Acoustic frequency combs using gas bubble cluster oscillations in liquids: a proof of concept
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p Acoustic frequency combs using gas bubble cluster oscillations in liquids
Ivan S. Maksymov ∗ and Bui Quoc Huy Nguyen Optical Sciences Centre, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
Sergey A. Suslov
Department of Mathematics, Swinburne University of Technology,Hawthorn, Victoria 3122, Australia
We propose a new approach to the generation of acoustic frequency combs (AFC)—signals withspectra containing equidistant coherent peaks. AFCs are essential for a number of sensing andmeasurement applications, where the established technology of optical frequency combs suffers fromfundamental physical limitations. Our proof-of-principle experiments demonstrate that nonlinearoscillations of a gas bubble cluster in water insonated by a low-pressure single-frequency ultrasoundwave produce signals with spectra consisting of equally spaced peaks originating from the interactionof the driving ultrasound wave with the response of the bubble cluster at its natural frequency.The so-generated AFC posses essential characteristics of optical frequency combs and thus, similarto their optical counterparts, can be used to measure various physical, chemical and biologicalquantities.
I. INTRODUCTION
Optical frequency combs—optical spectra composed ofequidistant narrow peaks—enable precision measurementin both fundamental and applied contexts [1, 2]. An op-tical frequency comb acts as a spectrum synthesizer thatenables the precise transfer of phase and frequency infor-mation from a stabilised reference to optical signals. Theso-generated signals can be used, for example, to obtainthe spectral response of a gas or liquid sample due tolinear or nonlinear absorption of light by the medium [3].One can also accurately measure distances by passing anoptical frequency comb signal through an interferome-ter and then analysing the resulting interference pattern,which is beneficial for the fields of satellite positioningand material science [4].However, using optical frequency combs is not alwayspossible because of a number of fundamental and tech-nical limitations. For example, in liquid samples such asbiological fluids light can be strongly reflected and ab-sorbed by the medium. Photoacoustic frequency combspectroscopy may help to partially resolve these prob-lems [5, 6], because this technique exploits absorption oflight and concomitant generation of acoustic waves thatcarry information about the absorption strength. Yet,more versatile and technologically simple approaches arestill required.Similar to optical frequency combs, acoustic(phononic) frequency combs (AFC)—purely acous-tic signals with spectra containing equidistant coherentpeaks—exploit the ability of acoustic waves to provideprecision information about the medium in whichthey propagate [7–13]. In contrast to light, acousticwaves can propagate in water and opaque liquids overlong distances, which underpins many acoustics-based ∗ [email protected] FIG. 1. (a) Schematic diagram of the suggested AFC gen-eration. The oscillations of the bubble cluster are drivenby a single-frequency ultrasound pressure wave. Acousticwaves scattered by the bubble cluster are recorded and post-processed to obtain a spectrum consisting of the equidistantpeaks. (b) Schematic of the experimental setup. Bubbles arecreated in a stainless steel tank using a bubble generator. Thedriving pressure wave is emitted by an ultrasonic transducer.Waves scattered by the bubble are detected by a hydrophone.(c) Photograph of typical gas bubbles emitted by the bubblegenerator in a water tank with transparent walls at otherwiseidentical experimental conditions to those in the stainless steeltank. The diffuser of the bubble generator and other elementsof the setup can be seen. technologies including sonar, underwater communicationand sensing [11] and marine biology [14]. Yet, eventhough acoustic frequency combs have already been usedto accurately measure distances between underwaterobjects [11], AFC research remains under-established.The development of new types of acoustic combs isneeded for sensing and imaging systems [7, 10], in
