Propagating and evanescent properties of duble-point defect in sonic crystals

Propagating and evanescent properties of double-point defects in sonic crystals This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 New J. Phys. 12 083024 (http://iopscience.iop.org/1367-2630/12/8/083024) Download details: IP Address: 85.55.36.148 The article was downloaded on 26/08/2010 at 17:08 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience The open–access journal for physics New Journal of Physics Propagating and evanescent properties of double-point defects in sonic crystals V Romero-García1,2,4, J V Sánchez-Pérez1and L M Garcia-Raffi3 1Centro de tecnologías físicas: Acústica, Materiales y Astrofísica, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain 2Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Científicas,Sor Juana lnés de la Cruz, 3, Cantoblanco, 28049, Madrid, Spain 3Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain E-mail: [email protected].es New Journal of Physics 12 (2010) 083024 (14pp) Received 22 April 2010 Published 10 August 2010 Online at http://www.njp.org/ doi:10.1088/1367-2630/12/8/083024 Abstract. Complex band structures and multiple scattering theory have been used in this paper to analyze the overlapping of the evanescent waves localized in point defects in sonic crystals (SCs). The extended plane wave expansion (EPWE) with supercell approximation gives the imaginary part of the Bloch vectors that produces the decay of the localized modes inside the periodic system. Double cavities can present a coupling between the evanescent modes localized in the defect, showing a symmetric or antisymmetric mode. When point defects are close, the complex band structures reveal a splitting of the frequencies of the localized modes. Both the real part and the imaginary values of kof the localized modes in the cavities present different values for each localized mode, which gives different properties for each mode. The novel measurements, in very good agreement with analytical data, show experimental evidence of the symmetric and antisymmetric localized modes for a double-point defect in SCs. The investigation of the localization phenomena and the coupling between defects in periodic systems has fundamental importance in both pure and applied physics. 4Author to whom any correspondence should be addressed. New Journal of Physics 12 (2010) 083024 1367-2630/10/083024+14$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 Contents 1. Introduction 2 2. Extended plane wave expansion (EPWE) with supercell approximation 3 3. EPWE results: localized modes 6 3.1. Splitting of localized modes ........................... 7 3.2. Symmetric and antisymmetric modes ....................... 9 3.3. Decay of the localized modes ........................... 11 4. Conclusions 13 Acknowledgments 13 References 13 1. Introduction Periodic distributions of elastic scatterers in an elastic host medium with different physical properties are known as phononic crystals (PCs) [1,2], and they are the elastic analogues of the well-known photonic crystals [3,4]. If one of the materials in PCs is a fluid, then the system is called a sonic crystal (SC) [5]. All of these systems present interesting physical properties and recently they have received increasing attention, mainly due to the great number of applications in several branches of physics and engineering [6]–[8]. One of the most important properties of these inhomogeneous materials is the so-called band gaps (BGs): frequency ranges where waves do not propagate through the periodic system. The existence of these BGs leads to several applications; for instance, in the case of SCs, as acoustic filters [9,10], acoustic barriers [11] or waveguides [12]. In periodic systems, Bloch’s theorem and Fourier expansion of the periodic physical properties transform the acoustic wave equation in an eigenvalue problem. The eigenfrequencies ω(k)for each Bloch’s vector kinside the irreducible part of the first Brillouin zone constitute the band structure. This methodology is usually called plane wave expansion (PWE) [13] and it can be used to obtain the so-called band structures, i.e. the propagating modes through the periodic system. The band structures reveal that BGs are ranges of frequencies where no real kexists. It has been shown that the eigenvalues of the problem have real value for the case of SCs [14]. One of the most important properties of the periodic structures is the emergence of localized modes within the BG when a point defect is introduced [9,15]. A widely used technique in the literature to obtain the effect of creation of point defects in crystals is the supercell approximation in PWE [10,16,17]. This approximation gives information only about the propagation nature of the localized modes in the point defects. In these cases, when periodicity is broken or when SCs have finite size, evanescent modes inside the periodic system may appear. Localized modes or modes inside the BG are characterized by evanescent behavior [7,18,19]. Then, a more accurate analysis is needed to characterize all the properties of the modes inside the periodic system. A wave impinging on a complete periodic system with a given frequency ωinside the BG is characterized by complex valued wave numbers k(ω), which represent the multi-exponential decay of the evanescent mode inside the periodic system [18]. Recent works [20]–[22] show an extension of the PWE (extended plane wave expansion (EPWE)) obtaining the complex part of New Journal of Physics 12 (2010) 083024 (http://www.njp.org/) 3 the Bloch’s waves, revealing that the decay of the modes inside the BG grows as the frequency reaches the center of the BG. In this sense, a localization factor has been defined recently to show this behavior [23]. The localization factor can also be related to recent results that show that, although the decay of these localized modes is multi-exponential, it can be approximated by an exponential-like decay considering only the first harmonic of the Bloch waves in SCs made of rigid cylinders [19]. On the other hand, Sainidou et al [24] have introduced a novel extension of the multiple scattering theory (MST) [25] for analyzing slabs that consist of slices of different materials as long as the periodicity parallel to the surface of the slab is preserved. The method, called layer multiple scattering (LMS), allows the study of the scattering problem of slabs that are finite in the direction parallel to the surface of the slab, but infinite in the normal directions to this surface. Alternatively, one can use this method to calculate the complex phononic band structures of an infinite crystal, associated with a given crystallographic plane. In this case, the method provides the propagating and evanescent Bloch waves of the elastic field in the given crystal, corresponding to a given kand a given frequency. LMS has been used to analyze the guidance and quasi-guidance of elastic waves in a glass plate coated on one side with a periodic monolayer of polymer spheres, immersed in water, observing the dispersion diagrams of the interacting modes of the composite slab [26]. The goal of the paper is to analyze the three main characteristics of defect modes in SCs: splitting, symmetry vibrational patterns and evanescent decay of the modes. In addition to PWE, to carry out this study we have used EPWE with supercell approximation, because it is fundamental for the complete understanding of the localized modes. We present the explicit matrix formulation of the supercell approximation in EPWE for Nppoint defects. From the complex and real band structures, we observe the splitting and the evanescent behavior of the localized modes inside the BG around the defect. We analyze the localized modes inside multipoint defects, especially in the double-point defect case. MST in finite SCs is used to analyze the vibrational patterns of the localized modes in a double-point defect. In this case, when the distance between both defects is low enough, it appears as symmetric and antisymmetric vibrational modes similar to the case of a system formed by two masses and three springs, or to the Zeeman effect in the atomic spectra [15]. The novel experimental data that are in good agreement with theory show for the first time the symmetry of the vibrational patterns of localized modes in such a double-point defect. Moreover, we observe the decay of the localized modes outside the double-point defect, in good agreement with the results obtained by EPWE with supercell approximation. The paper is organized as follows. First of all, we show the main ingredients of the EPWE as well as the explicit matrix formulation of the problem and the extension for a supercell with Nppoint defects. After that, the numerical, analytical