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Original Articles

Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability

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Pages 653-697 | Received 21 Feb 2007, Accepted 26 Jul 2007, Published online: 19 Oct 2007

References

  • Guenneau , S. 2001 . Homogénéisation des quasi-cristaux et analyse des modes dans des fibres optiques de type de cristal photonique , PhD thesis Université Aix-Mareille I .
  • Joannopoulos , J. , Meade , R. and Winn , J. 1995 . Photonic Crystals , Princeton : Princeton University Press .
  • De la Rue , R. C. and Kraus , T. F. 1996 . “ Strategies for the abrication of photonic microstructures in semiconductors ” . In Microcavities and Photonic Bandgaps , 175 – 192 . Dordrecht : Kluwer Academic .
  • Maystre , D. 1994 . Electromagnetic study of photonic band gaps . Pure Applied Optics , 3 : 975 – 993 .
  • Felbacq , D. , Tayeb , G. and Maystre , D. 1994 . Scattering by a random set of parallel cylinders . Journal of the Optical Society of America A-Optics Image Scienceand Vision , 11 : 2526 – 2538 .
  • Zolla , F. , Felbacq , D. and Guizal , B. 1998 . A remarkable diffractive property of photonic quasicrystals . Optics Communications , 148 : 6 – 10 .
  • Felbacq , D. , Guizal , B. and Zolla , F. 1998 . Wave propagation in one-dimensional photonic crystals . Optics Communications , 152 : 119 – 126 .
  • Felbacq , D. , Guizal , B. and Zolla , F. 1998 . Limit analysis of the diffraction of a plane wave by a stratified periodic medium . Journal of Mathematical Physics , 39 : 4604 – 4607 .
  • Bouchitté , G. , Felbacq , D. and Zolla , F. 2005 . Do Fresnel coefficients exist? . Wave Motion , 42 : 75 – 95 .
  • Yablonovitch , E. 1987 . Inhibited spontaneous emission in solid-state physics and electronics . Physical Review Letters , 58 : 2059 – 2062 .
  • Datta , S. , Chan , C. T. , Ho , K. M. and Soukoulis , C. M. 1993 . Effective dielectric constant of periodic composite structures . Physical Review B , 48 : 14936 – 14943 .
  • Halevi , P. , Krokhin , A. A. and Arriaga , J. 1999 . Photonic crystal optics and homogenization of 2D periodic composites . Physical Review Letters , 82 : 719 – 722 .
  • Krokhin , A. A. , Halevi , P. and Arriaga , J. 2002 . Long-wavelength limit (homogenization) for two-dimensional photonic crystals . Physical Review B , 65 : 1551
  • McPhedran , R. C. 1986 . Transport properties of cylinder pairs and of the square array of cylinders . Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences , 408 : 31 – 43 .
  • Bensoussan , A. , Lions , J. L. and Papanicolaou , G. 1978 . Asymptotic Analysis for Periodic Structures , Amsterdam : North-Holland .
  • Bouchitté , G. and Petit , R. 1985 . Homogenization techniques as applied in the electromagnetic theory of gratings . Electromagnetics , 5 : 17 – 36 .
  • Petit , R. and Bouchitté , G. 1988 . “ Replacement of a very fine grating by a stratified layer: homogenization techniques, and multiple-scale method theory ” . In Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena 431 Edited by: Lerner , J. 25 – 31 . Proceedings SPIE 815
  • Felbacq , D. 1994 . Étude théorique et numérique de la diffraction par des ensembles de tiges paralléles , PhD thesis Université Aix-Marseille III .
  • Felbacq , D. and Bouchitté , G. 1997 . Homogenization of a set of parallel fibers . Waves in Random and Complex Media , 7 : 245 – 256 .
  • Yablonovitch , E. 2001 . Photonic crystals: semiconductors of light . Scientific American , 285 : 34 – 41 .
  • Gralak , B. , Enoch , S. and Tayeb , G. 2000 . Anomalous refractive properties of photonic crystals . Journal of the Optical Society of America A-Optics Image Science and Vision , 17 : 1012 – 1020 .
  • Cherednichenko , K. D. and Guenneau , S. 2007 . Bloch wave homogenisation for spectral asymptotic analysis of the Maxwell operator . Waves in Random and Complex Media , (to appear)
  • Allaire , G. 1992 . Homogenization and two-scale convergence . SIAM Journal on Mathematical Analysis , 23 : 1482 – 1518 .
  • Guenneau , S. and Zolla , F. 2000 . Homogenization of three-dimensional finite photonic crystals . Journal of Electromagnetic Waves and Applications , 14 : 529 – 530 . Progress in Electromagnetics Research, PIER 27, 91–127. (doi:10.2528/PIER99071201)
