Anomalous transmission through periodic resistive sheets
AAnomalous transmission through periodic resistive sheets
Antonin Coutant, ∗ Yves Aur´egan, † and Vincent Pagneux ‡ Laboratoire d’Acoustique de l’Universit´e du Mans,Unite Mixte de Recherche 6613, Centre National de la Recherche Scientifique,Avenue O. Messiaen, F-72085 LE MANS Cedex 9, France (Dated: May 19, 2020)This work investigates anomalous transmission effects in periodic dissipative me-dia, which is identified as an acoustic analogue of the Borrmann effect. For this, thescattering of acoustic waves on a set of equidistant resistive sheets is considered. Itis shown both theoretically and experimentally that at the Bragg frequency of thesystem, the transmission coefficient is significantly higher than at other frequencies.The optimal conditions are identified: one needs a large number of sheets, whichinduce a very narrow peak, and the resistive sheets must be very thin compared tothe wavelength, which gives the highest maximal transmission. Using the transfermatrix formalism, it is shown that this effect occurs when the two eigenvalues ofthe transfer matrix coalesce, i.e. at an exceptional point. Exploiting this algebraiccondition, it is possible to obtain similar anomalous transmission peaks in more gen-eral periodic media. In particular, the system can be tuned to show a peak at anarbitrary long wavelength.
Keywords: Acoustic wave propagation, Absorption, Phononic Crystals.
I. INTRODUCTION
Periodic media, also known as phononic crystals are now a widely used tool to controlthe propagation properties of sound waves [1]. The first effect of periodicity is to induceband gaps: frequency ranges where waves become evanescent inside the material. Thisbands and gaps structure of the dispersion relation has found a large range of uses, amongwhich large decreases of transmission [2], wave localization and wave guiding [3, 4], or highquality resonances [5].On the contrary, the effect of absorption in periodic media has received much less atten-tion. Several authors have investigated the changes of the dispersion relation and whetherabsorption tends to enlarge or reduce band gaps [6–8]. However, in periodic media com-bination of absorption and Bragg scattering can lead to peculiar transmission properties.A well-known effect of that type in X-ray crystallography, called the Borrmann effect, isthe anomalous transmission of waves across a crystal slab at the Bragg frequency [9, 10].More recently, a similar anomalous transmission was observed in photonic crystals [11–13]. In these works, a light wave is sent on a slab made of successive pairs of layers madeof a high-absorbing and a low-absorbing material. When the Bragg condition is met(the normal wavelength is twice the spatial periodicity of the medium), an anomalouslyhigh transmission is observed. In acoustics, a similar anomalous transmission has beenreported for layered lossy structures in [14]. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] M a y In this work, we study theoretically and experimentally an acoustic realisation of theBorrmann effect, using plane resistive sheets placed equidistantly. We show that thereis a peak in transmission at the Bragg frequency (distance between successive sheets ishalf a wavelength), with a width decreasing rapidly with the number of sheets. Thiscontrasts with what is usually observed in phononic crystals, where the Bragg frequencyis associated to a low transmission. Moreover, we show that the anomalous transmissionis the highest when the resistive sheets of a given resistance are thin enough compared tothe wavelength.We also provide a detailed analysis of this effect by analyzing the Bloch wave solutionsinside the material using the transfer matrix formalism. In particular, we show that theanomalous transmission corresponds to an exceptional point (EP) of the 2 × II. THEORETICAL DESCRIPTION OF LAYERED DISSIPATIVE MEDIA
We consider acoustic waves propagating in air. The pressure field obeys the Hemholtzequation ∆ p + k p = 0 , (1)with k the natural wavenumber. Time dependence e − iωt is omitted, with ω = kc and c the speed of sound in air. The main ingredient of our setup is the use of purely resistivesheets [15] made of a very fine-meshed fabric. These sheets have acoustical propertiesmuch like porous materials: the characteristic size of the holes are such that viscouseffects dominate inside the fabric and the wave is attenuated [16]. Since their thickness isalso much smaller than the wavelength, an acoustic wave across such a sheet undergoesa pressure jump from one side to another. The pressure discontinuity is characterized bythe resistance γ such that [ p ] + − = − ρ c γu (0) , (2)where u is the acoustic velocity, which is continuous across the sheet [15], and ρ is theair density. Notice that the resistance γ is adimensionalized by the natural impedance ρ c of air. We now investigate how acoustic waves propagate in a layered medium madeof a number N of identical resistive sheets equidistantly spaced by (cid:96) (see Fig. 1). A. Periodic medium
