# Scattering by finite periodic $\mathcal{PT}$-symmetric structures

## TL;DR

This paper derives a closed-form expression for the transmission properties of finite one-dimensional $ ext{PT}$-symmetric periodic structures, revealing conditions for CPA-laser phenomena and enabling design of structures with high absorption or amplification.

## Contribution

It provides a simple formula for transmittance in finite $ ext{PT}$-symmetric systems and establishes conditions for CPA-laser operation independent of the number of cells.

## Key findings

- Derived a closed-form expression for transmittance.
- Identified conditions for CPA-laser in finite systems.
- Outlined design principles for high absorption and amplification.

## Abstract

In this work, we study the transmission properties of one dimensional finite periodic systems with $\mathcal{PT}$-symmetry. A simple closed form expression is obtained for the total transmittance from a lattice of N cells, that allows us to describe the transmission minima (maxima) when the system is in the $\mathcal{PT}$-unbroken (broken) phase. Utilizing this expression, we provide the necessary conditions, \textit{independent} of the number of cells, for the occurrence of a CPA-laser for any finite $\mathcal{PT}$-symmetric periodic potential. Under this conditions, we provide a recipe for building finite periodic structures with near perfect absorption and extremely large amplification.