Abstract
The optical dispersive prism is a wellstudied element, which allows separating white light into its constituent spectral colors, and stands in nature as water droplets. In analogy to this definition, the acoustic dispersive prism should be an acoustic device with capability of splitting a broadband acoustic wave into its constituent Fourier components. However, due to the acoustical nature of materials as well as the design and fabrication difficulties, there is neither any natural acoustic counterpart of the optical prism, nor any artificial design reported so far exhibiting an equivalent acoustic behaviour. Here, based on exotic properties of the acoustic transmissionline metamaterials and exploiting unique physical behaviour of acoustic leakywave radiation, we report the first acoustic dispersive prism, effective within the audible frequency range 800 Hz–1300 Hz. The dispersive nature, and consequently the frequencydependent refractive index of the metamaterial are exploited to split the sound waves towards different and frequencydependent directions. Meanwhile, the leakywave nature of the structure facilitates the sound wave radiation into the ambient medium.
Introduction
The advent of metamaterials has led to the emergence of diverse concepts in physics and engineering that were hardly envisaged previously. These artificial materials drew a new perspective for harnessing the physical laws of nature to achieve exotic phenomena, such as cloaking^{1,2}, super lensing^{3,4}, super absorbers^{5,6}, metasurfaces^{7,8}, nonreciprocal devices^{9,10}, etc. In addition to revolutionizing the field of electromagnetics, metamaterials have had a great impact in other fields such as optics^{11}, elastodynamics^{12,13}, thermodynamics^{14} and acoustics^{15,16}. Recently, a metamaterialbased reversed rainbow prism whose refractive index is a decreasing function of frequency, has been presented in the microwave regime. This brought artificial dispersive prisms into the limelight^{17} where the metamaterial is used to deal with causality and passivity constraints^{18}. While optical dispersive prisms continue to draw the attention of researchers^{19,20}, there is no acoustic dispersive prism reported to date, mainly due to the weak acoustic dispersive nature of matter.
This article proposes an acoustic dispersive prism structure in full analogy to the optical dispersive prism. The optical dispersive prism relies on its dispersive nature (frequency dependent refractive index) and low reflectivity (good impedance match between the prism and the ambient medium). For instance, a shaped glass in air satisfies both conditions for visible light. In acoustics, negligible dispersion^{21,22} and weak impedance matching between the ambient medium and the matter prevents spectral separation and efficient energy transfer. Thus, these are the two major problems that should be solved to design an acoustic dispersive prism.
To tackle the first problem, we propose an acoustic dispersive prism based on acoustic metamaterials. Indeed, the peculiar properties of metamaterials stem from their dispersion that can be harnessed and used in the design of dispersive prisms. The transmissionline (TL) metamaterial concept with positive/negative effective material properties (mass density and bulk modulus) can be a good candidate for acoustic dispersive prisms. Acoustic TL metamaterials exhibit frequencydependent refractive indices with positive/negative values which happen in the righthand/lefthand (RH/LH) regions, respectively^{23,24}. By analogy with the definitions of Electromagnetic RH/LH regions^{25}, the RH region of acoustic TL metamaterials is the frequency region where the phase velocity and the group velocity of the propagating acoustic wave have identical signs: , while the LH region is the part of the spectrum where their signs are opposite . Here, the phase velocity is defined by the ratio of the angular frequency to the wavenumber: , and the group velocity is the slope of the curve: . The RH behaviour of matter is a consequence of the positive material properties which are intrinsic to natural materials. However, the LH behaviour of the composite TL metamaterial stems from negative bulk modulus and negative mass density. Negative bulk modulus can be realized with shunt ducts^{26,27} and negative mass density can be achieved with clamped thin plates^{28,29}. Having effective dispersive material properties, the acoustic TL metamaterials with positive/negative acoustic refractive indices can solve the problem of negligible dispersion.
To tackle the second problem, the leakywave mechanism is used to overcome the impedance mismatch between the prism and the ambient medium. As in electromagnetics, leakywave radiation is the process of acoustic power leakage along a waveguiding structure^{24,30,31}. Matching the parallel components of the wavevector on the boundary between the leakywave structure and the external medium leads to wave leakage along the waveguide. This phase matching condition happens when the phase velocity of the wave in the waveguide is bigger than sound wave velocity in the ambient medium c. The frequency band where is referred to as fastwave (leakywave) region^{24,25}. The gradual energy leakage along a waveguide results in a directive beam radiated into the surrounding medium, with the beam direction depending on the frequency of operation.