As a first step, we describe how waves propagate inside such a layered medium ( N →∞ ). We consider for this a unit cell of size (cid:96) , with a resistive sheet in the center (seeFig. 1). We assume the problem to be one-dimensional, either by sending plane waveswith normal incidence, or because the sheets are placed in a waveguide of small cross N (cid:29) R T ... (cid:96) γ γ unit cell x Figure 1: Layered medium made of equidistantly spaced resistive sheets. section (monomodal propagation), as for the experiments of section III. In this case, itis very convenient to use the formalism of the transfer matrix [17]. The entries of thetransfer matrix are the forward and backward propagating components of the pressurefield (see Appendix A for details). The transfer matrix of a single cell is given by M c = t − r t r t − r t t , (3)where t (resp. r ) is the transmission (resp. reflection) coefficient of a single cell. Sincesuch a cell (and hence also a collection of them) is reciprocal and mirror symmetric, thetransmission and reflection are identical whether the incident wave comes from the left orthe right [17]. This justifies the specific form of the transfer matrix (3).By solving the Hemholtz equation (1) from − (cid:96)/ (cid:96)/ γ , size (cid:96) and wavenumber k : M c = (cid:16) − γ (cid:17) e ik(cid:96) γ − γ (cid:16) γ (cid:17) e − ik(cid:96) . (4)Eigenvectors of the transfer matrix of a cell represent Bloch wave solutions of the periodicsystem. The corresponding eigenvalue is the amplitude change after each cell. Since thesystem is reciprocal, the two eigenvalues are inverse, hence one can write them as e ± iq(cid:96) ,where q is in general complex. This amounts to looking for solutions in the form of Blochwaves with effective wavenumber q : p = e iqx ϕ ( x ) , (5)where ϕ has the periodicity of the medium ϕ ( x + (cid:96) ) = ϕ ( x ). Using the fact that the sumof both eigenvalues is the trace of M c , we obtain the dispersion relation of the periodicmedium: 2 cos( q(cid:96) ) = 2 cos( k(cid:96) ) − iγ sin( k(cid:96) ) . (6)To represent this relation, we show in Fig. 2 the trajectories of the eigenvalues of M c inthe complex plane as the frequency k varies. From equation (6), we immediately notice aremarkable property of the constructed medium: at the Bragg frequencies k(cid:96) = π the lastterm in (6) vanishes. In the following we focus on the Bragg frequency but we point outthe same is true for all multiple of the Bragg frequency k(cid:96) = nπ ( n ∈ N ), and thereforeso is the anomalous transmission we describe in this work. Equation (6) leads to twodistinct behaviors for the absorption of waves. At generic frequencies k(cid:96) (cid:54) = nπ , q has animaginary part governed by γ , and hence, a wave traveling in the medium in attenuatedexponentially with the distance. On the contrary, at the Bragg frequencies, the imaginarypart of q vanishes, and one has a double solution for q (see Fig. 2). -2 -1 0 1 2-1.5-1-0.500.511.5 (a) -1.2 -1 -0.8-0.3-0.2-0.100.10.2 (b)(a) Figure 1:
Figure 2: (a) Eigenvalues λ = e ± iq(cid:96) of the transfer matrix M c of a single cell in the complexplane. k(cid:96) runs from 0 to 2 π , and we took γ = 0 .
5. The arrows show the direction of increasingwavenumber k and the black dashed line is the unit circle. (b) Same as (a) but zoomed nearthe EP. We added the trajectory of the eigenvalues for a sheet of finite thickness (cid:96) = 0 . (cid:96) (inyellow); this will be discussed in Sec. II D. To fully understand the absorption properties at the Bragg frequency, we discuss indetails the two linearly independent wave solutions. At the Bragg frequency, the 2 × M c possesses a single eigenvalue, which corresponds to an EP andconsequently M c has a single eigenvector and a generalized eigenvector . The two areshown in Fig. 3. The eigenvector is a standing wave with nodes of velocity (equivalently,pressure extrema) placed on the resistive sheets. Hence, one directly sees from the pressurejump condition (2) that this solution does not interact with the resistive sheets, whichexplains why its amplitude is not affected by it (Fig. 3 left panel). To build the secondsolution, let us consider a real-valued standing wave in one cell, with pressure nodes placedon the resistive sheets (so it is by construction linearly independent with the eigenvector).While moving across a sheet, the wave acquires an imaginary part proportional to thederivative of the real part (see equation (2)). This imaginary part has its (pressure)extrema on the resistive sheets, and hence, it triggers no change of amplitude on the realpart. Hence, we just build a solution made of a real part being a standing wave notaffected by the resistive sheets, and an imaginary part having always the same increase inamplitude while passing across a sheet (Fig. 3 right panel). In other words, the amplitudevaries linearly with the distance, and therefore we anticipate a transmission coefficientdecreasing linearly with the size of the system.To conclude this section, we wish to underline two points. First, the occurrence ofan EP of the transfer matrix is a consequence of two features: half the periodicity of An EP is defined as a point in parameter space such that two eigen-values as well as the correspondingtwo eigen-vectors of a matrix coalesce (see [18] for details). Since a single eigen-vector exists at theEP, a basis can be obtained by adding a generalized eigen-vector. In our case (a 2 × V satisfies ( λ − M ) · ˆ V = 0 but ( λ − M ) · ˆ V (cid:54) = 0. Notice that because M c is 2 × the wave is the same as the distance between successive sheets and the symmetry of thewave is such that velocity nodes are aligned with the resistive sheets. Understandingthis will allow us to manufacture EPs in more general setups, and therefore have similaranomalous transmission effects. This will be explored in Sec. IV. Second, when one has anEP, the eigenvector is necessarily accompanied by a generalized eigenvector with a linearlychanging amplitude. Having this point in mind is central to understand the transmissionproperties of the medium. (a) (b) Figure 3: Wave solutions at the Bragg frequency ( k(cid:96) = π ) with γ = 0 .