FIG. 2. (a) Experimental spectra of a cluster of gas bubbles in water insonated with the 24.6 kHz sinusoidal signal of increasingpressure amplitude α =1.15, 3.75, 4, 4.2, 4.3, 7.5 and 11.5 kPa. The frequency axis is normalised with frequency f of the drivingfield. The scattered pressure values (in dB) are shown along the vertical axis with the vertical offset of 30 dB between spectra.(b) Calculated spectra of a single gas bubble with 1.95 mm radius at the same driving pressure frequency and amplitudes asin the experiment. The vertical offset between individual spectra is 100 dB. In both panels, the vertical dashed lines mark thepeaks at the natural frequency and its ultraharmonics (the left parts of the spectra) as well as the frequencies of the sidebandpeaks around the fundamental and second harmonic frequency of the driving signal. particular, biomedical imaging [12, 15, 16].In this work, we demonstrate the possibility of the AFCgeneration using a gas bubble cluster nonlinearly oscillat-ing in water [17], when it is driven by a single-frequencyultrasound wave [Fig. 1(a)]. Unlike in the scenario ofan optical frequency comb generation using a high-powerlaser light and exploiting fundamentally weak nonlinear-optical effects [12], we show that the application of low-pressure harmonic signals can trigger a strong nonlin-ear response of the cluster resulting in the generationof multiple ultraharmonic frequency peaks. The inter-action with the noise-induced bubble cluster oscillationsat its natural frequency, which is typically much lowerthan that of a driving ultrasound, results in the ampli-tude modulation of the the bubble cluster response andthe appearance of sidebands around the main peaks.Our current findings contribute to further developmentof an emergent field of AFC generation [7, 10–13]. Theyalso extend our previous observation of frequency combsoriginating from the onset of Faraday waves in verticallyvibrated liquid drops [18, 19]. However, in that systemthe spacing between the peaks of the comb was only 20–40 Hz. Whereas frequency combs with a Hz-range spac-ing can find certain applications [13], in the gas bubblesystem investigated in the present work we use the high-kHz range that can potentially be extended to the high- MHz range [20]. This opens up opportunities for usingacoustic combs instead of optical ones or in addition tothem in a number of practical situations where operationat higher frequencies may be required [12]. II. METHODSA. Experiment
Our experimental setup shown in Fig. 1(b) consistsof a 1.5 L thin-walled stainless steel tank filled with dis-tilled degassed water maintained at room temperature. Ageneric piezoceramic disc transducer with the measuredresonance frequency of 42 . ± . FIG. 3. (a) Measured acoustic response of the gas bubble cluster and (b) calculated acoustic response of a single equivalentbubble corresponding to a sinusoidal pressure wave with the frequency f = 24 . α = 11 . T = 1 /f nat ≈ . high voltage that in our case is produced by resonancetuning of the LC circuit on the frequency of interest.Hence, in our measurements we fix the frequency of thedriving ultrasound wave and change its pressure ampli-tude because this does not require re-tuning the inductorcoil.The hydrophone is based on a small piezoceramicdisc (type PIC155, PICeramic, Germany). Electric sig-nals produced by the piezo disc are first amplified usinga broadband voltage amplifier (BWD 603B, Australia)with the frequency response from 0 to 100 kHz. Then thesignal is sent to a digital oscilloscope (Rigol, DS-1202ZE,China) controlled via a laptop computer.We use a customised bubble generator consisting ofan air pump connected to a silicone tubing terminatingin a diffuser made of a piece of porous material. Thegenerator produces several single bubbles per second withthe radius of 1 . ± . R c = 20 ± χ ≈ . √ χR c is larger thanthe radius of the largest bubble in the cluster. Accord-ing to Eq. (6), this means that the detected frequency f c should be smaller than the natural frequency of indi-vidual bubbles, which is indeed confirmed by our exper-imental observations. B. Model