By merging a leakywave structure with an acoustic TL metamaterial, five distinct frequency regions are made to appear, namely: LH/RH guided regions where the wave propagates through the waveguide without any leakage, LH/RH leakywave region and the bandgap region. To obtain an acoustic prism, the LH/RH leakywave region should be designed to overlap the desired spectrum while the bandgap must be suppressed.
Results
Exploiting acoustic transmissionline metamaterial and leakywave radiation for the realization of an acoustic dispersive prism
The proposed structure is schematically presented in Fig. 1. The device is composed of a fluidfilled rigid waveguide with uniformly spaced, transversally attached, open channels (ducts). Finally, vibrating thin plates are added between each pair of consecutive duct, parallel to the cross section of the waveguide. It is assumed that a sound source feeds the structure from the left input while the output on the right is terminated by an anechoic condition.
Figure 2 presents the output of the acoustic dispersive prism visualized on the basis of frequency shifted standard CIE (Commission internationale de l’éclairage) observer color matching functions. The acoustic pressure wave in the frequency range of 800 Hz–1300 Hz, enters the structure on the left side while the output of the proposed acoustic device is expected to mimic the rainbow pattern within the frequency range of the input signal. As the wave travels along the structure, it leaks out through the side openings, radiating towards frequencymapped directions which are dictated by equation (1). In the frequency range of 800 Hz–1000 Hz the structure has a LH behaviour and the output wave radiates in the backward direction (left quadrant). However, between 1000 Hz and 1300 Hz the behaviour becomes RH and the output wave radiates in the forward direction (right quadrant). The proposed structure is inherently very simple. The dispersive nature of the TL metamaterial together with the directive radiation character of the leakywave antenna solve simultaneously the two invoked problems and therefore the structure behaves as a dispersive prism.
The proposed structure is based on the acoustic transmissionline metamaterial concept with subwavelength unitcells, which can be modelled as series mass and parallel compliance supporting the righthand propagation as well as series compliance and parallel mass supporting lefthand propagation (Fig. 3). Thus, this subwavelength structure exhibits acoustic bandpass filter characteristics allowing the acoustic wave to pass in the working frequency band while filtering the offspectrum frequency components. Furthermore, because of its dispersive nature, different phase velocities, and consequently distinct refractive indices, are associated with Fourier components of the input signal. Besides, the open, transverse ducts make the structure behave as a leakywave antenna. These open channels can be modelled as parallel acoustic radiation resistances, shunting the TL metamaterial to the static pressure. Although the radiation resistance does not affect the TL metamaterial’s function, it physically provides the appropriate condition for leaking acoustic waves towards the direction of radiation, depending on the frequency of the input. Thus, the leakywave mechanism favours a radiation direction that depends on the refractive index assigned by the TL metamaterial to each frequency component of the input wave.
To realize the circuit configuration of Fig. 3, the unitcell of Fig. 4 is proposed, where each mechanical element represents a specific acoustic lumped element. The airfilled waveguide, which is responsible for RH propagation, is modelled with series inductance and parallel capacitance. The vibrating thin plate alongside with the open channels are modelled with series capacitance and parallel inductance respectively, providing the condition for LH propagation. A detailed study of the dispersion diagram of the proposed unitcell reveals that, depending on the geometrical and material properties of the structure, five distinct regions may be identified, namely the LHguided and the LHleaky regions, the bandgap, the RHleaky and the RHguided regions^{24}. As the acoustic dispersive prism operates in both LH/RHleaky regions, the structure should be designed to discard the bandgap region by ensuring balanced condition between series and parallel resonances. Moreover, suppressing the bandgap enables the structure to radiate at broadside and to achieve near zero refractive index.
The performance of the whole structure can be summarized into a closed form expression which relates the main direction of the propagating output wave to the frequency (f) of the input wave,
where is the dispersive wave vector in the structure and is the wave vector of the ambient medium. Since depends on geometrical and physical properties of the TL metamaterial, the behaviour of the structure as a dispersive prism is conditioned by the material properties and geometrical dimensions.
Simulation and experimental validation
Based on the design procedure detailed in the Method section, an acoustic dispersive prism composed of 10 unitcells and with an approximate length of (including the input and output extremities) was designed, fabricated and tested (Fig. 5). The designed structure has geometrical dimensions of , , , , , , , and . Thin plates are chosen to be Kapton with material properties , and . The ambient medium is set to air with following parameters: , and . The fullwave simulation of the structure was carried out using COMSOL MultiPhysics and the results were compared to the theoretical TLbased model developed in Matlab. Moreover, the numerical and analytical simulations were validated through experiments.
