BBound States in the Continuum in Elasticity
O. Haq ∗ and S. V. Shabanov Department of Physics, University of Florida, Gainesville, FL 32611, USA Department of Mathematics, University of Florida, Gainesville, FL 32611, USA ∗ Corresponding author: omerhaq1@ufl.edu
Abstract
Diffraction of elastic waves is considered for a system consisting of two parallelarrays of thin (subwavelength) cylinders that are arranged periodically. The embed-ding media supports waves with all polarizations, one longitudinal and two transverse,having different dispersion relations. An interaction with scatters mixes longitudinaland one of the transverse modes. It is shown that the system supports bound statesin the continuum (BSC) that have no specific polarization, that is, there are standingwaves localized in the scattering structure whose wave numbers lies in the first opendiffraction channels for both longitudinal and transverse modes. BSCs are shown toexists only for specific distances between the arrays and for specific values of the wavevector component along the array. An analytic solution is obtained for such BSCs.For distances between the parallel arrays much larger than the wavelength, BSCs isproved to exist due to destructive interference of the far field resonance radiation, sim-ilar to the interference in a Fabry-Perot interferometer, that can occur simultaneouslyfor both propagating modes.
Bound States in the Continuum (BSC) have been studied in the context of many physicalsystem from photonics to quantum mechanics, since the seminal paper by von Neumannand Wigner [1, 2]. Examples of BSC have been explored in just about every field of wavephysics: quantum mechanics, electromagnetism, and acoustics. Systems supporting BSC canbe viewed as resonators with infinitely high quality factors. Due to these unusual physicalproperties BSC are intensively studied both theoretically and experimentally, especially inphotonics [3] and recently even in plasma-photonic systems [4]. In particular, acoustic BSChave been observed in “Wake Shedding’ experiment” [5] as well as in acoustic wave guideswith obstructions [6]. A relation between elastic BSC and photonic resonances in opto-mechanical crystal slabs [7] as well as surface acoustic waves on anisotropic crystals and non-periodic layered structures [8]-[9] has been investigated using a group-theoretical approach.Elastic systems offer a rich playground, both theoretically and experimentally, to investigateBSC especially in view of mechanical metamaterials [10]–[13] with unusual stiffness, rigidityand compressibility.There is no universal mechanism for formation of BSC, however there are a few classifi-cations which may not be mutually exclusive [3]. In some instances, the existence of BSCs1 a r X i v : . [ phy s i c s . c l a ss - ph ] J a n an be explained by a destructive interference of diffracted waves in the asymptotic region[18]. If a system of subwavelength scatterers that are arranged in a plane happens to bea resonator for incident plane waves, then two such systems separated by a distance forma Fabry-Perot interferometer with resonating interfaces. For a large enough distance, theinterfaces are interacting only through diffracted (propagating) modes, while an interactionvia evanescent modes is suppressed due to an exponential decay of the latter. Given qualityfactors of each resonating interface, it is then not difficult to compute the quality factor ofthe combined structure using the standard Fabry-Perot summation of transmitted and re-flected waves. It appears that the quality factor depends on the distance between the arraysand the wave speed in the media between the interfaces. There exist distances at whichthe quality factor becomes infinite, thus indicating the existence of BSC. At these distances,the diffracted wave from each interface undergo destructive interference in the asymptoticregion. In other words, each of the resonating Fabry-Perot interfaces acts as a mirror for astanding wave that becomes confined between the interfaces despite that its wave numberslies in open diffraction channels [19].Many of the photonic applications such as filters and sensor can be translated into thecontext of elastics with little efforts due to similarities