Abstract
AbstractA general mathematical framework is presented for modelling the pulling of optical glass fibres in a draw tower. The only modelling assumption is that the fibres are slender; cross-sections along the fibre can have general shape, including the possibility of multiple holes or channels. A key result is to demonstrate how a so-called reduced time variable $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\tau $ serves as a natural parameter in describing how an axial-stretching problem interacts with the evolution of a general surface-tension-driven transverse flow via a single important function of $\tau $, herein denoted by $H(\tau )$, derived from the total rescaled cross-plane perimeter. For any given preform geometry, this function $H(\tau )$ may be used to calculate the tension required to produce a given fibre geometry, assuming only that the surface tension is known. Of principal practical interest in applications is the ‘inverse problem’ of determining the initial cross-sectional geometry, and experimental draw parameters, necessary to draw a desired final cross-section. Two case studies involving annular tubes are presented in detail: one involves a cross-section comprising an annular concatenation of sintering near-circular discs, the cross-section of the other is a concentric annulus. These two examples allow us to exemplify and explore two features of the general inverse problem. One is the question of the uniqueness of solutions for a given set of experimental parameters, the other concerns the inherent ill-posedness of the inverse problem. Based on these examples we also give an experimental validation of the general model and discuss some experimental matters, such as buckling and stability. The ramifications for modelling the drawing of fibres with more complicated geometries, and multiple channels, are discussed.
Highlights
IntroductionMicrostructured optical fibres (MOFs) have revolutionised optical fibre technology, promising a virtually limitless range of fibre designs for a wide range of applications
Microstructured optical fibres (MOFs) have revolutionised optical fibre technology, promising a virtually limitless range of fibre designs for a wide range of applicationsDrawing of tubular optical fibres (Knight 2003; Monro & Ebendorff-Heidepriem 2006)
In this paper we describe a mathematical model for the drawing of MOFs, assuming that inertial effects are negligible and that there is no pressurisation of internal channels
Summary
Microstructured optical fibres (MOFs) have revolutionised optical fibre technology, promising a virtually limitless range of fibre designs for a wide range of applications. The model can be used to determine a preform geometry and draw parameters that will yield a desired fibre design: the ‘inverse problem’ Solution of both forward and inverse problems will be demonstrated for tubular fibres that may not be axisymmetric, and extension to more general fibre geometries will be discussed. The Lagrangian model, for both axial-stretching and transverse-flow problems, is first described in § 2 and the balance between fibre tension and surface tension needed to draw a fibre is discussed. This second case study is presented in some detail since it affords us the opportunity to discuss ideas about the so-called ‘forward’ and ‘inverse’ problems of fibre drawing Because it is a relatively simple solution which can be written down explicitly, the concentric annulus yields information that is difficult to obtain for fibres of other geometries. We conclude the paper with a more general discussion of how our results contribute to the challenging problem of solving the inverse problem for fibres having cross-sectional domains of higher connectivity
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