Slow phonon vortices and defect modes in periodic nano-waveguides

7.9 MB·

by  Yue Sun

Slow phonon vortices and defect modes in periodic nano-waveguides
Yue Sun,1,2,∗ Anton S. Desyatnikov,1,3 and Andrey A. Sukhorukov1
1Nonlinear Physics Centre, Research School of Physics and Engineering,
The Australian National University, Canberra, ACT 2601, Australia
2Laser Physics Centre, Research School of Physics and Engineering,
The Australian National University, Canberra, ACT 2601, Australia
3Department of Physics, School of Science and Technology,
Nazarbayev University, 53 Kabanbay Batyr Ave., Astana 010000, Kazakhstan
(Dated: January 28, 2016)
Weidentifyabroadclassofphononmodeswithpersistentvortexfluxesatarbitrarilyslowprop-
agating velocities in periodic nano-waveguides. Such phonon vortices are associated with the split
6 band-edgesindispersiondependencies,whichcanbeengineeredbywaveguidedesign. Modulations
1 introducedinsuchwaveguidescansupportapairofdefectcavityphononmodeswithanarbitrarily
0 small frequency splitting. These features can find applications for sensing and nano-manipulation.
2
PACSnumbers: 63.22.-m, 63.20.D-, 42.70.Qs
n
a
J
I. INTRODUCTION We predict that cavities created by modulating the ge-
6
ometry of dispersion-engineered waveguides can support
2
Periodically modulated nano-waveguides offer unique a pair of phonon modes with arbitrarily small frequency
] potential for controlling the wave propagation velocity splitting, analogous to optical cavities [9]. Monitoring of
s and trapping cavity modes at specially introduced de- this splitting can be applied for sensing. On the other
c
i fects, based on dispersion modification and formation of hand, the time scale of mode beating, defined as the in-
pt band-gaps through Bragg reflection effect. Remarkably, verse of frequency splitting, can be tuned for resonant
o itispossibletodesignastructuretosimultaneouslycon- regimes of nano-manipulation.
s. trol waves of different physical origins, including pho- This paper is organized as follows. In Sec. II, we
c tons and phonons [1–4]. Such phoxonic crystals can fa- present an approach for dispersion engineering in nano-
i
s cilitate strong acousto-optic interactions, opening new waveguides, and analyze the properties of slow phonon
y opportunities for frequency conversion, sensing, nano- modesassociatedwithsplitbandedges. TheninSec.III
h manipulation, and other applications [5]. we establish a correspondence between the slow-phonon
p
Thewaveinteractionscanbe enhancedforfrequencies vortices and field singularities. Finally, in Sec. IV we
[
close to band edges. In this regime the group velocity is discuss the features of pairs of phonon defect cavity

1 strongly reduced, which can lead to the increase of wave modes, supported by dispersion-engineered waveguides
v energy density [6]. Importantly, the actual dynamics of withmodulatedgeometricalparameters. Wepresentcon-
3
interactions depends on the nano-scale features of the clusions in Sec. V.
7
modeprofile. Inparticular,thestructureofenergyfluxes
3
can play a critical role in context of nano-manipulation.
7
0 In this paper, we demonstrate an approach for obtain-
. ing persistent vortex fluxes for phonons, in the regime
1 II. SLOW PHONON MODES AND DISPERSION
of slow group velocity propagation. We show that this
0 ENGINEERING
6 is realized when the phonon dispersion is specially engi-
1 neered to feature so-called split band edge (SBE), which
: is associated with wavevectors inside a Brillouin zone. Topresenttheprinciplesofdispersionengineeringand
v
The phenomenon of vortex flows is conceptually analo- slow phonon mode properties in nanowaveguides, we
i
X gous to the corresponding features of optical modes in consider a periodic structure configuration with exper-
r the SBE slow-light regime [7]. We establish that struc- imentally accessible geometry following Refs. [1, 2], see
a ture of vortices for slow phonon modes involving both Fig.1(a)forthesupercell. Whereashereweconsiderthe
transverse and longitudinal oscillations can be explained properties of phonon modes, the advantage of this struc-
byestablishingcorrespondencewiththetheoryofoptical tureisthatitcansimultaneouslyprovidephotonicband-
polarization singularities [8]. gaps at optical wavelengths. The structure consists of a
Furthermore, we demonstrate new possibilities for en- silicon strip waveguide with symmetric ’acoustic wings’
gineeringtheacousticdefectcavitymodes,byemploying attached at each side and circular air holes drilled at the
the similarity between photonic and phononic crystals. centre. The wings provides strong acoustic reflection,
such that the periodic waveguide has a very robust band
gap.
