Measuring monopole and dipole polarizability of acoustic meta-atomsJoshua Jordaan,1 Stefan Punzet,1, 2, 3 Anton Melnikov,4, 5, 6 Alexandre Sanches,1, 7 Sebastian Oberst,5 SteffenMarburg,4 and David A. Powell6, 1, a)

1)Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University,Canberra, ACT 2601, Australia.2)Faculty of Electrical Engineering and Information Technology, Ostbayerische Technische Hochschule Regensburg,Seybothstraße 2, 93053 Regensburg, Germany3)Department of Electrical and Computer Engineering, Technical University of Munich, Theresienstr. 90,80333 Munich, Germany4)Chair of Vibroacoustics of Vehicles and Machines, Technical University of Munich, Boltzmann Str. 15,85748 Garching, Germany5)Centre for Audio, Acoustics and Vibration, University of Technology Sydney, NSW 2007,Australia6)School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2610,Australia.7)School of Engineering, University of Sao Paulo, Av. Prof. Luciano Gualberto, 380 - Butanta, CEP 05508-010,Sao Paulo, SP, Brazil

We present a method to extract monopole and dipole polarizability from experimental measurements oftwo-dimensional acoustic meta-atoms. In contrast to extraction from numerical results, this enables allsecond-order effects and uncertainties in material properties to be accounted for. We apply the technique to3D-printed labyrinthine meta-atoms of a variety of geometries. We show that the polarizability of structureswith shorter acoustic path length agrees well with numerical results. However, those with longer path lengthssuffer strong additional damping, which we attribute to the strong viscous and thermal losses in narrowchannels.

Acoustic metasurfaces are metamaterial structureswith sub-wavelength thickness that can implement a richvariety of acoustic functions1,2. A promising approachfor metasurfaces is the design of structures with inter-nal labyrinthine configuration to slow down the acousticwave’s velocity to create compact resonators3,4. Struc-tures of this kind exhibit excellent wavefront shapingpotential1,5–7. Such meta-atoms can generate phaseshifts up to 2π by adjusting their geometry7. Thereby, awave manipulation function can be realized with a cor-responding phase gradient, which is then discretized toenable implementation with an array of meta-atoms.

Drawing inspiration from electromagnetism, the domi-nant design paradigm for acoustic metasurfaces has beenthe generalized Snell’s law5,8, where structures are de-signed for high amplitude, with spatially varying phase,both for transmission or reflection problems. However,in electromagnetism, it has been shown that the general-ized Snell’s law does not correctly account for impedancematching and energy conservation. Approaches based onsurface impedance must be used instead9,10 and equiv-alent electric and magnetic surface impedances needto be defined. Recently these more accurate surface-impedance models have also been applied to acousticmetasurfaces11,12. The impedances can be derived fromthe multipole moments of a single meta-atom13. In theacoustics of fluids, the fundamental moments are themonopole and dipole, corresponding to the net compres-sion and displacement of a fluid volume respectively. The

a)Electronic mail: david.powell@adfa.edu.au

acoustic response of sub-wavelength meta-atoms is well-approximated by their monopole and dipole polarizabil-ity coefficients. These coefficients relate the strength ofthe monopole and dipole moments to the incident pres-sure and velocity fields respectively. Developing a modelbased on polarizability can lead to great simplificationsin modelling, particularly for complex arrangements ofmeta-atoms.

An alternative to a continuously connected metasur-face is the use of sparse arrays of disconnected reso-nant meta-atoms1,14, which can enable highly efficientbeam refraction at large angles15. These elements mayfind their application in creating sound control struc-tures which also allow airflow. Here, the monopole anddipole polarizabilities of the meta-atoms are the mostnatural model to apply. To date these polarizabilitieshave not been directly measured; with most designs rely-ing on simulations or indirect observations of resonancesattributed to the monopolar and dipolar modes14,16.

In this work we present a technique for directly ex-tracting the acoustic monopole and dipole polarizabilityof two-dimensional meta-atoms from experimental mea-surements. In addition, the method can be applied tonumerically extracted data. Obtaining polarizability in-formation from experimental measurements is necessaryfor good accuracy, since numerical simulation may ne-glect viscous and thermal boundary layers, the excita-tion of vibration modes in thin structures, and it may bedifficult to obtain reliable material properties for rapidprototyping materials.

