Twists and turns for metamaterials

0.43 MB·English

by  Mingkai Liu

Twists and turns for metamaterials
Mingkai Liu,1 Yue Sun,1 David A. Powell,1 Ilya V. Shadrivov,1
Mikhail Lapine,2 Ross C. McPhedran,2 and Yuri S. Kivshar1
1Nonlinear Physics Centre and Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS),
Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia
2Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS),
School of Physics, University of Sydney, Sydney NSW 2006, Australia
We propose and verify experimentally a new concept for achieving strong nonlinear coupling be-
tweentheelectromagneticandelasticpropertiesinmetamaterials. Thiscouplingisprovidedthrough
3 anoveldegreeoffreedominmetamaterialdesign: internalrotationwithinstructuralelements. Our
1 meta-atoms have high sensitivity to electromagnetic wave power, and the elastic and electromag-
0 netic properties can be independently designed to optimise the response. We demonstrate a rich
2 range of nonlinear phenomena including self-tuning and bistability, and provide a comprehensive
experimental demonstration of the predicted effects.
n
a
J
Metamaterialsresearchhasgrownrapidlyoverthepast changes in the mutual orientation. Instead of a consid-
5
2 decade,exhibitingawidevarietyofnewwavephenomena erable displacement of an entire array [9], it is sufficient
[1, 2]. Being initially conceivedin the domain of electro- to move the crucial parts of the resonant particles with

] magnetics [3–5], the metamaterial concept also proved respectto eachother—forexample,the gapsofthe two
s
c to be fruitful in other areas of physics [6–8]. Until re- coupled split-ring resonators — which involves a more
i cently, however, direct interplay between different types gentle geometric alteration.
t
p of physical effects within the same metamaterial wasnot Let us now introduce our novel concept: nonlinear
o
considered,althoughmechanicalcontroloverelectromag- metamaterials with intrinsic rotation. As a building ele-
.
s neticmetamaterialpropertieswasemployedinstructural ment of the structure (see Fig. 1), we considertwo coax-
c
tuning [9, 10]. ialsplit ring resonators(SRRs) with elastic feedback be-
i
s
It turns out that introducing a mechanical degree of tween them. The rings are allowed to rotate about the
y
h freedom into electromagnetic metamaterials leads to an common axis, while the elastic feedback is provided by
p interestingrangeofnonlineareffects,givingrisetoanew connecting a thin elastic wire. Indeed, the use of elastic
[ class of magnetoelastic metamaterials [11] and to wide- wires has a prominent history in physics, being utilised

bandoperation[12]. Therangeofpossibleeffectsachiev- inthemilestoneachievementoftheexperimentaldemon-
1
v able in this way promisesto be richerthan inthe promi- strationoflightpressurebyP.N.Lebedev[18]. Here,we
0 nentareaofoptomechanics[13],becausethegreaterflex- employ the electromagnetic (EM) torque to construct a
6 ibility in metamaterial design overcomes the limits of “light-driven”meta-atom, which enables us to modulate
9
available material functionalities, and offers wider pos- the resonant frequencies by twisting the rotatable ele-
5
sibilities for optimisation. At the same time, the imple- mentdirectly withEMwaves. ThecomponentoftheEM
.
1
mentation of magnetoelastic metamaterials [11] remains forces which twists the rings with respect to each other
0
challengingandinsomecases,suchastheconformational
3
1 nonlinearityinresonantspirals[14],remainsinaccessible
: foroptics. Thereasonforthisisthatthemagneticforces,
v
employed in the initial designs, are relatively weak, so
i
X such materials require either high power or extremely
r small elastic restoring forces, which poses a considerable
a
manufacturing challenge.
