θ r = 30 ◦ and θ r = 70 ◦ , respectively.The results confirm the required performance of metasur-faces which anomalously reflect a perfect plane have withthe same amplitude as the incident wave. The surfaceimpedance given by Eq. (19) is a complex number as it isshown in Fig. 3(c) for θ i = 0 ◦ and θ r = 70 ◦ . The real partof the impedance takes positive values (modeling losses)over all the period showing that these metasurfaces arenecessarily lossy, which is a condition for keeping the am-plitude of the reflected wave equal to the incident wave.To illustrate this behavior we can analyze the efficiencyof the metasurface found from Eq. (12) when A = 1: η A =1 = cos θ r cos θ i . (20)This expression is represented in Fig. 3(d) as a functionof the reflection angle for θ i = 0 ◦ . Numerical resultshave been calculated in the same way as before, usingEq. (14). We can see that the efficiency dramatically de-creases when the reflection angle increases. When thereflection angle increases, the power sent into the desireddirection decreases and all the remaining energy is ab-sorbed in the metasurface. On the other hand, if θ i > θ r the real part of the surface impedance becomes nega-tive (gain), meaning that additional energy has to beintroduced in the system in other to obtain the desiredperformance. C. Active-lossy scenario and lossless non-localrealization
Obviously, the design approach defined by Eq. (19)presents important drawbacks in terms of power effi-ciency, although there are no parasitic reflections intounwanted directions. For perfect anomalous reflectionwhere all the impinging energy is sent into the desireddirection we have to ensure η = 1. From Eq. (12) it iseasy to find the amplitude coefficient which correspondsto the perfect performance: For perfect anomalous re-flection the amplitude of the reflected wave has to be A = (cid:112) cos θ i / cos θ r . Figure 4(d) shows a comparisonbetween the required amplitude for perfect performanceand the amplitude of conventional designs based on GSLwhen θ i = 0 ◦ . The difference between both approachesincreases with the angle of reflection, confirming our pre-vious conclusion about the poor efficiency of conventional -1 -0.5 0 0.5 100.511.522.53 -1.5-1-0.500.511.5 (a) -0.5 0 0.500.511.522.53 -1.5-1-0.500.511.5 (b) -0.5 0 0.5-4-20246 (c) (d) FIG. 4: Real part of the scattered pressure field for ametasurface designed according to Eq. (23) when: (a) θ i = 0 ◦ and θ r = 30 ◦ and (b) θ i = 0 ◦ and θ r = 70 ◦ . (c)Surface impedance described by Eq. (23) when θ i = 0 ◦ and θ r = 70 ◦ . (d) Amplitude of the reflected wave forperfect anomalous reflection (red symbols) comparedwith conventional designs based on GSL (blue line).design for large differences between incident and reflectedangles. In this scenario, the acoustic impedance of themetasurface can be calculated writing the pressure field p tot ( x,
0) = p (cid:32) (cid:114) cos θ i cos θ r e j Φ x (cid:33) e − jk sin θ i x (21)and the velocity at the metasurface (cid:126)v tot ( x,
0) = p Z (cid:32) sin θ i + (cid:114) cos θ i cos θ r sin θ r e j Φ x (cid:33) e − jk sin θ i x ˆ x + p Z (cid:16) − cos θ i + (cid:112) cos θ i cos θ r e j Φ x (cid:17) e − jk sin θ i x ˆ y. (22)Finally, the corresponding surface impedance reads Z s ( x ) = Z √ cos θ i cos θ r √ cos θ r + √ cos θ i e j Φ x √ cos θ i − √ cos θ r e j Φ x . (23)In Fig. 4 the numerical results obtained with Eq. (23)are represented. Figures 4(a) and 4(b) show numerical results for metasurfaces illuminated normally when thedesign reflection angles are θ r = 30 ◦ and θ r = 70 ◦ , respec-tively. We can see how the amplitude of three reflectedwave changes with the reflected angle according to thetheory. The surface impedance defined by Eq. (23) isa complex number [see Fig. 3(c)] whose real part takespositive (loss) and negative (gain) values. Obviously, theaveraged over one period normal component of the totalpower is zero, meaning that the macroscopic system islossless.Although active acoustic metamaterials have beenstudied in the literature [3], in general, the use of ac-tive and lossy elements is not desired in actual imple-mentations