Complex Band Structure of Phononic Crystals and the Diffraction Problem

  • Vincent Laude
  • Younes Achaoui
  • Sarah Benchabane
  • Abdelkrim Khelif
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 26)

Abstract

The diffraction of elastic waves on a two-dimensional finite phononic crystal is investigated by a plane wave technique. It is first remarked that a full solution to the phononic crystal problem requires that all modes of the periodic structure (Bloch waves) are indentified and incorporated in the solution, including evanescent Bloch waves. An extended plane-wave expansion (PWE) method is used to obtain the complex band structure of the phononic crystal, but also the band structure of diffracted waves in the incident and exit media. Complex isofrequency curves are presented and show sharp variations of the Bloch wave vector with the angle of propagation. Finally, the complex band structures are used to formulate a reflection/transmission problem similar to the one leading to Fresnel formulas for homogeneous media. Some examples of diffracted field computations are given.

Keywords

Evanescent Wave Phononic Crystal Diffraction Problem Bloch Wave Fresnel Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. .
    Kushwaha, M. S., Halevi, P., Dobrzynski, L., and Djafari-Rouhani, B.: Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71 2022-2025 (1993).CrossRefGoogle Scholar
  2. .
    Sigalas, M. M. and Economou, E. N.: Elastic and acoustic wave band structure. J. Sound Vib. 158 377-382 (1992).CrossRefGoogle Scholar
  3. .
    Hsue, Y.-C., Freeman, A. J., Gu, B. Y.: Extended plane-wave expansion method in threedimensional anisotropic photonic crystals. Phys. Rev. B 72 195118 (2005).Google Scholar
  4. .
    de Fornel, F.: Evanescent Waves. From Newtonian Optics to Atomic Optics. Springer (Berlin, 2001).Google Scholar
  5. .
    Sainidou, R., Stefanou, N., Modinos, A.: Formation of absolute frequency gaps in threedimensional solid phononic crystals. Phys. Rev. B 66 212301 (2002).Google Scholar
  6. .
    Sainidou, R., Stefanou, N., Psarobas, I. E. and Modinos, A.: A layer-multiple-scattering method for phononic crystals and heterostructures of such. Computer Physics Communications 166 197-240 (2005).CrossRefGoogle Scholar
  7. .
    Wilm, M., Ballandras, S., Laude, V., Pastureaud, T.: A full 3D plane-wave-expansion model for 1-3 piezoelectric composite structures. J. Acoust. Soc. Am. 112 943-952 (2002).CrossRefGoogle Scholar
  8. .
    Wu, T.-T., Huang, Z.-G., Lin, S.: Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy. Phys. Rev. B 69 094301 (2004).Google Scholar
  9. .
    Khelif, A., Aoubiza, B., Mohammadi, S., Adibi, A., Laude, V.: Complete band gaps in two-dimensional phononic crystal slabs. Phys. Rev. E 74 046610 (2006).Google Scholar
  10. 0.
    Laude, V., Achaoui, Y., Benchabane, S., Khelif, A.: Evanescent Bloch waves and the complex band structure of phononic crystals. Submitted (2009).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Vincent Laude
    • 1
  • Younes Achaoui
    • 1
  • Sarah Benchabane
    • 1
  • Abdelkrim Khelif
    • 2
  1. 1.Institut FEMTO-STUniversité de Franche-Comté, CNRS, ENSMM, UTBMBesançonFrance
  2. 2.GeorgiaTech-CNRSAtlantaUSA

Personalised recommendations