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Elliptical pore regularisation of the inverse problem for microstructured optical fibre fabrication

Published online by Cambridge University Press:  30 July 2015

Peter Buchak*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
Darren G. Crowdy
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
Yvonne M. Stokes
Affiliation:
School of Mathematical Sciences, The University of Adelaide, North Terrace, Adelaide, SA 5005, Australia
Heike Ebendorff-Heidepriem
Affiliation:
ARC Centre of Excellence for Nanoscale BioPhotonics, Institute for Photonics and Advanced Sensing, School of Chemistry and Physics, The University of Adelaide, North Terrace, Adelaide, SA 5005, Australia
*
Email address for correspondence: p.buchak@imperial.ac.uk

Abstract

A mathematical model is presented describing the deformation, under the combined effects of surface tension and draw tension, of an array of channels in the drawing of a broad class of slender viscous fibres. The process is relevant to the fabrication of microstructured optical fibres, also known as MOFs or holey fibres, where the pattern of channels in the fibre plays a crucial role in guiding light along it. Our model makes use of two asymptotic approximations, that the fibre is slender and that the cross-section of the fibre is a circular disc with well-separated elliptical channels that are not too close to the outer boundary. The latter assumption allows us to make use of a suitably generalised ‘elliptical pore model (EPM)’ introduced previously by one of the authors (Crowdy, J. Fluid Mech., vol. 501, 2004, pp. 251–277) to quantify the axial variation of the geometry during a steady-state draw. The accuracy of the elliptical pore model as an approximation is tested by comparison with full numerical simulations. Our model provides a fast and accurate reduction of the full free-boundary problem to a coupled system of nonlinear ordinary differential equations. More significantly, it also allows a regularisation of an important ill-posed inverse problem in MOF fabrication: how to find the initial preform geometry and the experimental parameters required to draw MOFs with desired cross-plane geometries.

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Papers
Copyright
© 2015 Cambridge University Press 

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