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Metadevices for the confinement of sound and broadband double-negativity behavior

Christensen, Johan; Liang, Z.; Willatzen, Morten

Published in:Physical Review B Condensed Matter

Link to article, DOI:10.1103/PhysRevB.88.100301

Publication date:2013

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Christensen, J., Liang, Z., & Willatzen, M. (2013). Metadevices for the confinement of sound and broadbanddouble-negativity behavior. Physical Review B Condensed Matter, 88(10), [100301].https://doi.org/10.1103/PhysRevB.88.100301

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PHYSICAL REVIEW B 88, 100301(R) (2013)

Metadevices for the confinement of sound and broadband double-negativity behavior

J. Christensen,1,2,* Z. Liang,3 and M. Willatzen2

1Institute of Technology and Innovation, University of Southern Denmark, DK-5230 Odense, Denmark2Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

3Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong(Received 30 May 2013; revised manuscript received 26 July 2013; published 10 September 2013)

We show that the acoustic response of perforated and elastically filled rigid screens can give rise to a broadlandscape of tunable devices. We begin presenting deep-subwavelength transmission properties of a structuredplate and demonstrate the immediate relationship to truly bound surface modes. We extend our theoretical modelto analyze structured metal-fluid-metal wave guides for the confinement of sound and present exact expressions forthe dispersion relations which describe the hybridization of resonances. We discuss the validity of our analyticalmodel by direct comparison to full-wave simulations and use this technique in the search for broadband responsein composite structures where the effective mass density and bulk modulus are simultaneously negative andexhibiting weak influences by viscous losses.

DOI: 10.1103/PhysRevB.88.100301 PACS number(s): 43.35.+d, 42.79.Dj, 81.05.Xj

Acoustic metamaterials with functionalities and propertiesdesigned by structural components in the form of resonatingmeta-atoms have been the subject of intense research inrecent years because of their applications and potential usefor the control of sound and elastic waves. It is difficultto generalize such a material in terms of its design andthe mechanisms responsible for inducing the desired waveproperties, as acoustic metamaterials have been created inmany different ways. The first type of metamaterials designedwas structures consisting of locally resonating building blocksin the form of rubber-coated metal spheres.1 When theseinertial metamaterials are excited at the natural frequency,the core oscillates strongly out of phase with the drivingforce, giving rise to a negative mass density. Other typesof metamaterials in the form of Helmholtz resonators andmembranes have been employed and it thus seems that, to alarge extent, design strategies have been inspired by microwavefilter components.2–4 Recently, we also witness the emergenceof perforated structures, which constitute flexible systemswhen designing broadband and all-angle negative refractionfor sound.5–8

In this paper we investigate acoustic metadevices inspiredby plasmonic building blocks and examine their feasibilityin the design of sound confinement structures and tunablenegative dispersion.9–11 One example of a plasmonic buildingblock is a perforated metal film, which has been the focus ofextensive research activities associated with the phenomenonof extraordinary optical transmission (EOT).12 Since EOT wasalso observed in perfect conducting films not supporting thepropagation of surface plasmons, it led to the considerationthat a textured perfect conducting screen could sustain spoofsurface EM modes very similar to plasmons in a bare metalfilm. In fact it has been shown that the effective permeabilityof those holey structures can be represented by a Drudeexpression dictated by the structure parameters.9 Recently,there has been much interest in related properties for the caseof sound in structured and elastically filled screens.13–15 Itwas shown that a Drude response can also be tailored inthese materials and this fact motivates the present study toseek other plasmon-like behavior in acoustical systems. In thiscontext we present an analytical study to describe coupled

acoustoelastic surface waves in finite slabs and their link totransmission resonances. Furthermore, we build an analogyto an insulating layer sandwiched between two semi-infinitemetals which can provide negative dispersion for surfaceplasmons. We demonstrate that a fluid layer between twostructured half-spaces hosting negative mass densities sustainsound guiding behavior. Finally, we present results for double-negativity (ρeff < 0 and 1/κeff < 0) over an extended range offrequencies.