We investigate the acoustic wave propagation in pe-
∗ [email protected] riodic nano-waveguides using the wave equation for the
2
(a) (b) dominant displacement in both x and y directions. We
(cid:27) - 6
la showthenormalisedxandy componentsofthedisplace-
ment mode profile at k l/2π = 0.15 in z = 0 plane in
x
Figs. 2(a-d). The displacement parallel to propagating
l direction, u , is symmetric with respect to a plane x=0
6 (cid:27) - x
and anti-symmetric to y = 0, while the y component
w (cid:19)(cid:19)(cid:19)7r wa is anti-symmetric with respect to x = 0 and symmet-
ric to y = 0. It is also worth mentioning that there
?
are clearly phase screw dislocations in both longitudinal
(x) and transverse (y) components, namely at (x = 0,
y 0.62µm)inu andat(x=0,y 0.29µm)inbothu
y x x
(cid:39) (cid:39)
6 ? and uy. We didn’t show the displacement in z direction
since it is three orders of magnitude smaller than these
- x two dominant components. For the band-edge mode at
k l/2π=0.85 the displacement profile has the same sym-
x
metry and magnitude but with an opposite phase.
FIG. 1. (a) Schematic top view of the periodic wave-
guide supercell. (b) Band diagram of eigenmode dispersion: For all kinds of band-edge modes the group velocity
grey shading highlights the band gap, and the split band- approaches zero, and therefore the total energy flux also
edge modes marked by cyan open circles. The geometry pa- has to vanish. However the local energy fluxes can be-
rameters are l=500nm, w=500nm, wa=1500nm, la=250nm, have differently, as these are linked to the local phase
r=150nm, and the height of the structure is h=220nm. gradients. A key distinctive feature of SBE modes is the
presenceofnontrivialphasegradientalongthewaveguide
accordingtotheBloch-Floquetconditions,incontrastto
mechanical displacement u,
the band-edge modes where phase can remain flat and
∂2u (cid:20) ( u)T + u(cid:21) only jump by π. It has been shown that optical SBE
ρ = C : (cid:53) (cid:53) . (1) modes in photonic crystals can sustain nonvanishing lo-
∂t2 (cid:53)· 2
cal energy fluxes at arbitrarily small group velocities [7],
This equation expresses the Newton’s second law, that and here we extend this analysis to phonon fields. The
theequilibriummassdensityρmultipliedbytheacceler- energy flow of an acoustic field is defined as
ation vector ∂2u/∂t2 equals to1the elastic restoring force
∂u (cid:20) ( u)T + u(cid:21)
at the right hand side, where C is the stiffness tensor. P= C : (cid:53) (cid:53) . (3)
We consider lossless regime with no damping. −∂t · 2
We seek solutions for periodically oscillating modes
We plot the streamlines of the time-averaged vector on
withfrequencyf asu(x,y,z,t)=u(x,y,z,0)exp(i2πft),
top of its time-averaged amplitude for the SBE mode in
(cid:20) ( u)T + u(cid:21) Fig. 2(e). We observe that non-zero local energy flow
Ω2ρu= C : (cid:53) (cid:53) . (2) has both forward (+x) and backward ( x) components
− (cid:53)· 2 −
at the boundary of x = l/2, similar to the photonic
±
Acoustic eigenmodes in periodic waveguides sat- SBE modes.