5. The locations ofthe resistive sheets are shown as black dashed lines. (a) Eigen-vector of M c . (b) Generalizedeigen-vector of M c . B. Slab of layered medium
We now discuss the scattering coefficients across a number N of cells as in Fig. 1. Wefirst notice that the total transfer matrix is given by the N -th power of M c . We then usethe same relation as (3) to obtain the scattering coefficients: M Nc = T − R T RT − RT T . (7)Because of the relative simplicity of the problem, all scattering coefficients can be obtainedexplicitly using the Chebyshev identity for M Nc (details are given in Appendix A). Theresults are shown in Fig. 4. We observe what was anticipated from the medium dispersionrelation (6): The transmission decreases exponentially with the number N of sheets,except at the Bragg frequency, where the decrease is instead linear. When fixing N largeand varying the frequency, this change of behavior manifests itself as a very sharp peakof transmission near k(cid:96) = π . The more sheets are taken, the finer the peak is.To further characterize this transmission peak, we estimate the maximum and min-imum transmission. When N is large enough, the transmission outside of the Braggfrequency is a plateau near its minimum (see Fig. 4). To estimate it, we can compute the -6 -4 -2 Figure 4: (a) evolution of the transmission for different values of the frequency k as a functionof N . (b) scattering coefficients for N = 30 resistive sheets with γ = 0 . transmission at the frequency k(cid:96) = π/ N (cid:29)
1. This gives us | T min | ∼ (cid:32) (cid:112) γ /
41 + (cid:112) γ / (cid:33) e − N argsh( γ/ . (8)We point out that | T | at k(cid:96) = π/ N , theamplitude of | T | oscillates around its plateau value, and hence, depending on the parity of N , it will be either a local maximum or a local minimum. However, these oscillations arevery small in the regime of interest, scaling as O ( e − N argsh( γ/ ). When γ is small enough,the transmission coefficient T min takes the simpler asymptotic form | T min | ∼ e − Nγ/ , (9a) ∼ (cid:16) γ (cid:17) − N . (9b)On the second line we recognize the product of N times the transmission of a single sheet t . In other words, outside of the Bragg frequency, the effects of the sheets multiply. Thisis strictly speaking true for γ (cid:28)
1, but we see from (8) that the behavior is similar alsofor large γ .The maximum transmission is reached at the Bragg frequency, and takes the simpleform | T max | = 11 + N γ/ . (10)This can be obtained directly from the general expression of M Nc (see Appendix A equa-tion (A5)), or by noticing that at the Bragg frequency, M c = − I + γ A with A = 0, andhence, M Nc = ( − N ( I − Nγ A ). We notice that this is the transmission of a single sheetof resistance N γ . In other words, at the Bragg frequencies, the effects of the differentsheets add up.To further understand the two characteristic behaviors of the transmission, we repre-sented the scattering solutions in Fig. 5. Outside the Bragg frequency, the amplitude ofthe wave decreases exponentially while the wave propagate in the material, as expectedwithin a dissipative medium. On the contrary, the structure of the solution at the Braggfrequency is rather unusual, with part of the wave (Im( p ) in Fig. 5) being seeminglyunaffected by the sheets, and another part (Re( p ) in Fig. 5) with a linearly decreasingamplitude. This can be readily understood by considering the two linearly independentsolutions of the periodic medium as described in Sec. II A. Since the scattering solutionwe consider is purely right moving at the end of the slab, it cannot be proportional to theeigenvector of M c , which is a standing wave. Hence it is a combination of the eigenvectorand generalized eigenvector, which explains that part of the wave is unaffected by theresistive sheets, while the other undergoes an amplitude decrease linear with the distance. Figure 5: Scattering solutions on a slab of N = 30 resistive sheets (marked by dashed blacklines) with γ = 0 .
