Propagating and evanescent properties of duble-point defect in sonic crystals

P opaga ing and e anescen p ope ies o double-poin de ec s in sonic c ys als This a icle has been downloaded om IOPscience. Please sc oll down o see he ull ex a icle. 2010 New J. Phys. 12 083024 (h p://iopscience.iop.o g/1367-2630/12/8/083024) Download de ails: IP Add ess: 85.55.36.148 The a icle was downloaded on 26/08/2010 a 17:08 Please no e ha e ms and condi ions apply. View he able o con en s o his issue, o go o he jou nal homepage o mo e Home Sea ch Collec ions Jou nals Abou Con ac us My IOPscience The open–access jou nal o physics New Jou nal o Physics P opaga ing and e anescen p ope ies o double-poin de ec s in sonic c ys als V Rome o-Ga cía1,2,4, J V Sánchez-Pé ez1and L M Ga cia-Ra i3 1Cen o de ecnologías ísicas: Acús ica, Ma e iales y As o ísica, Uni e sidad Poli écnica de Valencia, Camino de Ve a s/n, 46022 Valencia, Spain 2Ins i u o de Ciencia de Ma e iales, Consejo Supe io de In es igaciones Cien í icas,So Juana lnés de la C uz, 3, Can oblanco, 28049, Mad id, Spain 3Ins i u o Uni e si a io de Ma emá ica Pu a y Aplicada, Uni e sidad Poli écnica de Valencia, Camino de Ve a s/n, 46022 Valencia, Spain E-mail: [email p o ec ed].es New Jou nal o Physics 12 (2010) 083024 (14pp) Recei ed 22 Ap il 2010 Published 10 Augus 2010 Online a h p://www.njp.o g/ doi:10.1088/1367-2630/12/8/083024 Abs ac . Complex band s uc u es and mul iple sca e ing heo y ha e been used in his pape o analyze he o e lapping o he e anescen wa es localized in poin de ec s in sonic c ys als (SCs). The ex ended plane wa e expansion (EPWE) wi h supe cell app oxima ion gi es he imagina y pa o he Bloch ec o s ha p oduces he decay o he localized modes inside he pe iodic sys em. Double ca i ies can p esen a coupling be ween he e anescen modes localized in he de ec , showing a symme ic o an isymme ic mode. When poin de ec s a e close, he complex band s uc u es e eal a spli ing o he equencies o he localized modes. Bo h he eal pa and he imagina y alues o ko he localized modes in he ca i ies p esen di e en alues o each localized mode, which gi es di e en p ope ies o each mode. The no el measu emen s, in e y good ag eemen wi h analy ical da a, show expe imen al e idence o he symme ic and an isymme ic localized modes o a double-poin de ec in SCs. The in es iga ion o he localiza ion phenomena and he coupling be ween de ec s in pe iodic sys ems has undamen al impo ance in bo h pu e and applied physics. 4Au ho o whom any co espondence should be add essed. New Jou nal o Physics 12 (2010) 083024 1367-2630/10/083024+14$30.00 © IOP Publishing L d and Deu sche Physikalische Gesellscha 2 Con en s 1. In oduc ion 2 2. Ex ended plane wa e expansion (EPWE) wi h supe cell app oxima ion 3 3. EPWE esul s: localized modes 6 3.1. Spli ing o localized modes ........................... 7 3.2. Symme ic and an isymme ic modes ....................... 9 3.3. Decay o he localized modes ........................... 