We recall, however, that earlier research on struc-
turally tunable metamaterials [9] indicated that near-
field interactionmay significantly improvethe tunability
range,leadingtovariouseffectsassociatedwithnear-field
coupling [15]. In particular, changing the mutual orien-
tationbetweentheneighbouringelementshasaprofound
effectontheresonatorcouplingandthestructureoftheir
FIG. 1: Conceptual layout of a new metamaterial and its
modes [16, 17]. rotational “meta-atom”. Incident wave propagates along y
We therefore expect that the most efficient approach direction, having a linear polarisation with the electric field
toimplementdynamiccouplingbetweenelectromagnetic along xandmagnetic field along z. Theinducedelectromag-
and mechanical effects should rely on near-field interac- netictorquebetweentheresonatorschangesthemutualtwist
tion, which canhavea powerfulinfluence evenfor subtle angle θ between therings, connected by an elastic wire.
2
is normally not the strongest among the forces involved, Tostartwith,weuseasemi-analyticalmodeltostudy
buttheprominentadvantageofusingEMtorqueinstead the dynamics of an isolated meta-atom in free space. It
ofcollinearEMforcetodriveameta-atomisthattheef- will subsequently be demonstrated that this model ex-
fective leverarmofthe azimuthalEMforcecanbe much plains all qualitative features of an array. As shown in
larger than that of the azimuthal restoring force from a Fig.1,the twocoaxialidenticalSRRsareoffsetbya dis-
thinwire;thiscaneffectivelymagnifythedeformationin tance s in the z direction, and the twist angle between
azimuthal direction by orders of magnitude. them is θ. The incident wave propagates along the y
Technically, there are a number of ways to implement direction, with its magnetic field in the z direction and
this general scheme; in our design, one of the rings is electricfieldinthexdirection. Tostudythenonlinearbe-
fixed to a substrate and the other one is suspended on haviouroftherotatablemeta-atom,weutiliseanefficient
the long wire. The three symmetrically positioned wires analyticalmodelbasedonthesinglemodeapproximation
which attach the suspended ring to the string provide and the near-field interaction [16]. This model can pro-
stability against tilt. In this design, therefore, the only videareasonablepredictionoftheEMresponseaswellas
favourable movement is the rotation of the suspended the optomechanical properties of the structures [19]. As
ringwithrespecttothefixedringoverthe commonaxis, our previous studies showed [20], the current and charge
and all other mechanical degrees of freedom can be ne- of the SRR can be separated into frequency-dependent
glected. mode amplitudes and spatially-dependent distributions:
Suppose the initial position is such that the ring slits J(r,ω) = −jωQ(ω)j(r) and ρ(r,ω) = Q(ω)q(r). The
have a certain angle between them with respect to the mode amplitudes Q1,2 can be obtained after solving the
common axis (Fig. 1). An EM wave then induces a cer- coupled equations:
tain distribution of charges and currents in the two res-
onators, and the resonance of the system is determined Q1 =(E2Fm−E1Fs)/(Fs2−Fm2)
(1)
by their mutual orientation [16]. These chargesand cur- Q2 =(E1Fm−E2Fs)/(Fs2−Fm2),
rentsalsoresultinEMtorquebetweenthetworings[19],
which drives the suspended ring to rotate until the EM where E1 and E2 correspond to the effective voltage ap-
torque is compensated by the elasticity of the twisted plied to the lower (index 1) and top (index 2) SRRs by
wire. Meanwhile, the entire pattern of charges and cur- the external fields; Fs and Fm are the self and mutual
rents gets modified and the torque also changes, so the impedance terms, with their explicit forms given in the
final stable equilibrium is only achieved via a complex supplemental materials [21].
