Far-field and near-field directionality in acoustic scattering
FFar-field and near-field directionality in acoustic scattering
Lei Wei ∗ and Francisco J. Rodr´ıguez-Fortu˜no † Department of Physics, King’s College London, Strand, London, WC2R 2LS, United Kingdom (Dated: March 31, 2020)Far-field directional scattering and near-field directional coupling from simple sources have re-cently received great attention in photonics: beyond circularly-polarized dipoles, whose directionalcoupling to evanescent waves was recently applied to acoustics, the near-field directionality of modesin optics includes phased combinations of electric and magnetic dipoles, such as the Janus dipoleand the Huygens dipole, both of which have been experimentally implemented using high refrac-tive index nanoparticles. In this work we extend this to acoustics: we propose the use of highacoustic index scatterers exhibiting phased combinations of acoustic monopoles and dipoles withfar-field and near-field directionality. All solutions stem from the elegant acoustic angular spectrumof the acoustic source, in close analogy to electromagnetism. A Huygens acoustic source with zerobackward scattering is proposed and numerically demonstrated, as well as a Janus source achievingface-selective and position-dependent evanescent coupling to nearby acoustic waveguides.
I. INTRODUCTION
In electromagnetism and photonics, high index dielec-tric particles are becoming an important platform tostudy novel physical phenomena [1]. Unlike plasmonicnanoparticles, a high index dielectric particle can exhibitstrong magnetic Mie resonances [2, 3] that are of compa-rable strength to the electric ones. Sources such as theHuygens and Janus dipoles show interesting directionalscattering and coupling characteristics, both in the farfield and in the near field [4, 5], and they have been ex-perimentally demonstrated in high-index nanoparticles[6–9] with interfering electric p and magnetic m dipolemoments. In the far-field, the combination of orthogonal p and m dipoles following Kerker’s condition p = m/c gives rise to the Huygens dipole, exhibiting directionalscattering [6–8] with applications in reflectionless meta-surfaces [1, 10, 11] and optical metrology [12] amongothers. In the near-field, directional coupling of waveg-uided modes was initially predicted and demonstrated inelectromagnetism via the evanescent coupling of circu-larly polarised dipoles [13], relying on the transverse spinand spin-momentum locking in evanescent waves [14–16]. The analog acoustic scenario was recently demon-strated using circular acoustic dipoles [17, 18]. Thisshows that the transverse spin is a universal propertyof evanescent waves in any wave field including acous-tics [17–21], electromagnetism [14–16] and gravitationalwaves [22]. However, near-field directionality in electro-magnetism was generalised beyond circular dipoles to in-clude combinations of electric and magnetic dipoles thatachieve near-field directional coupling with different sym-metries [4, 5]: one example is the aforementioned Huy-gens dipole, which can be also applied for near-field di-rectionality, and another is the intriguing Janus dipole,whose combination between electric and magnetic dipoles ∗ [email protected] † francisco.rodriguez˙[email protected] requires a 90 degree phase difference to achieve a face-dependent or position-dependent coupling to the waveg-uide modes [4]. These sources exploit the amplitude andphase relations that exist between different componentsof the electric and magnetic fields in evanescent waves.In acoustics, similar amplitude and phase relations ex-ist between the scalar pressure and vector velocity fields,opening the possibility of Huygens and Janus-like direc-tional sources.High index materials are also sought after in acous-tics. Micro-sized air bubbles in liquid show strong reso-nances [23] and are widely used as a contrast agent forhigh resolution acoustic imaging [24]. Acoustic metama-terials [25, 26] made of high index materials includingair bubbles and porous silicone rubbers [27] are proposedto achieve exotic physical properties like negative effec-tive mass density and modulus [28, 29]. Mie-type acous-tic meta-atoms [30–32] have been proposed and demon-strated, which can have high effective acoustic index evenin the background of air. In this work we explore thepossibility to produce acoustic Huygens and Janus-typedirectional sources to achieve far and near-field direction-ality, using a high-index particle platform. II. THEORY