We begin by considering a structured half-space (takingh → ∞) as illustrated in the inset in Fig. 1. An elastic materialsuch as rubber is clamped to a rigid immovable frame andarranged into a periodic lattice along the x axis. The elasticmaterial is treated as an isotropic solid, i.e., the displacementsobey Navier’s equations,

ρ∂2ux

∂t2= μ

(∂2ux

∂z2+ ∂2ux

∂x2

)+ (λ + μ)

(∂2ux

∂x2+ ∂2uz

∂x∂z

),

(1)

ρ∂2uz

∂t2= μ

(∂2uz

∂z2+ ∂2uz

∂x2

)+ (λ + μ)

(∂2uz

∂z2+ ∂2ux

∂x∂z

),

where λ, μ and ρ are the modulus of incompressibility(first Lame coefficient), modulus of rigidity (second Lamecoefficient), and solid mass density, respectively. Due totranslation invariance along the y axis, we have uy = 0 and∂∂y

= 0 everywhere. Since the walls around the rubber layerare rigid, we have

uz(x = 0) = uz(x = ax) = 0,(2)

ux(x = 0) = ux(x = ax) = 0.

Further, at the fluid-rubber interface (z = 0), we must preservecontinuity of the displacement uz, the normal stress σzz =−p, and, invoking the absence of shear stresses, σxz = 0. Thisproblem is not separable and exact analytical solutions donot exist. Hence, we can only attempt to find approximateanalytical solutions. We do this by writing the ansatz

uz(x,z) = fz(z) · sin

(π

ax

x

)eiωt (3)

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FIG. 1. (Color online) Transmittance and effective mass densityspectra of the structured and elastically filled rigid screen surroundedby air. In all three cases the geometries are the same; we only varyct as indicated. All calculations are undertaken for normally incidentsound where the solid (dotted) lines represent data obtained by modalexpansions (COMSOL simulation).

and assuming that uz � ux such that the wave motion insidethe elastic slit cavity predominantly resembles the transversemotion of a thin plate fixed at the edges, x = 0 and x = ax . Tofurther justify this approximation, we have performed exactcalculations using the finite-element method in COMSOL,verifying that indeed uz � ux . Note that, in Eq. (3), we havetaken into consideration only the lowest order mode (m = 1),as it represents the field penetrating inside the cavity to a goodapproximation. Henceforth, neglecting a small contributionfrom ux , an approximate solution is sought in the form

fz(z) = A sin(βzz + φ), (4)

where φ is a phase, i.e.,

uz(x,z) = A sin(βzz + φ) · sin

(π

ax

x

)eiωt . (5)

In order to understand the wave nature of the displacementinside the elastic cavity we can solve for βz by substitutingEq. (5) into the second of Eqs. (1), which yields

βz = ω

√ρ

λ + 2μ

√1 − μ

ρ

π2

a2x

1

ω2= ω

cl

√1 − ω2

p

ω2. (6)

From Eq. (6) we can see that a cutoff frequency locates atωp =

√μ

ρπax

= ctπax

. This analytical finding is intriguing and

in fact different from acoustic hole or slit arrays where thefundamental eigenmode is a constant βz = nhk0.

In the following we derive an effective medium theory andmodal expansions by rigorously coupling the acoustoelasticfields of the present structure. Plane-wave expansions are usedin the reflection and transmission regions where the incidentplane wave points along the z axis. Inside the elastic cavities,the field is written as a linear combination of their eigenmodes.

However, as opposed to the approximate solution in Eq. (5),we consider the elastic material to be an anisotropic fluid. Wecan therefore write the z components of the fields in that regionas

uz =∑m

(Ameiβm

z z + Bme−iβmz z

)〈r|m〉,(7)

σzz = i(λ + 2μ)∑m

βmz

(Ameiβm

z z − Bme−iβmz z

)〈r|m〉,

where βmz = 1

cl

√ω2 − m2ω2

p (with m � 1) and 〈r|m〉 is thein-plane elastic field associated with mode m. As stated earlier,we impose continuity of the corresponding fields in orderto arrive at linear expressions of unknown coefficients. Webegin with a single interface [Bm = 0 in Eq. (7)] that accountsfor an incident fluid-borne sound wave radiating onto astructured half-space. We can easily extract the specular reflec-tion coefficient by taking the fundamental cavity eigenmode(m = 1) into account,

R0 = Zeff/Z0 − 1

Zeff/Z0 + 1=

βz(λ + 2μ) − ω2ρ0

kz

8ax

π2dx

βz(λ + 2μ) + ω2ρ0

kz

8ax

π2dx

, (8)

where kz =√k2

0 − k2x in free space. In other words, one can

say that the incident acoustic wave is “seeing” a materialof a given effective impedance Zeff = ρeffceff = ρeff

ωβz

, whichcan be derived from Eq. (8), when taking kx = 0. With theexpression of βz in the metamaterial limit, Eq. (6), we canderive an effective fluidic mass density ρeff = ρz = Zeff