isfy the Bloch-Floquet conditions, u(x + l,y,z) = Furthermore, thetime-averagedenergyfluxalsoforms
u(x,y,z)exp(ik l). In numerical simulations, we apply local circular and spiral energy flows. At the centre of
x
these conditions at x= l/2 cross-sections, and the rest such flux vortices, the energy flow density reaches its lo-
±
of the boundaries are treated as stress free. We nor- cal minima, see the energy flow across line x = 0 at
malize the eigenmode solutions such that the maximum z = 0 plane in Fig. 2(f). In order to clearly illustrate
displacement amplitude inside the structure is scaled to this feature with respect to the phase screw dislocations
unity, max(u) = 1. The simulations are performed in displacement fields, we label the local minima with
| |
by the eigen-frequency solver in structural mechanics open red circles and position vertical dashed lines at all
module provided by COMSOL Multiphysics Finite El- of the local minima positions in y. Positions y (y(cid:48)) and
1 1
ement Analysis Software. We treat silicon as a cubic y (y(cid:48)) give zero amplitude and correspond to the vor-
2 2
material with density ρ = 2330 kg/m3 and stiffness ten- tex centres, while the middle ones in between represent
sor C, which has three independent elastic coefficients the saddle points in the flux between two vortices and
c =166 GPa, c =64 GPa, and c =80 GHz [1]. have positive energy flow. At y 0.29µm both u and
11 12 44 1 x
∼
We optimize the structure geometry to obtain the dis- u feature phase singularities and they lead to vanishing
y
persion feature of split band edges (SBE), which are time-averaged energy flow at the vortex core, whereas at
marked by cyan open circles in Fig. 1(b). The corre- y 0.49µm neither u nor u shows phase screw but
2 x y
∼
sponding modes have wave-numbers inside the first Bril- the time-averaged energy flow inside vortex is also zero.
louin zone, while their group velocities can be vanish- We see that the two circulating flows of the same vor-
ingly small close to band-edge. The modes are a mix- ticities, e.g. clockwise at y and y , while appearing si-
1 2
ture of transverse and longitudinal waves as it has its multaneously at different spatial positions for the same
3
|u| @z=0 plane @ ka/2π = 0.15 @ freq = 4.0772 GHz
x placement profile do not correspond to each other spa-
0.2 (a) 0.6 tially, they suggest an analogy with optical polarization
x[]mµµx/m 0.01 000...345 sinintghuelaprhityiessic,awlhoircihginenoafbtlwesoupshtoonoenxpvlaoirntictehse.differences
−0.1 0.2
0.1
−0.2 ∠u @z=0 plane @ ka/2π = 0.15 @ freq = 4.0772 GHz
x
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 3 III. PHONON VORTICES AND FIELD
0.2 (b) y/µm
2 SINGULARITIES
]m 0.1 1
x[µµx/m 0 0 Wesuggestanapproachtotheanalysisofphononvor-
−0.1 −1
tices in analogy of similar treatment of transverse opti-
−2
−0.2 |uy| @z=0 plane @ ka/2π = 0.15 @ freq = 4.0772 GHz −3 cal fields [10]. We recall that the uz component of the
0.2 (c)−0.6 −0.4 −0.2 y/µ0m 0.2 0.4 0.6 0.7 displacement is three orders of magnitude smaller than
0.6
itsin-planecounterparts. Thesedisplacementsu form