nonlinearfeedback. Thetwistedwireprovidesarestoring Once the frequency-dependent mode amplitudes are
torque to balance the EM torque, so that we can control known, we can calculate the EM torque experienced by
thetwistangle(andthustheresonance)bychangingthe the SRRs. Here, we are particularly interested in the
external field signal. torque on the top rotating ring: MEM = R ρ(r2)r2 ×
V2
An additional feature of the proposed design is that E+r2×[J(r2)×B]dV2,wheretheintegrationisperformed
the nonlineardynamicsoftherotatableparticlenotonly over the volume V2 of the top SRR. We decompose the
depend on the parameters of the wire, but also on the total torque into two parts: external torque Mext con-
EMmode initially excitedinthe resonators,whichis de- tributed by the externalincident fields [19], and internal
termined by the starting angle between the gaps. The torqueMint duetothenear-fieldinteractionbetweenthe
lattercanbemadearbitrary,andanimplementationhas two SRRs. We assume that the lower SRR is fixed while
the possibility to deliberately adjust it, introducing tun- the top SRR is only allowed to rotate about the z axis.
ability to the system. See the supplemental materials [21] for explicit expres-
The above design, therefore, offers a tunable resonant sions for the internal and external torque.
nonlinear system with elastic feedback and, as we show Figures 2(a) and 2(b) depict the mode amplitude Q2
below,yields arichpatternofnonlinearresponseinclud- and the total EM torque MEM = Mext +Mint experi-
ing self-tuning and nonlinear bistability, which are much encedbythetopSRRasfunctionsoffrequencyandtwist
strongerthanthoseprovidedbyusingnonlinearsemicon- angle θ. Since radiation losses are taken into account in
ductor components. Below, we present a detailed theo- themodel[20],weareabletoaccuratelydescribetheevo-
retical analysis of rotational meta-atoms. We then pro- lution of the mode amplitudes, phases and lineshapes of
ceed to the experimental results obtained with a fabri- the resonances. As expected, this chiral meta-atom sup-
cated prototype of the rotational“meta-atom”placed in ports two hybrid resonances,which can be characterised
a rectangular waveguide, and confirm all important fea- as symmetric (lowerfrequency branch)and antisymmet-
tures predicted by theory. Finally, we perform full-wave ric (higher frequency branch) modes, according to the
numericalsimulationsofthe arrayof“meta-atoms”(i.e., symmetry of the H component [16].
z
metamaterial),anddemonstratethatallthenonlinearef- The directions of the EM torque at these two res-
fects observedforthe single meta-atominthe waveguide onances are also opposite. For the symmetric mode,
◦
are qualitatively the same in the array. θ = 0 corresponds to the configuration of highest po-
3
tential energy (unstable point), and thus the two repel
◦
each other once θ > 0 , until they come to the stable
◦
state at θ = 180 , while the reverse is true for the an-
tisymmetric mode [22]. The evaluated external torque
is about one order of magnitude smaller than the in-
ternal torque, and the total torque is of the order of
10−10Nm when the structure is pumped with a power
density PI=1mW/mm2. This is confirmed by the full-
wave simulation (CST Microwave Studio) followed by
calculation based on the Maxwell stress tensor, which
yields the EM torque through a surface integral of the FIG.3: Schematicoftheexperimentalset-up. Thepumpand
field components around the object [23]. probe signals are combined by a 3dB combiner. The sample
is positioned in the centre of the waveguide. 1: vector net-
The overall mechanism of achieving a nonlinear ef-
work analyser; 2: 3dB combiner; 3: rectangular waveguide;
fect is presented in Figs. 2(c,d). As an example, we 4: 20dB attenuator; 5: power amplifier; 6: signal generator;
choose a pump frequency (3.5GHz) at the symmetric 7: sample.