βz

ωthat

is nothing but

ρeff = π2dxρ

8ax

(1 − ω2

p

ω2

). (9)

In the following we wish to examine the scattering propertiesof structured slabs of finite thickness and demonstrate theimmediate relationship to the effective mass density aspresented in Eq. (9). By rigorously imposing continuity ofthe displacements and stresses at either side of the structuredslab, that is, imposing continuity of the elastic cavity modes[Eq. (7) with m = 1] to waves radiated, we arrive at a simplelinear set of equations of unknown modal fields ψ and ψ ′:[(

G + iβz(λ+2μ)tan(βzh)

)iβz(λ+2μ)

sin(βzh)

iβz(λ+2μ)sin(βzh)

(G + iβz(λ+2μ)

tan(βzh)

)][

ψ

ψ ′

]=

[2ω2ρ0

k0z

S0

0

].

(10)

In (10) we write the wave vectors knz =

√k2

0 − (knx )2 with kn

x =k0x + 2π

dxn and the sum G = ∑

n

ω2ρ0

knz

|Sn|2 with

Sn =√

8

axdx

π/ax

(knx )2 − π2/a2

x

cos

(knx

ax

2

). (11)

One can solve for the modal fields in Eq. (10) and then allscattering coefficients can be determined.16 In the metama-terial limit (n = 0 and kxax 1) we can reduce S2

n suchthat it only depends on geometrical parameters and weend up with a structure factor S2

0 = Sf = 8π2

ax

dx. Further to

this, we apply the derived complex transmission tn = −ψ ′Sn

and reflection rn = −ψSn + δ0n coefficients to compute the

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effective acoustic parameters based on a retrieval technique. Asrendered in Fig. 1 we plot the normal-incidence transmittanceand the effective mass density ρeff as a function of λ/dx

for a rigid plate containing a one-dimensional lattice of slitsfilled with a rubber material of various elastic parameters. Thegeometrical constants such as the slit width ax and thicknessh are held constant, while a rubber material of fixed solidmass density ρ and longitudinal speed of sound cl is chosen asindicated in Fig. 1. By modifying the value of the transversalspeed of sound ct , as predicted by Eq. (9), we expect tospectrally control the location of the resonant transmissionpeak where the effective mass density ρeff approaches 0.Figure 1 illustrates three different cases where the loweringof ct results in the corresponding resonance frequency beinglower as well. The spectral locations of full sound transmissiondo, in call cases, correspond to zero effective mass density,where ω = ωp as described by Eq. (9), the cutoff of asurface mode, which we explain later. Comparing the presenttechnique with full-wave simulations (COMSOL), the overalltrend of resonating frequencies is the same, although slightlyshifted to long wavelengths as shown in Fig. 1.

In the following we demonstrate that the properties respon-sible for the resonant transmission peaks are well explainedby the excitation of surface modes on and through the finitestructure. For this to be true we are seeking truly boundmodes by assuming waves to decay both into free space (kz =i√

k2x − k2

0) and into the elastic inclusion (βz = icl

√ω2

p − ω2).We recall that a bare rigid plate without any structuringdoes not support surface-bound modes of this kind.16 Oneneeds to pattern the structure either by perforation to supportguided modes or by excitation of plate modes in nonrigidstructures. The present theoretical formalism allows for theanalytical expression of a dispersion relation (frequency vskx) of acoustic surface waves controlled by elastically filledscreens. Imposing the above given conditions for bound modeswe can express kx by the zero determinant in Eq. (10), which,in the deep-subwavelength limit (n = 0), is

kx = ω

c0

√1 +

(ρ0c0

ρcl

)2ω2

ω2p − ω2

S2f γ ±2, (12)

with γ = (e− iβzh

2 − eiβzh

2 )/(e− iβzh

2 + eiβzh

2 ), where the ± signdetermines the symmetry of the mode. In Fig. 2 we plot thisdispersion relation, Eq. (12), with geometrical parameters asindicated in the caption. We predict a linear evolution of thisfunction close to the sound line, but when approaching thehole cutoff (the acoustoelastic plasma frequency, ω = ωp)we enter the flat and dispersionless regime at which sounddecays evanescently away from the structured slab to whichit is confined. Interestingly, at the cutoff frequency ρeff(ω =ωp) = 0, which could lay the foundation for properties similarto those of the ε near-zero materials in electromagnetism thatare known to support tunneling and supercoupling-relatedphenomena.17,18 Also, in Fig. 2 we plot the transmittanceversus normalized frequency and kx within (outside) the soundcone representing the excitation of leaky (bound) surfacemodes. When Bragg folding these coupled bound modes intothe radiative part of the dispersion relation we exactly mapthe transmittance peaks. This connection is also given when