Figure 6 depicts the farfield radiation pattern of the structure obtained by fullwave numerical simulation in 3 different frequencies: 870 Hz, 990 Hz and 1150 Hz radiating in backward, broadside and forward directions, respectively. For the acoustic prism along the y direction and fed from the left , it radiates backwards when in the LH region, which corresponds to negative values of refractive index. However, it radiates in the forward direction when the input frequency is in RH region, which corresponds to a positive refractive index. It radiates at broadside when curve crosses , which corresponds to the frequencies where the effective mass and bulk modulus have zero/nearzero values.
Figure 7a shows the reflection and transmission coefficients of the acoustic dispersive prism, obtained using TL modeling as well as numerical simulation. The reflection coefficient of the structure is below −10 dB in the frequency band of interest, that is to say less than 10% of power is reflected back to the source. This value can be compared to a glass/air interface, where about 4% of the incident power is reflected back. However, due to the short and finite length of the structure with 10 unitcells, which is merely without taking the input and output extremities into account, only a few percentage of the input power is radiated. The radiation efficiency of the structure can be increased by increasing number of unitcells. For a lossless structure, the distribution of power and radiation efficiency are formulated by and respectively. Thus, the lower and , the higher the efficiency η. While a low reflection is ensured by waveguide input impedance matching, the radiated power to the ambient medium depends on the amount of leaked energy. Consequently, radiation power depends on the leakage rate shown in Fig. 7b as well as on the length of the structure. As the leakage happens through the ducts, increasing the number of the unitcells will result in a higher radiation efficiency.
Figure 7c shows the dispersion diagram of the structure for the balanced case (without bandgap). The fastwave region, where the phase velocity of the wave in the prism is higher than the speed of sound in the ambient medium, is situated between 800 Hz and 1300 Hz, which corresponds to wavelength range of 26 m–42.5 cm. In the fastwave region, the energy leaks out due to the phase matching at the interface between the prism and the ambient medium. Extracting in the fastwave region from the dispersion diagram of Fig. 7c, and plugging it into equation (1), reveals the direction of radiation as a function of the frequency. Figure 7d depicts the data processed from measurements, fullwave simulation and TL modelling. Measurements follow qualitatively the expected trend observed on numerical and analytical methods, despite small divergences at higher frequencies. This is due to the bandgap resulting from the prototype fabrication. The bandgap is the consequence of an unbalanced RH/LH transition, mainly due to the misalignment of the vibrating thin plates resonances. Due to the very high sensitivity of the resonance frequency of the thin plate to the applied stress, the tuning process is difficult and small errors during the assembly of 10 thin plates in series can result in a misalignment of series and parallel resonances, thus creating an undesired bandgap.
Discussion
The proposed acoustic dispersive prism is smaller than . In terms of compactness, it is comparable to small water droplets of diameter 0.01 mm , which are natural optical prisms. While these small droplets produce rainbows with overlapping colors and poor contrast in optics, the proposed acoustic prism may be designed to achieve a better resolution.
Although the proposed acoustic dispersive prism directionally splits the sound wave according to its respective frequency components, a detailed study reveals other interesting features. The highly dispersive nature of the structure unveils a superprismlike behaviour, which leads to an extremely large angular dispersion. For light waves, a superprism sends optical beams with different wavelengths to considerably different angles (in space) and has been used to demonstrate wavelength demultiplexing. Having unitcells smaller than , the proposed structure acts as an acoustic superprism and works in the subwavelength regime, unlike its optical counterpart which are usually composed of photonic crystals with larger unitcells. Moreover, the superprism phenomenon is achieved in the audible frequency range (low frequencies), unlike the electromagnetic counterparts which operate in optical frequency bands (high frequencies).
While the nearfield study of the output spectrum of the proposed structure leads to acoustic prisming functionality, the farfield investigation uncovers a directive acoustic antenna with frequencydependent radiation directions which is the intrinsic behaviour of leakywave structures.
The study of the structure in the receiving mode unveils another exotic feature of the proposed configuration which is even more functional than prisming. Since the equation (1) is a bijective function within the range, there is only one direction angle that can be mapped to a specific frequency component. Then, the structure can be also used in reverse as a directionfinder device. Indeed, if a broadband noise source is located at a given direction θ, radiating towards the structure, there would be a single frequency which is received with higher amplitude with respect to other components of the signal. By inserting a microphone at one termination of the waveguide, this single frequency with higher amplitude can be measured and mapped to the direction of arrival according to equation (1). This procedure to find the direction of incoming sound using only one microphone is new to acoustics and can be very practical. Moreover, the directionfinding capability can be expanded to any arbitrary signal if prior knowledge of the transmitted signal is given or a second microphone is used for normalization. Used in receiving mode, the acoustic prism is then the analogue of the direction finding radar.