11 4. Conclusions 13 Acknowledgmen s 13 Re e ences 13 1. In oduc ion Pe iodic dis ibu ions o elas ic sca e e s in an elas ic hos medium wi h di e en physical p ope ies a e known as phononic c ys als (PCs) [1,2], and hey a e he elas ic analogues o he well-known pho onic c ys als [3,4]. I one o he ma e ials in PCs is a luid, hen he sys em is called a sonic c ys al (SC) [5]. All o hese sys ems p esen in e es ing physical p ope ies and ecen ly hey ha e ecei ed inc easing a en ion, mainly due o he g ea numbe o applica ions in se e al b anches o physics and enginee ing [6]–[8]. One o he mos impo an p ope ies o hese inhomogeneous ma e ials is he so-called band gaps (BGs): equency anges whe e wa es do no p opaga e h ough he pe iodic sys em. The exis ence o hese BGs leads o se e al applica ions; o ins ance, in he case o SCs, as acous ic il e s [9,10], acous ic ba ie s [11] o wa eguides [12]. In pe iodic sys ems, Bloch’s heo em and Fou ie expansion o he pe iodic physical p ope ies ans o m he acous ic wa e equa ion in an eigen alue p oblem. The eigen equencies ω(k) o each Bloch’s ec o kinside he i educible pa o he i s B illouin zone cons i u e he band s uc u e. This me hodology is usually called plane wa e expansion (PWE) [13] and i can be used o ob ain he so-called band s uc u es, i.e. he p opaga ing modes h ough he pe iodic sys em. The band s uc u es e eal ha BGs a e anges o equencies whe e no eal kexis s. I has been shown ha he eigen alues o he p oblem ha e eal alue o he case o SCs [14]. One o he mos impo an p ope ies o he pe iodic s uc u es is he eme gence o localized modes wi hin he BG when a poin de ec is in oduced [9,15]. A widely used echnique in he li e a u e o ob ain he e ec o c ea ion o poin de ec s in c ys als is he supe cell app oxima ion in PWE [10,16,17]. This app oxima ion gi es in o ma ion only abou he p opaga ion na u e o he localized modes in he poin de ec s. In hese cases, when pe iodici y is b oken o when SCs ha e ini e size, e anescen modes inside he pe iodic sys em may appea . Localized modes o modes inside he BG a e cha ac e ized by e anescen beha io [7,18,19]. Then, a mo e accu a e analysis is needed o cha ac e ize all he p ope ies o he modes inside he pe iodic sys em. A wa e impinging on a comple e pe iodic sys em wi h a gi en equency ωinside he BG is cha ac e ized by complex alued wa e numbe s k(ω), which ep esen he mul i-exponen ial decay o he e anescen mode inside he pe iodic sys em [18]. Recen wo ks [20]–[22] show an ex ension o he PWE (ex ended plane wa e expansion (EPWE)) ob aining he complex pa o New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/) 3 he Bloch’s wa es, e ealing ha he decay o he modes inside he BG g ows as he equency eaches he cen e o he BG. In his sense, a localiza ion ac o has been de ined ecen ly o show his beha io [23]. The localiza ion ac o can also be ela ed o ecen esul s ha show ha , al hough he decay o hese localized modes is mul i-exponen ial, i can be app oxima ed by an exponen ial-like decay conside ing only he i s ha monic o he Bloch wa es in SCs made o igid cylinde s [19]. On he o he hand, Sainidou e al [24] ha e in oduced a no el ex ension o he mul iple sca e ing heo y (MST) [25] o analyzing slabs ha consis o slices o di e en ma e ials as long as he pe iodici y pa allel o he su ace o he slab is p ese ed. The me hod, called laye mul iple sca e ing (LMS), allows he s udy o he sca e ing p oblem o slabs ha a e ini e in he di ec ion pa allel o he su ace o he slab, bu in ini e in he no mal di ec ions o his su ace. Al e na i ely, one can use his me hod o calcula e he complex phononic band s uc u es o an in ini e c ys al, associa ed wi h a gi en c ys allog aphic plane. In his case, he me hod p o ides he p opaga ing and e anescen Bloch wa es o he elas ic ield in he gi en c ys al, co esponding o a gi en kand a gi en equency. LMS has been used o analyze he guidance and quasi-guidance o elas ic wa es in a glass pla e coa ed on one side wi h a pe iodic monolaye o polyme sphe es, imme sed in wa e , obse