for practical reasons. In order to simplify thedesign and implementation, the active-lossy behavior canbe understood as a phenomenon of energy channeling, soit is not necessary to include active or lossy elements forimplementing these metasurfaces, and a lossless imple-mentation can be found. In order to overcome the fun-damental deficiency of all conventional reflective meta-surface and implement the required “gain-loss” responsedefined by Eq. (23), the metasurface has to receive en-ergy in the “lossy” regions, guide it along the surface,and radiate back in the “active” regions. The energychanneling along the metasurface corresponds to a non-local response. In non-local metasurfaces the behaviorof each element of the metasurface depends on the in-teraction with the neighbors, so traditional techniquesbased on the individual design of each meta-atom can-not be used. As it was demonstrated in [17] for electro-magnetic metasurfaces, properly designing the inhomo-geneous impedance of a lossless metasurface it is possibleto obtain the required non-local response. The operat-ing principle is similar to leaky waves antennas [20, 21],where periodical perturbations allow coupling betweenguided waves and propagating waves in free space.In what follows, we present a proof of concept of a non-local design with high efficient performance when θ i = 0 ◦ and θ r = 70 ◦ . For designing the non-local acoustic reflec-tor, we use as a first approximation the imaginary part ofthe complex impedance described by Eq. (23). This ap-proach allows one to design an array of lossless elementsby using conventional techniques which will produce alocal phase shift according to Γ = j Im( Z s ) − Z i j Im( Z s )+ Z i . Partic-ularly, each element is implemented by a rigidly endedwaveguide whose impedance can be calculated as Z stub = − jZ cot( kl n ) (24)where l n is the length of the n -th element. Once thelength of each stub has been fixed (15 particles in theexample presented in this work), we run a numerical op-timization setting as a goal full reflection in the desireddirection. The objective of this optimization is to tailorthe coupling effects between particle, using the evanes-cent fields as a mechanism for channeling the energy. Af-ter the optimization process the lengths of the stubs are0 . λ , 0 . λ , 0 . λ , 0 . λ , 0 . λ , 0 . λ ,0 . λ , 0 . λ , 0 . λ , 0 . λ , 0 . λ , 0 . λ , (a)(b) (c) FIG. 5: Non-local design of an anomalous reflector for θ i = 0 ◦ and θ r = 70 ◦ with 95% of efficiency: Real part(a) and magnitude (b) of the scattered field. (c)Bandwidth analysis for an actual implementation of theanomalous reflector designed at 3400 Hz.0 . λ , 0 . λ , and 0 . λ , respectively. The effi-ciency of the optimized metasurface is 95%. Figure 5(a)shows the real part of the scattered field, where we can seethe plane wave reflected in the desired direction. Figure5(b) represents the magnitude of the scattered field. Inthe vicinity of the metasurface, strong evanescent fieldsare created, as is required for theoretically perfect per-formance.The corrugated surface can be considered as alossless interface able to receive power in some regions,guide it by surface waves excited in the metasurface andradiate the energy back into the desired direction.For a more complete analysis of the non-local anoma-lous reflector, we study the efficiency as a function of fre-quency. In particular, we consider the previous design ofthe non-local metasurface when the operation frequencyis 3400 Hz. The results of this analysis are summarizedin Fig. 5(c). We obtain the maximum efficiency at thedesign frequency. The efficiency is defined as the powerredirected into the first diffracted mode ( n = 1) whichcorresponds to θ r = 70 ◦ at the design frequency. The ef-ficiency remains higher than 0.5 from 3350 Hz to 3550 Hz( ≈
6% of the operating frequency). 𝑝 𝐼 (𝑥, 𝑦) 𝑣 𝐼 (𝑥, 𝑦) 𝑝 𝐼𝐼 (𝑥, 𝑦) 𝑣 𝐼𝐼 (𝑥, 𝑦) 𝑥 𝑦 𝑛 𝜃 𝑖 𝜃 𝑡 (a) −ො𝑛 ∙ Ԧ𝑣 𝐼 ො𝑛 ∙ Ԧ𝑣 𝐼𝐼 𝑝 𝐼𝐼 𝑝 𝐼 𝑍 − Z 𝑍 − 𝑍 𝑍 = 𝑍 (b) FIG. 6: (a) Schematic representations of themetasurface behavior for the anomalous transmissionscenario. (b) Equivalent circuit for refractivemetasurfaces.