Methods
Model configuration
An acoustic waveguide, which naturally conveys the righthand behaviour, is taken as the host medium. The acoustic mass and acoustic compliance of the waveguide are and where the ρ and K are mass density and bulk modulus of the inner medium, d is the length and S is the cross section of the waveguide. Neglecting viscous losses, the acoustic waveguide is modelled by a series mass and parallel compliance, represented by the green elements in Fig. 3. The red elements, which are responsible for the lefthand behaviour are implemented using thin plates and transverse ducts.
The input acoustic impedance of a transverse duct of length L terminated by an acoustic load is:
where k and L are the wave vector and length of the duct, respectively. The acoustic characteristic impedance of the medium is defined as , where and are the mass density and sound wave velocity of the medium. For a duct radiating in a surrounding medium, , so will consist of real and imaginary parts given by equation (3) and (4). By properly choosing k, L and (cross section of the transverse duct), the parallel mass and radiation resistance values can be tuned. Represented with blue resistor in Fig. 3, the total resistance of the stub which contributes to the radiation is given by
and the mass representing the open duct is found as
The value of and for a flanged cylindrical open stub of radius a is given by equation (5). (supplementary notes)
The series compliance can be implemented with a thin plate, clamped on its edges. The vibrating thin plate physically behaves as a compliance (capacitance) and mass (inductance), in low and high frequencies respectively. The behaviour of the thin plate depends on its radius a, thickness , Young modulus E, Poisson’s ratio v and mass density . For a clamped circular thin plate with radius a the acoustic impedance is: (supplementary notes)
where is the wave number in the thin plate and it is given by and D is the flexural rigidity of the thin plate that is defined by and ^{24}. The 2D view of the proposed structure and circuit modelling of the elements are depicted for two adjacent unitcells in Fig. 4. If the proposed unitcell is periodically arranged and the homogeneity condition is satisfied, a composite right/lefthand TLbased leakywave antenna can be achieved which performs prisming role in acoustics.
Design principle
The structure is modelled with five main circuit elements: , , , and , where subscripts LH and RH represent lefthand and righthand elements respectively. The aforementioned elements are equivalent to physical dimensions and material properties , , d, , , , and (Fig. 1). The last four parameters (the thickness and material properties of the thin plate) are fixed and restricted by the available materials. Equation (6) can be used to find the impedance of the thin plate in the design process.
In composite right/lefthand (CRLH) leakywave structure the transition frequency where the transmissionline character changes from LH to RH is controlled by equation (7).
As the bandgap appears due to unbalanced resonances between series and shunt elements, the balanced condition to be satisfied to suppress this stopband, is
Equations (7) and (8) are the two necessary but not sufficient relations to design acoustic TL metamaterials with no bandgap. Two more equations are needed to define four main circuit elements of the CRLH in Fig. 3 and one more to define the value of leakywave radiation . Thus, the matching between input port and the CRLH transmissionline to decrease the reflected power, the bandwidth of leakywave (fastwave region bandwidth), and the radiation rate (leakagerate) are usually the most important factors which set enough number of constraints to design acoustic leakywave structure.
Using the circuit model of Fig. 3 and applying Floquet periodic boundary conditions the values of , and , which set the bandwidth, the radiation rate and the reflection coefficient of leakywave respectively, are found as a function of the circuit elements^{25},
where , , , are the elements of the transmission matrix [T] that, in the case of Πcircuit model of Fig. 4, is given by
and and are the total series impedance and parallel admittance of a unitcell, respectively. Equations (7)–(12), , , , , can be used to set the design and the optimization procedure to find the values of circuit elements^{24} in Fig. 4 and finally, equations (2)–(6), , , , are used to design the real prototype of Fig. 1.