In this work we consider the experimental configura-tion shown in Fig. 1, similar to that used in previousworks5,17. Two plates separated by a 66 mm gap form a

arX

iv:1

809.

0452

6v2

[ph

ysic

s.ap

p-ph

] 4

Dec

201

8

2

MicrophoneMicrophone belt

Anechoic foam boundary

LoudspeakerSample

Incident field

Inaccessible region

Accessible scan regions

x

y

Scattered field

(b)

FIG. 1. (a) Side-view of the parallel plate waveguide appa-ratus, where the lid has been raised to access the sample.(b) Schematic diagram of the apparatus and sample (not toscale), showing a top-down view of components within thewaveguide.

parallel-plate acoustic waveguide, with uniform pressuredistribution in the vertical direction, making it effectivelya two-dimensional (2D) system. A loudspeaker at oneend acts as the acoustic source while foam wedges forman absorbing boundary. A microphone mounted on a beltsystem can be scanned to any position within the x − yplane of the 2D acoustic system. A micro-controller de-velopment board with audio peripheral (Teensy 3.2 withAudio Adapter Board18) digitally generates and coher-ently detects the sinusoidal waves. Using the internallygenerated source as a phase reference eliminates the needfor a two microphone measurement. The amplitude andphase response of the speakers, microphone and amplifierare unknown, but are eliminated in the extraction proce-dure outlined below, by explicitly measuring the incidentfield as a reference. Details of the construction and initialcharacterization of this system can be found in Refs. 19and 20.

For the corresponding numerical analysis, we use a cus-tom 2D boundary element method (BEM) code. Thistreats all solids as acoustic hard boundaries, ignoreslosses in air, and solves for the scattered pressure onthe surface of the object. The code uses continuous el-ements with quadratic interpolation functions and dis-

cretization by collocation method with an adaptive inte-gration scheme21.

The experimental apparatus shown in Fig. 1 essentiallyyields the scalar pressure field throughout a plane. Theparallel plate waveguide is operated in a regime whereit supports only the fundamental mode with uniformpressure along the z direction. The first higher ordermodewith inhomogeneous pressure distribution in the zdirection22 becomes propagating at 2,598 Hz, setting theupper limit of measurement. The pressure field pinc of anarbitrary incident wave propagating within a 2D systemcan be expanded as

pinc(r, θ) =

∞∑n=−∞

βnJn(k0r)einθ, (1)

where Jn is the Bessel function of the first kind, k0 = ω/cis the propagation wavenumber, ω and c are the incidentwave’s natural frequency and the speed of sound in air,(r, θ) are the polar coordinates relative to the center ofthe sample, i is the imaginary unit and βn are the ex-pansion coefficients. These expansion coefficients can befound analytically only for specific incident field profilessuch as a plane wave23.

In our apparatus the incident field generated by thesingle speaker shows significant curvature of the wave-fronts, and a spatially inhomogeneous amplitude, thusit is not well approximated by a plane wave. Further-more the source amplitude varies strongly with frequencydue to the speaker response, requiring that βn are fit-ted to the measured acoustic field. For the calculationof polarizability, both the incident pressure and the ve-locity at the center of the scatterer can be directly re-trieved from the expansion coefficients as pinc(0) = β0

and vinc,y(0) = −β1+β−1

2cρ0, with ρ0 being the density of

air.The corresponding scattered pressure field pscat can be

expanded as

pscat(r, θ) =

∞∑n=−∞

γnH(1)n (k0r)e

inθ, (2)

where H(1)n is the Hankel function of the first kind. The

expansion coefficients γn can be related to the domi-nant monopole and dipole moments as M = 4i

ω2 γ0 and

Dy = 4cω3 (γ1 + γ−1). For a scatterer of arbitrary shape,

the scattering process is described by a full tensor γn =∑m Snmβn. However, for objects with approximate cir-

cular symmetry, all terms n 6= m are zero, and the rela-tionship becomes scalar: γn = Snnβn with S−n−n = Snn.