Modelling bubble clusters is a challenging task giventhat their geometry varies from experiment to experi-ment and with time. Therefore, models considering acluster as a single equivalent bubble of the size largerthan that of constituent bubbles are frequently used[21, 22, 24], especially when of interest is the natural os-cillation frequency of the cluster as a whole, which is thecase in the current work. However, when doing so, oneneeds to keep in mind that the ultrasound energy absorp-tion and scattering characteristics of a cluster may differfrom those of an equivalent single bubble [21, 25]. Thescattering ( σ scat ) and absorption ( σ abs ) cross-sections ofa single bubble placed in the field of an incident planeultrasound wave are defined as the ratios of the scat-tered and absorbed powers, respectively, to the power ofthe incident wave [26, 27]. The extinction cross-section σ ext = σ scat + σ abs characterises the incident wave en-ergy loss due to its absorption and scattering by the bub-ble. Generally speaking, a large gas bubble behaves as astrong acoustic scatterer with σ scat proportional to thesquare of the bubble radius [25]. Because σ abs also scaleswith the square of the radius squared [26, 27], largerbubbles have larger σ ext . However, the scattering cross-section of a bubble cluster is also proportional to the airfraction inside the cluster: σ scat c = χR c . Because inour case χ ≈ . R ¨ R + 32 ˙ R = 1 ρ (cid:16) P ( R, ˙ R ) − P ∞ ( t ) (cid:17) , (1)where P ( R, ˙ R ) = (cid:18) P − P v + 2 σR (cid:19) (cid:18) R R (cid:19) κ − µ ˙ RR − σR (2)and the expression P ∞ ( t ) = P − P v + α sin( ωt ) with ω = 2 πf represents the periodically varied pressure inthe liquid far from the bubble. The parameters R , R ( t ), µ , ρ , κ , σ , α , and f denote, respectively, the equilibriumand instantaneous bubble radii, the dynamic viscosityand the density of the liquid, the polytropic exponent ofa gas entrapped in the bubble, the surface tension of agas-liquid interface and the amplitude and the frequencyof a driving ultrasound wave. The diffusion of the gasthrough the bubble surface is neglected.In our model oscillations of the bubble are not affectedby fluid compressibility, and we can express the acoustic power scattered by the bubble into the far-field zone as[17] P scat ( R, t ) = ρRh (cid:16) R ¨ R + 2 ˙ R (cid:17) , (3)where h ≫ R is the distance from the centre of thebubble. The natural frequency of the bubble is [17] f nat = 12 π √ ρR s κ (cid:18) P − P v + 2 σR (cid:19) − σR − µ ρR ≈ f M (cid:18) κ − σR − µ κR ( P − P v ) (cid:19) , (4)where f M = p κ ( P − P v )2 π √ ρR (5)is the well-known Minnaert frequency [32]. We usethe following fluid parameters corresponding to waterat 20 ◦ C: µ = 10 − kg m/s, σ = 7 . × − N/m, ρ = 10 kg/m and P v = 2330 Pa. In our computationswe take the air pressure in a stationary bubble to be P = 10 Pa and the polytropic exponent of air to be κ = 4 / − and thus can be neglected. The natural frequencyof a bubble cluster is given by f c ≈ f M √ χ R R c , (6)where R c is the radius of the bubble cluster and χ is theair fraction in the liquid [24]. In our experiments, weestablished that √ χR c > R .Equation (1) was solved numerically using an explicitRunge-Kutta method [35] implemented in a standardsubroutine ode45 in the Octave software. The numer-ical solution was used to obtain the acoustic scatteringspectra calculated using Eq. (3). In the solver configura-tions, the numerical values of the absolute and relativeerror tolerances were set to machine accuracy. III. RESULTS
We demonstrate experimentally fundamental physicsbehind the principle of the AFC generation, which wesuggest, by studying a cluster of bubbles created using abubble generator. The natural frequency of such a clusteris smaller than that of constituent bubbles because thecluster effectively behaves as a single bubble of radius R c > R [see Eq. (6)]. This physical similarity also en-ables us to explain experimental findings by conductingnumerical modelling of nonlinear oscillations of a singleequivalent spherical bubble in water, see Sec. II B.Figure 2(a) shows the measured dependence of thescattering spectrum on the increasing amplitude ofthe driving pressure α at the driving frequency f =24 . f /f = 1 markedby the dashed lines. The distance between all side-band peaks is 1.67 kHz, which according to Eq. (6) corre-sponds to the natural frequency of a bubble cluster with √ χR c = 1 .