and experimental results of a double-point defect are shown, giving a complete explanation of the splitting, the symmetry of vibrational patterns and the decay of localized modes. Finally, we give a summary as well as the main conclusions of the work. 2. Extended plane wave expansion (EPWE) with supercell approximation The analysis of propagating modes can be done by the ω(E k)formulation, where the existence of BGs is indicated by the absence of bands in determined ranges of frequencies. The mechanism of creation of BGs in finite crystals could be understood by the evanescent behavior of the New Journal of Physics 12 (2010) 083024 (http://www.njp.org/) 4 modes inside it. At a given frequency ωinside the BGs, the evanescent wave is characterized by a complex valued Bloch vector E k(ω) that characterizes the decay of the mode inside the periodic structure. Based on the work of Hsue et al [20], recent work by Laude et al [21] shows the calculation of complex band structure for PCs. Recently, this work has been extended to the case of SCs for calculations using the supercell approximation [22], which is especially indicated for SCs with point defects. In this section, we present the explicit matrix formulation of the EPWE with supercell approximation to calculate the properties of SCs with Nppoint defects inside a supercell. We must take into account that PWE needs low interaction between supercells. ω(k)methods are characterized by the next eigenvalue problem, X E G0 ((E k+E G)σk(E G−E G0)(E k+E G0)−ω2η( E G−E G0))pE k(E G0)=0,(1) where E Gis the two-dimensional (2D) reciprocal-lattice vector, kis the Bloch vector and pkis the pressure. Equation (1) constitutes a set of linear, homogeneous equations for the eigenvectors pE k(E G)and the eigenfrequencies ω(E k). We obtain the band structures letting E kscan the irreducible part of the first Brillouin zone. Equation (1) can be expressed by the next matrix formulation [13], 3 X i=1 0i60iP=ω2P,(2) where i=1,2,3 and 6=   σ( E G1−E G1) . . . σ( E G1−E GN×N) . . ..... . . σ( E GN×N−E G1) . . . σ( E GN×N−E GN×N)   ,(3) =    η( E G1−E G1) . . . η( E G1−E GN×N) . . ..... . . η( E GN×N−E G1) . . . η( E GN×N−E GN×N)     ,(4) P=    P(E G1) . . . P(E GN×N)     ,(5) where E G=(G1,G2,G3)=(2πm/a1,2πn/a2,0)for the case of the 2D square arrays. If we chose m=n=(−M,...,M), the size of the previous matrices is N×N=(2M+1)×(2M+1). From equation (2), we define the next vector, 8i=60iP.(6) With this definition we can reformulate the eigenvalue problem (2) as the equations system, 8i=60iP, ω2P= 3 X i=1 0i8i.(7) New Journal of Physics 12 (2010) 083024 (http://www.njp.org/) 5 In order to obtain an eigenvalue problem for E k(ω), we write E k=kEα, where Eαis a unit vector. Then the 0imatrix can be written as 0i=00 i+kαiI,(8) where Iis the identity matrix and 00 i=      Gi0. . . 0 0Gi. . . 0 . . .. . ..... . . 0. . . . . . Gi       ,(9) αi=      αi0. . . 0 0αi. . . 0 . . .. . ..... . . 0. . . . . . αi       .(10) Then, equation (2) can be written in the form of (11), where 80=P3 i=1αi8i. ω2−P3 i=100 i600 i0 −P3 i=1600 iI!P 80=k P3 i=100 i6αiI P3 i=16αi0!P 80.(11) Equation (11) represents a generalized eigenvalue problem with 2Neigenvalues k, with possibly complex values. Complex band structures on the incidence direction Eαhave been obtained by solving the eigenvalue equation for a discrete number of frequencies and then sorted by continuity of k. In contrast to the ω(E k)method, in this formulation the periodicity is not relevant and k(ω) does not follow the first Brillouin zone. We consider an SC with primitive lattice vectors Eai(i=1,2,3). The supercell is a cluster of n1a×n2a×n3ascatterers periodically placed in the space. Then, the primitive lattice vectors in the supercell approximation are E a0i=niEai, and the complete set of lattices in the supercell approximation is {R0|R0=liE a0i}, where niand liare integers. The primitive reciprocal vectors are then E b0i=2πεi jk E a0j×E a0k E a01·(E a02×E a03),(12) where εi jk is the 3D Levi–Civita completely anti-symmetric symbol. The complete set of reciprocal lattice vectors in the supercell is {E G|E Gi=NiE b0i}, where Niare integers. The density ρiand the bulk modulus Biare the physical properties involved in the wave equation and, using the Fourier expansion and the geometry of the system, they can be expressed in terms of the structure factor for the PWE (EPWE) as