  The existence of flat bands is generally thought to be physically possible
only for dimensions larger than one. However, by exciting a system with
different orthogonal states this condition can be reformulated. In this work,
we demonstrate that a one-dimensional binary lattice supports always a trivial
flat band, which is formed by isolated single-site vertical dipolar states.
These flat band modes correspond to the highest localized modes for any
discrete system, without the need of any aditional mechanism like, e.g.,
disorder or nonlinearity. By fulfilling a specific relation between lattice
parameters, an extra flat band can be excited as well, with modes composed by
fundamental and dipolar states that occupy only three lattice sites.
Additionally, by inspecting the lattice edges, we are able to construct
analytical Shockley surface modes, which can be compact or present staggered or
unstaggered tails. We believe that our proposed model could be a good candidate
for observing transport and localization phenomena on a simple one-dimensional
linear photonic lattice.
