APPLIED PHYSICS LETTERS 100, 144103 (2012)
Omnidirectional broadband acoustic absorber based on metamaterials Alfonso Climente, Daniel Torrent, and Jose´ Sa´nchez-Dehesaa) Wave Phenomena Group, Universitat Polite`cnica de Vale`ncia, C/Camino de vera s.n. (Edificio 7F), ES-46022 Valencia, Spain
(Received 21 February 2012; accepted 21 March 2012; published online 6 April 2012) We present the design, construction, and experimental characterization of the acoustic analogue of the so called photonic black-hole. The fabricated sample has cylindrical symmetry and consists of two parts, a shell that bends the sound towards the center and a core that dissipates its energy. The shell is made of a metamaterial that perfectly matches the acoustic impedance of air and behaves like a gradient index lens. The experimental data obtained in a multi-modal impedance chamber demonstrate that the proposed acoustic black-hole acts like an onmidirectional broadband absorber with strong C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3701611] absorbing efficiency. V Research in electromagnetic (EM) and acoustic metamaterials has become a hot topic in the last years. One of the latest proposals in the EM realm is the optical black-hole, which was introduced by Narimanov and Kildishev1 as a structure working as an omnidirectional light absorber. The proposed structure consisted of a symmetric shell designed to bend the light rays towards the center and an inner core that perfectly absorbs the focused energy. This work has inspired several theoretical approaches2–5 as well as an experimental demonstration6 with EM waves in the microwave domain. In the acoustic realm, sonic crystals (SC) were initially exploited to develop lenticular lenses7 and interferometers.8 These functionalities are due to the properties of SC in the homogenization limit, i.e., in the range of (low) frequencies where they behave as uniform media with an effective refractive index. It has been shown that in this limit, their acoustic parameters mainly depend on the fraction of volume occupied by the scatterers.9–11 This property has been employed to develop gradient index (GRIN) acoustic lenses,12–14 which previously were only known in optics. This knowledge was recently employed to propose acoustic black holes15,16 whose performance for acoustic waves is equivalent to that proposed for EM waves. This letter reports the design, construction, and characterization of a cylindrical structure working as an omnidirectional broadband acoustic absorber of airborne sound. It consists of a cylindrically symmetric shell, with a radial variation of the acoustic refractive index n(r), and an inner core, where the acoustic energy is dissipated. The shell is a GRIN lens with n(r) matching the index of outer medium (air) and the internal core, respectively. Thus, the shell guides the energy sound to the inner core where it is dissipated by the designed core. Both the shell and the core are acoustic metamaterials based on SC. The shell contains cylindrical scatterers of circular section with different diameters. The core consists of cylinders with equal diameters distributed in a hexagonal lattice with a high filling fraction to produce dissipation by friction. The designed structure was constructed with a 3D prototyping machine and experimentally tested using a multi-modal impedance chamber (MMIC). a)
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[email protected].
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The radial dependence of n(r) has been obtained using the mapping existing between the EM and acoustic parameters for waves propagating in a two dimensional space.17 So, it is possible to translate the solution found for EM waves into the acoustic domain, leading to the following radially dependent index 8 Rs < r < nb (1) Rc < r < Rs ; nðrÞ ¼ Rrs nb : nc þ ic r < Rc where Rc and Rs are the radius of the core and the shell, respectively. They are related through Rs ¼ Rc nnbc ; nb being the refractive index of the air background and nc is the real part of the refractive index of the core material. Finally, c is a parameter representing the absorptive properties of the core. Both the shell and the core are designed using acoustic metamaterials based in SC. First, the shell is designed with five layers of rigid cylinders with diameters changing from the inner layer (the one closer to the center) to the outer layer. The diameter of cylinders in each row is properly determined to obtain the required local dependence of n(r) given by Eq. (1). Additionally, the matching of acoustic impedances between background and the core is also obtained. The procedure is described in Ref. 18 and has been previously employed to develop GRIN acoustical lenses with flat surfaces.13,14 Figure 1 shows the structure fabricated and characterized in this work. The cylinders are made of a plastic material, which can be considered acoustically rigid in the air background due to the high impedance mismatch between these two media. Cylinders in both the shell and the core are placed in an hexagonal lattice of lattice constant a ¼ 7.5 mm. The core is defined within the region with radius Rc ¼ 80 mm and contains cylinders with equal diameter dc ¼ 7.2 mm. Its effective refractive index nc ¼ 1:5 and the corresponding hexagonal lattice has a filling fraction fhex ¼ 2pp ffiffi3 ðdac Þ2 ¼ 83.6%. Note that this fraction of volume occupied by the sound scatterers is near to that corresponding to close-packing (CP), where CP ¼ 90.6%. Consequently, the air is forced to pass through fhex the narrow channels left between cylinders. The acoustic energy is strongly dissipated by friction and defines the
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FIG. 1. Photograph of the structure studied in this work. The outer shell is made of cylinders whose diameters increase with decreasing distance to the center. The inner core is made of identical cylinders in a hexagonal lattice with about 84% of filling fraction. The inset shows the ray trajectories of the sound traveling within the outer shell.