well as for the PWE (EPWE) with supercell approximation. The index i=(h,c)represents the host medium and the scatter, respectively. The filling fraction of a cylinder in a supercell is f=πr2/A, where Ais the area occupied by the supercell. If we consider that βirepresents the values (ρ−1 i,B−1 i)and that the New Journal of Physics 12 (2010) 083024 (http://www.njp.org/) 6 supercell has Nccylinders organized in an array of size n1a×n2a, then β(−→ G)=(βcNcf+βh(1−Ncf)if −→ G=−→ 0, (βc−βh)F(−→ G)if −→ G6= −→ 0,(13) where F(−→ G)is the structure factor of the supercell. In this approximation, the structure factor of the supercell has to be computed taking into account the size of the supercell. If we consider a 2D SC with cylindrical scatterers with radius rand size of the supercell n1a×n2a, the structure factor of the supercell is expressed by F(E G)= (n1−1)/2 X i=−(n1−1)/2 (n2−1)/2 X j=−(n2−1)/2 eı(ia|E G1|+ja|E G2|)P(E G), (14) where P(E G)=2f Gr J1(G), (15) and where ais the lattice constant inside the supercell and G= | E G|. Previous equations show the expressions for the approximation of the complete supercell. If the supercell presents Nppoint defects at the sites labeled (ls,ms)in the periodic system, with s=1,...,Np, then the Fourier coefficients of the expansions of the physical parameters involved in the problem follow the next equation, β(−→ G)=(βc(Nc−Np)f+βh(1−(Nc−Np)fif −→ G=−→ 0, (βc−βh)F(−→ G)if −→ G6= −→ 0. (16) The structure factor of such a supercell with Nppoint defects is F(E G)=  (n1−1)/2 X i=−(n1−1)/2 (n2−1)/2 X j=−(n2−1)/2 eı(ia|E G1|+ja|E G2|)− Np X s=1 eı(lsa|E G1|+msa|E G2|) P(E G). (17) The interaction of the defect points in the supercell approximation must be as low as possible between the neighboring supercells in order to decrease the overlap between defects. Thus the size of the supercell should be big enough to place the point defects separated in consecutive supercells. Introducing the previous expressions in the matrices of the PWE (2) or the EPWE (11), we can calculate the real and complex band structures. In the present paper, we analyze the case of a double-point defect in a square array at sites (1,0)and (−1,0)in a supercell of 11a×11a. In this situation, the distance between defects is equal to 2aand the distance between two doublepoint defects in different supercells is equal to 20a. 3. EPWE results: localized modes Since Sigalas [9] studied the defect mode produced by a point defect in periodic structures, several kinds of defects have been analyzed in the last few years, showing in all cases the localization of sound for frequencies inside the BG [15,27,28]. Experimental and numerical analyses of the localization in a point defect considered as a cavity inside the SC have been reported recently by Wu et al [29,30] and Zhao et al [17,31], showing the dependence of New Journal of Physics 12 (2010) 083024 (http://www.njp.org/) 7 localization on the size of the crystal and on the filling fraction (the bigger the size and the filling fraction, the bigger the localization in the cavity). Moreover, when we consider twopoint defects, coupling between localized modes in each defect point is possible [32,33]. An accurate interferometric setup has been used by Russell et al [33] for observing the coupled states in a double-point defect, noting the evidence for odd and even symmetry trapped states in a new class of ultra-efficient photosonic devices in which both sound and light are controlled with great precision and their interactions enhanced. For the case of double-point defects, the bigger the distance between cavities, the lower the coupling between defect points. In this section, we show novel results regarding the imaginary part of the Bloch vector for the localized modes inside the SC with multi-point defects. The localization of waves inside these defects is mainly characterized by three properties. Firstly, the modes are separated in the frequency domain, i.e. there is a splitting of the localization frequency if the point defects are close enough. Secondly, the modes present symmetries in the vibrational pattern depending on the number of vacancies in the crystal. Thirdly, the localized modes are evanescent and they decay outside the defect but inside the SC. Without loss of generality, we show the results of a double-point defect in very good agreement with the experimental data, showing the symmetric