For sub-wavelength meta-atoms, the monopolar anddipolar terms are expected to dominate scattering. Foranalytical modelling of collections of meta-atoms it ismore convenient to use monopole and dipole polarizabil-ity coefficients αpp and αvv satisfying

M = αpppinc(0), D = αvvvinc(0), (3)

3

where αvv becomes a scalar for the rotationally symmet-ric structures considered here. However, by normalizingthese polarizabilities, we find that they are trivially re-lated to the scattering coefficients:

α′pp =ω2

4iαpp = S00 (4)

α′vv =−ω3

8c2ρ0αvv = S11 = S−1−1 (5)

We use this normalization since it gives a simple physicalinterpretation of the strength of different types of polar-izability in terms of contribution to scattering, with amaximum magnitude of unity.

To experimentally measure the acoustic polarizability,we need to determine the incident field coefficients βnand the scattered field coefficients γn for n ∈ {0, 1,−1}.The incident field is measured on a circle of radius Rinc.Applying the orthogonality of exponential functions toEq. (1), we find the incident field coefficients as:

βn =1

2πJn(k0Rinc)

∫ π

−πpinc(Rinc, θ)e

−inθdθ. (6)

Note that the Bessel functions have zeros which makeEq. (6) singular, the first of which occurs at k0Rinc ≈2.4. Thus Rinc is chosen sufficiently small to remain wellbelow this singular condition at the highest frequency ofinterest.

For determination of the scattered field coefficients, wemeasure both the total field ptot and incident field pinc

at the same radius Rscat, with the scattered field givenby their difference pscat = ptot− pinc. Integrating Eq. (2)and applying orthogonality conditions, the scattered fieldcoefficients are given by

γn =1

2πH(1)n (k0Rscat)

∫ π

−πpscat(Rscat, θ)e

−inθdθ. (7)

As the Hankel function has no real zeros, there is morefreedom to choose Rscat. Referring to Fig. 1(b), we seethat when the sample is placed within the waveguide,there is an inaccessible region of width w where the fieldcannot be measured, as the belt on which the microphoneis mounted would collide with the sample. Therefore wemust measure over a reduced angular range, and approx-imate the angular integral as

12π

∫ 2π

0

· · · dθ ≈ 12π−4θin

(∫ π−θin

θin

· · · dθ +

∫ 2π−θin

π+θin

· · · dθ

),

(8)where θin = arcsin w

2Rscatis the angular half-width of the

inaccessible region. Since the range of inaccessible anglesreduces with increasing Rscat, larger values are preferredfor increased accuracy. This reduced angular range of in-tegration means that we do not have exact orthogonalitybetween different orders n. However, for sub-wavelengthmeta-atoms with dominant monopolar and dipolar radi-ation the scattered field will have relatively smooth an-gular variation, and we do not expect significant interfer-ence from higher order terms with |n| > 1.

I II

R

twIII IV

FIG. 2. Top row: Cross-section of each meta-atom design.Meta-atom I was taken from Ref. 14, IV from Ref. 24. Thediagram of meta-atom IV defines the parameters wall thick-ness t, channel width w and meta-atom radius R. For I:R = 50 mm, w = 4 mm and t = 1 mm. II: R = 25 mm,w = 2 mm, t = 0.5 mm. III: R = 25 mm, w = 4 mm,t = 1 mm. IV: R = 40 mm, w = 6 mm, t = 2 mm. Bot-tom row: Photographs of the 3D printed meta-atoms, madefrom PLA with 0.1 mm layer thickness to a height of 66 mm.

The ratio of the scattered field coefficients γn to theincident field coefficients βn gives the corresponding scat-tering coefficient Snn according to Eqs. (4) and (5). Sincewe have two equivalent expressions for the dipole polariz-ability α′vv, their average is taken to reduce the influenceof measurement uncertainties.