x[]mµµx/m 0.01 000...345 aprvoeaccthoersfideelvde,loapneddwfeorcathneatpwpoly-dtimheenmsiaotnhaelmtartaxinc,yasvlearpse-
−0.1 0.2 optical fields. Specifically, we can use optical theories by
−0.2 ∠uy @z=0 plane @ ka/2π = 0.15 @ freq = 4.0772 GHz 0.1 formally treating ux,y mechanical displacement compo-
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 3
0.2 (d) y/µm nents as the values of linearly polarized components of
2
the field u. The optical energy flow for the latter vector
]m 0.1 1
x[µµx/m 0 0 fiofelpdrhesaesnbceeeonfapnhaalyszeesdinegxutelanrsiitvieelsy[[1151,–1164]]iinntihtseccoonmtpexot-
−0.1 −1
−2 nentsaswellaspolarizationsingularities[8]inthespatial
−0.2 real(velocity*stress tensor*)@z=0 −3 mapofpolarizationdistribution. Ateachspatialpointof
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0.2 (e) y/µm 0.3 this map the state of the field vector u is represented by
]m 0.1 0.25 polarization ellipse, left- or right-handed, depending on
x[µµx/m 0 00..125 the temporal dynamics of u. Points in plane where the
ellipse is degenerated into a circle, i.e. the points where
−0.1 0.1
polarization is purely circular and the orientation of the
−0.2 0.05
0 ellipse is undeterminate (singular), are called C-points.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0.2 (f) y/µm EachC-pointcorrespondstooneofthethreetopological
.]u
[.aP||0.01 tsytapres[8o],fapsolwaerilzlaatsiotnwositnygpuelasriintitehs:elceomnotonu,rstcalar,ssainfidcamtioonn-,
−0.6 y02−0.4 y01−0.2 0 0.2 y1 0.4 y2 0.6 elliptic or hyperbolic [17].
y[µm] We introduce the circular polarization components of
u = u c + u c , namely u = (u iu )/√2 and
+ + − − ± x y
±
FIG. 2. (a-d) Mode profile of the split band-edge mode the eigenvectors of circular polarizations are given by
at kxl/2π=0.15 in z = 0 plane. (a,c) Amplitudes and c± =(x∓iy)/√2, here x and y are Cartesian unit vec-
(b,d)phasesofthe(a,b)u and(c,d)u displacementcompo- tors. We plot the amplitude and phase profiles of circu-
x y
nents. (e) The time-averaged energy density flux (cid:104)P(cid:105) shown lar polarization components in Figs. 3(a-d). The whole
withwhitestreamlinesonthebackgroundcolor-codedmagni- structure is summarized in the polarization map plotted
tude|(cid:104)P(cid:105)|(f)Theamplitudeo1ftime-averagedenergydensity in Fig. 3(e), where we also indicate with circles the posi-
flux along x=0 line in (e).
tions y of two vortices observed in Fig. 2(e).
1,2
The first type of vortex at y clearly appears as ze-
1
ros in both u , because the total field vanishes here,
±
eigenmodeasshowninFig.2(e),shouldhaveverydiffer- u(0,y1) = 0. The polarization ellipses degenerate into
ent structures. Indeed, the first one at y1, as already linearsegmentsaroundvortexaty1,i.e. thepolarization
noted above, corresponds to zero displacement ampli- is linear at each point in the vicinity of the vortex and
tudesandscrewdislocationsofthesamechiralityinboth it is directed tangential to concentric circles. This is a
linear components of the displacement u and u . Thus, typical picture for an “azimuthally polarized” field [18].
