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Photonic Band Structures of Atomic Lattices
[chapter]

Rudolf Sprik, A. D. Lagendijk, Bart A. Tiggelen

1996
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Photonic Band Gap Materials
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A calculation of the optical band structure of a three dimensional lattice of resonant two-level atoms in the dipole approximation is presented. The formation of band gaps is exhibited and confirmed by a calculation of the density of states. The band structure can be characterized by two dimensionless parameters. We find a longitudinal polarization mode as well as a class of vacuum modes that are unaltered by the interaction with matter. Numerical calculations are performed for a face centered
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... ubic lattice; other lattices can be evaluated as easily. [S0031-9007(96)01222-7] PACS numbers: 32.80.Pj, 42.25.Bs At present the study of photonic crystals [1], i.e., dielectric materials with a periodicity matching optical wavelengths, is a subject of active research. The periodicity induces an optical band structure quite analogous to the band structure in semiconductor physics. On both the theoretical (numerical) and experimental side, a search is going on for materials exhibiting a photonic band gap. Such a gap can give rise to the suppression of spontaneous emission of interstitial atoms and has promising consequences for applications. Moreover, from a fundamental point of view such materials, after some randomization, are interesting for the observation of the localization of light, and also the quantum electrodynamics of photonic crystals merits a further study [2] . Various authors have reported systems exhibiting photonic band gaps, depending on the type of unit cell, shape of the "atoms" [behavior of electric permeability´͑x͒ over a unit cell], and refractive index contrast [1,3-5]. A similar band structure can arise in atomic optical lattices. Atoms, cooled down to the microkelvin regime, can be trapped in their ac Stark shift potential wells in a one, two, or three dimensional interference pattern created by a combination of laser beams. Consequently, the lattice constant is essentially the wavelength of the trapping field. Already results on Bragg scattering have been reported [6] showing long range periodic order. The main difference with the photonic crystals is the sharp resonant character of the scatterers (the atoms on the lattice sites) near an optical resonance in the atom. Furthermore, in the limit of weak light fields and if recoil effects are ignored, the propagation of light is coherent [7] and without dissipation. As a consequence, in a twolevel approximation, atoms can be accurately described by classical damped linear point dipole oscillators possessing a sharp resonance. Here we calculate the band-structure properties of such a dipolar lattice. It leads to point interactions on the lattice sites which reduce the required computational effort immensely. Therefore our method is promising to be applied to more complicated lattices as well. The starting point is the set of Maxwell's equations for a static isotropic dielectric medium [permeabilitý ͑v, x͒ ϵ 1 1 4px͑v, x͒] without external charges and currents. Elimination of the magnetic field component in favor of the electric field E gives, after Fourier transformation with respect to time,

doi:10.1007/978-94-009-1665-4_39
fatcat:e7omm6oigzdhzcc6iverund2au