Created by sebastien.popoff on 25/02/2019

## 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.

Created by sebastien.popoff on 29/12/2018

## 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.

Created by sebastien.popoff on 13/12/2018

## Part 1: Straight Fibers

Under the weakly guided approximation, analytical solutions for the mode profiles of step-index (SI) and graded-index (GRIN) multimode fibers (MMF) can be found [1]. It also gives a semi-analytical solution for the dispersion relation in SI MMFs, and, by adding stronger approximations, an analytical solution for the parabolic profile GRIN MMFs [2] (note that those approximations do fail for lower order modes). An arbitrary index profile requires numerical simulations to estimate the mode profiles and the corresponding propagation constants of the modes. I present in this tutorial how to numerically estimate the scalar solution for the profiles and propagation constants of guides modes in multimode circular waveguide with arbitrary index profile and in the presence of bending. I released a beta version of the Python module pyMMF based on such an approach [3]. It relies on expressing the transverse Helmholtz equation as an eigenvalue problem. Solutions are found by finding the eigenvectors of a large but sparse matrix representing the equation on the discretized space.

Created by sebastien.popoff on 21/09/2018

## How to calibrate linearly aligned nematic liquid crystal based SLMs

I previously posted a tutorial presenting a technique to calibrate spatial light modulators (SLMs). The approach was based on measuring the interference between two paths that have been reflected off two different regions of the SLM. This technique is always valid but requires aligning a mask, using a lens, and capturing and processing images of interference patterns. Nowadays, most phase-only SLMs based on liquid crystals use linearly aligned nematic crystals. Unlike twisted nematic liquid crystals, they allow phase-only modulation on one polarization while not affecting the orthogonal polarization. This feature can be used to simplify the calibration setup to characterize the SLM with a common path interferometer, not requiring a precise alignment [1]. Furthermore, it only needs a photo-detector, compared to a digital camera in the previously presented approach. This is convenient to measure the inevitable phase fluctuations of an SLM, usually around a 100 to 400 Hz frequency. In this document, we briefly describe the principle of the characterization scheme as presented in [2] and show typical results of the calibration and phase fluctuations.

Written by Paul Balondrade and Sébastien M. Popoff