# Topological Boundary Modes from Translational Deformations

## TL;DR

This paper introduces a new mechanism for generating localized states in periodic potentials through translational deformations, supported by theoretical proof and experimental verification in microwave photonic crystals.

## Contribution

It presents a novel approach to create localized states via continuous translational deformations, with rigorous proof and experimental validation in photonic systems.

## Key findings

- Localized states emerge from translational deformations.
- Experimental verification in microwave photonic crystals.
- Observation of topological phase windings in reflected waves.

## Abstract

Localized states universally appear when a periodic potential is perturbed by defects or terminated at its surface. In this Letter, we theoretically and experimentally demonstrate a mechanism that generates localized states through continuous translational deformations of periodic potentials. We provide a rigorous proof of the emergence of the localized states under the deformations. The mechanism is experimentally verified in microwave photonic crystals. We also demonstrate topological phase windings of reflected waves for translated photonic crystals.