Characterization of anisotropic acoustic metamaterials

Transcription

1 INTER-NOISE 06 Characteriatio of aisotropic acoustic metamaterials Ju Hyeog PARK ; Hyug Ji LEE ; Yoo Youg KIM 3 Departmet of Mechaical ad Aerospace Egieerig, Seoul Natioal Uiversity, Korea Istitute of Advaced Machies ad Desig, Seoul Natioal Uiversity, Korea 3 Departmet of Mechaical ad Aerospace Egieerig ad Istitute of Advaced Machies ad Desig, Seoul Natioal Uiversity, Korea ABSTRACT A acoustic metamaterial is a artificially desiged structure which ca maipulate the propagatio of acoustic waves. It ca be treated as a ective medium whe the homogeeity coditio is satisfied. Cosiderig the acoustic metamaterial as a ective medium, its ective material properties ca be defied as the costitutive properties for materials i ature. I a aisotropic acoustic metamaterial havig arbitrarily shaped iclusios, the ective mass desity ca be defied as a tesor quatity ad its off-diagoal compoets should be cosidered to describe the aisotropic acoustic behavior. However, there have bee few studies related to the characteriatio of aisotropic acoustic metamaterials. I this wor, we propose a method to characterie aisotropic metamaterials. I the method, the ective properties of aisotropic acoustic metamaterials are determied by usig the reflectio ad trasmissio coiciets obtaied from a limited umber of ormal ad oblique plae wave icideces ad the spatial dispersio relatio of aisotropic acoustic metamaterials. Also, the proposed method hadles the aisotropic acoustic metamaterial slabs with both sigle ad multiple uit cells alog the thicess directio to evaluate the ective material properties more accurately. The validity of the proposed method is verified by performig full wave simulatios for the aisotropic acoustic metamaterials. Keywords: Acoustic Metamaterials, Aisotropy, Effective Properties Number(s): 3.6 I-INCE Classificatio of Subjects. INTRODUCTION A acoustic metamaterial is a periodically arraged subwavelegth structure that ca maipulate the soud propagatio. Because its uit cell structure is much smaller tha the wavelegth of iterest, a acoustic metamaterial ca be regarded as a ective homogeeous medium. The ective medium ca be characteried i terms of ective mass desity ad bul modulus. The ective material properties of a acoustic metamaterial gover its macroscopic wave behavior ad thus a acoustic system utiliig the acoustic metamaterial. There have bee various methods to characterie acoustic metamaterials. Amog others, the S-parameter retrieval method has bee vastly utilied to characterie various ids of metamaterials because it eables the characteriatio of metamaterials with complex geometry oly with the scatterig parameters such as reflectio ad trasmissio coiciets. I this wor, we exploit this method ad preset a ew method () that is suitable for geeral aisotropic acoustic metamaterials. I previous studies, a method was proposed to characterie isotropic acoustic metamaterials () i which both the ective mass desity ad bul modulus are defied as scalar quatities. Also, the method for the aisotropic acoustic metamaterials with tesorial ective mass desity ad scalar ective bul modulus was proposed (3) by simply extedig the method for the isotropic case ad verified experimetally (4). However, the method ca be applied oly for a aisotropic acoustic metamaterial if its ective mass desity tesor is diagoal. This limitatio should be overcome to aalye geeral aisotropic acoustic metamaterials with o-diagoal mass desity tesors, for juhyeogpar@su.ac.r hyugjilee@su.ac.r 3 yyim@su.ac.r 760

2 INTER-NOISE 06 which the pricipal axes are ot parallel to the slab iterfaces. I this wor, we preset a method to characterie aisotropic acoustic metamaterials that h ave a o-diagoal mass desity tesor which ca be determied by the S-parameter retrieval method. Previously proposed methods to characterie aisotropic metamaterials havig o-diagoal ective properties require may wave icideces (5). However, we preset a iciet characteriatio procedure requirig at most 8 plae wave icideces sice the spatial dispersio relatio of the metamaterials is explicitly utilied. Also, the proposed method treats the acoustic metamaterial slabs with both sigle ad multiple uit cells for more accurate retrieval of the ective properties. Whe characteriig the metamaterial slab with multiple uit cells, the couplig ects betwee adjacet cells are cosidered. Thereby a accurate evaluatio of ective properties of aisotropic acoustic metamaterial slabs with both sigle ad multiple uit cells is achieved. Fially, the validity of proposed method is verified through full wave simulatios for the aisotropic acoustic metamaterial slabs with Z-shaped iclusios are carried out. The ective properties of slabs are calculated i advace, ad the scatterig parameters for both origial metamaterial slabs ad equivalet homogeeous slabs are the compared.. CHARACTERIZATION METHOD. Theoretical Bacgroud Cosider a geeral aisotropic acoustic metamaterials of which ective mass desity ad bul modulus are defied as o-diagoal tesor quatity ad scalar quatity, respectively. I the D cofiguratio as show i Figure, a aisotropic acoustic metamaterial slab with thicess d, ective mass desity ρ, ective bul modulus B is surrouded by a homogeeous isotropic media with 0 ad B 0. For the field variables, pressure p ad the particle velocity v, the correspodig goverig equatios are expressed as p ρ p 0, (a) B t where x x. x xx x xx ρ (b) As described earlier, the spatial dispersio relatio of aisotropic acoustic metamaterials is utilied for retrieval of ective properties i our method. By substitutig a plae wave solutio ito Eq. (a), the spatial dispersio relatio of a two-dimesioal aisotropic acoustic metamaterial ca be obtaied as follows, x x xx x, () B which meas there ca exist values of two differet magitudes deoted by f ad b for a give wave icidece characteried by x. Eq. () will be exploited to evaluate ective properties of aisotropic acoustic metamaterial slabs i the followig subsectios. Figure A aisotropic acoustic metamaterial slab surrouded by a homogeeous medium. 76