95 mm. The peaks at this natural frequencyand its higher-order ultraharmonics are also distinguish-able and they are marked by the leftmost dashed line inFig. 2(a).We also observe that the nonlinearly induced higher-order ultraharmonics of the natural response of the bub-ble cluster result in the secondary sideband peaks around f /f = 1. In fact, the sideband peaks adjacent to thecentral peak originate from the natural frequency of thebubble cluster, but the other two are due to the interfer-ence with the first ultraharmonic of the cluster responseat the frequency equal twice the natural one. A qualita-tively similar sideband peak structure can be seen around f /f = 2.Figure 2(b) shows the calculated spectra obtained forthe experimental values of the frequency and ampli-tude. Consistently with the size of the bubble clusterinferred from the experiment, in the calculation we as-sume that the radius of the single equivalent gas bubbleis 1.95 mm (significantly, the calculated spectra are quali-tatively similar for the bubble radii in the 1–2 mm range).We note an overall good qualitative agreement betweenthe experimental and calculated spectra. In experiments,we can clearly see the primary and secondary sidebandsaround f /f = 1. The calculation also predicts the exis-tence of the tertiary sidebands at high values of α . How-ever, these are undetectable in our measurements due totheir low relative magnitude.Interestingly, the tertiary sideband peaks can be seenaround f /f = 1 at α = 4 . f /f = 2compared to the calculated values. This observation isconsistent with the fact that a response of a bubble clus-ter rather than of a single bubble is measured: clustersexhibit stronger acoustic nonlinearities [27, 28] that giverise to more energetic signals at the second harmonic fre-quency f /f = 2.Figure 3(a) shows the temporal far-field pressure pro-file corresponding to f = 24 . f /f = 1) at α = 11 . FIG. 4. (a) Measured acoustic bubble cluster response corre-sponding to a sinusoidal driving pressure wave with the fre-quency f = 49 . α = 4 . T = 1 /f nat ≈ . cause in the calculation the power of the driving ultra-sound wave is the same as in the experiment, the oscil-lations of the equivalent bubble at its natural frequencyare less energetic. Therefore, their interaction with thedriving ultrasound waves results in a weaker amplitudemodulation as indeed is seen in Fig. 3.Next we focus on the experimental sideband peakstructures at f = 49 . f /f = 2) at α = 4 . IV. DISCUSSION
AFC technique is an emerging metrological approachthat benefits from technological maturity of optical fre-quency combs. It opens up opportunities for accuratemeasurements in various physical, chemical and biologi-cal systems in situations, where using light poses techni-cal and fundamental limitations, for example, when pre-cise underwater distance measurement is required [11].In good agreement with our numerical predictions, ourexperimental results demonstrate that a signal producedby gas bubbles oscillating in water has a frequency spec-trum composed of equidistant peaks and is characterisedby amplitude modulation at the bubble cluster naturalfrequency. These features are similar to those of typicaloptical frequency combs and thus they demonstrate thefeasibility of the acoustic frequency combs generation byusing gas bubble oscillations in a liquid.The so-generated acoustic combs should find an ap-plication niche in the fields of underwater distance mea-surements and communication. However, their wider useis expected to be in the areas of biology and medicine,where there is a need for novel types of biomedical sen-sors. For example, AFC suggested here can be used tomeasure elastic properties of some biological tissues andliving cells and sensing biochemical processes inside themvia inducing elastic deformation in the proximity of anoscillating bubble [36, 37]. Such a local mechanical de-formation would affect the oscillation dynamics of thebubble [38] and lead to detectable modifications of thesideband spectral structure of the comb. Thus, it shouldbe possible to use bubbles oscillating in water contami-nated with pathogens (e.g. bacteria) to obtain informa- tion about their presence and concentration required forchoosing an adequate disinfection [39, 40] or removal [41]strategy.Our AFC can also be used to measure the resonancefrequency of a bubble of unknown size [42, 43]. Thusfar, a number of bubble sizing techniques using two-frequency excitation have been employed [42, 43]. There,two beams—a pump beam of variable frequency andan imaging beam of fixed frequency—are simultaneouslyused to scan across the expected resonance frequency ofthe bubble and to achieve the coupling between the twosignal, when the bubble undergoes nonlinear oscillationsat resonance. Using a frequency comb generated withjust one driving wave will extend the capability of thistechnique because, from the technical point of view, onlyone ultrasound transducer needs to be employed.
ACKNOWLEDGMENTS
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