We begin this work by deriving all possible combi-nations of an acoustic monopole M and dipole D thatachieve far- and near-field directionality. The complexpressure field of such a source is given by: p ( r ) = M e ik r k r + 1 ik D · ∇ (cid:18) e ik r k r (cid:19) , (1)where r = | r | is the distance to the source, assumed tobe at the origin, and k = 2 π/λ is the acoustic wave-number of free space. To analyse both far- and near-fielddirectionality we will apply a standard technique in elec-tromagnetism: the angular spectrum decomposition [33–36]. Such decomposition expands the fields as a superpo- a r X i v : . [ phy s i c s . c l a ss - ph ] M a r sition of momentum eigenmodes p ( r ) = (cid:82) k p ( k ) e i k · r d k .Each component p ( k ) e i k · r has a constant wave-vector k = ( k x , k y , k z ). Owing to the dispersion relation, the k z component of k can be derived from the in-plane momen-tum ( k x , k y ) via the dispersion relation k x + k y + k z = k .As is well-known in photonics, in the region k x + k y ≤ k the momentum eigenmodes correspond to propagatingplane waves with a real-valued k . However, in the region k x + k y > k , the component k z becomes imaginary, and e i k · r represents an evanescent wave, corresponding to thenear-field spectrum [37].The angular spectrum p ( k ) can be analytically calcu-lated via a partial Fourier transform of p ( r ) from Eq. 1,using Weyl’s identity [35], and it is given as (see supple-mentary information): p ( k ) = i πk k z (cid:16) M + ˆ k · D (cid:17) , (2)where ˆ k = k /k . Eq. 2 is the master equation fromwhich any type of directionality can be analysed or de-signed. Far-field directionality manifests itself as zeroesin the angular spectrum inside the circle k x + k y = k ,while near-field directionality manifests itself as zeroes outside of that circle [4, 5, 38]. For example, let’s startwith far-field directionality: to achieve directionality inthe forward x direction we may introduce a zero of p ( k )for the plane wave propagating along the negative x axis.Substituting ˆ k = ( − , ,
0) into Eq. 2 and equating it tozero, one immediately arrives at the acoustic analogue ofKerker’s condition M − D x = 0. An acoustic monopole M combined with an acoustic dipole D = ( M, ,
0) willresult in Kerker-like far-field directionality, in completeanalogy to a Huygens’ dipole. This is shown in the far-field diagrams of Fig. 1. Intuitively, the monopole sourceis expanding and contracting in an oscillating manner,creating an isotropic spherical pressure wave, while thedipolar source is vibrating back and forth, creating apeanut-like radiation diagram, with opposite pressurechanges and opposite velocities on opposite directions.Their coherent combination results in a very special vi-bration of the source: the source moves forwards whileexpanding, and then moves backwards while contracting,in such a way that the backward-facing surface does notmove, producing no pressure wave in the backward direc-tion. In the next section we show how to implement thisacoustic Huygens source in a realistic spherical or cylin-drical scatterer upon plane wave excitation, exhibitingno back-scattering, with interesting applications. E l ec t r o m a gn e t i s m Electric dipole Magnetic dipole Huygens source(Kerker’s condition) =+ p m pmE far E far E far A c ou s t i cs =+ v far v far v far D M Monopole Dipole D M FIG. 1. Analogy between far field directionality (Kerker’s condition) in electromagnetic and acoustic scattering.