ing he dispe sion diag ams o he in e ac ing modes o he composi e slab [26]. The goal o he pape is o analyze he h ee main cha ac e is ics o de ec modes in SCs: spli ing, symme y ib a ional pa e ns and e anescen decay o he modes. In addi ion o PWE, o ca y ou his s udy we ha e used EPWE wi h supe cell app oxima ion, because i is undamen al o he comple e unde s anding o he localized modes. We p esen he explici ma ix o mula ion o he supe cell app oxima ion in EPWE o Nppoin de ec s. F om he complex and eal band s uc u es, we obse e he spli ing and he e anescen beha io o he localized modes inside he BG a ound he de ec . We analyze he localized modes inside mul i- poin de ec s, especially in he double-poin de ec case. MST in ini e SCs is used o analyze he ib a ional pa e ns o he localized modes in a double-poin de ec . In his case, when he dis ance be ween bo h de ec s is low enough, i appea s as symme ic and an isymme ic ib a ional modes simila o he case o a sys em o med by wo masses and h ee sp ings, o o he Zeeman e ec in he a omic spec a [15]. The no el expe imen al da a ha a e in good ag eemen wi h heo y show o he i s ime he symme y o he ib a ional pa e ns o localized modes in such a double-poin de ec . Mo eo e , we obse e he decay o he localized modes ou side he double-poin de ec , in good ag eemen wi h he esul s ob ained by EPWE wi h supe cell app oxima ion. The pape is o ganized as ollows. Fi s o all, we show he main ing edien s o he EPWE as well as he explici ma ix o mula ion o he p oblem and he ex ension o a supe cell wi h Nppoin de ec s. A e ha , he nume ical, analy ical and expe imen al esul s o a double-poin de ec a e shown, gi ing a comple e explana ion o he spli ing, he symme y o ib a ional pa e ns and he decay o localized modes. Finally, we gi e a summa y as well as he main conclusions o he wo k. 2. Ex ended plane wa e expansion (EPWE) wi h supe cell app oxima ion The analysis o p opaga ing modes can be done by he ω(E k) o mula ion, whe e he exis ence o BGs is indica ed by he absence o bands in de e mined anges o equencies. The mechanism o c ea ion o BGs in ini e c ys als could be unde s ood by he e anescen beha io o he New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/) 4 modes inside i . A a gi en equency ωinside he BGs, he e anescen wa e is cha ac e ized by a complex alued Bloch ec o E k(ω) ha cha ac e izes he decay o he mode inside he pe iodic s uc u e. Based on he wo k o Hsue e al [20], ecen wo k by Laude e al [21] shows he calcula ion o complex band s uc u e o PCs. Recen ly, his wo k has been ex ended o he case o SCs o calcula ions using he supe cell app oxima ion [22], which is especially indica ed o SCs wi h poin de ec s. In his sec ion, we p esen he explici ma ix o mula ion o he EPWE wi h supe cell app oxima ion o calcula e he p ope ies o SCs wi h Nppoin de ec s inside a supe cell. We mus ake in o accoun ha PWE needs low in e ac ion be ween supe cells. ω(k)me hods a e cha ac e ized by he nex eigen alue p oblem, X E G0 ((E k+E G)σk(E G−E G0)(E k+E G0)−ω2η( E G−E G0))pE k(E G0)=0,(1) whe e E Gis he wo-dimensional (2D) ecip ocal-la ice ec o , kis he Bloch ec o and pkis he p essu e. Equa ion (1) cons i u es a se o linea , homogeneous equa ions o he eigen ec o s pE k(E G)and he eigen equencies ω(E k). We ob ain he band s uc u es le ing E kscan he i educible pa o he i s B illouin zone. Equa ion (1) can be exp essed by he nex ma ix o mula ion [13], 3 X i=1 0i60iP=ω2P,(2) whe e i=1,2,3 and 6=   σ( E G1−E G1) . . . σ( E G1−E GN×N) . . ..... . . σ( E GN×N−E G1) . . . σ( E GN×N−E GN×N)   ,(3) =    η( E G1−E G1) . . . η( E G1−E GN×N) . . ..... . . η( E GN×N−E G1) . . . η( E GN×N−E GN×N)     ,(4) P=    P(E G1) . . . P(E GN×N)     ,(5) whe e E G=(G1,G2,G3)=(2πm/a1,2πn/a2,0) o he case o he 2D squa e a ays. I we chose m=n=(−M,...,M), he size o he p e ious ma ices is N×N=(2M+1)×(2M+1). F om equa ion (2), we de ine he nex ec o , 8i=60iP.