Localized Function Method Based on the Galerkin Method Applying a Set of Sine Functions to Model Photonic Crystal Fibres

TL;DR

This paper introduces a localized function method based on the Galerkin approach using sine functions to model modes in photonic crystal fibers, reducing computational complexity and avoiding refractive index expansion inaccuracies.

Contribution

The paper presents a novel localized function method that simplifies modeling of photonic crystal fibers by reducing integrals and eliminating the need for refractive index expansion.

Findings

Method reduces the number of integrals needed for symmetrical hole shapes.

Analytical solutions are provided for circular hole geometries.

The approach avoids inaccuracies associated with refractive index expansion.

Abstract

A development and an application of the localized function method based on the Galerkin method applying a set of Sine functions to approximate the unknown mode fields of the localized modes propagating along the photonic crystal fibres is proposed. A way for considerably reducing the number of integrals in the case of symmetrical holes shapes with respect to the axes of coordinate systems located at the centers of the holes, circular, elliptical etc., is also presented. The method does not require an expansion of the refractive index and thus inaccuracies of the expansion can be avoided. In the case of a circular form of the holes all integrals are solved analytically.