Comptes Rendus
Negative bending mode curvature via Robin boundary conditions
Comptes Rendus. Physique, Volume 10 (2009) no. 5, pp. 437-446.

We examine the band spectrum, and associated Floquet–Bloch eigensolutions, arising in straight walled acoustic waveguides that have periodic structure along the guide. Homogeneous impedance (Robin) conditions are imposed along the guide walls and we find that in certain circumstances, negative curvature of the lowest (bending) mode can be achieved. This is unexpected, and has not been observed in a variety of physical situations examined by other authors. Further unexpected properties include the existence of the bending mode only on a subset of the Brillouin zone, as well as permitting otherwise unobtainable velocities of energy transmission. We conclude with a discussion of how such boundary conditions might be physically reproduced using effective conditions and homogenization theory, although the methodology to achieve these effective conditions is an open problem.

Nous étudions le spectre de bande associé aux modes de Floquet–Bloch dans des guides d'ondes planaires acoustiques périodiques. Nous imposons des conditions d'impédance homogènes (conditions de Robin) sur les bords du guide d'épaisseur finie, et observons dans certains cas une courbure négative de la première bande de dispersion (mode de flexion). Cette trouvaille est pour le moins inattendue, car une telle anomalie n'a pas été reportée à ce jour pour d'autres types de conditions limites dans nombre de problèmes physiques. Encore plus surprenante est l'existence de modes de flexions dans un segment de la zone de Brillouin ainsi que des vitesses de groupe extrêmes. Finalement, nous suggérons certaines pistes pour obtenir de telles conditions d'impédances, telle que la voie classique de l'homogénéisation, même si cela reste pour l'heure un problème ouvert.

Published online:
DOI: 10.1016/j.crhy.2009.03.009
Keywords: Robin conditions, Negative refraction, Sub-wavelength imaging
Mot clés : Conditions de Robin, Réfraction négative, Imagerie haute résolution

Samuel D.M. Adams 1; Richard V. Craster 2; Sébastien Guenneau 3

1 Department of Mathematics, Imperial College London, London SW7-2AZ, UK
2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1
3 Department of Mathematical Sciences, Liverpool University, Liverpool L69-3BX, UK
     author = {Samuel D.M. Adams and Richard V. Craster and S\'ebastien Guenneau},
     title = {Negative bending mode curvature via {Robin} boundary conditions},
     journal = {Comptes Rendus. Physique},
     pages = {437--446},
     publisher = {Elsevier},
     volume = {10},
     number = {5},
     year = {2009},
     doi = {10.1016/j.crhy.2009.03.009},
     language = {en},
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TI  - Negative bending mode curvature via Robin boundary conditions
JO  - Comptes Rendus. Physique
PY  - 2009
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EP  - 446
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PB  - Elsevier
DO  - 10.1016/j.crhy.2009.03.009
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%A Richard V. Craster
%A Sébastien Guenneau
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Samuel D.M. Adams; Richard V. Craster; Sébastien Guenneau. Negative bending mode curvature via Robin boundary conditions. Comptes Rendus. Physique, Volume 10 (2009) no. 5, pp. 437-446. doi : 10.1016/j.crhy.2009.03.009.

[1] V.G. Veselago The electrodynamics of substances with simultaneously negative value of ϵ and μ, Sov. Phys. Uspekhi, Volume 10 (1968), pp. 509-514

[2] J.B. Pendry Negative refraction makes a perfect lens, Phys. Rev. Lett., Volume 86 (2000), p. 3966

[3] D. Maystre; S. Enoch Perfect lenses made with left-handed materials: Alice's mirror?, J. Opt. Soc. Am. A, Volume 21 (2004), pp. 122-131

[4] S.A. Ramakrishna Physics of negative refraction, Rep. Prog. Phys., Volume 68 (2005), p. 449

[5] S.D.M. Adams; R.V. Craster; S. Guenneau Bloch waves in multi-layered acoustic waveguides, Proc. R. Soc. London A, Volume 464 (2008), pp. 2669-2692

[6] A. Figotin; I. Vitebskiy Slow light in photonic crystals, Waves Random Complex Media, Volume 16 (2006), pp. 293-392

[7] J.T. Mok; C.M. de Sterke; I.C.M. Littler; B.J. Eggleton Dispersionless slow light using gap solitons, Nat. Phys., Volume 2 (2006), pp. 775-780

[8] R.V. Craster; S. Guenneau; S.D.M. Adams Mechanism for slow waves near cutoff frequencies in periodic waveguides, Phys. Rev. B, Volume 79 (2009), p. 045129

[9] M. Saillard; D. Maystre Scattering from metallic and dielectric rough surfaces, J. Opt. Soc. Am. A, Volume 7 (1990), pp. 982-990

[10] C. Caloz; T. Itoh Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, John Wiley and Sons, 2006

[11] L. Brillouin Wave Propagation in Periodic Structures, McGraw–Hill, 1946

[12] J.R. Pierce Traveling-Wave Tubes, D. Van Nostrand, 1950

[13] J.B. Pendry; L. Martin-Moreno; F.J. Garcia-Vidal Mimicking surface plasmons with structured surfaces, Science, Volume 305 (2004), pp. 847-848

[14] J.B. Pendry; A.J. Holden; D.J. Roberts; W.J. Stewart IEEE Trans. Micr. Theory Techniques, 47 (1999), p. 2075

[15] W.B. Fraser Orthogonality relation for the Rayleigh–Lamb modes of vibration of a plate, J. Acoust. Soc. Am., Volume 59 (1976), pp. 215-216

[16] A.B. Movchan; N.V. Movchan; C.G. Poulton Asymptotic Models of Fields in Dilute and Densely Packed Composites, ICP Press, London, 2002

[17] R.L. Kronig; W.G. Penney Quantum mechanics of electrons in crystal lattices, Proc. R. Soc. London, Volume 130 (1931), pp. 499-531

[18] A.B. Movchan; S. Guenneau Split-ring resonators and localized modes, Phys. Rev. B, Volume 70 (2004), p. 125116

[19] F. Zolla; G. Bouchitte; S. Guenneau Pure currents in foliated waveguides, Q. J. Mech. Appl. Math., Volume 61 (2008) no. 6, pp. 453-474

[20] J.L. Zhang; H.T. Jiang; S. Enoch; G. Tayeb; B. Gralak; M. Lequime Two-dimensional complete band gaps in one-dimensional metal-dielectric periodic structures, Appl. Phys. Lett., Volume 92 (2008), p. 053104

[21] J.M. Harrison; P. Kuchment; A. Sobolev; B. Winn On occurrence of spectral edges for periodic operators inside the Brillouin zone, J. Phys. A – Math., Volume 40 (2007), pp. 7597-7618

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