Even more interesting solutions appear if we look atnear-field directionality. In this case, we must set theangular spectrum in Eq. 2 to be zero at some value of( k x , k y ) outside of the circle k x + k y = k . Followingan identical approach to the optical case [5, 34, 38], wecan study near-field directional coupling of a waveguided mode with an effective refractive index n eff . The evanes-cent wave near-field component that would couple to sucha mode, propagating in the ± x direction, is given byˆ k = ( ± n eff , , iγ ), where γ = ± (cid:112) n − k · ˆ k = 1. The sign of ± n eff will TABLE I. Elemental monopole and dipole combinations fornear-field directionality in planar waveguides.Source ConditionCircular D x = − i √ n − n eff D z Huygens M = − n eff D x Janus M = − i (cid:112) n − D z determine the direction of propagation of the mode, + x or − x , while the sign of γ will determine the position ofthe source, below or above the waveguide, respectively.Substituting this ˆ k into Eq. 2, and equating it to zero, weimmediately arrive at M + n eff D x + iγD z = 0. Three sim-ple solutions emerge when only two of the three sourcecomponents are allowed to be non-zero: (i) the circu-larly polarized dipole, (ii) the near-field Huygens dipole,and (iii) the Janus dipole. The three solutions are sum-marized in Table I and simulated numerically in Com-sol by placing the different sources near a waveguide,shown in Fig. 2. The sources are a clear mathematicalanalogy to their electromagnetic counterparts [4]. Whilein electromagnetism we could find two versions of eachsolution –corresponding to each of the two transversepolarizations–, in acoustics there is only one version ofeach solution, consistent with the fact that acoustic waveshave a single longitudinal polarisation. In the next sec-tion we show how high acoustic index particles can beused to achieve these solutions, with the required rela-tive amplitudes and phases between the monopole anddipole components, and numerically demonstrate Huy-gens and Janus behaviour in the far-field and near-field,respectively. (a)(b) Janus JanusHuygensCircular (c) “off”“on”–1 1Re[ p ( r )] (a.u.) (d) λ FIG. 2. Near-field directional coupling using acousticmonopole and dipole combinations. (a) Circular dipole, (b)Huygens source, (c-d) Janus source. The waveguide slab has¯ ρ = ¯ β = 2, thickness 0 . λ , and n eff ≈ .
31. The sourceis placed at a distance 0 . λ above (a-c) or below (d) thewaveguide. III. HIGH INDEX ACOUSTIC SCATTERERS
Consider a high index acoustic scatterer upon whichan external, time-harmonic sound wave with pressuredistribution p in ( r ) and velocity field v in = iωρ ∇ p in isincident. We assume the scatterer is located at r = 0in a background with mass density ρ and compressibil-ity β and only longitudinal sound waves with velocity c = 1 / √ ρ β considered. The monopole and dipole in-duced in the acoustic scatterer are given by: M = α M p in , D = α D (cid:114) ρ β v in , (3)where p in and v in are evaluated at r = 0, and α M and α D represent the acoustic monopolar and dipolar strength,solely determined by the scatterer and the backgroundmaterial. In the special case of plane wave or evanescentwave incidence p in ( r ) = p e ik ˆk in · r , the dipole momentis reduced to D = ˆk in α D p and the master equation forthe angular spectrum of the scattered field, Eq. 2, can besimplified as: p ( k ) = ip πk k z (cid:104) α M + α D (cid:16) ˆ k · ˆk in (cid:17)(cid:105) . (4)In order to illustrate our concept in a simple mannerbut without loss of generality, let’s assume the scatterer isa sphere of radius r and made of a material that supportslongitudinal sound waves only, and has a relative massdensity and compressibility ¯ ρ = ρ /ρ and ¯ β = β /β .The acoustic scattering of spheres and cylinders can beanalytically calculated (as detailed in the supplementary)in a similar way to Mie theory for optical scattering. Ahigh acoustic index n = (cid:112) ¯ ρ ¯ β corresponds to a strongcontrast in the speed of sound between the scatterer andthe background medium c = c /n . Just like the elec-tromagnetic case, where a high refractive index resultsin a spectral region (i.e. certain values of 2 πr /λ ) domi-nated by the electric and magnetic dipolar contribution,a high index in acoustics also results in a spectral regionof strong acoustic monopolar and dipolar responses, withthe higher order modes suppressed.The relative amplitude and phase between themonopolar and dipolar strength can be tuned with thematerial properties and size, enabling us to easily achievethe specific conditions required for Huygens and Janussources. Fig. 3 shows the relative amplitude and phaseof the acoustic monopolar and dipolar moments for asphere with an acoustic index n = 3, with varying rela-tive mass density ¯ ρ and compressibility ¯ β . In the rangeshown, 0 < πr /λ < .