of the two theories. However thereare drastic differences among the two such as the presence of longitudinally polarized wavemodes in elastics as well as the boundary conditions at the interface. Even for simple scat-tering geometries, like the aforementioned periodic arrays of cylinders, the normal tractionboundary condition [22] at the interface of the elastic rods (which is necessary for a mechan-ical equilibrium of the system) leads to a coupling between the longitudinal (compression)and one of the transverse (sheer) modes. Given that these modes have different dispersionrelations, the Fabri-Perot argument [18] becomes inapplicable to prove the existence of elasticBSCs, and a full analysis of the scattering problem is required.In this paper, BSC for elastic waves are investigated in a system of two periodic arrays ofcylindrical scatters separated by a distance. Due to the translational symmetry, transversewaves polarized parallel to the cylinders are decoupled from the other two polarization modes(one transverse and one longitudinal). The latter modes are coupled and have differentdispersion relations. The problem is solved by means of the Lippmann-Schwinger formalismin the dipole approximation. A confinement of an elastic wave between two arrays that couplethe transverse and longitudinal modes, due to normal traction boundary conditions, requiresthat both modes interfere destructively in the asymptotic region. It is not surprising that byadjusting the distance one can confine a particular polarization mode if it is not coupled toany other mode (e.g., the longitudinal one). It is remarkable that by adjusting geometricaland spectral parameters it is possible to create a standing wave consisting of both coupledpolarization modes. The conditions under which such elastic BSC exists comprise our mainresult. To our knowledge, this is the first analytic solution for a BSC containing coupledwaves with different dispersion relations. It is noteworthy that so far only electromagneticBSCs with no specific polarization were found in numerical studies [14]-[17] where bothtransverse modes propagates with the same group velocity. Although in the present studythe existence of such BSCs is established under several simplified assumptions such as a2ipole approximation in solving the scattering problem, a large distance between the arrays(to justify the use of the Fabri-Perot argument), and some simplifying conditions on theelastic properties of the scatterers, the stated formalism holds even if these assumptionsare dropped. However, the analysis of the Lippmann-Schwinger integral equation becomesmathematically involved and will be presented elsewhere. Elastic wave scattering on a periodic array of cylinders
A displacement field u in an elastic media has three components, one longitudinal (compres-sion wave) and two transverse (sheer waves). In general, the longitudinal and transversecomponents have different group velocities c l and c t , respectively, determined by the mediamass density ρ b and lame coefficients, λ b and µ b . A scattering structure is described bythe relative mass and lame coefficients in units of the corresponding background quantities,denoted here by ξ ρ,λ,µ . For example, ξ ρ = ( ρ s − ρ b ) /ρ b , where ρ s is the mass density of ascattering structure, and similarly for the relative lame coefficients, so that ξ ρ,λ,µ ( r ) = 0 atany point r where no scatterers are present.Let the scattering structure be a periodic array of cylinders, that is, ξ ρ,λ,µ ( r ) (cid:54) = 0 oncylinders whose axes are arranged periodically in a plane and all cylinders have the sameradius, which is much smaller than the period. Owing to the translational symmetry alongaxes of the cylinders, the field u depends only on two variables spanning the plane perpen-dicular to the cylinders. In what follows, a unit system is chosen so that the period of thearray is one, the z axis is chosen to be parallel to the cylinders, and the array is periodicalong the y axis so that u depends on x and y . The mode u z is decoupled from the twomodes in the xy plane due to the translational symmetry. The scattering problem for u z is fully analogous to the electromagnetic case studied in detail in [19]. So the existence