  • Guenneau , S. , Geuzaine , C. , Nicolet , A. , Movchan , A. B. and Zolla , F. 2004 . Low frequency electromagnetic waves in periodic structures . International Journal of Applied Electromagnetics and Mechanics , 19 : 479 – 483 .
  • Zolla , F. and Guenneau , S. . Artificial ferro-magnetic anisotropy: homogenization of 3D finite photonic crystals . Proceedings of IUTAM symposium held in Liverpool on Asymptotics, Singularities and Homogenization in Problems of Mechanics . Edited by: Movchan , A. B. pp. 375 – 385 . Dordrecht : Kluwer Academic .
  • Bouchitté , G. and Felbacq , D. 2006 . Homogenization of a wire photonic crystal: the case of small volume fraction . SIAM Journal on Applied Mathematics , 66 : 2061
  • Cherednichenko , K. D. , Smyshlaev , V. P. and Zhikov , V. V. 2006 . Non-local homogenized limits for composite media with highly anisotropic periodic fibres . Proceedings of the Royal Society of Edinburgh Section A-Mathematics , 136 : 87 – 114 .
  • Ke-Da , B. , Poulton , C. G. , Botten , L. C. , McPhedran , R. C. and Nicorovici , N. A. 2000 . The electromagnetic modes and homogenization for a cubic lattice of spheres . Physica B , 279 : 162 – 163 .
  • Wellander , N. 2001 . Homogenization of the Maxwell equations: Case I. Linear theory . Appl. Math. , 46 : 29 – 51 .
  • Zolla , F. and Guenneau , S. 2003 . Duality relation for the Maxwell system . Physical Review E , 67 : 026610
  • Zhikov , V. V. 2000 . On an extension of the method of two-scale convergence and its applications . Sb. Math. , 191 : 973 – 1014 .
  • Birman , M. and Suslina , T. 2001 . Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics . Operator Theory: Advances and Applications , 129 : 71 – 107 .
  • Allaire , G. and Conca , C. 1998 . Bloch wave homogenization and spectral asymptotic analysis . Journal de Mathematiques Pures et Appliquees , 77 : 153 – 208 .
  • Lions , J. L. 1985 . “ Remarques sur les problemes d'homogeneisation dans les milieux a structure periodique et sur quelques problemes raides ” . In Les Methodes d'homogeneisation: Theorie et Applications en Physique , Edited by: Bergman , D. , Lions , J. L. , Papanicolaou , G. , Murat , F. , Tartar , L. and Sanchez-Palencia , E. 129 – 228 . Paris : Eyrolles .
  • Artola , M. and Cessenat , M. 1991 . Quelques extensions de resultats d'homogeneisation en electromagnetisme . Comptes Rendus de l Academie des Sciences Serie I-Mathematique , 312 : 373 – 377 .
  • Alexandre , R. 1997 . Some results in homogenization tackling memory effects . Asymptotic Analysis , 15 : 229 – 259 .
  • Sanchez-Palencia , E. 1980 . Nonhomogeneous Media and Vibration Theory , Berlin : Springer . Lecture Notes in Physics, Number 127
  • Sanchez-Hubert , J. and Sanchez-Palencia , E. 1978 . Sur certains problemes physiques d'homogeneisation donnant lieu a des problemes de relaxation . Comptes Rendus de l Academie des Sciences Serie I-Mathematique , 286 : 286 – 903 .
  • Bakhalov , N. and Panasenko , G. 1989 . Homogenization: Averaging Processes in Periodic Media , Dordrecht : Kluwer .
  • Jikov , V. , Kozlov , S. and Oleinik , O. 1995 . Homogenization of Differential Operators , Berlin : Springer-Verlag .
  • Sanchez-Palencia , E. and Zaoui , A. 1987 . Homogenization Techniques for Composite Media , Berlin : Springer . Lecture Notes in Physics, Number 272
  • Conca , C. and Vanninathan , M. 1997 . Homogenization of periodic structures via Bloch decomposition . SIAM Journal on Applied Mathematics , 57 : 1639 – 1659 .
  • Amirat , Y. , Hamdache , K. and Ziani , A. 1992 . Homogenization of parametrized fanilies of hyperbolic problems . Proceedings of the Royal Society of Edinburgh Section A-Mathematics , 120A : 199 – 221 .
  • El Feddi , M. , Ren , Z. , Razek , A. and Bossavit , A. 1997 . Homogenization technique for Maxwell equations in periodic structures . IEEE Transactions on Magnetics , 33 : 1382 – 1385 .
  • Bottauscio , O. , Chiado Piat , V. , Chiampi , M. , Codegone , M. and Manzin , A. 2006 . Electromagnetic phenomena in heterogeneous media: Effective properties and local behaviour . Journal of Applied Physics , 100 : 044902
  • Milton , G. W. 2002 . The Theory of Composites , Cambridge : Cambridge University Press .