(6) Wi h his de ini ion we can e o mula e he eigen alue p oblem (2) as he equa ions sys em, 8i=60iP, ω2P= 3 X i=1 0i8i.(7) New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/) 5 In o de o ob ain an eigen alue p oblem o E k(ω), we w i e E k=kEα, whe e Eαis a uni ec o . Then he 0ima ix can be w i en as 0i=00 i+kαiI,(8) whe e Iis he iden i y ma ix and 00 i=      Gi0. . . 0 0Gi. . . 0 . . .. . ..... . . 0. . . . . . Gi       ,(9) αi=      αi0. . . 0 0αi. . . 0 . . .. . ..... . . 0. . . . . . αi       .(10) Then, equa ion (2) can be w i en in he o m o (11), whe e 80=P3 i=1αi8i. ω2−P3 i=100 i600 i0 −P3 i=1600 iI!P 80=k P3 i=100 i6αiI P3 i=16αi0!P 80.(11) Equa ion (11) ep esen s a gene alized eigen alue p oblem wi h 2Neigen alues k, wi h possibly complex alues. Complex band s uc u es on he incidence di ec ion Eαha e been ob ained by sol ing he eigen alue equa ion o a disc e e numbe o equencies and hen so ed by con inui y o k. In con as o he ω(E k)me hod, in his o mula ion he pe iodici y is no ele an and k(ω) does no ollow he i s B illouin zone. We conside an SC wi h p imi i e la ice ec o s Eai(i=1,2,3). The supe cell is a clus e o n1a×n2a×n3asca e e s pe iodically placed in he space. Then, he p imi i e la ice ec o s in he supe cell app oxima ion a e E a0i=niEai, and he comple e se o la ices in he supe cell app oxima ion is {R0|R0=liE a0i}, whe e niand lia e in ege s. The p imi i e ecip ocal ec o s a e hen E b0i=2πεi jk E a0j×E a0k E a01·(E a02×E a03),(12) whe e εi jk is he 3D Le i–Ci i a comple ely an i-symme ic symbol. The comple e se o ecip ocal la ice ec o s in he supe cell is {E G|E Gi=NiE b0i}, whe e Nia e in ege s. The densi y ρiand he bulk modulus Bia e he physical p ope ies in ol ed in he wa e equa ion and, using he Fou ie expansion and he geome y o he sys em, hey can be exp essed in e ms o he s uc u e ac o o he PWE (EPWE) as well as o he PWE (EPWE) wi h supe cell app oxima ion. The index i=(h,c) ep esen s he hos medium and he sca e , espec i ely. The illing ac ion o a cylinde in a supe cell is =π 2/A, whe e Ais he a ea occupied by he supe cell. I we conside ha βi ep esen s he alues (ρ−1 i,B−1 i)and ha he New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/) 6 supe cell has Nccylinde s o ganized in an a ay o size n1a×n2a, hen β(−→ G)=(βcNc +βh(1−Nc )i −→ G=−→ 0, (βc−βh)F(−→ G)i −→ G6= −→ 0,(13) whe e F(−→ G)is he s uc u e ac o o he supe cell. In his app oxima ion, he s uc u e ac o o he supe cell has o be compu ed aking in o accoun he size o he supe cell. I we conside a 2D SC wi h cylind ical sca e e s wi h adius and size o he supe cell n1a×n2a, he s uc u e ac o o he supe cell is exp essed by F(E G)= (n1−1)/2 X i=−(n1−1)/2 (n2−1)/2 X j=−(n2−1)/2 eı(ia|E G1|+ja|E G2|)P(E G), (14) whe e P(E G)=2 G J1(G), (15) and whe e ais he la ice cons an inside he supe cell and G= | E G|. P e ious equa ions show he exp essions o he app oxima ion o he comple e supe cell. I he supe cell p esen s Nppoin de ec s a he si es labeled (ls,ms)in he pe iodic sys em, wi h s=1,...,Np, hen he Fou ie coe icien s o he expansions o he physical pa ame e