3 INTER-NOISE 06. Characteriatio of metamaterial slab with a sigle uit cell To characterie a aisotropic acoustic metamaterial slab with a sigle uit cell, we eed three differet plae wave icideces, a ormal icidece alog the forward directio ad oblique icideces alog the forward ad bacward directios as show i Figure (see icidece A, B ad D i Figure ). Whe a plae wave icidece A illumiates a aisotropic metamaterial slab, the tagetial compoet of the wave vector iside surroudig medium ad the slab become 0, i.e. x 0x 0. Thus, the spatial dispersio relatio, Eq. (), is reduced to B, (3) ad this meas that the ormal compoets of the wave vectors iside the slab become f b. For a ormally icidet plae wave, the iversio relatio betwee the reflectio, trasmissio coiciets at ormal icidece ( R, T ) ad the wave parameters such as the ormal compoet of wave vector ad relative impedaces (, ) are derived as [see details i (,, 6)] T R T T R T R cos d,. (4a) I Eqs. (4a) ad (4b), the real part of with (4b) 0 0. should be positive ad the imagiary part of e should be j t egative for the harmoic time depedece because of the passivity costrait. I additio to the sig covetio of the wave parameters, oe also has to cosider the brach ambiguity (7) for the correct evaluatio of the ormal compoet of wave vector. After the evaluatio of wave parameters i Eq. (4a), oe compoet of the ective reciprocal mass desity tesor ad the ective bul modulus B ca be calculated by usig Eqs. (3) ad (4b). Figure The required plae wave icideces to characterie aisotropic acoustic metamaterial slabs with a sigle uit cell. From the scatterig coiciets obtaied from plae wave icideces B ad D, the remaiig compoets of the reciprocal mass desity tesor, are determied. To evaluate them, the magitude of the ormal compoets of the wave vectors f ad b for the specific oblique plae wave icideces eed to be evaluated at first. Ad, these ca be obtaied by usig the trasfer matrix for the plae wave icideces. The trasfer matrix for the specific plae wave icideces ca be costituted (8) with the reflectio ad trasmissio coiciets for two oblique plae wave icideces * havig the same icidet agle deoted by icidece B ad D i Figure [see the ic,f ic,b * * detailed procedure i ()]. Whe choosig a icidet agle, oe should select to be smaller tha the critical agle cr to avoid the total reflectio at the iterfaces. After the trasfer matrix for the specific wave icidece is costructed, the magitude of the ormal compoets of the wave vectors f ad x b ca be retrieved easily sice the eigevalues of the trasfer matrix are xx j fd e ad 76