0.5 1.0

0.5

1.0

1.5

kxdx/

Eq. (12)

/p

10-4 1.0

FIG. 2. (Color online) Analytical and numerical dispersion rela-tions. The band diagram, transmittance as a function of ω and kx , ismapped for a structure with the parameters ax/dx = 0.6 and h/dx = 0.5and contains filling parameters as in the latter example, though withct = 56 m/s. Additionally, we plot Eq. (12), the analytical dispersionrelation for the bound coupled acoustoelastic surface mode.

plotting the transmittance for kx > k0 which coincides withthe curve given by Eq. (12) plotted in Fig. 2.

Next, we wish to examine a configuration of two separatedstructured half-spaces which is an analog to an insulating filmbetween two semi-infinite metals known to support guidedsurface plasmons for the case of light. Those structures arecapable of supporting a negative index propagating modewith high figures of merit.10,11,19 These modes are not onlytunable but can extend over a wide range of frequencies.In the present case we analyze its acoustical analogy byusing the same theoretical formalism as presented above.Free-space scattering takes place in the region (air) separatingthe structured half-spaces (see the inset in Fig. 3), which iswhere sound is confined since waves decay evanescently intothe structured slits by their fundamental cavity mode as longas ωp > ω. We focus on the lowest possible guided modewithin this regime, where we attempt to mimic the behaviorof a fluid layer guided mode. Interestingly, we are able tocreate guided modes that decay away throughout the structurehalf-spaces whether we are within or outside the sound cone.In the deep-subwavelength regime, where the wavelength λ

is larger than all geometrical parameters, we can write thedispersion relation for these trapped modes,

kx = ω

c0

√1 +

(ρ0c0

ρcl

)2ω2

ω2p − ω2

S2f γ , (13)

where γ = tanh2(√

k2x − k2

0s2 ) for kx > k0 (propagating

waves), γ = −tan2(√

k20 − k2

xs2 ) for k0 > kx (evanescent

waves), and s is the width of the gap. Note that the expressiongiven by Eq. (13) is purely real whether we are outside orwithin the sound cone. We compare the exact dispersionrelation from Eq. (13) with full-wave simulations (COMSOL)

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FIG. 3. (Color online) Dispersion relation of structured metal-fluid-metal wave guides. An air layer of thickness s separates two half-spacesstructured by slits of width ax/dx = 0.6 and filled by an elastic material with ρ = 950 kg/m3, cl = 289 m/s, and ct = 150 m/s. The confinedmodes are plotted for four values of the separation s as indicated in each panel.

where all material and geometrical parameters apart from thegap separation s are held constant as indicated in Fig. 3. Inall plotted cases we predict a strong resonance dependencewith the gap separation s. The nature of the bands canbe understood by the odd and even spoof surface modes(induced by the structured half-spaces) hybridizing with thezeroth-order sound line and the first-order fluid guided mode,respectively [see Figs. 3(a)–3(c)]. In Fig. 3(d) we see a similarbehavior, but due to a larger gap separation s/dx = 2.50, we areable to excite higher ordered modes in the plotted range. Whilethe upper edge of the spoof surface modes for all simulatedcases asymptotically approaches the limit ω = ωp, the loweredge always coincides with a λ half-resonance, s = λ/2.When we enter into the spectrally narrow and flat regimearound ωp, it is possible to trap sound with the advantageof providing omnidirectional guiding. It is also shown thatwith an enlargement of s we can broaden the entire bandwidthof the corresponding mode, as the lower edge consequentlymoves towards lower frequencies. This in turn has the fortunateadvantage of leading to stronger confinement within the gapas a consequence of shorter penetration into the structuredregions, ω ωp.

