# Perfect localization on flat band binary one-dimensional photonic lattices

## TL;DR

This paper demonstrates that one-dimensional binary photonic lattices can support multiple flat bands and localized modes, including Shockley surface states, challenging the notion that flat bands require higher dimensions.

## Contribution

It reveals the existence of trivial and additional flat bands in 1D binary lattices and constructs analytical Shockley surface modes, expanding understanding of localization in simple photonic systems.

## Key findings

- A trivial flat band formed by isolated dipolar states exists in 1D binary lattices.
- An extra flat band can be excited with specific lattice parameters, involving three sites.
- Analytical Shockley surface modes, including compact and staggered types, are constructed.

## Abstract

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.