Guided modes of helical waveguides

TL;DR

This paper develops a mathematical and numerical framework to analyze guided modes in helical waveguides, reducing a complex 3D problem to a more manageable 2D eigenproblem, and demonstrates how coiling affects mode localization.

Contribution

It introduces a novel reduction of the 3D mode-finding problem in helical waveguides to a 2D eigenproblem, enabling efficient computation and analysis of guided modes.

Findings

Coiling shifts mode localization away from the center.

Reduction to 2D eigenproblem significantly saves computational resources.

Mode variations with coiling pitch are characterized for optical fibers.

Abstract

This paper studies guided transverse scalar modes propagating through helically coiled waveguides. Modeling the modes as solutions of the Helmholtz equation within the three-dimensional (3D) waveguide geometry, a propagation ansatz transforms the mode-finding problem into a 3D quadratic eigenproblem. Through an untwisting map, the problem is shown to be equivalent to a 3D quadratic eigenproblem on a straightened configuration. Next, exploiting the constant torsion and curvature of the Frenet frame of a circular helix, the 3D eigenproblem is further reduced to a two-dimensional (2D) eigenproblem on the waveguide cross section. All three eigenproblems are numerically treated. As expected, significant computational savings are realized in the 2D model. A few nontrivial numerical techniques are needed to make the computation of modes within the 3D geometry feasible. They are presented along with a procedure to effectively filter out unwanted non-propagating eigenfunctions. Computational results show that the geometric effect of coiling is to shift the localization of guided modes away from the coiling center. The variations in modes as coiling pitch is changed are reported considering the example of a coiled optical fiber.