5. The locations of the resistive sheets are shown as black dashed lines. (a) At k(cid:96) = π/ k(cid:96) = π where there is the EP associatedwith a linear decrease. C. Width of the anomalous transmission peak
A remarkable property of the anomalous transmission observed in Fig. 5 is that thepeak becomes increasingly narrow as N is taken larger. To see this, we analyze thetransmission coefficient in the vicinity of the EP. This is legitimate in the limit of a largenumber of resistive sheets N (cid:29)
1, since the peak becomes finer in that limit. For this wedefine k(cid:96) = π + δ, (11a) q(cid:96) = π + σ, (11b)and assume small δ and σ . The dispersion relation (6) then reduces to σ = iγδ. (12)We now look at the vicinity of the peak by assuming | σ | (cid:28) N (cid:29)
1, but withoutany restriction on
N σ . In this limit, the transmission coefficient simplifies into | T | ∼ | σ | γ | sin( N σ ) | ∼ | δ | γ sin (cid:16) N (cid:112) γ | δ | / (cid:17) + γ sinh (cid:16) N (cid:112) γ | δ | / (cid:17) . (13)We now define the width as the value δ = ± (cid:96) ∆ k such that | T | = T max /
2. From the aboveexpression we obtain (cid:96) ∆ k = 2 ζ γN , (14)where we introduced the numerical constant ζ ≈ . ζ = sin ( ζ ) + sinh ( ζ ). From equation (14), we see that the peak becomesrapidly finer as N increases. It is also interesting to notice that the maximum andminimum transmission are both essentially governed by N γ (see Eqs. (9) and (10)), soone can maintain those fixed while increasing the peak width by adding more cells.
D. Effect of thickness
We now analyze a second key property of the anomalous transmission peak of Fig. 4:the smaller the sheet thicknesses are compared to the wavelength, the stronger the trans-mission peak is. To illustrate this, we consider a dissipative slab made of open horizontalpores of length (cid:96) . This corresponds to a simple model of porous material [16]. Thenet effect of the slab can be described with an effective (complex) dimensionless density ρ eff ( k ) of fixed (independent of (cid:96) ) total resistance γρ eff = (cid:114) iγk(cid:96) . (15)We also assume that the effective bulk modulus is the same as in air (this amounts toneglecting visco-thermal effects in the material, which is legitimate at low frequencies).Using (15), the Helmholtz equation in the material reads ∂ x p + k ρ eff p = 0 . (16)We now put this slab inside a cell of length (cid:96) , similarly to Fig. 1. Imposing continuityof pressure and velocity at the interfaces allows us the extract the transfer matrix of thecell. From this we obtain the transmission and reflection coefficients of a single cell: t = 2 ρ eff ρ eff exp( i (( ρ eff − (cid:96) + (cid:96) ) k ) − i ( ρ eff − sin( ρ eff (cid:96) k ) e − i ( (cid:96) − (cid:96) ) k , (17a) r = i ( ρ −
1) sin( ρ eff (cid:96) k )2 ρ eff exp( i (( ρ eff − (cid:96) + (cid:96) ) k ) − i ( ρ eff − sin( ρ eff (cid:96) k ) e − i ( (cid:96) − (cid:96) ) k , (17b)A direct verification shows that the limit of equation (17) for (cid:96) → (cid:96) . This is done by imposingthat the scattering coefficients in the zero frequency limit verify (cid:12)(cid:12)(cid:12)(cid:12) r t (cid:12)(cid:12)(cid:12)(cid:12) k → = γ / , (18)independently of (cid:96) . When looking at the Bloch wave solutions, the EP at k(cid:96) = π is replaced by an avoided crossing (see Fig. 2 (b)). We therefore anticipate that thetransmission peak will be reduced. In Fig. 6, we show the transmission coefficient acrossa periodic piece of such porous slabs. We observe that the plateau of transmission outsidethe Bragg frequency has the same value for all (cid:96) , meaning that this value only dependson the total resistance γ . On the contrary, the maximum transmission at the anomalouspeak increases significantly as the dissipative slab becomes of sub-wavelength size. As wesee in Fig. 2, the optimal transmission is reached for (cid:96) /(cid:96) (cid:46) -3 -2 -1 Figure 6: (a) transmission coefficient against the frequency, near the Bragg frequency. We varythe thickness fraction (cid:15) = (cid:96) /(cid:96) while maintaining γ = 0 . N = 30 cells. (b)maximum transmission (for k(cid:96) = π ) as a function of the thickness fraction (cid:15) = (cid:96) /(cid:96) . III. EXPERIMENTAL RESULTS
For the experimental characterization of the Borrmann effect with resistive sheets, weused a rigid cylindrical duct of radius 3cm, with two measurement sections on each end.Each of them comprises two microphones and an acoustic source, allowing to extractreflection and transmission coefficients. A more detailed description of the experimentalsetup can be found in [19] (see also Fig. 7). Inside the duct, we put resistive sheets madeof a fine meshed fabric mounted on a metallic perforated plate. All sheets are separatedby a distance (cid:96) = 7 ± . . γ in the jump condition (2) by a complex (adimensional)impedance z = γ − iµk(cid:96) . γ is the resistance and µ the added mass. Both coefficients arereal and adimensional. We also take into account the losses due to friction on the wall ofthe waveguide (see for instance Sec. 4.5.3 in [20]). (a) (b) Figure 1:
Figure 7: Pictures of the experimental setup. (a) cylindrical duct with speakers and microphoneson both ends. (b) resistive sheets made of fine meshed fabric mounted on a perforated metallicplate. The radius of both the duct and the resistive sheets is 3cm. N = 10, 15, 20, and 25 resistive sheets.The results are shown in Fig. 8. We clearly see the anomalous transmission at the Braggfrequency. We also see that the peak becomes narrower as N is increased. The valueof the impedance is determined by comparing the theory with experimental data usinga least square method. We found γ = 0 . ± .
