Photonic Band Structure of Fibonacci Superlattices with Metamaterials. Part 1: The Algebraic Method of Calculation for Lossless Layers

Abstract

We study the properties of the photonic band structure (PBS) of infinite and binary Fibonacci superlattices (FS) containing metamaterial lossless layers. The transfer matrix method (TMM) and periodic boundary conditions are applied. The algebraic method of PBS calculations, based on the Bloch theorem, is described and used. The dispersion relations, for both type of light polarization, characterizing peculiarities of PBS are calculated numerically and presented. Fractal properties of a photonic band structure and the existence of so called zero-n gaps are established. Full Text: PDF References E. Macia,"The role of aperiodic order in science and technology", Rep. Prog. Phys. 69, 397 (2006) [CrossRef] W. Steurer, D. Suter-Widmer,"Photonic and phononic quasicrystals", J. Phys. D: Appl. Phys. 40, R229 (2007) [CrossRef] S. A. Ramakrishna, T. M. Grzegorczyk, "Physics and Applications of Negative Refractive Index Materials", SPIE Press and CRC Press 2009 J.E. Lugo et al,"Multiband negative refraction in one-dimensional photonic crystals", Optics Express, 17, 3042 (2009) [CrossRef] H. Zhang, X. Chen, Y. Li, Y. Fu, N. Yuan,"The Bragg gap vanishing phenomena in one-dimensional photonic crystals", Optics Express, 17, 7800 (2009) [CrossRef] M. de Dios-Leyva, J.C. Drake-Perez,"Zero-width band gap associated with the nŻ =0 condition in photonic crystals containing left-handed materials", Phys. Rev. E, 79, 036608 (2009) [CrossRef] M. de Dios-Leyva, O.E. Gonzales-Vasquez,"Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials", Phys. Rev. B, 77,125102 (2008) [CrossRef] R. Srivastava, K.B. Thapa, S. Pati, S.P. Ojha,"Negative refraction in 1D photonic crystals", Sol. State Comm., 147, 157 (2008) [CrossRef] Y. Fang, S. He,"Transparent structure consisting of metamaterial layers and matching layers", Phys. Rev. A, 78, 023813 (2008) [CrossRef] A. Bruno-Alfonso, E. Reyes-Gomez, S.B. Cavalcanti, L.E. Oliveira,"Band edge states of the ?n?=0 gap of Fibonacci photonic lattices", Phys. Rev. A, 78, 035801 (2008) [CrossRef] Eds. C. M. Krowne, Y. Zhang, "Physics of Negative Refraction and Negative Index Materials", Springer 2007 Y.-T Fang, J. Zhou, E.Y.B. Pun,"High-Q filters based on one-dimensional photonic crystals using epsilon-negative materials", Appl. Phys. B, 86, 587 (2007) [CrossRef] X. Hu, Z. Liu, Q. Gong,"A multichannel filter in a photonic crystal heterostructure containing single-negative materials", J. Opt. A: Pure Appl. Opt., 9, 877 (2007) [CrossRef] F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos,"Transmission spectra in photonic band-gap Fibonacci nanostructures", Surface Science, 601, 4492 (2007) [CrossRef] K. Tarnowski, W. Salejda, M. H. Tyc,"Propagation of polarized light through optical nanosuperlattices", Optica Applicata, 37, 387, 2007 M.H. Tyc, W. Salejda, A. Klauzer-Kruszyna, K. Tarnowski,"Photonic band structure of one-dimensional aperiodic superlattices composed of negative refraction metamaterials", SPIE Proceedings Series, 2007, 6581, 658112 [CrossRef] M.H. Tyc, W. Salejda, A. Klauzer-Kruszyna, K. Tarnowski,"Propagation of polarized light through superlattices composed of left- and right-handed materials", SPIE Proceedings Series, 2007, 6581, 658113 [CrossRef] P. Yeh, "Optical Waves in Layered Media", Wiley-Interscience 2005 H.X. Da, C. Xu, Z.Y. Li,"Omnidirectional reflection from one-dimensional quasi-periodic photonic crystal containing left-handed material", Phys. Lett. A, 345, 459 (2005) [CrossRef] W. Salejda, A. Klauzer-Kruszyna, M. H. Tyc, K. Tarnowski,"One-dimensional photonic quasicrystals: application to Bragg reflectors", SPIE Proceedings Series, 2005, 5950, 59501Q [CrossRef] W. Salejda, A. Klauzer-Kruszyna, M. H. Tyc, K. Tarnowski,"Electromagnetic wave propagation through aperiodic superlattices composed of left- and right-handed materials", SPIE Proceedings Series, 2005, 5955, 595514 [CrossRef] J. Li, D. Zhao, Z. Liu,"Zero n - photonic band gap in a quasiperiodic stacking of positive and negative refractive index materials", Phys. Lett. A, 332, 461 (2004) [CrossRef] A. Klauzer- Kruszyna, W. Salejda, M.H. Tyc,"Polarized Light Transmission through Generalized Fibonacci Multilayers. I. Dynamical maps approach", Optik, 115, 257 (2004) A. Klauzer- Kruszyna, W. Salejda, M.H. Tyc,"Polarized Light Transmission through Generalized Fibonacci Multilayers. II. Numerical Results", Optik, 115, 267 (2004) M. Kohomoto, B. Sutherland, K. Iguchi,"Localization of optics: Quasiperiodic media", Phys. Rev. Lett., 58, 2436 (1987) [CrossRef] J. Li, L. Zhou, C.T. Chan, P. Shang,"Photonic Band Gap from a Stack of Positive and Negative Index Materials", Phys. Rev. Lett., 90, 083901 (2003) [CrossRef]