The developed extraction procedure is applied to in-dividual 2D acoustic meta-atoms based on labyrinthinedesigns with eight-fold rotational symmetry. Four meta-atoms were fabricated and characterized, two of themhaving geometries previously reported in Refs. 14 and24. Diagrams of the designs are shown in Fig. 2, withphotographs of the fabricated meta-atoms shown below.All of the designs were fabricated by 3D printing usingPLA filament with a 0.1 mm layer thickness, to a heightof 66 mm. The top of each meta-atom was left open tosimplify fabrication, and to allow verification of the fabri-cation quality. Initial experiments with this configurationshowed poor agreement with the numerical results, dueto imperfect contact between the meta-atom and the topwaveguide plate. We attributed these poor initial resultsto sound leakage from a small gap between the meta-atomand top plate, and the excitation of vibrational modes inthe meta-atoms. To suppress these effects we inserted athin rubber sheet between the meta-atom and top plate,which greatly improved agreement, as detailed below.

The experimentally measured (dashed curves) and nu-merically calculated (solid curves) polarizabilities areshown in Fig. 3. Note that we consider only frequenciesabove 500 Hz, since data at lower frequencies are inaccu-rate due to the poor performance of the absorbing bound-aries at long wavelengths, as well as high backgroundnoise levels in that frequency range. Overall the agree-ment is reasonable, but the experimental results containsmall spurious peaks and ripples, and in certain casesresonant features predicted in the numerical results are

4

1.0

0.5

0.0

0.5I

Monopole polarizability ′pp Dipole polarizability ′

vv

1.0

0.5

0.0

0.5II

1.0

0.5

0.0

0.5III

Re (num)Im (num)Re (exp)Im (exp)

500 1000 1500 2000 2500f (Hz)

1.0

0.5

0.0

0.5IV

500 1000 1500 2000 2500f (Hz)

FIG. 3. Experimentally measured (dashed lines) and numeri-cally calculated (solid lines) normalized monopole and dipolepolarizability.

heavily suppressed. A possible explanation for the rip-ples are multiple reflections between the meta-atom andthe speaker. These multiple reflections result in an effec-tive incident field that differs from that field, measuredin the absence of the meta-atom. On the other hand, thesuppression of certain resonances is a genuine physicaleffect, as discussed below.

Consider first the numerical results for meta-atomI, corresponding to the geometry reported in Ref. 14.In that work, the authors performed a semi-analyticalderivation of the multipole expansion coefficients of thefield scattered by a plane-wave (comparable to Snn).They predicted a fundamental monopole resonance at518 Hz and a dipole resonance at 1,080 Hz, as well as asecond monopole resonance at 1,549 Hz. We see that ournumerical results are fairly consistent with these semi-analytical predictions, however the numerically deter-mined resonant frequencies are somewhat lower. Thishas the effect of pushing the fundamental monopole res-onance below our measurement range. To enable us

to characterize this resonance, we designed meta-atomII, where all geometric parameters are downscaled by1/2. The numerically predicted fundamental dipole andmonopole resonances are now moved to double the fre-quency, and the fundamental monopole resonance iswithin the measurable range.

Comparing the experimental results of meta-atoms Iand II to their numerical results, indicates the presenceof resonant peaks, which are suppressed to the level ofthe experimental uncertainties. This is most likely dueto the viscous and thermal boundary layers, which arenot included in our boundary element model. These lay-ers have been shown to dominate the acoustic response inchannels25, lowering the quality factor of resonances. Toreduce the influence of losses, we created meta-atom III.This had the same channel thickness w and wall thick-ness t as meta-atom I, but with the external radius beingscaled by 1/2. Due to the greatly reduced acoustic path-length, the fundamental monopolar resonance is therebyincreased to approximately 1,900 Hz, and the dipolar res-onance is outside the measurable frequency range. Forthis structure, it can be seen that the numerical resultsagree much better with the experiments.

To further investigate the applicability of our extrac-tion method, we also fabricated meta-atom IV, whichwas originally presented in Ref. 24 with a predicted fun-damental monopole resonance at 1,360 Hz. Meta-atomIV has thicker walls and wider channels than the othermeta-atoms; thus it is expected that this structure is lesssusceptible to boundary layer effects. It can be seen thatthe experimental and numerical results are in good agree-ment for this structure. The resonant frequency is lowerthan predicted in Ref. 24, being approximately 1,200 Hzin both numerics and experiment. The good agreementbetween numerical and experimental results for this sam-ple further validates our approach.