ofelastic BSC for this mode can readily be established. The compression mode and the sheermode polarized in the xy plane remains coupled through the scattering structure and, yet,they have different dispersion relations. The objective is to investigate whether elastic BSCsconsisting of the two coupled modes exist in a system of two parallel arrays of periodicallypositioned cylinders.Let u j ( r ), j = x, y , be the amplitude of a monochromatic elastic (displacement) field offrequency ω at a point r = ( x, y ); the indices i , j , and k are used to denote the x and y components of a vector in the coordinate system described above, while the indices a and b toindicate polarization states in the xy plane, a = l (longitudinal) and a = t (transverse). Forexample, the incident wave is u a,j ( r ) = u a e ik a,x x + ik y y ˆ e a,j where the unit polarization vector ˆ e a is parallel to the wave vector ( k a,x , k y ) for the longitudinal wave and perpendicular to it forthe transverse one, the length of ( k a,x , k y ) for each polarization mode is set by the dispersionrelation, c a k a = c a ( k a,x + k y ) = ω , thus defining k a,x via ω , u a corresponds to the incidentamplitude of polarization a . The standard basis ˆ x , ˆ y can always be converted to the basisˆ e a by a suitable rotation.The elastic field is a superposition of the incident and scattered fields, u j ( r ) = u j ( r ) +3 Sj ( r ). Adopting the Einstein rule for summation over repeated indices, the governing equa-tions for u j ( r ) have the form [22]:[( ω + c t (cid:52) ) δ jk + ( c l − c t ) ∇ j ∇ k ] u k ( r ) = 1 ρ b P j ( r ) (2.1) P j ( r ) = − ρ b ( ω ξ ρ ( r ) u j ( r ) + ∇ k σ kj ( r )) (2.2) σ kj ( r ) = ( c l − c t ) ξ λ ( r ) ∇ i u i ( r ) δ kj + c t ξ µ ( r )( ∇ k u j ( r ) + ∇ j u k ( r )) . The incident wave u j ( r ) satisfies the homogeneous equation (2.1) with P j ( r ) = 0. Withoutloss of generality, put ρ b = 1. The quantity P j ( r ) has a simple physical interpretation as aninduced dipole moment per unit area in the xy plane. The scattered field at a position r , du Sj ( r ), is a result the radiation field produced by an infinitesimal induced dipole momentcentered at position r , P i ( r ) d r ; it has the standard form du Sj ( r ) = G ji ( r − r ) P i ( r ) d r where G ji ( r ) is the Green’s function for the differential operator in the left side of (2.1)satisfying Sommerfeld radiation conditions at spatial infinity: G ji ( r ) = i c l (cid:32) ∇ j ∇ i k l (cid:33) H (1)0 ( k l | r | ) − i c t (cid:32) ∇ j ∇ i k t + δ ji (cid:33) H (1)0 ( k t | r | )where k a = ωc a are the wave numbers of the longitudinal and transverse modes, and H (1)0 isthe Hankel function of the first kind. It follows that the total displacement field reads u j ( r ) = u j ( r ) + (cid:90) Ω G ji ( r − r ) P i ( r ) d r , (2.3)where Ω is the region occupied by scatterers.To simplify the discussion further and avoid complicated technicalities associated withsolving the Lippmann-Schwinger integral equation (2.3), the simplest case with two opendiffraction channels, one transverse and one longitudinal, is considered so that the frequencyrange is limited to c l k y < ω < c t ( k y − π ) ≡ ω , (2.4)0 < k y < πα α , where k y is the y component of the wave vector, and the materials are assumed isotropic,in this case, α = c t /c l < / √ ω , k y ) and its polarization state a .The theory is symmetric under k y → − k y so that k y > k y is necessary in order for the longitudinal continuum edge to liebelow the first diffraction threshold for the transverse mode. The special case k y = 0 willbe discussed later. The relative density, ξ ρ = ρ s − ρ b ρ b = ρ s −
