Photonic band gaps in periodic dielectric structures: The scalar-wave approximation

Using a plane-wave expansion method we have computed the band structure for a scalar wave propagating in periodic lattices of dielectric spheres (dielectric constant ${\mathrm{\epsilon }}_{\mathit{a}}$) in a uniform dielectric background (${\mathrm{\epsilon }}_{\mathit{b}}$). All of the lattices studied (simple cubic, bcc, fcc, and diamond) do possess a full band gap. The optimal values of the filling ratio f of spheres and of the relative dielectric contrast for the existence of a gap are obtained. The minimum value of the relative dielectric contrast for creating a gap is also obtained. These results are applicable to the problem of classical-wave propagation in composite media and relevant to the problem of classical-wave localization.