III. ACOUSTIC METASURFACES FORANOMALOUS TRANSMISSIONA. Design based on the generalized Snell’s law
As in the above study of reflective metasurfaces, westart with the analysis of the conventional refractive gra-dient index metasurfaces based on the generalized Snell’slaw. If we consider the scenario illustrated in Fig. 6(a),the pressure field above and beyond the metasurface canbe written as p I ( x, y ) = p e − jk sin θ i x e jk cos θ i y , (25) p II ( x, y ) = Ap e − jk sin θ t x e jk cos θ t y , (26)where p is the amplitude of the incident plane wave, θ i and θ t are the incidence and transmission angles, re-spectively, k = ω/c is the wavenumber at the operationfrequency, and A is the coefficient which relates the am-plitudes of the incident and transmitted waves. The ve-locity vectors at both sides of the metasurface can beexpressed as (cid:126)v I ( x, y ) = p I ( x, y ) Z (sin θ i ˆ x − cos θ i ˆ y ) (27) (cid:126)v II ( x, y ) = p II ( x, y ) Z (sin θ t ˆ x − cos θ t ˆ y ) (28)with Z = cρ being the characteristic impedance of thebackground medium. Pressure and velocity at both sideof the metasurface can be related by using the specificimpedance matrix as (cid:20) p I ( x, p II ( x, (cid:21) = (cid:20) Z Z Z Z (cid:21) (cid:20) − ˆ n · (cid:126)v I ( x, n · (cid:126)v II ( x, (cid:21) . (29)In the most general linear case and assuming reciprocity( Z = Z ), the relation between the acoustic field atboth side of the metasurfaces can be modeled by theequivalent circuit represented in Fig. 6(b). Conventionalrefractive metasurfaces are designed in such a way thateach meta-atom introduces a local phase-shift in trans-mission according to t = p II ( x, p I ( x,
0) = e j Φ x , (30)where Φ x = k sin θ i x − k sin θ t x is the linearly-varyingphase of the local transmission coefficient t . In the knowndesigns symmetric meta-atoms ( Z = Z ) are used forthe implementation of the desired transmission coefficientat every point of the metasurface. The relation betweenthe Z -matrix elements and the local transmission coeffi-cient can be expressed as [23] Z = Z = Z i t − t = j Z cos θ i cot(Φ x ) , (31) Z = Z i t − t = j Z cos θ i x ) . (32)These impedances define the behavior of conventional de-signs. We can see that the impedances are purely imagi-nary, meaning that lossless implementations are possible[9–13].In this work and as a proof of concept we propose a sim-ple implementation of the meta-atoms based on clampedrectangular membranes. Each membrane can be mod-eled as a series LC -resonator controlled by its acousticmass and compliance [20, 21]. Particularly, each meta-atom consists of three membranes separated by distance l [see Fig. 7]. The period of the metasurface is dividedinto N unit-cells (the width of the meta-atoms is D/N )and rigid walls are introduced between the meta-atomsto avoid coupling between them and prevent excitationof guided modes between membranes. By independentlytuning the response of each membrane we can obtain thedesired response of the meta-atoms. 𝑥 𝑦 P e r i o d i c B o u n d a r y C o n d i t i o n P e r i o d i c B o u n d a r y C o n d i t i o n 𝑍 (𝑥)𝑍 (𝑥)𝑍 (𝑥) 𝑛 −1 𝑑 𝑛𝑑 W a ll W a ll 𝐷 (a) ො𝑥 ො𝑦 P e r i o d i c B o u n d a r y C o n d i t i o n P e r i o d i c B o u n d a r y C o n d i t i o n 𝑍 (𝑥)𝑍 (𝑥)𝑍 (𝑥) 𝑛−1 𝑑 𝑛𝑑 W a ll 𝐷 −ො𝑛 ∙ Ԧ𝑣 𝐼 −ො𝑛 ∙ Ԧ𝑣 𝐼𝐼 𝑝 𝐼𝐼 𝑝 𝐼 𝑍 (𝑥)𝑍 (𝑥)𝑍 (𝑥) 𝑍 ,𝑙𝑍 ,𝑙 (b) FIG. 7: (a) Schematic representations of themetasurface topology (b) Equivalent circuit for theproposed meta-atoms.The equivalent circuit for the proposed implementationis shown in Fig. 7(b), where the membranes are mod-eled as reactive impedances in series. For designing themembranes we need to know the relation between theirsheet impedances