1, is required to be positive.4ince the density in elastics plays the same role as the permittivity in electromagnetism, thepositivity of ξ ρ is required for the existence of BSC. A negative relative density correspondsto a repulsive potential in quantum mechanics or conductors in electromagnetism (in eithercase, BSC cannot form). If the wavelength of the incident wave is much smaller than theradius R of the cylinders, that is, ωR (cid:28) c t , then variations of the field u within each cylindercan be neglected so that the integral in (2.3) is reduced to the sum over cylinders: u j ( r ) = u j ( r ) + (cid:88) n G ji ( r − n ˆ y ) p i ( n ) (2.5)where p ( n ) = (cid:15)P ( r n ) is the induced dipole moment of the cylinder at r n = (0 , n ), n =0 , ± , ± , ... , and (cid:15) = πR is the area of the cross section of the cylinder.To simplify calculation of the induced dipoles (2.2), the lame coefficients of the cylindersand the background media are assumed to be the same, ξ λ,µ = 0. In this case, p j ( n ) = − ω (cid:15)ξ ρ u j ( n ˆ y ) = − ω (cid:15)ξ ρ u j (0) e ik y n where the Block periodicity of u j ( r ) was used. Setting r = 0 in (2.3), a consistency conditionon the the induced dipole moment of the central scatter, p (0), is obtained, which, in turn,determines the field on the central scatter: u i (0) = u i (0)1 + ω ξ ρ (cid:80) n e ik y n (cid:82) | s |
0. In the spectral range un-der consideration, there are only two open diffraction channels, one for each polarization a . The amplitude of the scattered wave in the asymptotic region x → −∞ is given by u Ra = R ab ( ω ) u b , where R ab is the reflection matrix, whereas u Ta = T ab ( ω ) u b is the amplitudeof the transmitted wave when x → ∞ . Since all induced dipoles are proportional to u j (0),the frequency dependence of the reflection (or transmission) matrix R ( ω ) (or T ( ω )) is fullydetermined by the frequency dependence of u j (0). In the complex plane ω , R ( ω ) and T ( ω )5ave a pole if the scattering structure has a resonance. Near the pole the reflection matrixhas the following form: R ( ω ) ∼ ˜ Rω − ω + i Γ + K , (2.7)where ˜ R is the residue matrix, and K is the analytical part of R ( ω ) evaluated at the pole (itdescribes the so called background scattering), the transmission matrix T ( ω ) has a similarform near a resonance. Note that the graph of | u Ra | has the standard Lorentzian shape as afunction of real ω for both in-plane polarizations.Figure 1: Left panel: Plot of | (cid:15)u x (0) u x (0) | vs s = ω/c t for the following parameter values: ( k y , (cid:15), ξ ρ , α ) =(1 . , . , . , . ω , Γ) = (4 . c t , . · − c t );Right panel: Plot of (cid:15)u y (0) u y (0) vs s for the same parameter values as the left panel, the black verticalline corresponds to the resonance position specified by the Lorentzian fit. The solid black line in Figure 1 shows (cid:15) | u x (0) || u x (0) | (left panel) and (cid:15) | u y (0) || u y (0) | (right panel) cal-culated from (2.6) as a function of s = ω/c t . The red dashed line shows a numerical fitto the theoretical curve by a Lorentzian profile (the resonance frequency ω , width Γ, andthe maximal value are the fitting parameters). Two important observations follows fromthis numerical analysis. First, the contribution of u y (0) to the induced dipole is negligible(right panel) as | u x (0) | / | u y (0) | ∼ − (if | u x | ∼ | u y | ), and it contributes only to the back-ground scattering. Second, the scattering is dominated by a resonance (left panel) so thatthe background scattering K can be neglected near the resonance in (2.6).The conclusion holds for significant variations of the system parameters, although theparameters of the Lorentzian profile may change significantly. Figures 2 and 3 display (cid:15) | u x (0) || u x (0) | and its fit by a Lorentzian profile for various values of the parameters ( k y , (cid:15), ξ ρ , α ) (indicatedin the captions). One can see that the resonance remains relatively narrow under increasing α or k y , while increasing (cid:15) and ξ ρ by an order of magnitude results in a drastic change inthe resonance width (as might be seen in Fig. 4). It should be noted, however, that as (cid:15) and ξ ρ increase, the dipole approximation becomes inapplicable so that Eq. (2.6) is no longervalid and one must solve (2.3) by other means. As a point of fact, that the scattering inthe system considered is resonance dominated can be proved analytically without all thesimplifying assumptions made above. However mathematical details of this study are ratherinvolved and are omitted. 6igure 2: Left panel: Plot | (cid:15)u x (0) u x (0) | vs s for ( k y , (cid:15), ξ ρ , α ) = (2 , . , . , . ω , Γ) = (4 . c t , . · − c t ).; Right panel: Plot | (cid:15)u x (0) u x (0) | vs s for ( k y , (cid:15), ξ ρ , α ) = (1 . , . , . , .
