Created by sebastien.popoff on 19/05/2019
## Complex-Valued Neural Networks for Physics Applications## An implementation in PyTorch
Artificial neural networks are mainly used for treating data encoded in real values, such as digitized images or sounds. In such systems, using complex-valued tensors would be quite useless. This is however different for physic related topics. When dealing with wave propagation in particular, using complex values is interesting since the physics typically has linear, hence more simple, behavior when considering complex fields. This is sometimes true even when the inputs and the outputs of the system are real values. For instance, consider a complex media that you excite using an amplitude modulator, such as a DMD (Digital Micromirror Device) and you measure the output intensity. You manipulate only real values, but if you want to characterize the system, you have to keep in mind that the phase is a hidden variable as the effect of propagation is represented by the multiplication by a complex matrix on the optical field. |

Created by sebastien.popoff on 25/02/2019
## pyMMF: Simulating Multimode Fibers in Python## Part 1: Step Index BenchmarkI 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
## Numerical Estimation of Multimode Fiber Modes and Propagation Constants:## Part 2: Bent FibersWe 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
## Numerical Estimation of Multimode Fiber Modes and Propagation Constants:## Part 1: Straight FibersUnder 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. |