and the Z -matrix. The response of a meta-atom can be expressed in terms of the transmis-sion matrices of membranes and empty spacings betweenthem: (cid:20) p I ( x, − ˆ n · (cid:126)v I ( x, (cid:21) = (cid:20) A BC D (cid:21) (cid:20) p II ( x, − ˆ n · (cid:126)v II ( x, (cid:21) , (33)where (cid:20) A BC D (cid:21) = M Z M T M Z M T M Z (34)with M Zi = (cid:20) Z i (cid:21) , i = 1 , , M T = (cid:20) cos( kl ) jZ sin( kl ) j Z sin( kl ) cos( kl ) (cid:21) . (36)The three elements of the ABCD-matrix needed for thedefinition of meta-atoms are A = cos ( kl ) − sin ( kl ) (cid:18) Z Z Z (cid:19) + j cos( kl ) sin( kl ) (cid:18) Z + Z Z (cid:19) (37) C = j Y cos( kl ) sin( kl ) − Z Z sin ( kl ) (38) D = cos ( kl ) − Z Z Z sin ( kl )+2 j Z + Z Z cos( kl ) sin( kl ) . (39)On the other hand, we can write the desired responsemodeled by the Z -matrix in terms of the ABCD-matrix[23] as (cid:20) A BC D (cid:21) = (cid:34) Z Z | Z | Z Z Z Z (cid:35) (40)with | Z | = Z Z − Z . Finally, equating each ele-ment of both ABCD-matrices, we can obtain the rela-tions which define the membranes: Z ( x ) = Z + Z + jZ cot( kl ) , (41) Z ( x ) = j Z cot( kl ) − Z Z ( kl ) , (42) Z ( x ) = Z + Z + jZ cot( kl ) . (43)Figure 8 shows the results of the numerical analysis foran acoustic metasurface based on the generalized Snell’slaw. Particularly, the study has been done for θ i = 0 ◦ , θ t = 70 ◦ , and l = λ/ -1.5-1-0.500.511.5 (a) -0.5 0 0.5-1001020 (b) (c) FIG. 8: Refractive metasurface based on a linear phasegradient: (a) Real part of the total pressure field for a conventionalmetasurface when θ i = 0 ◦ and θ t = 70 ◦ ; (b) Impedancewhich model the three membranes of the meta-atoms; (c)Bandwidth of the metasurface designed for 3400 Hz. that the wavefront of the transmitted wave is not perfectand some perturbations due to parasitic waves appear.For comparison, it is interesting to analyse also con-ventional refractive metasurfaces based on a linear phasegradient as a function of the frequency. Let us first con-sider a refractive metasurface made of non-dispersive ele-ments designed for implementing a linear phase gradientaccording to Eq. (30). The theoretical response of thedesigned structure as a function of the frequency can becalculated by considering the change in the impedancedue to the change in the direction of the diffracted modefor different frequencies θ r ( f ) = arcsin k sin θ i +2 π/Dk . Fol-lowing the same approach as for reflective metasurfaceswe can calculate the amplitude of the transmitted waveby analyzing the impedance mismatch between the inci-dent and transmitted waves, T ( f ) = θ i cos θ i +cos θ r ( f ) . Con-sequently, the efficiency of the metasurface defined as thepercentage of power sent into the desired diffracted mode can be calculated as η ( f ) = T ( f ) θ r ( f )cos θ i . This efficiencyis represented in Fig. 8(c) with the red line. In actualimplementations, the dispersion of the elements has tobe taken into account in the analysis of the bandwidth.Figure 8(c) shows the results of a numerical study ofthe bandwidth for a metasurface designed for operationat 3400 Hz (yellow symbol) according to Eqs. (31) and(32) and implemented with the 3-membranes topologydescribed above. We can clearly see that the theoreticalperformance of the metasurface is perturbed by the dis-persive behavior of the constituent elements reducing thebandwidth of the designed metasurface. B. Asymmetric acoustic metasurfaces for perfectanomalous refraction