5, the higher order multi-poles are negligible. We begin by looking at the con-ditions required for a Huygens-type far-field directionalparticle. Following Eq. 4, the scattering pressure ofthe particle in the forward/backward direction, relative π r / λ ∞ ρ β ∞ ρ β π r / λ abs( α M / α D )arg( α M / α D ) -1 -2 π /2 π−π /2 −π abs( α M / α D ) = α M / α D ) = π /2(a)(b) FIG. 3. Relative amplitude (a) and phase (b) of the acous-tic monopolar and dipolar moments α M and α D for a spherewith acoustic index n = 3. The black dashed lines indi-cate the parametric locations where | α M /α D | = 1, and thewhite dashed lines indicate the parametric locations wherearg { α M /α D } = π/ to the incident plane wave, is given by p ( ± ˆk in k ) ∝ α M + α D (cid:104) ( ± ˆk in ) · ˆk in (cid:105) . Owing to the dispersion rela-tion, we know that ˆk in · ˆk in = 1, and so, the condition toachieve zero forward/backward scattering becomes: p ( ± ˆk in k ) = ip πk k z [ α M ± α D ] = 0 . (5)The condition for no backward scattering is therefore α M = α D , and it can be easily implemented with a sub-wavelength particle. Consider the behavior shown in Fig.3(b) in the limit of small particles ( r /λ → ± π out of phase (preferredbackward scattering) in the ranges 0 < ¯ β < < ¯ ρ <
1, while they are in phase (preferred for-ward scattering) in the overlapping region where both¯ β and ¯ ρ are larger than 1. This result can be derivedanalytically: in the limit r /λ →
0, the two moments can be approximated as α M ≈ ( ¯ β − k r ) / α D ≈ (¯ ρ − k r ) / (2¯ ρ + 1). In the region where bothmonopole and dipole are in phase, there is a point wherethey also have equal amplitudes, marked with a blackdashed line, and so α M = α D . This condition repre-sents the acoustic analog of the electromagnetic Huy-gens dipole with zero back-scattering in the far field,easily achievable with high-index spheres or cylinders.An acoustic Comsol simulation of such a particle (in thesimpler two-dimensional case of a cylindrical scatterer) isshown in Fig. 4(a), clearly showing the forward scatter-ing. (a)
01 Abs[ p sca ( r )](a.u.) λλ k inc k inc k inc (b)(c) AcousticJanuscylinderslabwaveguideAcousticHuygenscylinder k inc k inc k inc FIG. 4. (a) Scattered pressure distribution of a cylinder (ra-dius r = 0 . λ/ (2 π ), relative acoustic index n = 3 and relativemass density ¯ ρ = 4 .
2) upon plane wave incidence along the z direction, where the Huygens condition α M = α D for zerobackward scattering is met. (b, c) Scattered pressure distribu-tion of a cylinder (radius r = 0 . λ/ (2 π ), relative acousticindex n = 8 .
082 and relative mass density ¯ ρ = 1 .
