Evidences of evanescent Bloch waves in phononic crystals (original) (raw)

We show both experimentally and theoretically the evanescent behavior of modes in the band gap of finite phononic crystal PC. Based on experimental and numerical data we obtain the imaginary part of the wave vector in good agreement with the complex band structures obtained by the extended plane wave expansion. The calculated and measured acoustic field of a localized mode out of the point defect inside the PC presents also evanescent behavior. The correct understanding of evanescent modes is fundamental for designing narrow filters and waveguides based on PCs with defects. During the past few years, there has been a great deal of interest in studying propagation of waves inside periodic structures. These systems are composites made of inhomoge-neous distribution of some material periodically embedded in other with different physical properties. Phononic crystals PC 1,2 are one of the examples of these systems. PCs are the extension of the so-called photonic crystals 3 when elastic and acoustic waves propagate in periodic structures made of materials with different elastic properties. When one of these elastic materials is a fluid medium, then PC are called sonic crystals SC. 4,5 For these artificial materials, both theoretical and experimental results have shown several interesting physical properties. 6 In the homogenization limit, 7 it is possible to design acoustic metamaterials that can be used to build re-fractive devices. 8 In the range of wavelengths similar to the periodicity of the PC a, multiple scattering process inside the PC leads to the phenomenon of so called band gaps BGs, which are required for filtering sound, 5 trapping sound in defects, 9,10 and for acoustic wave guiding. 11 Propagating waves inside a periodic media are a set of solutions of the wave equations satisfying the translational symmetry property. However, periodic media with point defects where the translational symmetry is broken, or finite periodic media, can support evanescent modes as well. Recently Laude et al. 13 have analyzed the evanescent Bloch waves and the complex band structure of PC. Complex band structures show bands that are simply not revealed by the traditional k method. By means of the complex band structures, BG can be defined as ranges of frequencies where all Bloch waves must be evanescent. The goal of the paper is to characterize the evanescent behavior of waves with frequencies in the BG inside of PC. Analytical, numerical, and experimental data show the evi-dences for the exponential-like decay of these modes. En-gelen et al. 12 shown that modes inside BG in photonic crystals decay multiexponentially. Supercell approximation in extended plane wave expansion EPWE 13–16 has been used in the present work to determine the imaginary part of the wave vector of evanescent modes. Specifically, we have deduced both analytically and experimentally the imaginary part of the wave vector observing that only the first harmonic contributes substantially to the decay of the acoustic field inside complete SC. In all cases we have obtained a very good agreement between theoretical and experimental results. We have performed experiments in an echo-free chamber of dimensions 8 6 3 m 3. The finite two-dimensional SC used in this paper forms a square array with lattice constant a = 22 cm. The size of the SC is 5a 5a and the radius of the cylinders is r = 10 cm. A prepolarized free-field 1 / 2 microphone Type 4189 B&K has been used throughout the experiments. The diameter of the microphone is 1.32 cm, which is approximately 0.06a. Our system 3DReAMS three-dimensional Robotized e-Acoustic Measurement System is capable of sweeping the microphone through a 3D grid of measuring points located at any trajectory inside the echo-free chamber. Motion of the robot is controlled by NI-PCI 7334. Figure 1 shows the complex and real band structures for the SC with a point defect. The complex band structures and the value of the k number for the modes inside the BG can be obtained by EPWE and it becomes in a purely real value for the localized mode. We can observe the localized mode at 920 Hz continuous line. That value exactly coincides with the value obtained by PWE with supercell approximation. We have compared these results with experimental data measuring the insertion loss IL behind the SC with and without the point defect. In Fig. 1, we can observe that the experimental IL for the localized mode at frequency 920 Hz continuous line is lower than the case of the complete SC dashed line, i.e., it can be concluded that there is a passing mode. This results because the localized mode is not killed completely by the SC around the point defect see also Ref. 10. In fact, although the localized mode has an evanescent behavior, as we will see later, in this case there is not enough number of rows around the point defect to kill it. For frequencies in the BG, the borders of the point defect act as perfect mirrors producing the localization in this a Electronic mail: virogar1@mat.upv.es.