Ultra-thin Acoustic Metasurface-Based Schroeder Diffuser

Transcription

1 Ultra-thin Acoustic Metasurface-Base Schroeer Diffuser Yifan Zhu, Xuong Fan, Bin Liang *, Jianchun Cheng *, an Yun Jing * Key Laboratory of Moern Acoustics, MOE, Institute of Acoustics, Department of Physics, Collaborative Innovation Center of Avance Microstructures, Nanjing University, Nanjing 93, P. R. China Department of Mechanical an Aerospace Engineering, North Carolina State University, Raleigh, North Carolina 7695, USA These authors contribute equally to this work. * s: liangbin@nju.eu.cn (B.L.); jccheng@nju.eu.cn (J.C.C.); yjing@ncsu.eu (Y.J.)

2 Supplementary Note. Analytical erivation of phase responses of the meta-structure unit-cell for normally incient waves. As shown in Fig. S(a), the meta-structure unit consists of a short pipe with length l an a cavity with volume V. Acoustic pressure in the cavity satisfies the relationship p ( r, ω) + k p ( r, ω) =, r V p ik β( r, ω) p =, r S n p = iρ (, ), ckvn r ω r S n (S) where k = w/ c is the wave number. ρ an c are ensity an soun spee in air, respectively. v (, ) r ω is normal vector component of volume velocity. β( r, ω) is n inuce by the impeance at bounary S, an here har bounary leas to β( r, ω ) =. Then, by employing the Green s function, the acoustic pressure at arbitrary position of S region can be expresse as p(, r ω) = G (, ') p (', ω) S' s r r r s n p ( r', ω) n = iρ c k G (, rr' ) v ( r', ω) S' G (, rr' ) n (S) The frequency-omain Green s function can be expresse as G (, rr', ω) = ψ ( r', ω ) ψ (, r ω ), (S3) = k k where is the number of eigenmoe, ψ is the eigenmoe, ω is the eigenfrequency, an k is the wave number at the corresponing eigenfrequency. Subsequently, we obtain the acoustic pressure p(, r ω) = iρck ψ (', ) (, ) v n( ', ) S' k k s ω ψ ω ω r r r. (S4) =

3 The average normal vector component of volume velocity is expresse as vn = v (, ) s n ' ω S' S r, (S5) where S is the area of S surface. We reserve the > term to take high orer eigenmoes into account, an by assuming k = k, Eq. S4 leas to ρ c ψ (, r ω ) p r i S v i c k v r ' S'. (S6) (, ω) n + ρ n ψ (, ) kv S ω = k The acoustic pressure at ( xyl,, ) is ρ c ψ (, x y,, l ω ) pxyl i Sv + i cksv, (,,, ω) n ρ n ψ ( ) ω kv = k (S7) where ψ is the average eigenmoes at z = l ψ = ( x', y', l, ) x'y' S ψ S ω. (S8) We obtain the average acoustic pressure at z (, ) = (,,, ) n n S + S kv = l ρ c plω pxylω S'=-i S v iρ cks vε, (S9) where ε = S ψ ( ω ) k (S) which contains high orer moes ue to the reason that the cavity is not eep-subwavelength. The acoustic impeance at z = l is Z l pl (, ω) ρc ρck = = i ε. (S) Sv n kv S By using the acoustic impeance transfer formula

4 Z ρ c S / ρ c iz tan( k l), (S) S Z i( S / c ) tan( k l) l = l ρ an consiering the raiating impeance at the orifice, we obtain the acoustic impeance at z = (surface of the meta-structure) as ( l+ εω ) V / Sc Z iρc + Z ω V + ( + Vω ε / Sc ) ls p, (S3) where Z is the acoustic impeance of the piston with an area of p S, which is expresse as Z p ρc J( kw) is( kw) S kw kw, (S4) where J an S are st orer Bessel function an th orer Struve function, respectively. We then calculate the value of ε. As shown in Fig. S(b), the cavity size is L L L. The opening is a square with the area of S = w w. The eigenmoes can be expresse as ψ pqr εεε p q r pπx qπy rπz (, xyz,, ω) = cos cos cos, (S5) V l x l y lz where the coefficient is ε p, q, r, ( p, q, r = ) =. (S6), ( p, q, r > ) From Eq. S8, we obtain ψ pqr = ψ (,,, ) w w pqr xy ωpqr xy w εεε p q r ( L+ w)/ pπ ( L+ w)/ pπ cos xx cos ( L w)/ ( L w)/ = yy w V L L (S7) εεε p q r pπ qπ w w = cos cos sin pπ sin qπ V pπw/ L qπw/l L L Since in our ultra-thin structure, the z-irection size L is much smaller than L, we only

5 consier the r = terms in our esigne frequency range, which yiels ε = S ( ψ ) + ( ψ ) q S pq q= kq p, q= kpq. (S8) Notice that k pqr pπ qπ rπ = + +. (S9) l l l x y y Combing Eqs. S7 an S8 with Eq. S9, we have Sw sin x Sw sin x sin y ε = + V x x V x y x y m= m, n= +, (S) where x= pπ w/l an y = qπ w/l. By substituting Eq. S into Eq. S3, we can estimate Z at z =. Finally, the phase for the reflecte wave is calculate by [ Z c S Z c S ] φ = arg ( ρ / ) / ( + ρ / ). (S) The comparison of analytical an simulate results for five frequencies use in MSD an BMSDs are shown in Fig. S(c), which shows a goo agreement.

