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/)