65) the fitted parameters of the Lorentzian curve are( ω , Γ) = (4 . c t , . · − c t ). Figure 3:
Left panel: Plot | (cid:15)u x (0) u x (0) | vs s for ( k y , (cid:15), ξ ρ , α ) = (1 . , . , . , . ω , Γ) = (4 . c t , . c t ).; Right panel: Plot | (cid:15)u x (0) u x (0) | vs s for ( k y , (cid:15), ξ ρ , α ) = (1 . , . , . , . ω , Γ) = (4 . c t , . c t ). A Fabry-Perot interferometer with two propagating coupled modes
Consider a Fabry-Perot resonator made of two resonating scattering interfaces such as theone discussed above. If the distance between the interfaces is large enough so that theevanescent fields near each interface do not contribute to the field on the other interface,then the reflection matrix of this composite structure is: R F P ( ω ) = R ( ω ) + T ( ω ) D ( ω, d ) R ( ω ) D ( ω, d ) × ( I − ( R ( ω ) D ( ω, d )) ) − T ( ω ) (3.8) T F P ( ω ) = T ( ω ) D ( ω, d )( I − ( R ( ω ) D ( ω, d )) ) − T ( ω ) , where I is the unit diagonal matrix, D ( ω, d ) is the matrix defined by the amplitude ofthe wave after propagation through a distance d , in this case d is the distance betweenthe interfaces. It is a diagonal matrix with elements being phase factors corresponding todifferent group velocities of the modes. A resonance position of the composite system (a7ole of R F P ( ω )) is defined by the conditiondet[ I − ( R ( ω ) D ( ω, d )) ] = 0 . (3.9)The scattering on each interface is further assumed to be resonance dominated (the back-ground scattering can be neglected). This assumption is always justified if the scatteringinterface is composed of small identical (subwavelength) scatters separated by distances thatare much larger than the size of scatterers as in the example presented above. Then the an-alytic part of the single interface reflection matrix will scale with the volume of the scatterso that the background scattering K can be neglected in (2.7) as compared to the resonantpart. For the array considered above this approximation holds because u y (0) is negligible ascompared to u x (0). In this case, Eq. (3.9) is further simplified todet[( ω − ω + i Γ) I − ( ˜ RD ( ω , d )) ] = 0 (3.10)If the polarization modes are decoupled ( ˜ R is diagonal), then this equation is reduced to thecase of light scattering on a dielectric double array [18] where it was shown that, for specificvalues of the distance d , there exists a real root ω that lies in the radiation continuum,thus indicating a BSC as a resonance with the vanishing width. Here the situation is morecomplicated as the residue matrix ˜ R is not diagonal due to the coupling of the compressionand sheer modes, and the phase factors in D are different for each mode because c t (cid:54) = c l . The existence of BSC depends on the structure of the residue matrix ˜ R . Its calculationgenerally requires solving the Lippmann-Schwinger integral equation. If the dipole approxi-mation is applicable, this task is simplified to summation of the dipole radiation. The lattercan be done analytically for either a single or double periodic array of thin, long, cylindricalscatter where the longitudinal and transverse polarizations in the plane perpendicular to thecylinders are coupled at the surface of the cylinders. As noted, the transverse mode polarizedparallel to the cylinders is decoupled. Therefore the three-dimensional reflection matrix isblock-diagonal. The 1 × × e l = ˆ k l = ( k l,x ˆ x + k y ˆ y ) /k l and ˆ e t = − (ˆ k t × ˆ z ) = − ( k y ˆ x − k t,x ˆ y ) /k t ,8here k a,x = (cid:113) k a − k y . The position of the resonance pole ( ω and Γ) is calculated in theleading order of (cid:15) . Expanding u x (0) to leading order in (cid:15) one infers that˜ R ll = p l,x i Γ p