Regularization for the inversion of Fibre Bragg Grating spectra

TL;DR

This paper investigates the nonlinear inverse problem of Fibre Bragg Grating spectra inversion, highlighting challenges like non-uniqueness and regularization issues, and proposes initial steps toward improved information extraction from single measurements.

Contribution

It introduces the core nonlinear inverse problem for Fibre Bragg Gratings and explores regularization approaches, addressing challenges like non-uniqueness and parameter selection.

Findings

Identifies non-uniqueness in the inverse problem

Highlights lack of parameter selection rules

Proposes initial regularization strategies

Abstract

Fibre Bragg Gratings have become widespread measurement devices in engineering and other fields of application. In all but a few cases, the relation between cause and effect is simplified to a proportional model. However, at its mathematical core lies a nonlinear inverse problem which appears not to have received much attention in the literature. In this paper, we present this core problem to the mathematical community and provide a first report on opportunities and limitations of a regularization approach. In particular, we show that difficulties arise from non-uniqueness and the absence of established parameter selection rules for nonlinear inverse problems with multiple regularization parameters. Nevertheless, the paper takes a first step toward extracting more information from a single FBG measurement.