Bloch waves in periodic multi-layered acoustic waveguides

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 464, Issue 2098, 2008, Pages 2669-2692

  Adams, Samuel D M ^a   Craster, Richard V ^a   Guenneau, Sebastien ^b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF LIVERPOOL   (United Kingdom)

Author keywords

Bragg waveguide; Finite element method; Imaging; Localized modes; Multi modal matrix method; Stop bands

Indexed keywords

ACOUSTICS; DISPERSION (WAVES); ELECTROMAGNETISM; WAVEGUIDES;

ACOUSTIC WAVEGUIDES; ANOMALOUS DISPERSIONS; ASYMPTOTIC TECHNIQUES; B LOCH WAVES; BAND SPECTRUM; BRAGG WAVEGUIDE; BRILLOUIN ZONE; DIRICHLET; DISPERSION CURVES; EIGEN-SOLUTIONS; ELECTROMAGNETIC WAVEGUIDES; FINITE WIDTH; FINITE-ELEMENT METHOD; FLOQUET; GEOMETRIC DEFECTS; IMAGING; IMAGING APPLICATIONS; LOCALIZED MODES; LOW FREQUENCIES; MULTI-LAYERED; MULTI-MODAL MATRIX METHOD; NEUMANN CONDITIONS; STOP BANDS; SUB-WAVELENGTH RESOLUTIONS; TRAPPED MODES; ULTRA SLOW LIGHT; ZERO-FREQUENCY;

PERIODIC STRUCTURES;

EID: 49749145642     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2008.0065     Document Type: Article

References (40)

