Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-06T06:15:50.675Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  30 July 2015

Peter Buchak ,
Darren G. Crowdy ,
Yvonne M. Stokes  and
Heike Ebendorff-Heidepriem
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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 5 - 38
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boyd, K., Ebendorff-Heidepriem, H., Monro, T. M. & Munch, J. 2012 Surface tension and viscosity measurement of optical glasses using a scanning CO2 laser. Opt. Mater. Express 2 (8), 1101–1110.CrossRefGoogle Scholar
Buchak, P. & Crowdy, D. G.2014 Surface-tension-driven Stokes flow: a numerical method based on conformal geometry. J. Comput. Phys. (submitted).Google Scholar
Chakravarthy, S. S. & Chiu, W. K. S. 2009 Boundary integral method for the evolution of slender viscous fibres containing holes in the cross-section. J. Fluid. Mech. 621, 155–182.CrossRefGoogle Scholar
Chen, M. J., Stokes, Y. M., Buchak, P., Crowdy, D. G. & Ebendorff-Heidepriem, H.2015 Microstructured optical fibre drawing with active channel pressurisation. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Chen, Y. & Birks, T. A. 2013 Predicting hole sizes after fibre drawing without knowing the viscosity. Opt. Mater. Express 3 (3), 346–356.CrossRefGoogle Scholar
Crowdy, D. G. 2003a Compressible bubbles in Stokes flow. J. Fluid Mech. 476, 345–356.CrossRefGoogle Scholar
Crowdy, D. G. 2003b Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores. Eur. J. Appl. Maths 14, 421–445.CrossRefGoogle Scholar
Crowdy, D. G. 2004 An elliptical-pore model of late-stage planar viscous sintering. J. Fluid Mech. 501, 251–277.CrossRefGoogle Scholar
Crowdy, D. G. & Or, Y. 2010 Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys. Rev. E 81, 036313.CrossRefGoogle ScholarPubMed
Cummings, L. & Howell, P. D. 1999 On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity. J. Fluid Mech. 389, 361–389.CrossRefGoogle Scholar
Cummings, L., Howison, S. & King, J. R. 1999 Two-dimensional Stokes and Hele-Shaw flow with free surfaces. Eur. J. Appl. Maths 10, 635–680.CrossRefGoogle Scholar
Ebendorff-Heidepriem, H., Schuppich, J., Dowler, A., Lima-Marques, L. & Monro, T. M. 2014 3D-printed extrusion dies: a versatile approach to optical material processing. Opt. Mater. Express 4 (8), 1494–1504.CrossRefGoogle Scholar
Fitt, A. D., Furusawa, K., Monro, T. M., Please, C. P. & Richardson, D. A. 2002 The mathematical modelling of capillary drawing for holey fibre manufacture. J. Engng Maths 43, 201–227.CrossRefGoogle Scholar
Fornberg, B. 1980 A numerical method for conformal mappings. SIAM J. Sci. Stat. Comput. 1 (3), 386–400.CrossRefGoogle Scholar
Gospodinov, P. & Yarin, A. L. 1997 Draw resonance of optical microcapillaries in non-isothermal drawing. Intl J. Multiphase Flow 23, 967–976.CrossRefGoogle Scholar
Griffiths, I. M. & Howell, P. D. 2007 The surface-tension-driven evolution of a two-dimensional annular viscous tube. J. Fluid Mech. 593, 181–208.CrossRefGoogle Scholar
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181–208.CrossRefGoogle Scholar
Griffiths, I. M. & Howell, P. D. 2009 The surface-tension-driven retraction of a viscida. SIAM J. Appl. Maths 70 (5), 1453–1487.CrossRefGoogle Scholar
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity of a free surface. J. Fluid Mech. 213, 349–375.CrossRefGoogle Scholar
Issa, N. A., van Eijkelenborg, M. A., Fellew, M., Cox, F., Henry, G. & Large, M. C. J. 2004 Fabrication and study of microstructured optical fibers with elliptical holes. Opt. Lett. 29 (12), 1336–1338.CrossRefGoogle ScholarPubMed