l,x + k y p − t,x , ˜ R lt = − αk y i Γ p l,x + k y p − t,x , ˜ R tt = k y p t,x i Γ p l,x + k y p − t,x , ˜ R tl = − p l,x k y αp t,x i Γ p l,x + k y p − t,x where Γ = (cid:15)ξ ρ βω ( p l,x + k y p − t,x ), β = (cid:15) ξ ρ ( k y − π ) , p l,x = (cid:113) α ( k y − π ) − k y , and p t,x = (cid:113) ( k y − π ) − k y .It should be noted that ˜ R lt , ˜ R tl (cid:54) = 0, as a result the two in-plane polarization modes arecoupled at the interface. It follows from the structure of ˜ R that det( ˜ R ) = 0. Physically, thisis because near the resonance frequency, the elastic field on the scatter lies almost completelyparallel to the ˆ x direction, this means that the reflection amplitudes, u Rl , u Rt , are proportionalto u x (0) because u y (0) is of higher order in the volume of the scatters (as shown by the rightpanel of Fig. 1): u Rl , u Rt ∝ u x (0) ∝ u x (0) ∝ u l ˆ e l,x + u t ˆ e t,x This feature of the scattering process is easily understood in the considered dipole approx-imation. In this case the induced dipole moment of each scatterer is parallel to ˆ x by thereflection symmetry about the position of each scatterer so that the rows of the reflectionmatrix are linearly dependent. Therefore, neglecting K , det( ˜ R ) = det( R ( ω )) = 0 nearthe resonance frequency. Equation (3.10) can be viewed as an eigenvalue problem for the2 × RD ( ω , d ). Since det( ˜ RD ( ω , d )) = det( D ( ω , d )) det( ˜ R ) = 0, ω = ω − i Γ isalways a solution. However, it does not define a pole in (3.8) if R ( ω ) has the form (2.7) (allsingular factors ( ω − ω + i Γ) − are cancelled out, this can be seen explicitly by using theunitary condition on the scattering matrices). The other roots determine the poles ω ± ofthe composite structure: ω ± − ω + i Γ = ± (cid:113) Tr( ˜ RD ( ω , d )) = ± i Γ p l,x e ip l,x d + k y p − t,x e ip t,x d p l,x + k y p − t,x , (4.1)A BSC occurs if the imaginary part of the pole can be driven to zero by adjusting parametersof the system. This happens if the phase factors in the numerator in the right side of (4.1)become unit, e ip l,x d = e ip t,x d = ±
1, that is, both modes satisfy the quantization conditions: p l,x d = πM < p t,x d = πN where M and N are both either mutually even or odd integers. It should be noted thatthe system has a parity symmetry in the x direction. The even/odd integers in the phasefactors above corresponds to even/odd parity BSC just as in the scalar case. In contrastto electromagnetic case [18], the quantization conditions cannot be satisfied by adjusting9he distance d between the arrays because the modes have different dispersions (note theparameter α in p l,x ). Both conditions can only be fulfilled for a generic parameter α if thespectral parameter k y is such that 0 < p l,x p t,x = nm < n and m . It follows from (4.1) that under the stated conditions one of theresonances turns into a BSC at the resonance position of the single array ω BSC = ω while the other resonance has a double width 2Γ and the same position ω BSC , just like inthe electromagnetic case.A BSC is not coupled to radiation modes, hence, cannot be excited by incident waves. Itis a solution to the homogeneous equation (2.3) for the double array that is square integrablein x and Bloch-periodic in y (Sommerfeld conditions at infinity | x | → ∞ are satisfied due tosquare integrability). If the cylinders in a double array are located at r = r ± n = ± d ˆ x + n ˆ y , n being an integer, then a BSC has a real frequency that lies in the radiation continuum, ω = ω BSC > c a k y . Just as in the case of the single array, the scattering matrices aredetermined by the dipole moment of the central scatter of each array ˜ u i ( r ± ), (˜ u ( r ) is theelastic field of the double array). The consistency condition on the dipole moment of eachcentral scatter (located at r ± ) results in a system of equations which has a non-trivial solutiononly if its determinant vanishes, which defines ω BSC .It was shown that in the Fabri-Perot limit, k a,x d (cid:29) a , suchfrequency does exist and ω BSC = ω , where ω is the resonance position for a single arrayif the distance d between arrays and k y satisfy the conditions stated above. The BSC field˜ u ( r ) is given by a similar expression as the scattered field in the right side of (2.5), withnecessary modifications in order to account for the second array. By construction of theinduced dipoles ˜ u ( r ± ) (without an incident wave), the amplitudes of propagating modes in(2.5) should vanish in the far field when ω = ω BSC (by the energy flux conservation) so thata BSC solution ˜ u ( r ) to the double array decays exponentially | ˜ u ( r ) |∼ e − (cid:15) | ξρ | ( ky − π )2 | x | (4.2)as | x | → ∞ , and, hence, has a finite energy per each period of the array.The energy density of a BSC is shown in the middle panel of Figure 4 in unit of ρ b c t | ˜ u x ( d ˆ x ) | .The left panel shows Re[ ˜ u x ( r )˜ u x ( d ˆ x ) ]. It is also worth noting that the y − component of the BCSsolution, ˜ u y ( r ) ∼ O ( (cid:15) ), so its contribution to the energy density is of higher order in (cid:15) andcan be neglected. The parts of the array shown in the middle and right panels correspondsto a region indicated by a rectangle in the left panel where a pictorial representation of thearray is shown. As one can see, the energy and the field is concentrated near each array. The10isplayed BSC has an even parity so that ˜ u x ( r ) is odd in x and d (cid:113) k BSC − ( k y − π ) (cid:29) Left panel: Schematic of the double array of periodically positioned elastic cylinders,the x − axis is horizontal, the y − axis is vertical; Middle panel: The energy density of a BSC in avicinity of the right arrays for parameters ( k y , (cid:15), d, ξ ρ , α ) = (1 . , . , , . , . ω BSC = 4 . c t . The vertical grid is defined in the text andthe horizontal scale is measured in units of the period of the array; Right panel: The x − componentof the displacement field of this BSC, the grid legend and parameters are identical to the former. In the frequency range under consideration, the two polarizations decouple for normalincident k y = 0, the scattering matrix is proved to become diagonal, that is, the longitu-dinal and transverse wave are decoupled in the first open diffraction channel (which canbe understood from the parity symmetry in x direction). The transmission coefficient forthe transverse mode becomes T tt ( ω ) = 1 − O ( (cid:15) ), which can be realized from the induceddipole radiation analysis presented above. It is therefore clear that a transverse BSC cannotexist in this spectral range. However the longitudinal mode has a resonance, and the lon-gitudinal reflection coefficient is shown to have the Breit-Wigner form near this resonance,hence, BSCs exist. These are single-mode BSCs and have already been discussed in manycontexts. For higher diffraction channels the longitudinal and transverse modes are coupledeven for k y = 0. However, the analysis of BSCs in higher diffraction channels becomesmathematically involved even in the single mode case [19] and will not be given here. It was shown that elastic meta-interfaces can be used to obtain BSC via tuning of the cavitywidth and Bloch phase. Elastic BSC are shown to contain coupled transverse and longitudi-nal modes that have different dispersion relations. In contrast to previously reported BSC,such as the case