03 and µ = 0 . ± .
01. The value ofthe resistance was measured using an independent method, and coincides within errorbars. Additionally, we notice that the transmission peak becomes asymmetric comparedto Fig. 4. This is due to the non-zero imaginary part of the impedance and can beunderstood in the following way: when z is purely imaginary (Im( z ) > N is increased (Fig. 8 right panel). We can see that the decrease in transmissionis slower at the Bragg frequency. -3 -2 -1 (b) Figure 8: (a) transmission coefficient against the frequency for different number of resistivesheets. Continuous lines are experimental data and dashed lines are theory. (b) maximumtransmission and minimum transmission (taken for lower frequencies than Bragg) as a functionof the number of sheets N . Diamonds are experimental data and continuous lines are theory. IV. ANOMALOUS TRANSMISSION IN MORE GENERAL PERIODICMEDIA
In this section, we show how to obtain a similar anomalous transmission by placingresistive sheets inside a (lossless) structured periodic medium. This has two main interests.First, by tuning the base medium, we can change the frequency at which the anomaloustransmission peak occurs. In particular, this allows us to obtain a peak at a wavelengthsignificantly larger than the size of the unit cell. Second, the value of the maximumtransmission is in general increased compared to an array of resistive sheets alone (thatis compared to equation (10)).
A. Manufacturing exceptional points
For our base medium we use a waveguide with periodic changes of cross section: a firsttube of cross section S and length (cid:96) followed by one of cross section S and length (cid:96) ,1and repeated on a spatial period (cid:96) = (cid:96) + (cid:96) (see Fig. 9). This corresponds to an arrayof Helmholtz resonators in series, the small cross section playing the role of the neck andthe large one being the cavity. (cid:96) Unit cell γ S S (a) (cid:96) Unit cell γ S S (b) Figure 9: Design of a configuration with resistive sheets and resonators in series. (a) Resistivesheets placed in the middle of the large cross-section parts. (b) Resistive sheets placed in themiddle of the small cross-section parts.
To simplify, we consider that the transverse lengths of the guide are much smaller thanthe cell length (i.e. S , (cid:28) (cid:96) ) and hence the typical wavelengths. In this limit, evanescentmodes excited near each cross-section change can be safely neglected. Therefore, acousticpropagation reduces to that of the plane mode. This means that the system is effectivelyone-dimensional and can be describe using the same transfer matrix formalism as before.At cross section changes, pressure and debit are continuous. Using these conditions, weobtain the transfer matrix (see Appendix A), which leads to the dispersion relation2 cos( q(cid:96) ) = (1 + ν ) ν cos( k(cid:96) ) − (1 − ν ) ν cos( k ( (cid:96) − (cid:96) )) , (19)where ν = S /S is the cross section ratio. This dispersion relation presents bands andgaps, as shown in Fig. 10 (a). When looking at the eigenvalues e ± iq(cid:96) of the transfer matrix,each edge of a band gap is an EP (see Fig. 10 (b)). Upon adding dissipation, eigenvaluesare pushed away from the unit circle, and EPs are avoided. This encodes the fact thatwaves cannot propagate without attenuation. The idea is to place resistive sheets in thebase medium such that a specific EP stays exceptional.To do so we use mirror symmetric cells, with either the large cross section tube orthe small cross section tube in the middle (see Fig. 9). At an edge of a band gap, the(unique) eigenvector of the transfer matrix is either symmetric or anti-symmetric [21].This corresponds to a standing waves with pressure nodes either in the center (symmetricvector) of the cell or on the edges (anti-symmetric vector). Now, by adding resistivesheets at the location of the velocity nodes (pressure extrema), the same argument asin Sec. II A shows that this standing wave is insensitive to the presence of the sheets.2Hence, one obtains an EP for the transfer matrix, with similar properties as in Sec. II A(a detailed proof of this statement is given in Appendix B). This is shown in Fig. 10 forour configuration with an array of resonators in series and resistive sheets (configurationof Fig. 9). The wave at the lower edge of the first gap (marked as crosses in Fig. 10) hasits velocity nodes in the center of the large cross section tube, while that of the upperedge has its velocity nodes in the center of the small cross section tube. (a) -3 -2 -1 0 1 2 302468 -2 -1 0 1 2-1.5-1-0.500.511.5 (b) -2 -1 0 1 2-1-0.500.511.5 (c) -2 -1 0 1 2-1.5-1-0.500.51 (d) Figure 10: Dispersion relation and eigenvalues λ = e ± iq(cid:96) of the transfer matrix. We used ν = 2, (cid:96) /(cid:96) = 0 .