mode [regime denoted by the black dashed line in Fig. 2
(b)]. It can be seen that MEM is a Lorentz-like func-
tion of the twist angle, while the restoring torques MR gle,itnaturallyleadstononlinearsolutions. Asshownin
under different initial twist angles θ0 are approximated Fig.2(d),the power-dependenttwistanglesunderdiffer-
by linear functions (Hooke’s law). The intersections of entinitialanglesdemonstratetheevolutionfromsmooth
these two functions, MEM(θe,PI)+MR(θe) = 0, corre- nonlineartobistable responseasθ0 departsfromthe an-
spond to the equilibrium angles θe. However, only the gle of maximum EM torque. In principle, as θ0 moves
angles with ∂∂θ [MEM(θe)+MR(θe)] < 0 are stable. As further away from the resonance, more noticeable rota-
the pumppowerPI increasesfromzerotomaximumand tionandhysteresiseffectsareexpected,buthigherpump
◦
then reduces, the stable angles also change accordingly. powerisrequired(seethecaseforθ0 =45 ). Suchevolu-
Withthis method, wecannumericallyfinda sequenceof tion of the power-dependent nonlinear response can also
stable angles under different pump power PI. be observed by fixing the initial twist angle but chang-
ing the pump frequency, as will be demonstrated in the
SincetheEMtorqueisanonlinearfunctionoftwistan-
experiment below.
To confirm the feasibility of the proposed nonlinear
(cid:1) e(cid:5)H(cid:7)Hq (cid:31) e(cid:13)!(cid:1)"q rotatable meta-atoms we carry out a pump-probe mi-
563 4 563 7(cid:31)
crowave experiment. To experimentally realise a strong
5g3 5g3 nonlinear or even bistable effect, the restoring torque
(cid:11)(cid:25)q543 (cid:11)(cid:25)q543 from the wire has to be sufficiently small so that the
(cid:11) (cid:11)
(cid:24)(cid:10) 93 (cid:24)(cid:10) 93 structure can be twisted by a large enough angle within
(cid:11) (cid:11)
(cid:22) e(cid:23) s3 (cid:22) e(cid:23) s3 the maximum available power. We found that rubber is
θ3 θ3 a good candidate, since the shear modulus of rubber is
3 3 of the order of 0.2MPa – 2.4MPa [24], which is at least
θ θHg d dHg g θ θHg d dHg g
three orders of magnitude smaller than it is for other
(cid:9)(cid:10)(cid:11)(cid:12)(cid:7)(cid:11)(cid:13)(cid:14)(cid:15) e/(cid:17)(cid:18)q (cid:9)(cid:10)(cid:11)(cid:12)(cid:7)(cid:11)(cid:13)(cid:14)(cid:15) e/(cid:17)(cid:18)q
polymers.
(cid:1)(cid:3)(cid:4)(cid:5)(cid:1) 543
(cid:1)(cid:31) e(cid:13)!"q& 33HH54 (cid:1)(cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)(cid:3)(cid:5)(cid:6)(cid:7)(cid:5)(cid:2)(cid:5)(cid:1)(cid:1)(cid:1) (cid:31)#7$(cid:31) (cid:24)(cid:10)(cid:11)(cid:11)(cid:25)q53633 (cid:1)(cid:1)(cid:2)(cid:2)==(cid:3)(cid:3)==(cid:7)(cid:6)(cid:5)(cid:2)==(cid:1)(cid:1) FraidgA.iu3ss.crhWe=meah3ta.i2vcmeoumfs,etdhtretawcekoxpwseeirpdiamtrhaent1etmdalmcos,eptcpuoepprpSiesRrRsthhsoicw(kinnnneeisnrs
f G(cid:31) 3 (cid:22) e(cid:23)(cid:11) s3 (cid:1)(cid:1)(cid:2)==(cid:3)(cid:3)==(cid:5)(cid:4)(cid:5)(cid:5)==(cid:1)(cid:1) 0R.4003053µsmubasntrdatselist(wǫrid=th3.g5,=los0s.2tamnmge)ntpr0i.n0t0e2d7,osnubRsotrgaetres
(cid:31)7 G(cid:31)& e(cid:14)q d3 (cid:2) e(cid:23)q thickness0.5mm). ThelowerSRRisfixedandpositioned
g3 0g 533 54g 3 3H4g 3Hg 3H0g 5 at the centre of a WR229 rectangular waveguide with
◦
(cid:22) e(cid:23)(cid:11)(cid:24)(cid:10)(cid:11)(cid:11)(cid:25)q # e")z""4q Φ = 0 , and the top SRR is suspended with a thin rub-
$
ber wire (radius a = 50µm, length d = 20mm), so that
it canrotate aboutthe commonaxis. The twoSRRs are
FIG. 2: The principle of nonlinear response in rotatable
alignedcoaxially,withafacetofacedistanceof0.75mm,
meta-atoms. (a) The mode amplitude Q2 and (b) the EM
torque MEM of the top rotatable ring. (c) The EM torque separated by air. The horizontal positions of the SRRs
at 3.5GHz for different pump powers from 0 to 1 mW/mm2 are carefully adjusted and the initial twist angle θ0 is
in 0.2mW/mm2 steps, and the restoring torque for different set at around 70◦. The mass of the suspended sample
initial twist angle θ0; (d) the corresponding paths of power- is 101mg, which leads to a 6.1% elongation of the wire.