and antisymmetric vibrational patterns of the localized modes. Evidently, the oscillation modes of N-point defects with N>2 will present more complicated vibrational patterns than the ones appearing in the double-point defect; then they cannot be classified into such simple modes as symmetric and antisymmetric ones. The complex and real band structures reveal that the values of kfor the localized modes are characterized by a real value of kand it is related to the localization frequency. However, this localized mode presents evanescent behavior out of the defect, because it is surrounded by a perfect periodicity; thus, the excited mode in the surrounding crystal by the localized mode presents an imaginary kthat is related to the evanescent behavior of the mode outside the point defects [19]. The rest of the modes inside the BG only present the imaginary part; then they are killed inside the crystal because of their evanescent behavior. 3.1. Splitting of localized modes In order to analyze the splitting of the localized modes, we have calculated the real and complex band structures of an SC with a double-point defect using EPWE with supercell approximation using a supercell of size 11a×11a. We consider a 2D SC consisting of PVC cylinders of radius rin air background arranged in a square lattice with lattice constant a. The material parameters employed in the calculations are ρair =1.23 kg m−3,ρPVC =1400 kg m−3,cair =340 m s−1and cPVC =2380 m s−1. We consider a filling fraction f=πr2/a2≃0.65. For the calculations, we have used N=(2×15 + 1)2=961 plane waves. Several calculations have been carried out in order to obtain a good convergence of the solution. This number of plane waves is bigger than the one used in previous works [21] and provides a good convergence of the solution of the eigenvalue problem. A mode within the BG in an infinite SC without defects is characterized by a pure imaginary value of k=ikim (where kim =Im(k)) [19,21,23]. In figure 1(a) (left panel), we can observe the dependence of Im(k)on kfor a complete SC within the BG for the 0X direction. We can observe a maximum value of Im(k)for the frequency in the midgap (926 Hz), which means that the imaginary part of the wave number for frequencies inside the BG grows with values of frequency closer to the center of the BG and disappears at the edges of the BG, i.e. the New Journal of Physics 12 (2010) 083024 (http://www.njp.org/) 8 Figure 1. Real and complex band structures for a sonic crystal (SC) with point defects. (a) Left: complex band structure of a complete SC calculated by extended plane wave expansion (EPWE) with supercell approximation. Center: band structures calculated by PWE with supercell approximation of a SC with a point defect; the continuous red line represents the defect mode. Right: band structures for a SC with a double-point defect; the dashed green line represents the defect modes of a double-point defect. Insets show the supercell used in the calculations. (b) Complex and real band structures of a double-point defect. rate of decay is bigger for frequencies closer to the center of the BG [7,19,23]. Modes within the BG decay inside the SC because of their evanescent behavior [19]. In contrast to the modes in the BG, localized modes can travel up to the point defect where the wave is localized. Figure 1(a) (central and right panels) represents the real band structures calculated by PWE with supercell approximation for both a SC with a point defect (central) and a SC with a double-point defect (right). We can observe the localized mode generated by a point defect in a SC at the frequency ν0=932 Hz, whereas the frequencies of the localized modes of a double-point defect have been split (right panel of figure 1(a)). The frequencies of the two localized modes due to the double-point defect split around the localized mode of a single defect: one with a lower frequency, ν1=910 Hz, than the corresponding frequency of the localized mode in a single defect, and another one, ν2=958 Hz, with a higher frequency than the single defect. This phenomenon is analogous to the splitting of the degenerate atomic levels in diatomic molecules. The splitting in two peaks may be understood qualitatively by considering that the double cavity in the double-point defect is coupled forming a large cavity with two resonant New Journal of Physics 12 (2010) 083024 (http://www.njp.org/)