x y
the energy flux and the kinetic energy, under the restric- In addition to this expected defect, we also observe in
tion of finite energy, naturally vanish at this point. It Figs. 3(b,d) phase dislocations at different locations y
3,4
is not the case for the second type of vortex at y , as intwocomponentsu ,alsoclearlyvisibleaszerosofcor-
2 ±
there are no zeros and no phase singularities in the com- responding amplitudes in Figs. 3(a,c). Polarization map
ponentsu . Note, however, thatthereisanotherphase inFig.3(e)illustratesthiscomplexspatialstructure. Let
x,y
dislocation in the u component at about y 0.62µm, us start with the rightmost zero of the u component at
x −
(cid:39)
visually shifted further to the right from the vortex at y 0.71µm. Since the u component vanishes and u
4 − +
(cid:39)
y 0.49µm. Although these two features of the dis- dominates,thefieldisright-handed(redcrosses),andthe
2
(cid:39)
(a)
0.2 0.8
0.1 0.6
]m
0
[µ 0.4
x
0.1
� 0.2
0.2
�
(b) 3
0.2
(a)
0.2 0.8
0.1
]m 0.1 02.6
µ
]mx[ 0
0
x[µ 0.1 01.4
�
0.1
�0.2 0.2 4
�0.2 0
�
(((((baaac)))))(a)00000..0...22.02222.52 |u+| @z=0 plane @ ka/2π = 0.15 @ freq = 4.0772 GHz 03.0008...888 rFeisgu.lt3i(neg).C-Tphoienttoispomloagrikceadl binydaexlef+t-1p/o2inotfintghtisriaCn-gploeinint
((((((((((((((((((((xxxcdexxxxxx[][][]mmmbcdabcdµµµxxxxxxxxxx[][][][][][] mmmmmmbcdbcddeabµµµµµµ[][][][][][][][][][]mmmmmmmmmmµµµµµµµµµµ )) A)))))) ))))))))) )))(|(|(((((����uu������������xxx��aacrbcd����������������[][][]bmmmµµµ����gx/mx/mx/m++µµµ)))))))0000000000||000000000000000000000000000(00000000000000000000000000000000000 0000−−−−−−−−−000������u000000..........0000000000...........................−−−−−−0000000000000000000.......................................11212212121121221212122..........121212121212.........00000000000012+1121221121221121212121212121212121212010120122102102101212............0000000000000)5555555555555555555000211212212121 00000 000............. 1212122121212 0 y 000,0. ....111µ11 m ∠∠ | u uu 0− −+| @ .@@0000 2 ....z2222 zz = == 0 00 p pp lll a aa nnn e eey y000 0@ @@1....y3333 1 [k kk aµ aa / //2 22 π ππyyym /// ===µµµ 0 mmm 0 00] 0000 .......1 .1144444 555 @ @@ f ff r rr eeey qqq 2y 000=0== 2... . 5555444 ... 0 00y 777 73y 77 2 223 G GG0 H HH 0000. z....zz 66666 0 000 y....7 777 4y 4 230000000−−−01230000000001201230...............00001233000001233000012330000123301230000000003321123456712345678......................8246246824624624682468000001233....2468 iignspapioplbsclpcbccctlchtnmsiaohoppiuluauetpyaheooeeedewldnneaoapWAsrrlecttpysipaayeapnwerriteintniesorooe3nnttteelxiedthrbreedeieawsiix,rnnnogodabnoeoaixeeitlaiasnottbpttaliflennstnasaai,swptyoheclnesodontncitcwtopraoeeoochhcidteygatcclmsnfrrliaadwfbfyohltee4huneesnvtytreptiyutrn.oeaipCoiylte.svaahlrheraoudbn,seloapdmoxa-tmNv+eorfecTrteiurpierhliiro[onosceeiatiiasto1c1thtaeonyftnarrnohyuotehlta2/laieijt-ptepetitxeinfetacolcal]2ednilisrddhahs.eshprnrtaorlxrtideoao,rseeeaebbsahsnoMabeaabfilatnccoteascusuitssrrnhrcktxlsoysree-ittiopooitatuttocooewenielrbpimyrispehsiyfiangonrrstntceesu3iotv2reioliehyhnsuioatslmhlesxtiofilpleceiioaasitfmsi+daionvtsesrflusdcrniprtc,crscottpietatiorstogpnlligovi,zhrchcLbtreyhzotwcdm[urtnoeitaoheii1tfaaealene-oeeseorltittanhmle1ttftpalneeytx-[oiinoooriwtbie1,onhetrlntooflrutpilryroii3rcnzealepnhnnl1btboerr,oiomtlia]pl;e,niseree4elowe.iiinawseatlntnedtofansdss]isiulsioeraCs.ransipieserieIoteimcnnofsetilclpinp,antnd-anhtwlnutTtgoieippedonaomflgiinr.coltrnollowaaertohscoeteloa,lrtineadelhiarennCeiuFwepehre[lIpCtnnpttsd1ltietdsnrtthie-Clpitttisae7pg,-sionpiiodacdietpfspcflffrc]c-.ofoidagrnorwneopasttoooedrnyunht3ici.Cdtioteowrecrinehcwitnseyy(cdnnipersvtli-ttoedetopptlnfieocttphCtoeasti)uinhnisosttheeptnseoa]elaolnr-pi.e.cosfr.cnasnldopibietcmscbosoigsnetehC[snrvTtTtooor1snoi,thrfttpicaeoeocviriaoo9ghhlhhdnaunxrigioosearieillnnn]yeeeceessstt--------ff.l,
(((((xxxxx[]meeµxxxxxx[][][][] mmmmeecµµµµ[][][][][][]mmmmmmµµµµµµ A A)) ))) ((|(((urxx��cre��ed�����[][]bmmµµ������ggx/mµ)-))))|0(00000000(000000000000 000000 000000−−−u0u0000.��000000........−−0000........................1-112112....00012+1121121212121211210210)121121...000)555501210000 00 00... ... 112 121 0 y . 0,0 1..1µ1 m 0 . 00 2 ..22 y y 001 ..y33[1 µ 1 m 1y /0 µ ]m . 00 .4 .44 y y 2 002 .. 55 y 3 y 3 0 .00 ..6 66 0 0y ..7 7 4 y 4 −−−010000012000001200001200000120000012321..................246246846824682468012000...468 tcgtamosnehloinfaonxaneAnBwyitInoosiferVsptwffdpareulsr.ichsaircrhyttivotartriehntcftipePtgnouieovheoAugrlnedtanaiashIsdosrlftRoeefeiarisnfsomtntSia,gdthmrdnuhieeeOtndaleaphtavmFlthhpestohcpeeorlrsiPaieittsintpnsesomHhseeeoxneHesnisOsolcaicaoaittfnoNnerfiilaoofldiqoatOnnuteicfCeh,ffvNiiopsrre-woefcyphraceuSCaototnileBsenniAaddnxxtoErfetVoaodncfCmprioumIiffsnTo-rrppipeomtrnshYireeiodneteeornieindnetnotcMshsctuonC[tiee1nrOsnaisn-r3pstspeDbgp]hent.[oee1rohEetitvTr0nnsffewiSe]otohea.soe,dnucnreaitiisindnccss-,
�������000000��......12222200000.200000....11110.10000....22220.2yyyyyyy1111[[[yµµµ1mmm0000]]]....44440.4yyyy2222yyyyy23333y00003....66660.6 yyyy4444y 4 00000.200.2 dcasuoepncfehtaceitraxntcoaaoflvfydipsteiihsfeeosfctotoornrmipigchoincdoraneytsoisnntcagafilnefcrladaovpmsi.ptieteahsr,ei[tS9B]w,Eaasnmpdroewddeeisc.tpeeIdrnftohtrhamet
(d) 0.2 y[µym[µ]m] 3 We introduce the defect modes and create the grad-
111 ual structural modulation by tapering the width of the
FIG.3. 0(.1a-d)Profilesofthec1irc1ularpolarizationcomponents ’acoustic wings’ la linearly from outside (la = 250 nm)
(a,b) u and (c,d) u . Show are the (a,c) amplitud2es and
(b,d)]m p(dh+) ases. (e)Pola−rizationmapshownwithmajoraxesof to the center (la = 240 nm) along N = 8 periods,
polaxr[µAizrag(tui0o-)n ellipses; blue and red for left- and right-handed see Fig. 4(a). Such type of structures can be fabri-
catedexperimentallywithexcellentprecision[2]. Weper-
ellipses,r0e.1spectively. Circlesindicatepositionsoftwov1ortices
at y �[cf. Fig. 2(e)], and triangles indicate the positions of form simulations using COMSOL solid mechanics eigen-
1,2
frequency solver, considering fixed ends and stress-free
C-points0a.2t y3,4.