As it was explained for reflective metasurfaces, for aperfect performance of the refractive metasurface we haveto ensure that all the energy of the incident plane waveis carried away by the transmitted plane wave propagat-ing in the desired direction. This condition gives us thefollowing relation: p Z cos θ i = A p Z cos θ t . (44)The amplitudes of the incident and transmitted waveshave to be different, and the relation between them reads A = (cid:114) cos θ i cos θ t . (45)If we impose this amplitude for the transmitted waveand look for a lossless reciprocal solution ( Z = jX , Z = jX , and Z = Z = jX ), the boundaryconditions given by Eq. (29) simplify to1 = jX cos θ i Z − jX A cos θ r Z e j Φ x , (46) Ae j Φ( x ) = jX cos θ i Z − jX A cos θ r Z e j Φ x . (47)Separating the real and imaginary parts, can write:1 = X A cos θ r Z sin (Φ x ) , (48)0 = X cos θ i Z − X A cos θ r Z cos (Φ x ) , (49) A cos Φ( x ) = X A cos θ r Z sin (Φ x ) , (50) A sin Φ( x ) = X cos θ i Z − X A cos θ r Z cos (Φ x ) . (51)Solving this system of equations, we finally find the ele-ments of the corresponding Z -matrix: Z = j Z cos θ i cot (Φ x ) , (52) Z = j Z cos θ t cot (Φ x ) , (53) Z = j Z √ cos θ i cos θ t x ) . (54)The first difference as compared with the conventionaldesign is that the meta-atoms are not symmetric ( Z (cid:54) = Z ). This asymmetric response is comparable with thebianisotropic requirements described for the electromag-netic counterpart [14]. The same three-membranes topol-ogy can be used for the implementation of this new de-sign. However, as we can see from Fig. 9(a)-9(b), Z (cid:54) = Z in order to realize the required asymmetry. Figure 9(a)shows the results of numerical simulations when θ i = 0 ◦ , θ t = 70 ◦ . Clearly, the proposed design generates a per-fect plane wave in the desired direction.Figure 9(c) shows the results of a numerical study ofthe bandwidth and the comparison between a symmetricdesign [according to Eqs. (31) and (32)] and an asym-metric design [according to Eqs. (52), (53), and (54)].The operational frequency for both designs is 3400 Hz.The efficiency of the asymmetric design is higher thanof the GSL-based design in the frequency range between3315 Hz and 3470 Hz; then the efficiency of the asymmet-ric design decrease faster. The efficiency of the proposeddesign is higher than 0.5 in the frequency range between3295 Hz and 3585 Hz (8.5 % of the operating frequency).It is also interesting to consider the local transmissionand reflection coefficients in this scenario. Due to theasymmetry, the response of the particles when they areilluminated from media I (forward illumination) and II(backward illumination) will be different [see Fig. 10(a)and Fig. 10(b)]. Transmission and reflection coefficientscan be calculated using the impedance matrix as follows: R + = ( Z − Z )( Z + Z ) − Z ∆ Z , (55) T + = T − = 2 Z Z ∆ Z , (56) R − = ( Z + Z )( Z − Z ) − Z ∆ Z , (57)where ∆ Z = ( Z + Z )( Z + Z ) − Z and the ± signscorrespond to the forward and backward illuminations.Calculated results are plotted in Fig. 10. We can seehow individually the meta-atoms are not matched andgenerate forward and backward reflections.The above results demonstrate a possibility to changethe refraction angle θ r when the metasurface is illumi-nated normally. The method can be easily applied forother incidence angles. Figure 11 illustrates two differ-ent designs for a metasurface illuminated at θ i = 10 ◦ .In the first scenario the incident plane wave is refractedinto θ r = − ◦ . The relation between the amplitudes of -1.5-1-0.500.511.5 (a) -0.5 0 0.5-1001020 (b) (c) FIG. 9: Asymmetric refractive metasurface. (a) Realpart of the total pressure field for a perfect metasurfacewhen θ i = 0 ◦ and θ t = 70 ◦ ; (b) Impedance which modelthe three membranes of the meta-atoms; (c) Bandwidthof the metasurface designed for 3400 Hz.the incident and reflected waves is defined by Eq. (45)being A = 1 .
