  We study the photonic spectrum of a one-dimensional optical lattice
possessing a double primitive cell, when the atoms are well localized at the
lattice minima. While a one-dimensional lattice with a simple Wigner-Seitz cell
always possesses a photonic bandgap at the atomic resonance, in this
configuration the photonic transmission spectrum may exhibit none, double or
multiple photonic bandgaps depending on the ratio between the interparticle
distance $\varrho$ inside the cell and the cell size $a$. The transmission
spectra of a weak incident probe are evaluated when the atoms are trapped in
free space and inside an optical resonator for realistic experimental
parameters.