04) placedat a distance of λ/ λ/
6, relative acoustic index n = 1 . ρ = 2). A sound plane wave is inci-dent along the z direction. The cylinder scatterer fulfils theJanus condition with α M /α D = 0 . i and the slab waveguidesupports a guided mode with effective index n eff ≈ . The Janus condition can also be achieved, but notin the limit of small particles. The white dashed linesin Fig. 3 indicate the locations where the monopoleand dipole moments are on quadrature phase difference α M /α D = i | α M /α D | as required by Janus sources. Thespecific required amplitude ratio | α M /α D | will dependon the effective index of the modes we want to couple to.Assume that the incident sound wave is propagating inthe +ˆ z direction, i.e. ˆk in = +ˆ z . Following Eq. (4), thescattering angular spectrum for evanescent wavevectors k ev with ( k x + k y ) / = k n eff > k , and corresponding k z = ± ik γ , is given by: p ( k ev ) = ip πk k z (cid:20) i (cid:12)(cid:12)(cid:12)(cid:12) α M α D (cid:12)(cid:12)(cid:12)(cid:12) ± iγ (cid:21) , (6)where we introduced the Janus condition α M /α D = i | α M /α D | , and where γ = ( n − / is always takenas the positive square root, while the ± sign corre-sponds to scattering in the z > z < k x + k y ) / = k n eff = k (1 + | α M /α D | ) / , will have azero amplitude in the lower half space ( z < n eff when the waveguide is placed inthe z < z > z > α M /α D = i | α M /α D | – constitutes thesignature of a Janus source, with its charcteristic face-dependent behaviour, in perfect analogy to electromag-netism. Fig. 4(b-c) shows a cylindrical particle designedin this way, simulated in Comsol, and placed at eitherside of a planar slab, showing the side-dependent on-offcoupling to the waveguide modes. IV. CONCLUSION
We have extended the analogy of far and near-field di-rectionality from dipolar sources in electromagnetism tothe domain of acoustics. On one hand, we theoreticallydescribe and numerically demonstrate the acoustic anal-ogy of a far-field directional Huygens dipole with zero back-scattering, implemented with high acoustic indexspheres or cylinders. This may have interesting applica-tion for reduced reflection materials in acoustics engineer-ing. The possibility of zero forward-scattering Huygensscatterers could also lead to the design of novel soundbarriers.On the other hand, we also theoretically describednear-field directional coupling of waveguided modes us-ing monopole and dipole combinations. We show thatthe three solutions: circular dipole, Huygens source, andJanus source, in perfect analogy to the electromagneticcase, appear naturally as independent solutions to thesimple angular spectrum of an acoustic source. We the-oretically and numerically propose a simple realistic wayof achieving a Janus scatterer with spherical or cylindri-cal high acoustic index materials, clearly exhibiting thecharacteristic position-dependent near-field coupling be-haviour. This has clear implications for the understand-ing and control of the near-fields of sound waves.During the writing of this manuscript we noticed therecent publication of the theoretical and experimentalRef. [39] which describes and experimentally demon-strates the same set of three acoustic near-field direc-tional sources described here. The work elegantly derivesthe three sources from fundamental symmetry considera-tions and follows a Fermi Golden rule approach. The keydifference of our work is our angular spectrum approachand our proposed realization of the sources using the scat-tering from spherical or cylindrical particles made of highacoustic index materials, instead of phased combinationsof acoustic monopoles.
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In this work, we consider an acoustic scatterer withdominant monopolar and dipolar responses, subject toan external time-harmonic sound wave with a pressuredistribution p in ( r , t ) = p in ( r )e − iωt and p in ( r ) = p e i k r .We assume that the scatterer is located at the origin r = 0 in a fluidic background with mass density ρ andcompressibility β and only longitudinal sound waves areconsidered. The longitudinal sound velocity of the back-ground medium is c = 1 / √ ρ β = ω/k with k = 2 π/λ being the wave-number of the background medium and λ the wavelength of the sound wave.The contribution of the monopolar and dipolar re-sponses of the acoustic