# Clustering in particle chains - summation techniques for the periodic Green's function

## TL;DR

This paper derives and summarizes efficient summation formulas for the periodic Green's function in 1D lattice structures, applicable to photonic and antenna array problems, using polylogarithms and Poisson summation techniques.

## Contribution

It provides a comprehensive set of summation formulas for the 1D lattice Green's function, covering both on-axis and off-axis cases, facilitating computations in photonics and antenna design.

## Key findings

- Derived polylogarithmic expressions for on-axis summations
- Applied Poisson summation for off-axis cases
- Enhanced computational methods for periodic Green's functions

## Abstract

1D lattice summations of the 3D Green's function are needed in many applications such as photonic crystals, antenna arrays, and so on. Such summations are usually divided into two cases, depending on the location of the observer: Out of the summation axis, or on the summation axis. Here, as a service for the community, we present and summarize the summation formulas for both cases. On the summation axis, we use polylogarithmic functions to express the summation, and Away from the summation axis we use Poisson summation (equivalent to the expansion of the field to cylindrical harmonics)