  • McPhedran , R. C. , Poulton , C. G. , Nicorovici , N. A. and Movchan , A. B. 1996 . Low frequency correction to the static effective dielectric constant of a two-dimensional composite material . Proc. Royal Soc. Lon. A , 452 : 2231 – 2245 .
  • Zolla , F. , Renversez , G. , Nicolet , A. , Kuhlmey , B. , Guenneau , S. and Felbacq , D. 2005 . Fondations of Photonic Crystal Fibres , London : Imperial College Press .
  • McPhedran , R. C. , Derrick , G. H. and Botten , L. C. 1980 . “ Theory of crossed gratings ” . In Electromagnetic Theory of Gratings , Edited by: Petit , R. Berlin : Springer Verlag .
  • Cessenat , M. 1996 . Mathematical Methods in Electromagnetism , Singapore : World scientific .
  • Bouchitte , G. and Felbacq , D. 2000 . “ Low frequency scattering by a set of parallel metallic rods ” . In Mathematical and Numerical Aspects of Wave Propagation , Edited by: Joly , P. and Bermudez , A. 226 – 230 . Philadelphia : SIAM .
  • Pendry , J. B. , Holden , A. J. , Robbins , D. J. and Stewart , W. J. 1999 . Magnetism from conductors and enhanced nonlinear phenomena . IEEE Transactions on Microwave Theory and Techniques , 47 : 2075 – 2084 .
  • Poulton , C. G. , Guenneau , S. and Movchan , A. B. 2004 . Non-commuting limits and effective properties for electromagnetism in conical incidence . Physical Review B , 69 : 195112
  • Van Bladely , J. 1985 . Electromagnetic Fields , Washington : Hemisphere .
  • Nicorovici , N. A. , McPhedran , R. C. and Ke-Da , B. 1995 . Propagation of electromagnetic waves in periodic lattices of spheres: Green's function and lattice sums . Physical Review E , 51 : 690 – 702 .
  • Lorentz , L. 1952 . The Theory of Electrons , New York : Dover .
  • Keller , J. B. 1964 . A theorem on the conductivity of a composite medium . Journal of Mathematical Physics , 5 : 548 – 549 .
  • Dykhne , A. M. 1970 . Conductivity of a two-dimensional two-phase system . Soviet Physics JETP , 32 : 63 – 65 .
  • Dular , P. , Geuzaine , C. , Henrotte , F. and Legros , W. 1998 . A general environment for the treatment of discrete problems and its application to the finite element method . IEEE Transactions on Magnetics , 34 : 3395 – 3398 .
  • Mortola , S. and Steffe , S. 1991 . “ A two-dimensional homogenization problem ” . In Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie VIII 77 – 82 . Volume 1
  • Craster , R. V. and Obnosov , Y. V. 2000 . Four-phase checkerboard composites . SIAM Journal of Applied Mathematics , 61 : 1839 – 1856 .
  • Milton , G. W. 2001 . Proof of the conjecture on the conductivity of checkerboards . Journal of Mathematical Physics , 42 : 4873 – 4882 .
  • Zolla , F. , Petit , R. and Cadilhac , M. 1994 . Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources . Journal of the Optical Society of America A-Optics Image Science and Vision , 11 : 1087 – 1096 .
  • Hashin , Z. and Shtrikman , S. 1962 . A variational approach to the theory of effective magnetic permeability of multiphase materials . Journal of Applied Physics , 33 : 3125 – 3131 .
  • Bossavit , A. 1990 . Solving Maxwell's equations in a closed cavity and the question of spurious modes . IEEE Transactions on Magnetics , 26 ( 2 ) Mar : 702 – 705 .
  • Conca , C. , Planchard , J. and Vanninathan , M. 1995 . Fluids and Periodic Structures , Paris : RMA, J. Wiley and Masson .
  • Nicolet , A. , Guenneau , S. , Geuzaine , C. and Zolla , F. 2004 . Modelling of electromagnetic waves in periodic media with finite elements . Journal of Computational and Applied Mathematics , 168 : 321 – 329 .
  • Mossotti , O. F. 1850 . Analytical discussion of the influence which the action of a dielectric medium exerts on the distribution of the electricity on the surfaces of several electric bodies dispersed in it (translated from the 1836 paper in italian) . Memorie di Matematica e di Fizica della Societa Italiana dellee Scienze Residente in Modena , 24 : 49 – 74 .
  • Poulton , C. G. , Botten , L. C. , McPhedran , R. C. , Nicorovici , N. A. and Movchan , A. B. 2001 . Non-commuting limits in electromagnetic scattering: asymptotic analysis for an array of highly conducting inclusions . SIAM Journal on Applied Mathematics , 61 : 1706 – 1730 .

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