scatterer to the total scatteringis purely determined by the monopole moment M anddipole moment D : M = α M p in ( r = 0) , (A1) D = α D (cid:114) ρ β v in ( r = 0) , where v in = iωρ ∇ p in is the velocity field of the incidentsound wave, α M = a /i and α D = 3 a /i represent themonopole and dipole strength of the scatterer, and where a and a are coefficients for the monopole and dipole,solely determined by the scatterer itself. For the specialcase of a spherical object, its scattering of sound wavescan be analytically treated like Mie theory for opticalscattering. Consider a spherical scatterer with a radiusof r and made of materials with mass density ρ andcompressibility β , supporting longitudinal sound waveswith a velocity c = 1 / √ ρ β . The coefficients a and a can be determined by the following expression: a n = (cid:112) ¯ β/ ¯ ρj (cid:48) n ( k r ) j n ( k r ) − j n ( k r ) j (cid:48) n ( k r ) j n ( k r ) h (1) (cid:48) n ( k r ) − (cid:112) ¯ β/ ¯ ρj (cid:48) n ( k r ) h (1) n ( k r ) , (A2)where j n ( kr ) is the spherical Bessel function of the firstkind and h (1) n ( kr ) is the spherical Hankel function of thefirst kind, j (cid:48) n ( kr ) and h (1) (cid:48) n ( kr ) are their first order deriva-tives with respect to the argument variable kr . Therelative mass density and compressibility are defined as¯ ρ = ρ /ρ and ¯ β = β /β , and k = ω/c = k n with n = (cid:112) ¯ ρ ¯ β being the acoustic refractive index.The scattering pressure distribution due to themonopole and dipole contribution can be expressed as: p ( r ) = M e ik r k r + 1 ik D · ∇ (cid:18) e ik r k r (cid:19) . (A3)In order to obtain the angular spectrum of this source,we need to perform a partial Fourier transform in the xy plane. The angular spectrum p ( k x , k y ) is defined suchthat: p ( r ) = (cid:90) (cid:90) + ∞−∞ p ( k x , k y )e i ( k x x + k y y + k z | z | ) d k x d k y , (A4)where the z direction is an arbitrarily defined directionin space, so that k z is taken to be the dependent variable k z = ( k − k x − k y ) / while the angular spectrum is de-fined in the two dimensional domain of transverse wave-vectors p ( k ) = p ( k x , k y ). As is well-known in electromag-netism, strictly speaking two different spectra p + ( k x , k y )and p − ( k x , k y ) have to be defined, corresponding to thefields in the z > z < k z , respectively. In order to write Eq. (A3) in the form of Eq. (A4), wemake use of Weyl’s identity [35]:e ik r r = (cid:90) (cid:90) + ∞−∞ i πk z e i ( k x x + k y y + k z | z | ) d k x d k y . (A5)Substituting Weyl’s identity into the correspondingterms in Eq. (A3), and applying the linearity of the in-tegration and gradient operations (the gradient operatorbecomes a multiplication times i k inside the integral) wearrive at: p ( r ) = (cid:90) (cid:90) + ∞−∞ i πk k z (cid:104) M + (cid:16) ˆ k · D (cid:17)(cid:105) (A6)e i ( k x x + k y y + k z | z | ) d k x d k y , where ˆ k = k /k . By comparing Eq. (A6) with the def-inition of the angular spectrum in Eq. (A4), we finallyidentify the expression for the angular spectrum: p ( k x , k y ) = i πk k z (cid:104) M + (cid:16) ˆ k · D (cid:17)(cid:105) , (A7)as given in the main text. Note that the two angular spec-tra p + ( k x , k y ) and p − ( k x , k y ), corresponding to the twohalf spaces z > z < z component of the vector ˆ k . Also note that Eq. (A7)is a complete and exact analytical form of the angularspectrum, revealing not only the far-field directionality ofthe source (for k x + k y ≤ k ) but also its near-field direc-tionality (corresponding to the evanescent wave spectrumwhen k x + k y > k ) associated to the coupling behaviourbetween this source and nearby bound waveguide modes. Appendix B: Scattering coefficients of cylindricalparticles
The monopole and dipole strength of a cylindrical scat-terer can be determined as α M = a /i and α D = 2 a /i ,with the acoustic Mie coefficients a and a determinedby the following expression: a n = (cid:112) ¯ β/ ¯ ρJ (cid:48) n ( k r ) J n ( k r ) − J n ( k r ) J (cid:48) n ( k r ) J n ( k r ) H (1) (cid:48) n ( k r ) − (cid:112) ¯ β/ ¯ ρJ (cid:48) n ( k r ) H (1) n ( k r ) , (B1)where J n ( kr ) is the Bessel function of the first kind and H (1) n ( kr ) is the Hankel function of the first kind, J (cid:48) n ( kr )and H (1) (cid:48) n ( kr ) are their first order derivatives with respectto the argument variable krkr