Negative Birefraction of Acoustic Waves in a Sonic Crystal

Transcription

1 Negatve Brefracton of Acoustc Waves n a Sonc Crysta Mng-Hu Lu 1, Chao Zhang 1, Lang Feng 1, * Jun Zhao 1, Yan-Feng Chen 1, Y-We Mao 2, Jan Z 3, Yong-Yuan Zhu 1, Sh-Nng Zhu 1 and Na-Ben Mng 1 1 Natona Laboratory of Sod State Mcrostructures and Departent of Materas Scence and Engneerng, Nanjng Unversty, Nanjng , Peope s Repubc of Chna 2 Insttute of Acoustcs, Nanjng Unversty, Nanjng , Peope s Repubc of Chna 3 Natona Laboratory of Surface Physcs, Fudan Unversty, Shangha , Peope s Repubc of Chna * Present address: Departent of Eectrca and Coputer Engneerng, Unversty of Caforna, San Dego, CA Correspondence shoud be addressed. E-a: yfchen@nju.edu.cn

2 Suppeentary Inforaton Suppeentary Texts We anayze the acoustc waves propagatng n the SC n the second band and thrd band by the cacuaton of the Pane Wave Expanson ethod (PWE) wth 901 pane waves, and Mutpe Scatterng Theory (MST) n whch the anguar oentu odes of Besse and Hanke functons are 15. A the suatons are cacuated usng the Fnte-Dfference Te-Doan ethod (FDTD). The FDTD ethod s a drect and effcent ethod for deang wth the wave transsson probes. The ethod s frst deveoped for sovng the eectroagnetc wave propagatng probes, and t aso can be odfed to sove other sar probes such as acoustc probes wthout dffcuty. The FDTD agorth perfored here s qute sar wth that wdey used n the cacuatons of photonc crystas, except that the acoustc wave equatons are used n the cacuaton nstead of the Maxwe equatons. In our cacuaton, the FDTD grds are unfor and orthogona. The spata grd sze s set to be Δ x=δ y = D 36, where D s the perod of the sonc crysta. Ths grd sze settng s far saer than 1/10 of the ncdent waveength thus the nuerca dsperson effect can be negected. The te-step s set to be Δx Δ t =, where c s the acoustc veocty n the stee. The second-order 2 2c Mur absorbng boundary condtons are epoyed to enate the non-physca refecton at the boundares of cacuaton, and 240,000 te steps are executed n order to obtan a convergent resut.

3 The PWE and MST are the cassca ethods to sove the egen-vaue probe of waves propagatng n the perodc structures. Here, we w show the detas of the PWE and MST ethod for sonc crystas (SCs). In SCs, Laé coeffcents λ ( r), ( r) μ and the (ass) densty ρ () r are oduated perodcay and acoustc band structures can be estabshed. The wave equaton can be wrtten as: 2 u 2 t 1 = ρ x u λ x + x u μ x u + x, = 1,2,3, (1) where u ( = 1,2, 3 ) are the Cartesan coponents of the dspaceent vector u () r and x ( = 1,2, 3 ) are the Cartesan coponents of the poston vector. For the SC consstng of stee and ar, ony the ongtudna waves are aowed. Then Eq. (1) coud be spfed as foows: 2 1 φ φ = λ 2, (2) t ρ where ρ u = φ. By usng the pane wave expanson (PWE) ethod and appyng the Boch theore, Eq. (2) coud yed the egen-vaue equaton as: r r r r [ λ r v ρ r v ( K + G ) ( K + G )] u r = 0 G G G G G G ω, (3) where K v s effectve Boch wave vector, whch s restrcted wthn the frst Broun zone and G r s the recproca vector. By usng 901 pane waves, the band-structure s cacuated as shown n Fg. 1a (eft) (Red Hoow crque). Based on ths band structure, the acoustc equ-frequency surface (EFS) (Fg. 1a rght) s constructed for the acoustc waves propagatng fro ar to the SC wth the nterface nora aong Γ-M drecton n k space.

