Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal

Abstract

By the use of the plane-wave method, we calculate the photonic band structure for electromagnetic waves of E and H polarization propagating in a system consisting of an infinite array of identical, infinitely long, parallel cylinders of circular cross section, embedded in vacuum, whose intersections with a perpendicular plane form a square or triangular lattice. The cylinders are fabricated from GaAs, which represents a cubic, polar crystal material, containing two atoms in a primitive unit cell, for which the dielectric function has the form \ensuremath{\epsilon}(\ensuremath{\omega})=${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}^{2g}$-${\mathrm{\ensuremath{\omega}}}^{2}$ )/(${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}^{2g}$-${\mathrm{\ensuremath{\omega}}}^{2}$ ), where ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$ is the optical frequency dielectric constant, while ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ are the frequencies of the longitudinal optical and transverse optical vibration modes of infinite wavelength, respectively. For electromagnetic waves of both polarizations the problem of obtaining the photonic band structure is reduced to the solution of a generalized eigenvalue problem. In comparison with the dispersion curves of electromagnetic waves in vacuum, the photonic band structures are the most significantly affected by the particular form of the dielectric function we have assumed for frequencies in the vicinity of the polariton gap ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ . For small filling fractions of the GaAs cylinders the photonic band structure for each polarization, except within the frequency range of the polariton gap, is a slightly perturbed version of the dispersion curves of electromagnetic waves in vacuum. For higher values of the filling fraction the dispersion curves deviate substantially from the dispersion curves of electromagnetic waves in vacuum within a broader frequency range, and in the case of E polarization, the photonic band structures reveal the existence of absolute band gaps. As the most striking feature, the calculated photonic band structures yield additional, nearly dispersionless bands that for waves of both polarizations occur at and below ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ and in the case of H polarization occur also within the frequency range ${\mathrm{\ensuremath{\omega}}}_{\mathrm{T}}$ ${\mathrm{\ensuremath{\omega}}}_{\mathrm{L}}$ where \ensuremath{\epsilon}(\ensuremath{\omega}) is negative. The results for H polarization and small filling fractions of the GaAs cylinders are reproduced by an approach in which the zeros of a determinant are sought. A possible origin of the flat bands obtained in this polarization by both approaches is discussed.