To confirm that the viscous and thermal boundary lay-ers are responsible for suppressing the scattering peaks ofmeta-atoms I and II, we apply the theory of Stinson26 tocalculate the complex wavenumber keff within the narrowchannels. We consider one of the channels in each meta-atom, and approximate it as a straight rectangular pipewith length leff given by the acoustic path length fromthe center cavity to the exterior. The damping withinthe pipe yields its dissipative quality factor as22

Qdiss =Re{keff}2Im{keff}

. (9)

We also take into account the leakage of energy throughthe external aperture by each channel. This can be cal-culated from the radiation resistance of an unflangedpipe22, to yield the radiative quality factor

Qrad =4πleff

SRe{keff}, (10)

where S is the cross-sectional area of the aperture.These quality factors were calculated for the lowest or-

5

TABLE I. Estimated dissipative and radiative quality factorsof the lowest frequency monople resonances shown in Fig. 3.

Structure I II III IV

Qdiss 36 12 43 43

Qrad 146 142 37 39

der monopole mode of each meta-atom within our exper-imental frequency range, and are listed in Table I.

The dissipative and radiative quality factors canbe combined to find the total quality factor Qtot =(Q−1

diss +Q−1rad

)−1, which determines the total rate of en-

ergy loss from the cavity. However, to interpret the scat-tering response, we need to consider the individual con-tribution from each of these terms. By reciprocity, theradiative quality factor Qrad also determines how long ittakes an incident wave to couple into the structure. Qdiss

determines the time scale over which energy is dissipatedinternally. If Qdiss � Qrad, then internal dissipation willdominate over radiation of energy, and resonant scatter-ing will be suppressed. As can be seen in Table I, formeta-atoms I and II, internal dissipation dominates overradiation, which explains the strong suppression of theresonant scattering peaks. In contrast, for meta-atomsIII and IV, radiative and dissipative losses are compara-ble, thus the experimentally observed scattering is onlymoderately suppressed compared to the simulated values.

In conclusion, we presented a method for extractingthe monopole and dipole polarizability from experimentalmeasurements of two-dimensional acoustic meta-atoms.We applied this method to labyrinthine meta-atoms pre-viously reported in the literature. For structures withthin walls and long acoustic path length, the resonancespredicted numerically were highly damped, and were es-sentially unobservable in the experiment. We attributethis to the viscous and thermal boundary layers, whichhave thickness comparable to the width of the narrowchannels. When applying our method to structures withshorter acoustic path lengths and wider channels, wefound good agreement with numerical results.

We acknowledge useful discussions with Andrea Aluand Li Quan. AM acknowledges the financial supportprovided by SO over the UTS Centre for Audio, Acous-tics and Vibration (CAAV) international visitor funds.DP acknowledges funding from the Australian ResearchCouncil through Discovery Project DP150103611.

1J. Zhao, Manipulation of Sound Properties by Acoustic Meta-surface and Metastructure, Springer Theses (Springer Singapore,Singapore, 2016).

2S. A. Cummer, J. Christensen, and A. Alu, “Controlling soundwith acoustic metamaterials,” Nature Reviews Materials 1, 16001(2016).

3Z. Liang and J. Li, “Extreme acoustic metamaterial by coilingup space,” Physical Review Letters 108, 114301 (2012).

4Y. Xie, A. Konneker, B.-I. Popa, and S. A. Cummer, “Taperedlabyrinthine acoustic metamaterials for broadband impedancematching,” Applied Physics Letters 103, 201906 (2013).

5Y. Li, X. Jiang, R.-q. Li, B. Liang, X.-y. Zou, L.-l. Yin, andJ.-c. Cheng, “Experimental realization of full control of reflectedwaves with subwavelength acoustic metasurfaces,” Physical Re-view Applied 2 (2014), 10.1103/PhysRevApplied.2.064002.

6Y. Xie, W. Wang, H. Chen, A. Konneker, B.-I. Popa, and S. A.Cummer, “Wavefront modulation and subwavelength diffractiveacoustics with an acoustic metasurface,” Nature Communica-tions 5, 5553 (2014).