imaginary part c of n(r). The surrounding shell has external radius Rs ¼ 120 mm, and it is separated from the core by the line of defects seen in Fig. 1. It is designed as a SC GRIN lens in which the cylinders closer to the core have the same radius while the external row of cylinders has the minimum radius. The resulting refractive index depends on the distance to the center n(r) and has the ability of bending the impinging sound waves towards the center of the structure as it was proposed in its EM counterpart.1 The inset in Fig. 1 schematically depicts the expected behavior in terms of ray tracing. The broadband performance of the structure shown in Fig. 1 is based on effective medium theory, which is valid for any wavelength large enough to satisfy the homogenization condition, that is, k 4a.10 By substituting the value of a, the corresponding frequencies are 11430 Hz, and, therefore, above this cutoff the effective medium approach breaks down. Figure 2 depicts a schematic view of the impedance chamber and details of the experimental setup employed in the characterization of the fabricated sample, the acoustic black-hole. The MMIC consisting in a rectangular prism
Appl. Phys. Lett. 100, 144103 (2012)
made with height h, width D, and length L. A speaker located at the chamber’s left side excites a sound field that propagates along the positive x-axis. This sound field leaves Region 1 and enters into Region 2 where it interacts with the black-hole sample and it is reflected at the chamber’s right end. Since the chamber is made of aluminum walls 1 cm thick, almost no losses are expected through the walls of the cavity. Therefore, in principle, we assume that all the energy dissipated during the process is absorbed by the sample. By properly measuring the sound field, the dissipated energy can be estimated, and the absorbing efficiency of the structure is then completely characterized. The pressure field P in the chamber can be represented as a linear combination of plane waves propagating along the x-axis (Fig. 2). Moreover, since we consider that the chamber behaves as a waveguide with rectangular section and rigid walls, in principle, all the modes in the YZ plane must be also taken into account. However, measurements are performed for frequencies below 3470 Hz, which is the condition for mono-mode propagation along the z-axis. So, we work under the simplification assumption that multi-modes along the y-axis are the only ones here considered. Therefore, the pressure field P ¼ P(x, y) can be expressed as follows: Pðx; yÞ ¼
(2)
where the propagation constants are bm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 =c2b ðmp=DÞ2 , with x ¼ 2p the angular frequency, cb the speed of sound in air, and D the chamber’s width. Though the summation is infinite, as we increase m the propagation constant bm becomes complex and the waves are evanescent; the contribution of these modes to P is negligible and, therefore, P only depends on a few number of coefficients Am and Bm . To measure the A and B coefficients a set of microphones are employed at selected positions ðxa ; ya Þ in the chamber. An additional microphone is used as a reference (Ref. Mic. in Fig. 2) and is located at position ðx0 ; y0 Þ. The chamber is then excited with a speaker by injecting an additive white Gaussian noise (AWGN), and the pressure field is measured by all the microphones located at the different positions. From the auto-spectrum of the reference microphone S00 ¼ P0 P0 and the cross-spectrum between the other measurements and the reference measure H0a ¼ Pa P0 =S00 ¼ Pa =P0 , the coefficients Am =P0 and Bm =P0 are obtained by solving the following linear system of equations:
H0a ¼
M X Am m¼0
FIG. 2. Scheme of the multimodal impedance chamber and the experimental setup employed in the characterization of the acoustic black-hole. The chamber has a width D ¼ 30 cm, a length L ¼ 150 cm, and height h ¼ 5 cm. The speaker (S) at the left excites an acoustic flow represented by coefficients A, while the backscattered flow is given by coefficients B. Black dots define the 9 pairs of microphones used to record the signal. Another microphophone (Ref. Mic.) is employed as the reference. The sample is placed in the right hand side region, which is accessible by a removable tap.