dependenttwist angles underdifferent θ0. TheYoung’smodulusisthusestimatedas2.06MPa,and
4
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:4)v(cid:10)(cid:4)(cid:2)(cid:11)(cid:6)(cid:12)(cid:4)(cid:13)(cid:4)(cid:14)(cid:15) 5(cid:12)(cid:12)(cid:2)(cid:17)v(cid:18)(cid:7)(cid:13)(cid:6)(cid:19)(cid:2)(cid:15)(cid:7)(cid:20)(cid:14) which indicates that the twist angle is increased. When
(cid:28)rv )y(c Gl the pump frequency is at the red tail of the resonance,a
w
(cid:25)(cid:17)vsP )y(( GHql &vs(cid:8)(cid:4)(cid:5) llainregweisdptehc)trcaaln“jbuemopb”se(ravbeoduwtthherneethteimpeusmopftphoewreesropnaasnscees
(cid:6)(cid:4)(cid:14) )yoH Hq (cid:12)(cid:4)(cid:4) a certain threshold value [Fig. 4(a)]. The thresholds are
b(cid:12)(cid:4)(cid:24) )yoF s(cid:2)r s)r zl (cid:11)r different for increasing and decreasing pump powers. As
v oq oH (o (F (G oq oH (o (F (G the pump frequency approaches to the initial resonance,
the spectral “jump” becomes smaller [Fig. 4(c)] and fi-
w(cid:28)rv )y(q G( nally disappears [Fig. 4(e)]. We also observed similar
(cid:25)(cid:17)vsP )y(( GHcl &vs(cid:8)(cid:4)(cid:5) ereffdecttasil(noofttshheowannt)iswymhemnetthriecpmuomdpe,frienquwehniccyh icsaaset tthhee
(cid:4)(cid:24)(cid:6)(cid:4)(cid:14) )yoz s(cid:25)r s(cid:8)r HF (cid:12)(cid:4)(cid:4)(cid:11)r tdwiroecrteisoonnoafnctheseaEpMprotoarcqhueea.ch other due to the opposite
b(cid:12) )yoc HH
v o( oG (( (Go( oG (( (G Tofurthervalidatetheobservedeffect,wenumerically
rv simulated the exact experimental geometry and found
(cid:28)
w )y(q G( excellent agreement[21]. The estimated optimum initial
(cid:17)vsP Gq &vs(cid:8)(cid:4) twist angle is around 72.5◦, and the maximum twist an-
(cid:4)(cid:24)(cid:6)(cid:4)(cid:14)(cid:25) )y(( s(cid:4)r s-r GHHo (cid:5)(cid:12)(cid:4)(cid:4)(cid:11)r g3l.e2s1GobHtazinaneddf3o.2r3tGheHtzh)raeerepaurmoupndfre9q0u◦e,n8c5i◦esan(3d.1882G◦,Hrez-,
b(cid:12) )yoz spectively [21]. Finally we remap these angles back to
v ol oF oH (( (c ol oF oH (( (c the corresponding resonant frequencies according to the
*(cid:20)+(cid:4)(cid:12)vs(cid:8)7(cid:13)r *(cid:20)+(cid:4)(cid:12)vs(cid:8)7(cid:13)r
simulation spectra (see Fig. S3 (b), (d) and (f) of sup-
plementary file).