4 INTER-NOISE 06 jbd e. Based o the retrieved values, of ective properties x, xx, are obtaied as B, f, ad x( 0x), the remaiig compoets b x f b, xx fb x x B (5) with the fact that the retrieved values give x. f ad b satisfy the spatial dispersio relatio () for.3 Characteriatio of metamaterial slab with multiple uit cells For accurate characteriatio of metamaterial slabs with multiple uit cells, a ovel approach is required because the ective properties retrieved from the slab with a sigle uit cell caot always describe multilayered metamaterial slab. Thus, we propose to icorporate the method for the slab with a sigle uit cell i a previously preseted ihomogeeous model approach (9). I detail, we regard the metamaterial slab with multiple uit cells as a ihomogeeous material slab with two differet homogeeous uit cells, edge cell ad ier cell, to tae ito accout the couplig ects betwee the adjacet cells i the slab as show i Figure 3. Also, 8 times of plae wave icideces are required for the characteriatio of the slab with multiple uit cell, icludig ormal icideces alog forward ad bacward directios ad oblique icideces alog forward ad bacward directios to slabs made of both double ad triple uit cells, respectively as show i Figure 3. Figure 3 The required plae wave icideces to characterie aisotropic acoustic metamaterial slabs with multiple uit cells. The metamaterial slab with double uit cells ca be cosidered as a equivalet homogeeous material slab with two edge cells, ad the metamaterials slab with triple uit cells ca be cosidered as a equivalet ihomogeeous slab with two edge cells ad a ier cell. As depicted i [step ] i Figure 3, the characteriatio will begi with the double uit cell case. The reflectio ad trasmissio coiciets at icideces A, B, C ad D should be calculated at first. Whe the coiciets are obtaied, the ective properties of a edge cells are determied usig the method for the slab with a sigle uit cell. Ad, the trasfer matrices of a edge cell for the ormal icidece plae waves T e, ad for the oblique icidece plae waves T eo ca be obtaied from the scatterig coiciets at icidece A, B ad C, D, respectively [see details i ()]. For the triple uit cell case, we also costruct the trasfer matrices T ad T o of the metamaterial slab made of triple uit cells by usig the coiciets at icidece A, B, C ad D i [step ] of Figure 3. Because the trasfer matrices T ad T o ca be expressed as T Te TiT e ad To Teo TioT eo where T i ad T io are trasfer matrices of a ier cell for ormal ad oblique icideces, respectively, oe ca simply obtai T T T T, T T T T. (6) i e e io eo o eo Usig the covetioal relatio betwee the reciprocal desity ad the trasfer matrix (, 8), 763

5 INTER-NOISE 06 for a ier cell ca be calculated by usig Eq. (7) below,, (7) i, j T where T i, is the -compoet of T i. Ad, the ective bul modulus from Eq. (3). Because we ca retrieve the ormal compoets of wave vectors B ca be determied f ad b from T io, the remaiig ective properties x, xx ca be also determied by usig Eq. (5). The ective properties determied by the proposed procedure ca be applied to geeral aisotropic acoustic metamaterial slabs regardless of the umber of uit cells sice the multiple uit cells ca be divided ito two edge cells ad the remaiig ier cells. 3. VERIFICATION 3. Numerical Simulatio Full umerical simulatios are performed for two-dimesioal aisotropic acoustic metamaterial slabs by the fiite elemet method with COMSOL Multiphysics. For the simulatios, the uit cell with Z-shaped iclusio show i Figure 4 is chose because ay of the pricipal axes of the metamaterial are ot aliged with the slab iterfaces. Thus, the acoustic metamaterial with Z-shaped iclusios shows aisotropic behavior which caot be described without o-diagoal mass desity tesor. The 3 Z-shaped iclusio of alumium ( 700 g m, cal 500 m s, BAl 70.3 GPa ) is embedded Al 3 i air ( 0.5 g m, c0 343 m s, B Pa ), ad the geometric specificatios are selected as d 7 mm, L 3 mm, L mm, L3. mm, L4 0.9 mm, L5 5.5 mm ad rot 5 [deg]. Based o the geometric dimesios, the simulatios are performed from 500 to 3000 H i which the homogeeity coditio is satisfied. The scatterig coiciets are obtaied from the pressure fields calculated by the simulatios, ad the ective properties are determied by usig the methods proposed i the previous sectio. The, the coiciets for the origial metamaterial slabs ad the equivalet homogeied slabs made of the determied ective properties are compared at vari ous plae wave icideces. Figure 4 Uit cell with a Z-shaped iclusio. 3. Acoustic Metamaterial Slab with a Sigle Uit Cell We coduct full umerical simulatios for a metamaterial slab with a sigle uit cell havig Z-shaped iclusio i the frequecy rage betwee 500 ad 3000 H. The ective properties are calculated from the reflectio ad trasmissio coiciets for 3 plae waves, a ormal icidece wave ( ic,f 0 [deg] ) ad two oblique icidece waves ( ic,f ic,b 0 [deg] ) accordig to the proposed method i the previous sectio. Figure 5 shows the retrieved ective properties of the metamaterial slab with a sigle uit cell. 764

