  This work presents a rigorous theory for topological photonic materials in
one dimension. The main focus is on the existence and stability of interface
modes that are induced by topological properties of the bulk structure. For a
general 1D photonic structure with time-reversal symmetry, the associated Zak
phase (or Berry phase) may not be quantized. We investigate the existence of an
interface mode which is induced by a Dirac point upon perturbation.
Specifically, we establish conditions on the perturbation which guarantee the
opening of a band gap around the Dirac point and the existence of an interface
mode. For a periodic photonic structure with both time-reversal and inversion
symmetry, the Zak phase is quantized, taking only two values $0, \pi$. We show
that the Zak phase is determined by the parity (even or odd) of the Bloch modes
at the band edges. For a photonic structure consisting of two semi-infinite
systems on the two sides of an interface with distinct topological indices, we
show the existence of an interface mode inside the common gap. The stability of
the mode under perturbations is also investigated. Finally, we study resonances
for finite topological structures. Our results are based on the transfer matrix
method and the oscillation theory for Sturm-Liouville operators. The methods
and results can be extended to general topological Sturm-Liouville systems in
one dimension.