s in ol ed in he p oblem ollow he nex equa ion, β(−→ G)=(βc(Nc−Np) +βh(1−(Nc−Np) i −→ G=−→ 0, (βc−βh)F(−→ G)i −→ G6= −→ 0. (16) The s uc u e ac o o such a supe cell wi h Nppoin de ec s is F(E G)=  (n1−1)/2 X i=−(n1−1)/2 (n2−1)/2 X j=−(n2−1)/2 eı(ia|E G1|+ja|E G2|)− Np X s=1 eı(lsa|E G1|+msa|E G2|) P(E G). (17) The in e ac ion o he de ec poin s in he supe cell app oxima ion mus be as low as possible be ween he neighbo ing supe cells in o de o dec ease he o e lap be ween de ec s. Thus he size o he supe cell should be big enough o place he poin de ec s sepa a ed in consecu i e supe cells. In oducing he p e ious exp essions in he ma ices o he PWE (2) o he EPWE (11), we can calcula e he eal and complex band s uc u es. In he p esen pape , we analyze he case o a double-poin de ec in a squa e a ay a si es (1,0)and (−1,0)in a supe cell o 11a×11a. In his si ua ion, he dis ance be ween de ec s is equal o 2aand he dis ance be ween wo double- poin de ec s in di e en supe cells is equal o 20a. 3. EPWE esul s: localized modes Since Sigalas [9] s udied he de ec mode p oduced by a poin de ec in pe iodic s uc u es, se e al kinds o de ec s ha e been analyzed in he las ew yea s, showing in all cases he localiza ion o sound o equencies inside he BG [15,27,28]. Expe imen al and nume ical analyses o he localiza ion in a poin de ec conside ed as a ca i y inside he SC ha e been epo ed ecen ly by Wu e al [29,30] and Zhao e al [17,31], showing he dependence o New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/) 7 localiza ion on he size o he c ys al and on he illing ac ion ( he bigge he size and he illing ac ion, he bigge he localiza ion in he ca i y). Mo eo e , when we conside wo- poin de ec s, coupling be ween localized modes in each de ec poin is possible [32,33]. An accu a e in e e ome ic se up has been used by Russell e al [33] o obse ing he coupled s a es in a double-poin de ec , no ing he e idence o odd and e en symme y apped s a es in a new class o ul a-e icien pho osonic de ices in which bo h sound and ligh a e con olled wi h g ea p ecision and hei in e ac ions enhanced. Fo he case o double-poin de ec s, he bigge he dis ance be ween ca i ies, he lowe he coupling be ween de ec poin s. In his sec ion, we show no el esul s ega ding he imagina y pa o he Bloch ec o o he localized modes inside he SC wi h mul i-poin de ec s. The localiza ion o wa es inside hese de ec s is mainly cha ac e ized by h ee p ope ies. Fi s ly, he modes a e sepa a ed in he equency domain, i.e. he e is a spli ing o he localiza ion equency i he poin de ec s a e close enough. Secondly, he modes p esen symme ies in he ib a ional pa e n depending on he numbe o acancies in he c ys al. Thi dly, he localized modes a e e anescen and hey decay ou side he de ec bu inside he SC. Wi hou loss o gene ali y, we show he esul s o a double-poin de ec in e y good ag eemen wi h he expe imen al da a, showing he symme ic and an isymme ic ib a ional pa e ns o he localized modes. E iden ly, he oscilla ion modes o N-poin de ec s wi h N>2 will p esen mo e complica ed ib a ional pa e ns han he ones appea ing in he double-poin de ec ; hen hey canno be classi ied in o such simple modes as symme ic and an isymme ic ones. The complex and eal band s uc u es e eal ha he alues o k o he localized modes a e cha ac e ized by a eal alue o kand i is ela ed o he localiza ion equency. Howe e , his localized mode p esen s e anescen beha io ou o he de ec , because i is su ounded by a pe ec pe iodici y; hus, he exci ed mode in he su ounding c ys al by he localized mode p esen s an imagina y k ha is ela ed o he e anescen beha io o he mode ou side he poin de ec s [19]. The es o he modes inside he BG only p esen he imagina y pa ; hen hey a e killed inside he c ys al because o hei e anescen beha io . 