of the electromagnetic double array, this elastic BSC requires very peculiarcondition on the phases of both the transverse and longitudinal phase, such conditions are11ecessary in order to tune the far field radiation to zero. This fine tuning is an interestingartifact of the elastic double array which is a major distinction from the BSC previouslystudied in photonic and acoustic systems. An analytic solution is obtained for such BSCfor the first time. Although the stated results have been proved under simplifying assump-tions ( ξ λ,µ = 0 and the Fabri-Perot limit), the same dipole formalism can be used in thegeneral case, however this is mathematically involved and will be discussed in a subsequentpublication.Elastic BSCs can be used as elastic wave guides or as resonators with high quality factorsin a broad spectral range, especially in view of that elastic systems supporting BSC can bedesigned using mechanical metamaterials (as materials with desired elastic properties). Inparticular, owing to a high sensitivity of the quality factor to geometrical and physicalproperties of a resonating system, elastic BSCs can be used to detect impurities in solidsfrom variations of the density. Understanding the parallels and distinction among wavephenomena in a variety of physical systems is necessary for construction of sensor, filters,lasers, etc. This sort of artificial wave construction will allow one to analyze non-linearphenomena in elastic material in a way similar to the studies done in photonics [25]. Theenergy density of a high quality resonance (near-BSC state) has shown to exhibit “hot” spotswhere it exceeds the energy density of the incident wave by orders in magnitudes and wouldallow one to amplify non-linear effects in solids in a controlled manner. References [1] J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929)[2] F.H. Stillinger and D.R. Herrick Phys. Rev. A 11, 446 (1975)[3] C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljaˇc´ıc, Nat. Rev. Mater.1, 16048 (2016).[4] S.I. Azzam, V.M. Shalaev, A. Boltasseva, and A.V. Kildishev Phys. Rev. Lett. 121,253901 (2018).[5] R. Parker, J. Sound Vib. 4, 62 (1966)[6] M.D. Groves, Math. Method. Appl. Sci. 21, 479 (1998)[7] M. Zhao and K. Fang, Optic Express 7, 27, 10138 (2019)[8] T.Lim and G.Farnell, J.Acoust.Soc.Am.45, 845 (1969)[9] A.Maznev and A.Every, Phys.Rev.B 97,014108 (2018)[10] G.W. Milton and A.V. Cherkaev, J. Eng. Mater. Technol., 117, 483 (1995).1211] K. Bertoldi, V. Vitelli, J. Christensen, and M. van Hecke, Nat. Rev. Mater. 2, 17066(2017).[12] X. Yu, J. Zhou, H. Liang, Z. Jiang, and L. Wu, Prog. Mater. Sci., 94, 114 (2018).[13] J.U. Surjadi, L.Gao, H. Du, X. Li, X. Xiong, N.X. Fang, and Y. Lu, Adv. Eng. Mater.,21, 1800864 (2019)[14] C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljaˇc´ıc, NanoLett. 14, 2783 (2014).[15] A. Taghizadeh and I.-S. Chung, Appl. Phys. Lett. 111, 031114 (2017).[16] H.M. Doeleman, F. Monticone, W. Hollander, and A. Al, A. F. Koenderink, NaturePhotonics. 12, 397 (2018)[17] E. N. Bulgakov and D. N. Maksimov, Phys. Rev. Lett. 118, 267401 (2017).[18] D. C. Marinica, A. G. Borisov, and S. V. Shabanov Phys. Rev. Lett. 100, 183902 (2008)[19] R.F. Ngandali and S.V. Shabanov, J. Math. Phys. 51, 102901 (2010)[20] O.A. Bauchau and J. I. Craig, Structural Analysis With Applications to AerospaceStructures, Springer (2009)[21] J.N. Reddy Wiley, Energy Principles and Variational Methods in Applied Mechanics2nd Edition, (2002)[22] L.D. Landau, E.M. Lifshitz Course of Theoretical Physics, 7, 101 (1959)[23] V. Twersky, J. Opt. Soc. Amer. 52, 145-171 (1962)[24] F. J. Garc´ıa de Abajo, Rev. Mod. Phys. 79, 1267 (2007)[25] R.F. Ngandali and S.V. Shabanov,