Throughout this work we have analyzed the complexinteraction between fluid-borne sound and resonators in theform of elastic cavities. We have treated the elastic materialas an anisotropic fluid and learned that this assumption holdsfairly well when making comparisons to full-wave simulationsas shown in Figs. 1 and 3. Naturally, when imposing theseassumptions one needs to deal with limitations, e.g., we tookthe in-plane cavity motion to be sinusoidal [Eq. (3)], but forvery thin structures it could as well have been a bending-typemotion. In fact we show by full-wave simulations (COMSOL)that the choice of thin plates (h = 0.25dx , for example)can lead to broader resonances. As an example, we take a

composite metamaterial consisting of two facing structuredrigid plates that host resonances with a negative index causedby double-negativity behavior.15 As shown in Fig. 4, we obtainan extended pass band of full transmission which is the broadregion where both the effective mass density ρeff and the bulkmodulus 1/κeff are negative. By the specific choice of a verythin structure, we believe that the excitation of bending modesinside the cavities leads to the expected phenomena of negative

4.0 4.5 5.0 5.5-10

-5

0

5

10

Re(

1/ef

f) and

Re

eff

0.0

0.2

0.4

0.6

0.8

1.0

Re(1/eff

) < 0

Re(eff

) < 0

= 2300kg/m3

cl = 700m/s

ct = 530m/s

ax/d

x = 0.5

h/dx = 0.25

hg/d

x = 0.1

/dx

Transm

ittance

4 5 60.0

0.5

1.0

0 - 50°

FIG. 4. (Color online) Broadband double-negativity. A compositemetamaterial consisting of two separated structured and elasticallyfilled plates is immersed in water and tuned for maximal broadbandresponse with negative effective mass density ρeff < 0 and bulk mod-ulus 1/κeff < 0 (shaded region). The structure and filling parametersare as indicated and inside the gap we take a fluid with ρg/ρ0=1and cg/c0=0.27. Dotted lines in the spectrum represent data obtainedin the presence of viscous losses. Inset: Transmittance spectrum forvarious angles in the range 0–50◦.

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index behavior, however, with an extended bandwidth. Thedotted lines in Fig. 4 represent calculations of the transmittanceand the effective parameters comprising realistic viscous lossesin the fluid regions.7 Interestingly and fortunately, the influenceof losses is marginal and the same can be said for the angularresponse as shown in the inset in Fig. 4. Supporting broadbandbehavior for many angles of incidence together with theinsensitivity to viscous losses proves the robustness of thisdouble-negativity scheme.

In summary, we have demonstrated that an acoustic surfacewave equivalent to spoof surface plasmons can be engineeredin rigid structures when they are pierced with holes and filledwith an elastic inclusion. These surface waves, when designed

for finite plates, are responsible for transmission resonancesthrough tiny holes and can likewise be guided between twofacing structured half-spaces. These new types of acousticmetadevices can thus be tuned to produce broadband reso-nances or omnidirectional propagation for a narrow selectiverange of frequencies useful for many interesting sound controlapplications.

All authors acknowledge Jensen Li for stimulating discus-sions. J.C. gratefully acknowledges financial support fromthe Danish Council for Independent Research and a SapereAude grant (No. 12-134776). Z.L. thanks the GRO programof Samsung Advanced Institute of Technology for support.

*jochri@fotonik.dtu.dk1Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, andP. Sheng, Science 289, 1734 (2000).

2N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, andX. Zhang, Nat. Mater. 5, 452 (2006).

3S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, J.Phys.: Condens. Matter 21, 175704 (2009).

4S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, Phys.Rev. Lett. 104, 054301 (2010).

5J. Christensen and F. J. Garcia de Abajo, Phys. Rev. Lett. 108,124301 (2012).

6J. Christensen and F. J. Garcia de Abajo, Phys. Rev. B 86, 024301(2012).

7Z. Liang, T. Feng, S. Lok, F. Liu, K. B. Ng, C. H. Chan,J. Wang, S. Han, S. Lee, and J. Li, Sci. Rep. 3, 1614(2013).

8Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, Phys. Rev.Lett. 110, 175501 (2013).

9J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, Science305, 847 (2004).

10H. Shin and S. Fan, Phys. Rev. Lett. 96, 073907 (2006).11H. J. Lezec, J. A. Dionne, and H. A. Atwater, Science 316, 430

(2007).12T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff,

Nature 391, 667 (1998).13S. Yao, X. Zhou, and G. Hu, New J. Phys. 12, 103025 (2010).14R. Hao, C. Qiu, Y. Ye, C. Li, H. Jia, M. Ke, and Z. Liu, Appl. Phys.

Lett. 101, 021910 (2012).15Z. Liang, M. Willatzen, J. Li, and J. Christensen, Sci. Rep. 2, 859

(2012).16J. Christensen, L. Martin-Moreno, and F. J. Garcia-Vidal, Phys.

Rev. Lett. 101, 014301 (2008).17M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006).18R. Fleury and A. Alu, Phys. Rev. Lett. 111, 055501 (2013).19J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, Opt.

Express. 16, 19001 (2008).

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