69. The Z -matrix components of this struc-ture are represented in Fig. 11(c). We can implementit by using the 3-membranes topology as it is shown inFig. 11(a). The second designed metasurface changesthe direction of the transmitted wave from θ i = 10 ◦ to θ r = 70 ◦ . The relation between the amplitudes ofthe incident and transmitted waves is the same as inthe previous example, and the Z -matrix components arerepresented in Fig. 11(d). An important difference be-tween both designs is the period of the metasurface whichequals D = λ/ | sin θ r − sin θ i | . IV. CONCLUSIONS
This work introduces a new approach for the synthe-sis of acoustic metasurface for anomalous transmissionand reflection. We have explained the main ideas of the0 𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 ) 𝑛 − 1 𝑑 𝑛𝑑 𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 ) 𝑛 − 1 𝑑 𝑛𝑑 𝑇 + 𝑅 + 𝑅 − 𝑇 − (a) 𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 ) 𝑛 − 1 𝑑 𝑛𝑑 𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 )𝑍 (𝑥 𝑛 ) 𝑛 − 1 𝑑 𝑛𝑑 𝑇 + 𝑅 + 𝑅 − 𝑇 − (b) -0.5 0 0.5-1-0.500.51 (c) -0.5 0 0.5-1-0.500.51 (d) -0.5 0 0.500.20.40.6 (e) FIG. 10: Schematic representation of the forward (a)and backward (b) illumination scenarios. Forward (+)and backward (-) local transmission and reflectioncoefficients for a perfect refractive metasurface. (c)Forward reflection, (d) forward and backwardtransmission, and (e) backward reflection.method by using a simple model based on the inhomoge-neous surface impedance of the metasurface. This modelhas allowed direct comparison between conventional de-signs based on the generalized reflection and Snell’s lawsand the introduced new approach, showing drastic im-provements in power efficiency. The fundamental ad-vance introduced by our method is the full suppressionof parasitic reflections in non-desired directions which re-duce the total efficiency of the metasurface.Applying the introduced design method to the reflec-tion and transmission scenarios we have identified dif-ferent physical phenomena which need to be realized forthe ideal performance. Metasurfaces for controlling re-flection must exhibit non-local response, which allows en-ergy channeling along the metasurface. In contrast, forfull control of plane-wave transmission local but asym-metric response is required for each particle.Based on the general synthesis theory, we have iden-tified appropriate topologies of acoustic unit cells whichallow realization of perfect reflection and refraction. Forperfect reflection, arrays of acoustical stubs can be used,and for perfect refraction, one can use three-membranesunit cells. To realize the required non-local properties ofreflecting surfaces, the unit cells in each super-cell needto be optimized together, including near-field couplingsbetween the cells. We have exemplified this procedurewith the design of an acoustic anomalous reflector with95% of efficiency when θ i = 0 ◦ and θ r = 70 ◦ . In the (a) (b) -0.5 0 0.5-1001020 (c) -0.5 0 0.5-1001020 (d) FIG. 11: Anomalous refractive metasurface for θ i = 10 ◦ and θ r = − ◦ : (a) Real part of the total pressure fieldfor the 3-membranes implementation and (c) Z -matrixcomponents. Anomalous refractive metasurface for θ i = 10 ◦ and θ r = 70 ◦ : (b) Real part of the totalpressure field for the 3-membranes implementation and(d) Z -matrix components.transmission scenario, the required asymmetry of unitcells can be realized if all three membranes of each unitcell are different. Since the proposed synthesis methodallows complete suppression of parasitic reflection, theonly factor which will limit the power efficiency is thepower dissipation due to inevitable losses in the materi-als from which the metasurface is made.We hope that this work will motivate future exper-imental demonstrations of perfect anomalous reflectiveand refractive metasurfaces. To conclude, let us stressthe importance of this advance for the improvement ofother acoustic systems where perfect performance in thesense of suppression of parasitic scattering is desired. Theproposed design methodology can be extended for ar-bitrary manipulations of multiple plane waves allowingmore complex functionalities. In general, by designingamplitudes and phases of different waves and ensuringthe local conservation of the power, it will be possible toovercome the efficiency drawbacks of the existing solu-tions for arbitrarily transformations of acoustic fields.1 ACKNOWLEDGMENT
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