3.1. Spli ing o localized modes In o de o analyze he spli ing o he localized modes, we ha e calcula ed he eal and complex band s uc u es o an SC wi h a double-poin de ec using EPWE wi h supe cell app oxima ion using a supe cell o size 11a×11a. We conside a 2D SC consis ing o PVC cylinde s o adius in ai backg ound a anged in a squa e la ice wi h la ice cons an a. The ma e ial pa ame e s employed in he calcula ions a e ρai =1.23 kg m−3,ρPVC =1400 kg m−3,cai =340 m s−1and cPVC =2380 m s−1. We conside a illing ac ion =π 2/a2≃0.65. Fo he calcula ions, we ha e used N=(2×15 + 1)2=961 plane wa es. Se e al calcula ions ha e been ca ied ou in o de o ob ain a good con e gence o he solu ion. This numbe o plane wa es is bigge han he one used in p e ious wo ks [21] and p o ides a good con e gence o he solu ion o he eigen alue p oblem. A mode wi hin he BG in an in ini e SC wi hou de ec s is cha ac e ized by a pu e imagina y alue o k=ikim (whe e kim =Im(k)) [19,21,23]. In igu e 1(a) (le panel), we can obse e he dependence o Im(k)on k o a comple e SC wi hin he BG o he 0X di ec ion. We can obse e a maximum alue o Im(k) o he equency in he midgap (926 Hz), which means ha he imagina y pa o he wa e numbe o equencies inside he BG g ows wi h alues o equency close o he cen e o he BG and disappea s a he edges o he BG, i.e. he New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/) 8 Figu e 1. Real and complex band s uc u es o a sonic c ys al (SC) wi h poin de ec s. (a) Le : complex band s uc u e o a comple e SC calcula ed by ex ended plane wa e expansion (EPWE) wi h supe cell app oxima ion. Cen e : band s uc u es calcula ed by PWE wi h supe cell app oxima ion o a SC wi h a poin de ec ; he con inuous ed line ep esen s he de ec mode. Righ : band s uc u es o a SC wi h a double-poin de ec ; he dashed g een line ep esen s he de ec modes o a double-poin de ec . Inse s show he supe cell used in he calcula ions. (b) Complex and eal band s uc u es o a double-poin de ec . a e o decay is bigge o equencies close o he cen e o he BG [7,19,23]. Modes wi hin he BG decay inside he SC because o hei e anescen beha io [19]. In con as o he modes in he BG, localized modes can a el up o he poin de ec whe e he wa e is localized. Figu e 1(a) (cen al and igh panels) ep esen s he eal band s uc u es calcula ed by PWE wi h supe cell app oxima ion o bo h a SC wi h a poin de ec (cen al) and a SC wi h a double-poin de ec ( igh ). We can obse e he localized mode gene a ed by a poin de ec in a SC a he equency ν0=932 Hz, whe eas he equencies o he localized modes o a double-poin de ec ha e been spli ( igh panel o igu e 1(a)). The equencies o he wo localized modes due o he double-poin de ec spli a ound he localized mode o a single de ec : one wi h a lowe equency, ν1=910 Hz, han he co esponding equency o he localized mode in a single de ec , and ano he one, ν2=958 Hz, wi h a highe equency han he single de ec . This phenomenon is analogous o he spli ing o he degene a e a omic le els in dia omic molecules. The spli ing in wo peaks may be unde s ood quali a i ely by conside ing ha he double ca i y in he double-poin de ec is coupled o ming a la ge ca i y wi h wo esonan New Jou nal o Physics 12 (2010) 083024 (h p://www.njp.o g/)