1. Adams, S. D. M., Craster, R. V. & Guenneau, S. In preparation. Guided and standing Bloch waves in periodic elastic strips.
2. Ania-Castanon, J. D., Ellingham, T. J., Ibbotson, R., Chen, X., Zhang, L. & Turitsyn, S. K. 2006 Ultralong Raman fiber lasers as virtually lossless optical media. Phys. Rev. Lett. 96, 023 902. (doi:10.1103/ PhysRevLett.96.023902)
3. Barnes, C. & Pendry, J. B. 1991 Multiple scattering of waves in random media a transfer matrix approach. Proc. R. Soc. A 435, 185-196. (doi:10.1098/rspa.1991.0137)
4. Ben-Menahem, A. 1987 Green's tensor and its potentials for inhomogeneous elastic media. Proc. R. Soc. A 409, 287-327. (doi:10.1098/rspa. 1987.0018)
5. Botten, L., White, T., Asatryan, A., Langtry, T., De Sterke, C. M. & Mcphedran, R. 2004 Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory. Phys. Rev. E 70, 056 606. (doi:10.1103/PhysRevE.70.056606)
6. Chen, A.-L. & Wang, Y.-S. 2007 Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals. Physica B 392, 369-378. (doi:10.1016/j.physb.2006.12.004)
7. Conca, C., Planchard, J. & Vanninathan, M. 1995 Fluids and periodic structures Research in applied mathematics. Paris, France: Masson.
8. Cullum, J. K. & Willoughby, R. A. 2002 Lanczos algorithms for large symmetric eigenvalue computations, vol. 1: theory. SIAM Classics in Applied Mathematics. Philadelphia, PA: SIAM Publications.
9. Dowling, J. P., Everitt, H. & Yablonovitch, E. 2008 Photonic and sonic band-gap bibliography. See http://phys.lsu.edu/~jdowling/pbgbib.html.
10. Fang, N., Xi, D., Xu, J., Ambati, M., Srituravanich, W., Sun, C. & Zhang, X. 2006 Ultrasonic metamaterials with negative modulus. Nature 5, 452. (doi:10.1038/nmat1644)
11. Felbacq, D., Guizal, B. & Zolla, F. 1998 Limit analysis of the diffraction of a plane wave by a stratified periodic medium. J. Math. Phys. 39, 4604-4607. (doi:10.1063/1.532526)
12. Figotin, A. & Vitebskiy, I. 2006 Slow light in photonic crystals. Waxes Random, Complex, Media 16, 293-392. (doi:10.1080/ 17455030600836507)
13. Godin, Y. 2006 Propagation of longitudinal waves in a random binary rod. Waves Random Complex Media 16, 409-416. (doi:10.1080/ 17455030500527370)
14. Gralak, B., Enoch, S. k Tayeb, G. 2000 Anomalous refractive properties of photonic crystals. J. Opt. Soc. Am. A 17, 1012-1020. (doi:10.1364/JOSAA.17.001012)
15. Gridin, D., Adamou, A. T. I. & Craster, R. V. 2004 Electronic eigenstates in quantum rings: asymptotics and numerics. Phys. Rev. B 69, 155 317. (doi:10.1103/PhysRevB.69.155317)
16. Guenneau, S., Poulton, C. G. & Movchan, A. B. 2003 Oblique propagation of electromagnetic and elastodynamic waves for an array of cylindrical fibres. Proc. R. Soc. A 459, 2215-2263. (doi:10.1098/rspa.2002.1114)
17. Guenneau, S., Movchan, A. B., Petursson, G. & Ramakrishna, S. A. 2007 Acoustic metamaterials for sound focusing and confinement. New J. Phys. 9, 399. (doi:10.1088/1367-2630/9/11/399)
18. Harrison, J. M., Kuchment, P., Sobolev, A. & Winn, B. 2007 On occurrence of spectral edges for periodic operators inside the Brillouin zone. J. Phys. A: Math. 40, 7597-7618. (doi:10.1088/1751-8113/40/27/011)
19. Lekner, J. 1994 Light in periodically stratified media. J. Opt. Soc. Am. A 11, 2892-2899.
20. Luo, C., Johnson, S. G., Joannopoulos, J. D. & Pendry, J. B. 2002 All-angle negative refraction without negative effective index. Phys. Rev. B 65, 201104. (doi:10.1103/PhysRevB.65.201104)
21. Mok, J., De Sterke, C., Littler, I. & Eggleton, B. J. 2006 Dispersionless slow light using gap solitons. Nat. Phys. 2, 775-780. (doi:10.1038/nphys438)
22. Moler, C. B. & Stewart, G. W. 1973 An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241-256. (doi:10.1137/0710024)
23. Movchan, A. B., Movchan, N. V. & Poulton, C. G. 2002 Asymptotic models of fields in dilute and densely packed composites. London, UK: ICP Press.
24. Neviere, M. 1994 Bragg-Fresnel multilayer gratings: electromagnetic theory. J. Opt. Soc. Am. A 11, 1835-1845.
25. Nicolet, A., Guenneau, S., Geuzaine, C. & Zolla, F. 2004 Modeling of electromagnetic waves in periodic media with finite elements. J. Comp. Appl. Math. 168, 321-329. (doi:10.1016/j.cam.2003.07.002)
26. Nicorovici, N. A., Mcphedran, R. C. & Botten, L. C. 1995 Photonic band gaps: non-commuting limits and the 'acoustic band'. Phys. Rev. Lett. 75, 1507-1510. (doi:10.1103/PhysRevLett.75.1507)
27. O'Brien, S. & Pendry, J. B. 2002 Photonic band gap effects and magnetic activity in dielectric composites. J. Phys. Condens. Matter 14, 4035-4044. (doi:10.1088/0953-8984/14/15/317)
28. Pagneux, V. & Maurel, A. 2002 Lamb wave propagation in inhomogeneous elastic waveguides. Proc. R. Soc. A 458, 1913-1930. (doi:10.1098/rspa.2001.0950)
29. Pendry, J. B. 2000 Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966-3969. (doi:10.1103/PhysRevLett.85.3966)
30. Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. 1996 Extremely low frequency Plasmons in metallic mesostructures. Phys. Rev. Lett. 76, 4773-4776. (doi:10.1103/PhysRevLett.76.4773)
31. Postava, K., Vanwolleghem, M., Thourhout, D. V., Baets, R., Visnovský, S., Beauvillain, P. & Pistora, J. 2005 Modeling of a novel InP-based monolithically integrated magneto-optical waveguide isolator. J. Opt. Soc. Am. B 22, 261-273. (doi:10.1364/JOSAB.22.000261)
32. Postnova, J. & Craster, R. V. 2006 Trapped modes in topographically varying elastic waveguides. Wave Motion 44, 205-221. (doi:10.1016/j.wavemoti.2006.10.002)
33. Poulton, C. G., Movchan, A. B., Mcphedran, R. C., Nicorovici, N. A. & Antipov, Y. A. 2000 Eigenvalue problems for doubly periodic elastic structures and phononic band gaps. Proc. R. Soc. A 456, 2543-2559. (doi:10.1098/rspa.2000.0624)
34. Poulton, C. G., Botten, L. C., Mcphedran, R. C., Nicorovici, N. A. & Movchan, A. B. 2001 Noncommuting limits in electromagnetic scattering: asymptotic analysis for an array of highly conducting inclusions. SIAM J. Appl. Math. 61, 1706-1730. (doi:10.1137/S0036139999352262)
35. Reed, M. & Simon, B. 1978 Methods of modern, mathematical physics IV: analysis of operators. New York, NY: Academic Press.
36. Russell, P. S., Marin, E., Diez, A., Guenneau, S. & Movchan, A. B. 2003 Sonic band gap PCF preforms: enhancing the interaction of sound and light. Opt. Express 11, 2555.
37. Skelton, E. A., Adams, S. D. M. & Craster, R. V. 2007 Guided elastic waves and perfectly matched layers. Wave Motion 44, 573-592. (doi:10.1016/j.wavemoti.2007.03.001)
38. Sternberg, E. 1960 On the integration of the equations of motion in the classical theory of elasticity. Arch. Rat. Mech. Anal. 6, 34-50.
39. Wilcox, C. 1978 Theory of Bloch waves. J. Anal. Math. 33, 146-167.
40. Wiltshire, M., Hajnal, J. V., Pendry, J. B., Edwards, D. J. & Stevens, C. J. 2003 Metamaterial endoscope for magnetic field transfer: near field imaging with magnetic wires. Opt. Express 11, 709-715.