Fast numerical estimations of axisymmetric multimode fibers modes

Estimating the propagation constants and the transverse mode profiles of multimode fibers is not as easy as it sounds. In our recent work we highlighted here, we needed to estimate the mode profiles for a standard graded-index fiber. It turned out that many standard approximations done in the literature to estimate the propagation constants do not give results accurate enough for the mode profile. The general approach we introduced in a previous tutorial to numerically find the fiber modes for any index profile using a 2D scalar finite differences approach is still valid. However, to provide accurate results, it needs a fine discretization of the space that leads to important memory and computational time requirements when the fiber core increases. If we consider an axisymmetric fiber, we can obtain a 1D formulation of the problem, that is unfortunately unstable under naive finite differences approaches. We detail here a stable formulation that leads to accurate and fast estimations of the mode profiles.

Compare Different Methods of Modes Estimation of Bent Multimode Fibers with pyMMF

In a previous tutorial, I explained how to calculate the modes of a bent multimode fibers. I introduced two methods, following the approach published in [M. Plöschner, T. Tyc, and T. Čižmár, Nat. Photon. (2015)]. In this short tutorial, I show how to use pyMMF to simulate bent fibers and compare the two different methods. A Jupyter notebook can be found on my Github account: compare_bending_methods.ipynb.

pyMMF:  Simulating Multimode Fibers in Python

Part 1: Step Index Benchmark

I recently published a two-part tutorial on how to find the modes of an arbitrary multimode fiber without or with bending. Based on this tutorial, I published a (still experimental) version of a Python module to find the modes of multimode fibers and calculate their transmission matrix: pyMMF. The goal of this module is not to compete with commercial solutions in terms of precision but to provide a way to easily simulate realistic fiber systems. To validate the approach, I use step-index multimode fibers as a benchmark test as the dispersion relation is analytically known (see my tutorial here) and for which the Linearly Polarized (LP) mode approximation yields good results. I focus my attention here on the precision of the numerically found propagation constants.

Numerical Estimation of Multimode Fiber Modes and Propagation Constants: 

Part 2: Bent Fibers 

We saw in the first part of the tutorial that the profiles and the propagation constants of the propagation modes of a straight multimode fiber can easily be avulated for an arbitrary index profile by inverting a large but sparse matrix. Under some approximations [1], a portion of fiber with a fixed radius of curvature satisfies a similar problem that can be solved with the same numerical tools, as we illustrate with the PyMMF Python module [2]. Moreover, when the modes are known for the straight fiber, the modes for a fixed radius can be approximate by inverting a square matrix of size the number of propagating modes [1]. It allows fast computation of the modes for different radii of curvature.