1 mp X y ; ½Am eibm x þ Bm eibm x cos D m¼0
P0
eibm xa þ
mp Bm ibm xa ya : e cos P0 D
(3)
Note that we can determine the coefficients only relative to the pressure field at the reference position, that is, Am =P0 and Bm =P0 . However, these quantities still allow us to obtain the reflectance in Region 1 since this reflectance is given by the ratio of the energy that leaves the region UB by the energy that enters it UA . They are given, respectively, by
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UA ¼
M Dh X b jAm j2 ; 4xqb m¼0 m
(4)
UB ¼
M Dh X b jBm j2 : 4xqb m¼0 m
(5)
These expressions have been obtained by integrating the acoustic intensity across the chamber cross section (the area being D h). The reflectance is obtained from R ¼ UB =UA , and it is independent of any multiplicative factor appearing in the coefficients. Finally, the energy absorbed by the sample is a ¼ 1 R:
(6)
Measurements have been performed for both the core and the whole structure (core surrounded by the shell). Figures 3 and 4 show the data for coefficients Am =P0 and Bm =P0 of the core and the complete structure, respectively. The frequency region analyzed allows the propagation of modes m ¼ 0 to m ¼ 5. However, due to the symmetry of the chamber, the odd modes are not excited (their coefficients are negligible), and, consequently, modes m ¼ 0, 2, and 4 are the only ones depicted. The vertical lines define the frequencies at which the second ( 2 ¼ 1143 Hz) and the fourth ( 4 ¼ 2287 Hz) modes along the y-axis start propagating inside the chamber. Figure 5 shows the frequency dependence of a for the core (continuous line) and for the complete black-hole (dashed line). Note that for almost any frequency the absorption due to the core is strongly enhanced when the complete structure (coreþshell) is considered, which demonstrates the functionality of the shell. As a parameter defining the absorptive properties of a given structure let us introduce Qa as
FIG. 3. Coefficients Am (a) and Bm (relative to a reference pressure P0 ) of the 3 principal modes propagating in the chamber and characterizing the absorbing behavior of the metamaterial core. The vertical lines are guides for the eye and define the cutoffs at which the modes m ¼ 2 and m ¼ 4, respectively, start propagating inside the chamber.
FIG. 4. Coefficients Am (a) and Bm (relative to a reference pressure P0 ) of the 3 principal modes characterizing the behavior of the acoustic black-hole (shell and core). The vertical lines are guides for the eye and define the cutoffs at which the modes m ¼ 2 and m ¼ 4, respectively, start propagating inside the chamber.
Qa
1 D
ð f aðÞd;
(7)
i
where D ¼ f i is the bandwidth. Quality factors have been determined from the spectra shown in Fig. 5 in the frequency interval from i ¼580 Hz to f ¼ 3400 Hz. Table I shows the results obtained for the acoustic black-hole (coreþshell) and the core alone. For comparison purposes the absorptive factor of a core with the size of the black-hole was also obtained. It is observed that the shell produces a strong enhancement of the core Qa factor due to the embedded guiding mechanism. The enhancement is about 20% and indicates that we have room for improving the absorption power of the proposed structure by better designs of the core and the shell. In summary, we have designed, fabricated, and characterized the acoustic analogue of the photonic black-hole
FIG. 5. Absorption produced by the core of the black-hole sample (continuous line) and by the complete black-hole (dashed line).
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TABLE I. Absorptive factor Qa (see Eq. (7)) of the samples analyzed. The coreþshell defines the complete acoustic black-hole. Sample Core Core Coreþshell
Radius(mm)
Qa (%)
80 120 120
59.0 62.7 79.6
introduced by Narimanov and Kildishev.1 We have demonstrated that the sample constructed acts like a broadband omnidirectional acoustic absorber where a 80% of the impinging acoustic energy is dissipated. This structure has been designed by considering an outer shell that guides the sound energy to the core center and a core that dissipates the incoming energy by friction. Both parts were designed using metamaterials based on sonic crystals. Metamaterial structures like the ones studied here are potentially applicable as acoustic invisibility devices based on total absorption as well as practical structures to attenuate environmental noise. Work supported by the U.S. Office of Naval Research. The authors acknowledge V.M. Garcı´a-Chocano and F. Cervera for useful discussions and technical help. D.T. acknowledges a fellowship provided by the program Campus de Excelencia Internacional 2010 UPV.
1
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