Kostecki, R., Ebendorff-Heidepriem, H., Warren-Smith, S. & Monro, T. 2014 Predicting the drawing conditions for microstructured optical fiber fabrication. Opt. Mater. Express 4, 29–40.CrossRefGoogle Scholar
Kropf, E.2009 A Fornberg-like method for the numerical conformal mapping of bounded multiply connected domains, Master’s thesis, Wichita State University.Google Scholar
Langlois, W. 1964 Slow Viscous Flow. Macmillan.Google Scholar
Lyytikäinen, K. J.2004 Control of complex structural geometry in optical fibre drawing, PhD thesis, University of Sydney.Google Scholar
Pearson, J. R. A. & Matovich, M. A. 1969 Spinning a molten threadline; stability. Ind. Engng Chem. Fundam. 8, 605–609.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flows. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2001 Expansion of a compressible gas bubble in Stokes flow. J. Fluid Mech. 442, 171–189.CrossRefGoogle Scholar
Pozrikidis, C. 2003 Computation of the pressure inside bubbles and pores in Stokes flow. J. Fluid Mech. 474, 319–337.CrossRefGoogle Scholar
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. Appl. Maths 3, 193–207.CrossRefGoogle Scholar
Scheid, B., Quiligotti, S., Tranh, B., Gy, R. & Stone, H. A. 2010 On the (de)stabilization of draw resonance due to cooling. J. Fluid Mech. 636, 155–176.CrossRefGoogle Scholar
Stokes, Y. M., Buchak, P., Crowdy, D. G. & Ebendorff-Heidepriem, H. 2014 Drawing of hollow-core fibres: circular and non-circular tubes. J. Fluid Mech. 755, 176–203.CrossRefGoogle Scholar
Tanveer, S. & Vasconcelos, G. L. 1995 Time-evolving bubbles in two-dimensional Stokes flow. J. Fluid Mech. 301, 325–344.CrossRefGoogle Scholar
Taroni, M., Breward, C. J. W., Cummings, L. J. & Griffiths, I. M. 2013 Asymptotic solutions of glass temperature profiles during steady optical fibre drawing. J. Engng Maths 80, 1–20.CrossRefGoogle Scholar
van de Vorst, G. A. L. 1993 Integral method for a two-dimensional Stokes flow with shrinking holes applied to viscous sintering. J. Fluid Mech. 257, 667–689.CrossRefGoogle Scholar
Voyce, C. J., Fitt, A. D. & Monro, T. M. 2004 Mathematical model of the spinning of microstructured fibres. Opt. Express 12 (23), 5810–5820.CrossRefGoogle ScholarPubMed
Voyce, C. J., Fitt, A. D. & Monro, T. M. 2008 The mathematical modelling of rotating capillary tubes for holey-fibre manufacture. J. Engng Maths 60, 69–87.CrossRefGoogle Scholar
Xue, S. C., Large, M. C. J., Barton, G. W., Tanner, R. I., Poladian, L. & Lwin, R. 2005a Role of material properties and drawing conditions in the fabrication of microstructured optical fibres. J. Lightwave Technol. 24, 853–860.CrossRefGoogle Scholar
Xue, S. C., Tanner, R. I., Barton, G. W., Lwin, R., Large, M. C. J. & Poladian, L. 2005b Fabrication of microstructured optical fibres – Part I: problem formulation and numerical modeling of transient draw process. J. Lightwave Technol. 23, 2245–2254.CrossRefGoogle Scholar
Xue, S. C., Tanner, R. I., Barton, G. W., Lwin, R., Large, M. C. J. & Poladian, L. 2005c Fabrication of microstructured optical fibres – Part II: numerical modeling of steady-state draw process. J. Lightwave Technol. 23, 2255–2266.CrossRefGoogle Scholar
Yarin, A., Rusinov, V. I., Gospodinov, P. & St. Radev 1989 Quasi one-dimensional model of drawing of glass micro capillaries and approximate solutions. Theor. Appl. Mech. 20 (3), 55–62.Google Scholar
Yarin, A. L. 1995 Surface-tension-driven flows at low Reynolds number arising in optoelectronic technology. J. Fluid Mech. 286, 173–200.CrossRefGoogle Scholar
Yarin, A. L., Gospodinov, P. & Roussinov, V. I. 1994 Stability loss and sensitivity in hollow fiber drawing. Phys. Fluids 6, 1454–1463.CrossRefGoogle Scholar