Resonant effects in random dielectric structures
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 217-246.

In [G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. Paris 339 (2004) 377–382; D. Felbacq and G. Bouchitté, Phys. Rev. Lett. 94 (2005) 183902; D. Felbacq and G. Bouchitté, New J. Phys. 7 (2005) 159], a theory for artificial magnetism in two-dimensional photonic crystals has been developed for large wavelength using homogenization techniques. In this paper we pursue this approach within a rigorous stochastic framework: dielectric parallel nanorods are randomly disposed, each of them having, up to a large scaling factor, a random permittivity ε(ω) whose law is represented by a density on a window Δ h =[a - ,a + ]×[0,h] of the complex plane. We give precise conditions on the initial probability law (permittivity, radius and position of the rods) under which the homogenization process can be performed leading to a deterministic dispersion law for the effective permeability with possibly negative real part. Subsequently a limit analysis h0, accounting a density law of ε which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure.

Reçu le :
DOI : 10.1051/cocv/2014026
Classification : 35B27, 35Q60, 35Q61, 35R60, 60H25, 78M35, 78M40
Mots clés : Stochastic homogenization, photonic crystals, metamaterials, micro-resonators, effective tensors, dynamical system
Bouchitté, Guy 1 ; Bourel, Christophe 2 ; Manca, Luigi 3

1 IMATH, Université du Sud Toulon-Var, 83957 La Garde cedex, France.
2 LMPA, Université du littoral côte d’Opale, 62228 Calais cedex, France.
3 LAMA, Université de Marne la Vallée, 77454 Marne la Vallée cedex 2, France.
@article{COCV_2015__21_1_217_0,
     author = {Bouchitt\'e, Guy and Bourel, Christophe and Manca, Luigi},
     title = {Resonant effects in random dielectric structures},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {217--246},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {1},
     year = {2015},
     doi = {10.1051/cocv/2014026},
     zbl = {1315.35020},
     mrnumber = {3348421},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014026/}
}
TY  - JOUR
AU  - Bouchitté, Guy
AU  - Bourel, Christophe
AU  - Manca, Luigi
TI  - Resonant effects in random dielectric structures
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 217
EP  - 246
VL  - 21
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2014026/
DO  - 10.1051/cocv/2014026
LA  - en
ID  - COCV_2015__21_1_217_0
ER  - 
%0 Journal Article
%A Bouchitté, Guy
%A Bourel, Christophe
%A Manca, Luigi
%T Resonant effects in random dielectric structures
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 217-246
%V 21
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2014026/
%R 10.1051/cocv/2014026
%G en
%F COCV_2015__21_1_217_0
Bouchitté, Guy; Bourel, Christophe; Manca, Luigi. Resonant effects in random dielectric structures. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 217-246. doi : 10.1051/cocv/2014026. http://archive.numdam.org/articles/10.1051/cocv/2014026/

G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. | DOI | MR | Zbl

G. Bouchitté and C. Bourel, Multiscale nanorod metamaterials and realizable permittivity tensors. Commun. Comput. Phys. 11 (2012) 489–507. | DOI | MR | Zbl

G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics. C. R. Math. Acad. Sci. Paris 339 (2004) 377–382. | DOI | MR | Zbl

G. Bouchitté and D. Felbacq, Homogenization of a wire photonic crystal: the case of small volume fraction. SIAM J. Appl. Math. 66 (2006) 2061–2084. | DOI | MR | Zbl

G. Bouchitté and B. Schweizer, Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model. Simul. 8 (2010) 717–750. | DOI | MR | Zbl

G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism. C. R. Math. Acad. Sci. Paris 347 (2009) 571–576. | DOI | MR | Zbl

A. Bourgeat, A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456 (1994) 19–51. | MR | Zbl

D. Cioranescu and J.S.J. Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590–607. | DOI | MR | Zbl

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory. Vol. 93 of Appl. Math. Sci. Springer-Verlag, Berlin, 2nd edition (1998). | MR | Zbl

D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes. General theory and structure. Vol. 2 of Probab. Appl. Springer, New York, 2nd edition (2008). | MR | Zbl

D. Felbacq and G. Bouchitté, Theory of mesoscopic magnetism in photonic crystals. Phys. Rev. Lett. 94 (2005) 183902. | DOI

R.M. Dudley. Real analysis and probability, Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1989). | MR | Zbl

D. Felbacq, G. Tayeb and D. Maystre, Scattering by a random set of parallel cylinders. J. Optim. Soc. Am. A 11 (1994) 2526–2538. | DOI | MR

D. Felbacq and G. Bouchitté, Negative refraction in periodic and random photonic crystals. New J. Phys. 7 (2005) 159. | DOI

G. Bouchitté and D. Felbacq, Homogenization of a set of parallel fibers. Waves in Random Media 7 (1997) 1–12. | MR | Zbl

A.A. Houck, J.B. Brock and I.L. Chuang, Experimental observations of a left-handed material that obeys snell’s law. Phys. Rev. Lett. 90 137401 (2003). | DOI

V.V. Jikov, S.M. Kozlov and O.A. Oleĭnik, Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994). | MR | Zbl

R.V. Kohn and S.P. Shipman, Magnetism and homogenization of microresonators. Multiscale Model. Simul. 7 (2008) 62–92. | DOI | MR | Zbl

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. | DOI | MR | Zbl

S. O’Brien and J.B. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals. J. Phys.: Condensed Matter 14 (2002) 6383–6394.

S. O’Brien and J.B. Pendry, Photonic band-gaps effects and magnetic activity in dielectric composites. J. Phys.: Condensed Matter 14 (2002) 4035.

J.B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart, Magnetism from conductors and enhanced nonlinear phenomena. Microwave Theory and Techniques. IEEE Trans. 47 (1999) 2075–2084.

R.A. Shelby, D.R. Smith and S. Schultz, Experimental verification of a negative index of refraction. Science 292 (2001) 77–79. | DOI

D.R. Smith, Willie J. Padilla, D.C. Vier, S.C. Nemat-Nasser and S. Schultz, Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84 (2000) 4184–4187. | DOI

V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ϵ and μ. Soviet Phys. Usp. 10 (1968) 509. | DOI

V.V. Zhikov and A.L. Piatnitski, Homogenization of random singular structures and random measures. Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) 23–74. | MR | Zbl

Cité par Sources :