6 FIG. S. Analytical moel. (a) A -D schematic iagram of the meta-structure unit-cell. (b) A 3-D schematic iagram of the meta-structure unit-cell. (c) The comparison between simulate an analytical phase responses of a.5cm thick unit cell for 577Hz, 646Hz, 853Hz, 969Hz, an 57Hz, respectively. Supplementary Note. Phase responses of the meta-structure unit-cell for ifferent incient angles. At a eep-subwavelength scale, the phase responses of the unit cell shoul be inepenent of the incient angle. Figure S shows the simulate phase responses of the unit

7 cells at ifferent angles of incience. The phase ifferences are small for small an large w an are relatively large for intermeiate w. The phase ifferences between ifferent angles of incience are resulte from the fact that the unit cells are not in eep-subwavelength size. These phase ifferences, however, oes not significantly reuce the iffuse reflection for obliquely incient waves, as can be seen in Fig. 4 of the manuscript. Finally, it is note that the unit cells of conventional SDs also have phase responses that epen on the angle of incience []. FIG. S. The simulate phase responses of unit cells at ifferent angles of incience. Supplementary Note 3. Scalability of the MSD The simulate phase responses of unit cells (5cm an.5cm thick) at 343Hz an 686Hz are shown in Fig. S3(a). The corresponing simulate 3-D far-fiel scattering patterns are shown in Fig. S3(b). The results suggest that the MSD can be scale up/own to an arbitrary wavelength/frequency with thickness of /.

8 FIG. S3. (a) The simulate phase responses of unit cells at 343Hz an 686Hz (5cm an.5cm thick) for 343Hz an 686Hz, respectively. (b) 3-D far-fiel scattering patterns of two MSDs with center frequencies of 343Hz an 686Hz, respectively. Supplementary Note 4. The influence of perio number of QRS. Figure S4 shows the simulate normalize iffusion coefficients of MSD with ifferent number of perios:,, 4 4, an 6 6 with normal incience. The results show that the iffusion coefficient slightly eteriorates as the perio number increases for both SD an MSD, which agrees with the conclusion in Refs. an 3. In aition, once the perio number excees, the tren of the iffusion coefficient curve becomes stable, although with ifferent amplitues at ifferent unit-cell numbers. This result, therefore, valiates the comparability between SD an MSD at a perio number of.

9 FIG. S4. The influence of perio number. The normalize iffusion coefficients with ifferent perio numbers of,, 4 4, an 6 6 for MSD an SD with normal incience. Supplementary Note 5. The influence of thermal-viscous effect. visc The bounary-layer thickness can be approximately calculate by =.mm. (S) f At f = 686Hz, we have visc =.66mm, which is / 57 of the inner thickness of the cavity an is small enough so that the viscous effect shoul be also small. As shown in Fig. S5, we have compare the lossy an lossless moels by numerically simulating the phase response, reflection amplitue an the acoustic pressure fiel pattern in Acoustic-Thermoacoustic Interaction, Frequency Domain moule an Pressure Acoustics, Frequency Domain moule, respectively, in COMSOL Multiphysics. For the lossy moel, both thermal an viscous losses are consiere. Figures S5(a) an S5(b) show the phase an amplitue responses for loss an lossless moels at 686Hz an 343Hz, respectively. The largest phase an amplitue shift occurs at about w =., at which the meta-structure

10 resonates. The maximum energy absorption coefficient is.6 for 686Hz an.4 for 343Hz, respectively. Thermal-viscous effect is frequency-epenent [] an loss is more negligible at lower frequencies (e.g., -5Hz). Figures S5(c) an S5() shows the simulate scattere acoustic pressure fiels for non-viscous an viscous moel. The comparison of fiel patterns suggest that the influence of viscosity is not significant in this case. Finally, we briefly compare the Helmholtz-like resonator with the wiely use space-coiling metasurface unit cell. The space-coiling unit cell consists of a perforate panel an coile-up cavity (Fig. S6). Our simulation inicates that, the space-coiling metasurface unit cells seem to be more susceptible to the thermal an viscous losses. The space-coiling unit cell has a maximum absorption coefficient at.3 (corresponing to the lowest pressure reflection coefficient at.8), which is about twice as high as the Helmholtz-like resonator. There are primarily two reasons: ) the space-coiling units also nee to operate aroun its resonance frequency (possibly Fabry Pérot resonance) in orer to achieve the esire phases at a eep subwavelength thickness, which maximizes the loss; ) the very long length of the coile-up path naturally enhances the viscous an thermal losses. The space-coiling units also seem to be more complicate to manufacture than the Helmholtz-like resonators.

11 FIG. S5. The influence of thermal-viscous effect. (a-b) The phase an amplitue responses of unit cell for lossless moel an lossy moels at 686Hz an 343Hz. The simulate scattere acoustic pressure fiels of MSD at 686Hz with normal incience (Left) an 45o -incience angles (Right), respectively, for (c) lossless moel an () lossy moel.

12 FIG. S6. The comparison between a Helmholtz-base resonator an a space-coiling metasurface unit cell. The space-coiling structure is similar to that in [4]. References. Kinsler L, Funamentals of Acoustic (Wiley, New York, 98).. T. Cox, P. D Antonio, Acoustic Absorbers an Diffusers: Theory, Design an Application, 3r eition (CRC Press) (6). 3. T. Cox, B. Dalenback, P. D Antonio, J. Embrechts, J. Jeon, E. Mommertz, an M. Vorläner, A tutorial on scattering an iffusion coefficients for room acoustic surfaces. Acta Acustica unite with Acustic. 9, -5 (6). 4. Y. Li, an B. M. Assouar, Acoustic metasurface-base perfect absorber with eep subwavelength thickness. Appl. Phys. Lett. 8, 635 (6).