FIG. 4: Comparisons of resonant frequencies demonstrated
The single meta-atom used in the waveguide exper-
in a waveguide experiment and numerically calculated for an
iment induces image currents in the waveguide walls.
array. (a) and (b) pump at 3.18GHz; (c) and (d) pump at
Thus, our experimental system is analogous to an array
3.21GHz; (e) and (f) pump at 3.23GHz. The stable twist
angles forthearray system areshown on theright axes. The where neighbouring elements interact. Since the nonlin-
power in thearray is thepower incident on each unit cell. earity arises from individual rotatable meta-atoms, the
behaviourinanarraywillnotshowqualitativedifference
from a single meta-atom as predicted in the analytical
the shear modulus follows as G≈0.69MPa. The exper- model, as long as the neighbour interaction is relatively
imental realization varies slightly from Fig. 1, however weak. Weusefullnumericalsimulationtomodelanarray
the underlying physical mechanism of the nonlinear re- with thickness of a single cell and periodicity of 45mm
sponseduetothedynamiccouplingbetweenelectromag- in the transverse directions, arranged as in Fig. 1. The
neticandelasticpropertiesisthesame. Thisequivalence parametersofthe meta-atomsusedinthe simulationare
is verified in the supplemental material [21]. thesameasintheexperiment. Thearrayandthewaveg-
The transmission spectrum measurements are per- uide experiment show quite good qualitative agreement
formedbyavectornetworkanalyser(RohdeandSchwarz (see Fig. 4(b), (d) and (f) for the power-dependent reso-
ZVB-20). The CW pump signal is generated by a signal nanceandFig. S1(b)fortheEMtorque),thusjustifying
generator(HP8673B)andisfurtheramplifiedbyapower that the nonlinear behaviour observed in the waveguide
amplifier (HP 83020A)before being sentinto the waveg- experiment is similar to that of the analogous array sys-
uide. Three pump frequencies (3.18, 3.21 and 3.23GHz) tem.
are chosen in order to capture the evolution of the non- The novel nonlinear “meta-atom” described in this
linear responseatdifferentdistances fromthe initial res- workprovedtopossessasensitiveelasticfeedbackbring-
onance. The pump power is increased in 1dB steps; for ing nonlinearity to the interaction of EM modes of the
each step, the sample reaches steady state after 30 sec- resonators. The resulting nonlinearity and bistability of
onds. The mechanical rotation is quite significant and the response was successfully observed in experiments
can be visually observed in the experiment, thus ruling anditturnsoutthatthese resultscanbe accuratelypre-
outotherpossiblenonlinearmechanismssuchasheating. dictedwiththeoreticalmodelling. Wealsonotethatthis
The experimentally observed transmission spectra are structure is chiral (except in the high symmetry cases
◦ ◦
shown in Fig. S2 of the supplementary file, and the cor- of 0 and 180 angle between the rings), it should also
respondingresonantfrequenciesaredepictedinFig.4(a), exhibit nonlinear optical activity, an effect which is rela-
(c)and(e),wherethepredictedevolutionfrombistability tivelyweakinnaturalmedia[25],butcanbequitestrong
tosmoothnonlinearityisclearlyshown. Theinitialreso- in metamaterials [26].
nance(symmetricmode)withoutpumpislocatedaround Althoughtheexperimentaldemonstrationinthiswork
3.256GHz,anditred-shiftsasthepumppowerincreases, was performed in the microwave frequency range, the
5
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