Topological States in 1D Photonic Quasi-crystals
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Topological States in 1D Photonic Quasi-crystals M.S. Vasconcelos1 and E.L. Albuquerque2 1
Escola de Ciência e Tecnologia, UFRN, 59072-970, Natal – RN, Brazil
2
Departamento de Biofísica e Farmacologia, UFRN, 59072-970, Natal – RN, Brazil
ABSTRACT We investigated theoretically the transmission spectra in one-dimensional photonic quasicrystals (1DPQ) made up from dielectric materials organized in accordance to a discrete varying electric permittivity profile that obeys an analogous of the quasiperiodic potential in the so-called AudryAndré (AA) model, in order to modulate the refraction index. Our results show that due to the incommensurate dielectric distribution, the spectrum splits into a fractal set of pass- and forbidden-band structure. By studying the transmission spectra as a function of the modulation phase φ, we found boundary states lying within the gaps localized either on the left or on the right boundary of the system, characterizing the so-called topological states. INTRODUCTION The discovery of topological insulators has started considerable interest in the study of the topological phases of matter [1]. Topological phases consist of various band insulators or superconductors that have gaps in their spectrum. In these insulators, spin-orbit effects take the role of an external magnetic field, with spins of opposite sign counter-propagating along the edge [2-3]. The main characteristic of these novel phases is the emergence of topologically protected boundary phenomena, e.g., quantum pumping, surface states related to exotic models from particle physics, and quasiparticles with non- Abelian statistics [1]. On the other hand, recently Kraus et. al. [4] found the connection between quasicrystals and the topological phases of matter. The question is interesting, because quasicrystals may be constructed by taking a cut through a standard crystal in a higher dimension, thus opening up the way of probing higher-dimensional topological phases in lower-dimensional systems. It is the aim of this work to investigate theoretically the transmission spectra in onedimensional photonic crystal, made up from dielectric materials organized in accordance to a discrete varying electric permittivity profile that obeys an analogous of the quasiperiodic potential in the so-called Audry-André (AA) model [5]. It describes electrons hopping on a onedimensional chain, with a spatially modulated on-site potential, in such a way that the period of the potential is incommensurate with the lattice period. In order to verify that the spectrum splits into an analogous fractal set of pass- and forbidden-band structure, known as Hofstader butterfly [6], we study the bulk modes in the incommensurate dielectric distribution by using a transfer matrix approach.
THEORY Consider that the light of frequency ω and s-polarization (TE waves) is incident from a transparent medium C at normal direction of the layered system. The layered system is formed by an incommensurate array of slabs of different materials (ε1, ε2, ε3,…). The reflectance