Numerical simulation
The simulation was performed with the commercial finite element analysis software COMSOL Multiphysics. Unitcell simulations to retrieve dispersion relation is performed using AcousticSolid Interaction module in frequency domain where , , , parameters of the unitcell is derived using [T] matrix retrieval method^{32} and equation (9) is used to find the dispersion relation . AcousticShell Interaction physics module in frequency domain is used for simulations of full structure. The open end of the stubs are flanged to an acoustic hard surface and the full structure and spherical simulation area is bounded by a Perfectly Matched Layer (PML) to realize nonreflecting boundaries. In the transmitting mode the leakywave input and output are set to radiation boundary condition and incident acoustic pressure is applied to the input port. Then, the scattering matrix [S] of the full structure is numerically calculated using a 4microphones measurement procedure^{32} by numerically measuring the pressure field in 2 distinct points near the input and output of the structure. Moreover, the nearfield and farfield radiation pressure field can be evaluated in transmitting mode. In the receiving mode, the acoustic background pressure field is applied to the same spherical simulation area bounded with (PML) and the received signal is numerically measured in one point near the input of the leakywave. Applying the background pressure field every 10 degrees, the frequency corresponding to the maximum of received signal is recorded and diagram is assessed.
The TL model of the structure is numerically evaluated in Matlab environment. Based on Fig. 4, the Πcircuit model of the unitcell is developed, and the [T] matrix of one unitcell is evaluated then the dispersion diagram as well as other characteristic parameters of the unitcell are computed using the equation (9, 10, 11, 12). Cascading 10 unitcells with , the total transmission matrix of the full structure is calculated and it is converted to scattering matrix to obtain the reflection and transmission coefficients of the structure.
Fabrication and Assembly
The structure is fabricated by stacking 10 unitcells, which are assembled using four screws located at the corners. Two of the screws are used for alignment propose and the other pair for adjusting the clamping of each thin plate to the two adjacent unitcells. Unitcells are machined out of aluminium blocks. DuPont^{TM} Kapton^{®} FPC is used for the thin plate. To ensure the clamping criteria of the thin plate, the resonant frequency of the last assembled thin plate is checked at each assembling stage. For this, stepbystep assembly process of the structure is simulated in COMSOL and the velocity of the thin plate is normalized to a reference signal (input pressure). At each step (n) of the tuning process, the frequency response function is measured, where and are the thin plate velocity and front pressure. Then, the achieved resonance frequency is compared to the one specified by COMSOL simulations. According to this measurement, the tension of the thin plate is tuned using the four screws until the desired resonance is achieved for each thin plate.
Measurement
Figure 5 shows the experimental setup to measure the performance of the proposed structure. To make the measurement process easier, the proposed prism is measured in the receiving mode. Based on reciprocity, for white noise produced by loudspeaker the spectrum of the received signal will depend on the direction from which the signal is received.
In the transmitting mode, the structure performance follows equation (1) which means each frequency is radiated in a specific direction defined by this relation. In the receiving mode, the reciprocity dictates the reception of the specific frequency of the spectrum with higher amplitude which will distinguish this frequency with respect to others, depending on the direction of arrival.
All the measurements are performed in an acoustic anechoic chamber. A loudspeaker is used as a source and the structure is flanged using a hard acoustic plate (wooden plate). The structure is mounted on a turntable and received signals are measured every 10 degrees. White noise signal is used as source signal. The two extremities of the waveguide are filled with a layer of mineral wool to achieve anechoic conditions. An additional hole is drilled near the input to insert a microphone transversely into the waveguide. A second microphone is installed on top of the flanged plate to measure the ambient noise, and for normalization.
Additional Information
How to cite this article: Esfahlani, H. et al. Acoustic dispersive prism. Sci. Rep. 6, 18911; doi: 10.1038/srep18911 (2016).
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Acknowledgements
This research was supported by the Swiss National Science Foundation (SNSF) under grant agreement 200020138086/1. The authors would like to thank Mohsen Yousefbeiki for proof reading the article and JeanPaul Brugger, Roland Dupuis and Karim Collomb of the AEM workshop for their help in the construction of the prototype.
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Affiliations
Ecole Polytechnique Fédérale de Lausanne, Laboratoire de Traitement des Signaux LTS2, Lausanne, Switzerland
 Hussein Esfahlani
 , Sami Karkar
 & Herve Lissek
Ecole Polytechnique Fédérale de Lausanne, Laboratoire d’Electromagnétisme et d’Antennes LEMA, Lausanne, Switzerland
 Hussein Esfahlani
 & Juan R. Mosig
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Contributions
H.E. proposed the idea and performed the design, numerical simulations and experimental verifications. H.E., S.K., H.L. and J.M. prepared the manuscript and discussed the data. S.K. and H.L. supervised the work closely and J.M. supervised the group. All authors contributed to review the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Herve Lissek.
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