7C. Shen, Design of Acoustic Metamaterials and Metasurfaces,Ph.D. thesis, North Carolina State University, Raleigh, NorthCarolina (2016).

8K. Tang, C. Qiu, M. Ke, J. Lu, Y. Ye, and Z. Liu, “Anomalous re-fraction of airborne sound through ultrathin metasurfaces,” Sci-entific Reports 4 (2014), 10.1038/srep06517.

9K. Achouri, M. Salem, and C. Caloz, “General metasurface syn-thesis based on susceptibility tensors,” IEEE Transactions onAntennas and Propagation 63, 2977–2991 (2015).

10N. M. Estakhri and A. Alu, “Wave-front transformation withgradient metasurfaces,” Physical Review X 6, 041008 (2016).

11A. Dıaz-Rubio and S. A. Tretyakov, “Acoustic metasurfaces forscattering-free anomalous reflection and refraction,” Physical Re-view B 96, 125409 (2017).

12J. Li, C. Shen, A. Dıaz-Rubio, S. A. Tretyakov, and S. A.Cummer, “Systematic design and experimental demonstrationof bianisotropic metasurfaces for scattering-free manipulation ofacoustic wavefronts,” Nature Communications 9, 1342 (2018).

13E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Hol-loway, “Averaged transition conditions for electromagnetic fieldsat a metafilm,” IEEE Transactions on Antennas and Propagation51, 2641–2651 (2003).

14Y. Cheng, C. Zhou, B. G. Yuan, D. J. Wu, Q. Wei, and X. J. Liu,“Ultra-sparse metasurface for high reflection of low-frequencysound based on artificial Mie resonances,” Nature Materials 14,1013–1019 (2015).

15L. Quan, Y. Ra’di, D. L. Sounas, and A. Alu, “Maximum Williscoupling in acoustic scatterers,” Physical Review Letters 120(2018), 10.1103/PhysRevLett.120.254301.

16A. O. Krushynska, F. Bosia, M. Miniaci, and N. M. Pugno,“Spider web-structured labyrinthine acoustic metamaterials forlow-frequency sound control,” New Journal of Physics 19, 105001(2017).

17L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Design and mea-surements of a broadband two-dimensional acoustic lens,” Phys-ical Review B 84, 024305 (2011).

18“Audio adaptor board for Teensy 3.0 - 3.6,” https://www.pjrc.

com/store/teensy3_audio.html.19J. Jordaan, Acoustic Meta-Atoms: An Experimental Determina-

tion of the Monopole and Dipole Scattering Coefficients, Finalyear project thesis, Australian National University (2017), avail-able online from http://dx.doi.org/10.13140/RG.2.2.23539.

43043.20S. Punzet, Design and Construction of an Acoustic Scanning

Stage, Internship report, Ostbayerische Technische HochschuleRegensburg University of Applied Science (2016), available onlinefrom http://dx.doi.org/10.13140/RG.2.2.36426.06081.

21S. Marburg, “Boundary element method for time-harmonicacoustic problems,” in Computational Acoustics, edited byM. Kaltenbacher (Springer International Publishing, Cham,2018) pp. 69–158.

22L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders,Fundamentals of Acoustics (John Wiley & Sons, 2000).

23C. C. Mei, Mathematical Analysis in Engineering: How to Usethe Basic Tools (Cambridge University Press, Cambridge, 1997).

24G. Lu, E. Ding, Y. Wang, X. Peng, J. Cui, X. Liu, and X. Liu,“Realization of acoustic wave directivity at low frequencies with asubwavelength Mie resonant structure,” Applied Physics Letters110, 123507 (2017).

25G. P. Ward, R. K. Lovelock, A. R. J. Murray, A. P. Hibbins, J. R.Sambles, and J. D. Smith, “Boundary-layer effects on acoustictransmission through narrow slit cavities,” Physical Review Let-

6

ters 115, 044302 (2015).26M. R. Stinson, “The propagation of plane sound waves in narrow

and wide circular tubes, and generalization to uniform tubes ofarbitrary cross-sectional shape,” The Journal of the AcousticalSociety of America 89, 550–558 (1991).