6 INTER-NOISE 06 Figure 5 Retrieved ective properties for the acoustic metamaterial slab with a sigle uit cell havig Z-shaped iclusio: (a) compoets of the ective reciprocal mass desity tesor, (b) the ective bul modulus. We compared the coiciets of the origial metamaterial slab ad those of the equivalet homogeied slab of the same thicess for a wide rage of icidet agles to show the validity of the proposed method. As show i Figure 6, the calculated coiciets of the equivalet slab for a rage of icidet agles from 70 [deg] to 70 [deg] at 000, 000 ad 3000 H trace those of the origial metamaterial slab accurately. Therefore, oe ca ote that the proposed method for a aisotropic metamaterial slab made of a sigle uit cell is valid for the aisotropic acoustic metamaterial slab made of sigle uit cell. Figure 6 Compariso of the reflectio ( R ) ad trasmissio (T ) coiciets of the origial metamaterial slab with a sigle uit cell (deoted by the subscript ms ) ad its equivalet homogeied slab of the same thicess (deoted by the subscript es ). Both real ad imagiary parts of the coiciets are compared. 3.3 Acoustic Metamaterial Slab with Multiple Uit Cells I multiple uit cells case, we also performed the umerical simulatios for the slab with the same uit cell i the frequecy rage betwee 500 ad 3000 H. To evaluate their ective properties metamaterial slabs with double uit cells ad triple uit cells are illumiated by ormal icidece waves ( ic,f ic,b 0 [deg] ) ad oblique icidece waves ( ic,f ic,b 0 [deg] ) as described i the previous sectio. The retrieved ective properties are show i Figure 7. Because a slab with more tha 3 uit cells ca be cosidered as beig composed of differet homogeeous uit cells, the ective properties of the edge cell ad ier cell metamaterials are estimated separately. 765

7 INTER-NOISE 06 Figure 7 Effective properties retrieved for the metamaterial slab made of multiple uit cells havig Z-shaped iclusios: (a) compoets of the ective reciprocal mass desity tesor, (b) the ective bul modulus. Blac solid lies ad blue dashed lies represet the ective properties of ier cell ad edge cell, respectively. To see if the retrieved ective parameters ca be valid to model a metamaterial slab havig more tha 3 uit cells, we perform the umerical simulatios for the slab with 0 uit cells havig Z-shaped iclusio. To model the equivalet slab, the origial metamaterial slab with 0 idetical uit cell is cosidered to be composed of edge cells ad 8 ier cells. As i the sigle uit cell case, the coiciets are compared for a rage of icidet agles from 70 [deg] to 70 [deg] at 000, 000 ad 3000 H as show i Figure 8. The good agreemet betwee the reflectio ad trasmissio coiciets of the origial metamaterial slab ad those of its equivalet slab is verified. Figure 8 Compariso of the reflectio ( R ) ad trasmissio (T ) coiciets of the origial metamaterial slab with 0 uit cells (deoted by the subscript ms ) ad its equivalet homogeied slab of the same thicess (deoted by the subscript es ). As i the sigle uit cell case, oe ca see that the proposed method eables to characterie the aisotropic acoustic metamaterial slab made of multiple uit cells accurately based o the results. 4. CONCLUSIONS This wor preseted a characteriatio method for aisotropic acoustic metamaterial slabs havig a o-diagoal mass desity tesor. Because the proposed method eabled the characteriatio of aisotropic acoustic metamaterial slabs by usig 8 plae wave icideces at most, the retrieval procedure ca be adopted for experimets. It ca be also used for desig optimiatio. Full umerical simulatios supported that the proposed method is valid regardless of the umber of uit cells i a slab. 766

8 INTER-NOISE 06 ACKNOWLEDGEMENTS This wor was supported by the Samsug Research Fudig Ceter of Samsug Electroics (No: SRFC-MA40-06). REFERENCES. Par JH, Lee HJ, Kim YY. Characteriatio of aisotropic acoustic metamaterial slabs. Joural of Applied Physics. 06;9(3): Foi V, Ambati M, Su C, Zhag X. Method for retrievig ective properties of locally resoat acoustic metamaterials. Phys Rev B. 007;76(4): Popa B-I, Cummer SA. Desig ad characteriatio of broadbad acoustic composite metamaterials. Phys Rev B. 009;80(7): Zigoeau L, Popa B-I, Starr AF, Cummer SA. Desig ad measuremets of a broadbad two-dimesioal acoustic metamaterial with aisotropic ective mass desity. J Appl Phys. 0;09(5): Castaié A, Mercier J-F, Félix S, Maurel A. Geeralied method for retrievig ective parameters of aisotropic metamaterials. Opt Express. 04;(4): Smith D, Schult S, Maroš P, Sououlis C. Determiatio of ective permittivity ad permeability of metamaterials from reflectio ad trasmissio coiciets. Physical Review B. 00;65(9): Arslaagic S, Hase TV, Mortese NA, Gregerse AH, Sigmud O, Ziolowsi RW, et al. A review of the scatterig-parameter extractio method with clarificatio of ambiguity issues i relatio to metamaterial homogeiatio. IEEE Ateas Propag Mag. 03;55(): Sog BH, Bolto JS. A trasfer-matrix approach for estimatig the characteristic impedace ad wave umbers of limp ad rigid porous materials. J Acoust Soc Am. 000;07(3): Hou LL, Chi JY, Yag XM, Li XQ, Liu R, Xu FY, et al. Advaced parameter retrievals for metamaterial slabs usig a ihomogeeous model. J Appl Phys. 008;03(6):