5, and γ = 0 . γ = 0).The crosses (resp. circles) shows the values of k at the lower edge (resp. upper) of the gap (alsoshown in (b-d)). (b-d) Eigenvalues of the transfer matrix for the setup of Fig. 9 and k(cid:96) from0 to 2 π : (b) without resistive sheets, (c) resistive sheets in the large cross section tube (Fig. 9(a)), and (d) resistive sheets in the small cross section tube (Fig. 9 (b)). B. Transmission peak
Once an EP is obtained, the transmission properties of the medium (with resistivesheets) follow as in Sec. II A: generic frequencies (not associated with an EP of the transfermatrix) are attenuated exponentially with the distance either due to dissipation or becausethey are in the gap and hence are evanescent. On the contrary, at the EP, the transmissionis decreasing linearly with the distance due to a combination of a standing wave insensitiveto the dissipation and a generalized eigenvector with linearly changing amplitude. The3transmission as a function of the frequency is shown in Fig. 11. We see that dependingon the location of the resistive sheets in the base medium, the transmission peak is foundon one edge or the other of the band gap.
Figure 11: Transmission coefficients as a function of the frequency for the configurations ofFig. 9. The upper and lower edge of the first gap are indicated by dashed lines. We used N = 30unit cells, ν = 40, (cid:96) /(cid:96) = 0 . γ = 2. For this cross-section ratio, the wavelength of thelower edge is ten times smaller than the size of the unit cell. (a) We placed the resistive sheetsin the center of the large cross section tube (Fig. 9 (a)). (b) We placed the resistive sheets inthe center of the small cross section tube (Fig. 9 (b)). To better understand the properties of the transmission peak, and in particular itsmaximum value, we analyze the transmission in the vicinity of the EP. We show this inFig. 12 for the configuration where the anomalous transmission is at the lower edge ofthe first gap, at a frequency noted k = k (cid:63) . The first thing to notice is that the peakis asymmetric: frequencies above the EP are highly suppressed because they correspondto the location of the gap of the base medium. On the contrary, frequencies below theEP are less efficiently suppressed. Moreover, the transmission oscillates while decreasingwhen the frequency moves away below the EP. As a result, the maximum transmissionis not reached at the EP, but at a frequency close below, and the transmission value isalways higher than that at the EP (see Fig. 12 left panel). When one looks at the scalingof the transmission as a function of N , the behavior is similar to the array of resistivesheets: at a fixed frequency, the transmission decreases exponentially with N , except atthe EP, where it decreases linearly. However, the maximum transmission is reached at afrequency k − k (cid:63) = O (1 /N ), which gets closer to the EP when N increases. This leadto a maximum transmission also decreasing linearly as a function of N (see Fig. 12 rightpanel).As a last remark, we point out that the symmetry of the wave function at the edges ofa gap is directly related to the topological properties of the system [22]. In particular, anoticeable change occurs when the base medium undergoes a topological phase transition.Indeed, it was observed in [13] that the location of the anomalous transmission peakschanges across a topological phase transition. Such a transition is characterized by a gapclosing and reopening when varying a parameter (such as the cross-section ratio ν here),while the wave functions at the edges of the gap exchange symmetry [21]. Therefore,from what precedes, we conclude that the anomalous transmission frequency will changefrom the lower edge of the gap to the upper edge of the gap, in agreement with what wasobserved in [13].4 -0.02 -0.01 0 0.01 0.0200.020.040.060.080.1 -4 -3 -2 -1 Figure 12: Transmission over a N cells of the configuration of Fig. 9 with resistive sheets in thelarge cross-section tube as a function of the frequency centered on the lower edge of the gap δ = k − k (cid:63) . We used γ = 2, ν = 40, (cid:96) /(cid:96) = 0 . δ max ( N ) is the frequency shift giving themaximum transmission. V. CONCLUSION