� conditions at the rest of the boundaries. We find that
0

(e)
0.2 0.8
0.1
0.6
]m
x[µ|(ue)| 0 0.4
-
0.1
� 0.2
0.2
� 0
0 0.1 0.2 y1 0.4 y2y30.6 y4
y[µm]
1
5
tween the paired defect modes depends on the gradient
(a)
of defect introduced to the periodic waveguide to cre-
250
la/nm224405 atatel cthaveiltoycafrliezqeudenrecsioesnaanncde.tWheefsrheoqwuetnhceypshpolnitotninicgcarsysa-
−4 −2 x/μm 2 4
function of the number of periods in the linear taper N,
see Figs. 4(d,e). The frequency splitting between the
paireddefectmodesdecreasesasthedefectgradientgets
(cid:8)(cid:25)(cid:8) z
(cid:8)(cid:8)* la yZ}ZZ(cid:8)6(cid:8)(cid:8)*x smmoadlelesris(Nvergyetssenlasirtgivere).toTthheedfreefqecutengcraydsiepnlitt,tianngaloofgothues
to their photonic counterparts [9]. One can obtain arbi-
(b) (c)
trarily close resonant frequencies by designing the defect
gradientinvolvingtaperingofseveralgeometricalparam-
eters, such as width of the strip waveguide w, length of
(d) (e) the’acousticwings’waorallthegeometricalparameters
4.04 2 together.
1.75
Hz Hz1.12.55
G M 1
f/4.034 df/0.75
0.5 V. CONCLUSIONS
4.032 0.25
10 11 12 14 15 16 10 11 12 14 15 16
N N
To summarize, we demonstrated that persistent slow
phonon vortices can appear in periodic nanowaveguides,
FIG. 4. Defect phonon modes of a modulated waveguide.
whichdispersionisengineeredtofeaturesplitbandedges.
(a) 3D schematic with linearly tappered wing width la, as
We identify two types of vortices, associated with phase
shown in the inset. (b,c) Top view of the total displacement
singularityandvanisingdisplacementatthevortexcore,
for defect modes at resonant frequencies (b) 4.046 GHz and
(c) 4.048 GHz. (d) Resonant frequencies and (e) frequency or with characteristics reminiscent of polarization singu-
splitting of two cavity modes vs. the number of tapering larities in optics. Furthermore, we find that the wave-
holes. guide modulations lead to the appearance of a pair of
cavity defect phonon modes with small and controllable
frequencysplitting. Theseresultssuggestnewopportuni-
such structure indeed supports two defect modes which tiesforthedevelopmentofacousticandacousto-opticde-
originatefromtheslow-phononvorticesattheband-edge, vices, including applications for frequency shifting, sens-
see Figs. 4(b,c) for the total displacement field of mode ing, and nano-manipulation.
1 and 2, respectively. We1 show here only the modes’
top view since, similar to their originating SBE modes,
the out-of-plane displacements of both the defect modes
ACKNOWLEDGMENTS
are substantially weaker than the displacement in other
directions. Defect mode 1 has symmetric u and anti-
x
symmetricu withrespecttox=0, whiledefectmode2 Authors thank K. Yu. Bliokh for useful discussions.
y
has antisymmetric u and symmetric u with respect to This work was supported by the Australian Research
x y
x=0. These two cavity-mode resonances have their fre- Council (ARC) Discovery Project DP130100086. Nu-
quencies very close to each other. Defect mode 1, shown merical simulations were performed with the assistance
inFig.4(b),oscillatesat4.046GHzwhiledefectmode2, of resources provided at the NCI National Facility sys-
shown in Fig. 4(c), oscillates at 4.048 GHz. temsattheAustralianNationalUniversitysupportedby
Furthermore, we find that the frequency splitting be- the Australian Government.
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