4 Beow we gve a bref dervaton of the MST n a for that appes to acoustc waves. By substtute t e ω φ = φ and 2 2 ( + ) φ = 0 2 ρ 2 k = ω nto Eq. (2), a Hehotz equaton coud be obtaned as: λ k. (4) Consder a syste of N dentca cynders of radus R and eastc paraeters λ and ass densty ρ. Here cyndrca coordnate s adopted. Let us frst consder the scatterng of a v partcuar cynder j ocated at r = (, θ ) j r j j. The ncdent waves coe fro both the externa source and the radaton of other cynders. The tota fed outsde and nsde the cynder j can be wrtten as the su of the ncdent and scattered feds,.e. φ r r r 0 r r ( ) = [ B H ( k j ) + A j, J ( k j )] ( θ θ j ), e. (5) where J and H are, respectvey, the Besse functon and the Hanke functon of the frst knd. Wth the use of the Graff addton theore, a set of sef-consstent equatons for B, can j be obtaned: B D = q B, q H q r r ( ) ( θ j ) k j e θ + A 0, j, (6) where the rato D = B D = has the for: A ( k / ρ ) J ( k0 R) J ( kr) ( k0 / ρ 0 ) J ( k0 R) J ( kr) ( k / ρ ) H ( k R) J ( kr) ( k / ρ ) H ( k R) J ( kr) , (7) To sove the egen vaue of the perodc structure, A 0, and appyng the Boch theore, Eq.(6) = can yed the egen-vaue equaton as: M = M r Qn, ( k, K) B = 0, (8) r 1 r r where Q ( k K ) = δ + S ( k, K ). ( k, K ) A n,, n, n Dn S n A, s the attce su of Hanke functon of the frst knd, whch can be converted to ore rapdy convergng sus wth Ewad s ethod. To

5 sove the Eq. (8), we can get the acoustc band structure as shown n Fg.1 (a) (Back sod crque), whch s we consstent wth the resuts cacuated by PWE.

6 Suppeentary Fgures and Legends Suppeentary Fgure S1 Absorber Γ Μ drecton X Y Ettng Transducer Recevng Transducer Or fter fat X%0.6 x-y Y%0.8 nora anyss hard phase Functon Generator Oscoscope Lock n apfer Suppeentary Fgure S1. A Scheatc of the experenta setup. The experenta setup s used to easure the transsson of utrasonc waves through a SC, consstng of two transducers, a fat rectanguar sab of stee cynders, a functon generator, and an oscoscope for puse easureent or a ock n apfer for contnuous wave easureent. The SC, the rectanguar sab of stee cynders shown as the dde-botto nset, s paced between two transducers.

7 a b Suppeentary Fgure S2. The aptude and phase dstrbutons of the pressure fed of an acoustc pont source. To get a pont source, ost of the area of the ettng transducer was bocked and ony a pn hoe at the center was eft wth a daeter of 6, on the order of the waveength of the acoustc waves. We easured the fed dstrbuton of the pont source usng the experent setup (Fg.S1) wthout the SC. Fg. a and Fg. b are the dstrbutons of the aptude and the phase, respectvey. It s convnced that a good pont source s achevabe usng ths technque based on Fg. S2 b, n whch the equ-phase surface s neary secrcuar. Coors changng fro brght to dark ndcate the vaue of the ntensty or phase, changng fro the hghest to the owest.

8 a b Suppeentary Fgure S3. The pressure fed of Boch egen states of the 2 nd and 3 rd bands at 73 khz of a 7 o ncdent bea. Two Boch egen states of the 2 nd and 3 rd bands at 73 khz are cacuated by PWE ethod. Fg. a and b show the aptude of the pressure fed of the 2 nd band Boch egen state and the 3 rd band Boch egen state n the SC at 73 khz when a 7 o ncdent bea pnges the SC, correspondng to dfferent Boch wave vectors respectvey. Coors changng fro red to bue ndcate the vaue of the pressure fed aptude, changng fro the hghest to the owest Whte crques denote the stee cynders. Copare the fgure wth the Fg 1.b n the paper, we can obvousy fnd the Boch wave propagaton n the SC, and can estate ore ceary whch band s the refractve bea beong to.

9 Suppeentary Fgure S4. The suaton of b-refracton of acoustc waves n a 2D square-attce sonc crysta. In a square attce sonc crysta wth stee cynders ebedded n ar, where the attce constant s 2.5, we suate the propagaton for the acoustc wave wth ncdent ange of 15 o at 95 khz by FDTD ethod. Two refractve beas are ceary shown n the fgure: one s postve refracton wth backward wave effect (arked by yeow arrow); and the other s negatve refracton wth backward wave effect as we (arked by bue arrow). Heren, the back arrow denotes the ncdent bea, and the red arrow denotes the refectve bea. Coors changng fro brght to dark ndcate the vaue of the ntensty, changng fro the hghest to the owest. In addton, two refractve beas can be both postve as we.