In this work, we use a periodic set of resistive sheets to realize an acoustic analogue ofthe Borrmann effect. This effect manifests itself as an anomalously high transmission atthe Bragg frequency of the system. This anomalous transmission is theoretically predictedfor our system and observed experimentally. Using the transfer matrix formalism, weshow that this anomalous transmission corresponds to an exceptional point at the Braggfrequency. As a result, the transmission coefficient decrease linearly with the number N of sheets, while it decreases exponentially for other frequencies. This change of scalingimplies that at large N , the transmission shows a very narrow peak at the Bragg frequency.In addition, we show that the subwavelength size of the resistive sheets is a key ingredient.When taking into account a finite thickness, the exceptional point is replaced by anavoided crossing, and the maximum transmission is lowered accordingly.Moreover, the exceptional point approach of the effect allows us to show that a similartransmission peak can be obtained at the edge of a band-gap of a lossless periodic mediumby placing resistive sheets at the velocity nodes of the eigensolution of the problem. Thiscomplements similar observations in photonic crystals [11–13] by providing a constructiveway of obtaining a Borrmann anomalous transmission and relating it to the topologicalproperties of the system.5 Appendix A: Transfer matrices and scattering coefficients
In this section we give explicit formulas for the scattering coefficients over a slab of N cells. For this we use the Chebyshev identity [17], which gives the N -th power of a2 × p ( x ) = p + ( x ) + p − ( x ) [17]. Each component are easilyexpressed in terms of pressure and velocity: p + = p + u , and p − = p − u , (A1)so that in the absence of scatterer p ± ( x ) = A ± e ± ikx . Then, the transfer matrix from apoint x to a point x is defined by (cid:18) p + ( x ) p − ( x ) (cid:19) = M · (cid:18) p + ( x ) p − ( x ) (cid:19) . (A2)We now consider the general form of a transfer matrix for the unit cell: M c . = (cid:18) α β ˜ β ˜ α (cid:19) , (A3)We assume reciprocity (which gives det( M ) = 1), and hence, we can write the eigenvaluesof M c as λ = e ± iq(cid:96) . The N -th power of M c is given by M Nc = (cid:32) sin( Nq(cid:96) )sin( q(cid:96) ) α − sin(( N − q(cid:96) )sin( q(cid:96) ) sin( Nq(cid:96) )sin( q(cid:96) ) β sin( Nq(cid:96) )sin( q(cid:96) ) ˜ β sin( Nq(cid:96) )sin( q(cid:96) ) ˜ α − sin(( N − q(cid:96) )sin( q(cid:96) ) (cid:33) . (A4)We now consider the scattering on an array of resistive sheets made of N unit cells,see Fig. 1. Using this and the relation between the transfer matrix and the scatteringcoefficients (7), we obtain1 T = (cid:16) γ (cid:17) sin( N q(cid:96) )sin( q(cid:96) ) e − ik(cid:96) − sin(( N − q(cid:96) )sin( q(cid:96) ) , (A5a) R = − γ N q(cid:96) )sin( q(cid:96) ) e − ik(cid:96) × T. (A5b)Evaluating this at the Bragg frequency k(cid:96) = q(cid:96) = π gives us the maximum transmission(10). To evaluate the plateau of minimum transmission, we evaluate it at the frequency k(cid:96) = π/
2, as in the core of the text. First, we see from the dispersion relation (6) thatthe Bloch wavenumber is given by q(cid:96) = π i argsh (cid:16) γ (cid:17) . (A6)Then, using it in the above expressions (A5) we obtain1 | T min | = (cid:32) (cid:112) γ / (cid:112) γ / (cid:33) e N argsh( γ/ + ( − N +1 (cid:32) − (cid:112) γ / (cid:112) γ / (cid:33) e − N argsh( γ/ . (A7)Taking the limit N (cid:29)
1, we obtain equation (8). We also give the transfer matrix of acell of the array of resonators (Fig. 9) in the monomode approximation: M (0) c = (cid:32) (1+ ν ) ν e ik ( (cid:96) + (cid:96) ) − (1 − ν ) ν e ik ( (cid:96) − (cid:96) ) i − ν ν sin( k(cid:96) ) − i − ν ν sin( k(cid:96) ) (1+ ν ) ν e − ik ( (cid:96) + (cid:96) ) − (1 − ν ) ν e − ik ( (cid:96) − (cid:96) ) (cid:33) . (A8)6The superscript (0) is here to recall that it is lossless. When ν > M (0) c describesFig. 9 (a), while when ν < M (0) c by the transfer matrix of a single resistive sheet M γ on the right. The latteris given by equation (4) with (cid:96) = 0: M γ = − γ γ − γ γ . (A9)In the core of the text, we mainly focused on transmission properties of the differentmedia. But it is also instructive to look at the reflection coefficient and the absorption A = 1 − | T | − | R | . This is done in Fig. 13 for the array of resistive sheet (configurationof Fig. 1) and the resistive sheets inside the array of resonators (configuration Fig. 9).Notice that in both cases, the increase of transmission due to the Borrmann effect is alsoaccompanied by an increase in reflection, and therefore, a decrease of absorption. Figure 13: (a) scattering coefficients over N equidistant resistive sheets as in Fig. 1. We used N = 30 and γ = 0 .
5. (b) scattering coefficients over N cells of resonators and resistive sheets inthe middle of the large cross section tube (configuration of Fig. 9 (a)). We used N = 30, ν = 40, (cid:96) = (cid:96) , and γ = 2. Appendix B: General condition to obtain an exceptional point with resistive sheets
In Sec IV, we saw how one can obtain an anomalous transmission using a base medium(lossless) and adding resistive sheets. Although we used in this work a base medium madeof series resonators, the construction of EPs in terms of the symmetry of the eigenvectorof the cell transfer matrix is very general. To illustrate this, we consider a base mediumdescribed by a transfer matrix of the form (A3). We only assume reciprocity and mirrorsymmetry, which impose β + ˜ β = 0 , (B1) α ˜ α + β = 1 . (B2)Now, since 2 cos( q(cid:96) ) = Tr( M ), on the edge of the first gap (the construction is similar forhigher gaps) we must have Tr( M ) = −
2. This gives us a last relation α + ˜ α = − . (B3)7We can now solve equations (B1), (B2) and (B3), and we obtain the general form of thetransfer matrix at an exceptional point: M (0) c = − − i µ is ± µ − is ± µ − i µ , (B4)where µ is defined by d − a = iµ , and s ± = ±
1. Notice that because the base mediumis assumed lossless, µ ∈ R . The important factor is s ± , because this sign dictates thesymmetry of the eigenvector. Indeed, we see that the (unique) eigenvector of the matrix(B4) is (cid:18) AB (cid:19) = (cid:18) s ± (cid:19) . (B5)If s ± = 1, the eigenvector is symmetric, if s ± = −
1, the eigenvector is anti-symmetric.Since the transfer matrix M γ of a single resistive sheet is at an EP, it commutes with M (0) c if and only if they have the same eigenvector. Since the eigenvector of M γ is symmetric,we see that if s ± = 1, the product M (0) c · M γ is at an EP. Translated in terms of wavesolutions, this symmetry condition means that the resistive sheets must be placed on thevelocity nodes of the eigen-solution of M (0) c . [1] P. A. Deymier, ed., Acoustic metamaterials and phononic crystals , vol. 173. Springer,Berlin, 2013.[2] J. Vasseur, P. A. Deymier, B. Chenni, B. Djafari-Rouhani, L. Dobrzynski, and D. Prevost,“Experimental and theoretical evidence for the existence of absolute acoustic band gaps intwo-dimensional solid phononic crystals,”
Phys. Rev. Lett. no. 14, (2001) 3012.[3] M. Kafesaki, M. Sigalas, and N. Garcia, “Frequency modulation in the transmittivity ofwave guides in elastic-wave band-gap materials,” Phys. Rev. Lett. no. 19, (2000) 4044.[4] C. Kane and T. Lubensky, “Topological boundary modes in isostatic lattices,” NaturePhysics no. 1, (2014) 39.[5] P. Kalozoumis, G. Theocharis, V. Achilleos, S. F´elix, O. Richoux, and V. Pagneux,“Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev.
A 98 no. 2, (2018) 023838.[6] I. Psarobas, “Viscoelastic response of sonic band-gap materials,”
Phys. Rev.
B 64 no. 1,(2001) 012303.[7] C.-Y. Lee, M. J. Leamy, and J. H. Nadler, “Frequency band structure and absorptionpredictions for multi-periodic acoustic composites,”
Journal of Sound and Vibration no. 10, (2010) 1809–1822.[8] M. I. Hussein and M. J. Frazier, “Metadamping: An emergent phenomenon in dissipativemetamaterials,”
Journal of Sound and Vibration no. 20, (2013) 4767–4774.[9] G. Borrmann, “ ¨Uber extinktionsdiagramme der r¨ontgenstrahlen von quarz,”
Phys. Z (1941) 157–162.[10] B. W. Batterman and H. Cole, “Dynamical diffraction of X rays by perfect crystals,” Reviews of modern physics no. 3, (1964) 681.[11] A. Vinogradov, Y. E. Lozovik, A. Merzlikin, A. Dorofeenko, I. Vitebskiy, A. Figotin,A. Granovsky, and A. Lisyansky, “Inverse Borrmann effect in photonic crystals,” Phys.Rev.
B 80 no. 23, (2009) 235106. [12] V. Novikov and T. Murzina, “Borrmann effect in photonic crystals,” Optics letters no. 7, (2017) 1389–1392.[13] V. Novikov and T. Murzina, “Borrmann effect in Laue diffraction in one-dimensionalphotonic crystals under a topological phase transition,” Phys. Rev.
B 99 no. 24, (2019)245403.[14] A. Cebrecos, R. Pic´o, V. Romero-Garc´ıa, A. Yasser, L. Maigyte, R. Herrero, M. Botey,V. J. S´anchez-Morcillo, and K. Staliunas, “Enhanced transmission band in periodic mediawith loss modulation,”
Applied Physics Letters no. 20, (2014) 204104.[15] U. Ingard,
Noise reduction analysis . Jones & Bartlett Publishers, 2009.[16] J. Allard and N. Atalla,
Propagation of sound in porous media: modelling sound absorbingmaterials 2e . John Wiley & Sons, 2009.[17] P. Markos and C. M. Soukoulis,
Wave propagation: from electrons to photonic crystalsand left-handed materials . Princeton University Press, 2008.[18] T. Kato,
Perturbation theory for linear operators , vol. 132, p. 64. Springer, Berlin, 2013.[19] P. Testud, Y. Aur´egan, P. Moussou, and A. Hirschberg, “The whistling potentiality of anorifice in a confined flow using an energetic criterion,”
Journal of Sound and Vibration no. 4-5, (2009) 769–780.[20] S. W. Rienstra and A. Hirschberg, “An introduction to acoustics,”
Report IWDE (2001)92–06.[21] M. Xiao, Z. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phasesin one-dimensional systems,”
Phys. Rev.
X 4 no. 2, (2014) 021017, arXiv:1401.1309[cond-mat.mes-hall] .[22] G. Ma, M. Xiao, and C. Chan, “Topological phases in acoustic and mechanical systems,”
Nature Reviews Physics1