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

  • Peter Buchak (a1), Darren G. Crowdy (a1), Yvonne M. Stokes (a2) and Heike Ebendorff-Heidepriem (a3)

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.

Copyright

Corresponding author

Email address for correspondence: p.buchak@imperial.ac.uk

References

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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), 11011110.
Buchak, P. & Crowdy, D. G.2014 Surface-tension-driven Stokes flow: a numerical method based on conformal geometry. J. Comput. Phys. (submitted).
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, 155182.
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).
Chen, Y. & Birks, T. A. 2013 Predicting hole sizes after fibre drawing without knowing the viscosity. Opt. Mater. Express 3 (3), 346356.
Crowdy, D. G. 2003a Compressible bubbles in Stokes flow. J. Fluid Mech. 476, 345356.
Crowdy, D. G. 2003b Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores. Eur. J. Appl. Maths 14, 421445.
Crowdy, D. G. 2004 An elliptical-pore model of late-stage planar viscous sintering. J. Fluid Mech. 501, 251277.
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.
Cummings, L. & Howell, P. D. 1999 On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity. J. Fluid Mech. 389, 361389.
Cummings, L., Howison, S. & King, J. R. 1999 Two-dimensional Stokes and Hele-Shaw flow with free surfaces. Eur. J. Appl. Maths 10, 635680.
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), 14941504.
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, 201227.
Fornberg, B. 1980 A numerical method for conformal mappings. SIAM J. Sci. Stat. Comput. 1 (3), 386400.
Gospodinov, P. & Yarin, A. L. 1997 Draw resonance of optical microcapillaries in non-isothermal drawing. Intl J. Multiphase Flow 23, 967976.
Griffiths, I. M. & Howell, P. D. 2007 The surface-tension-driven evolution of a two-dimensional annular viscous tube. J. Fluid Mech. 593, 181208.
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181208.
Griffiths, I. M. & Howell, P. D. 2009 The surface-tension-driven retraction of a viscida. SIAM J. Appl. Maths 70 (5), 14531487.
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity of a free surface. J. Fluid Mech. 213, 349375.
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), 13361338.
Kostecki, R., Ebendorff-Heidepriem, H., Warren-Smith, S. & Monro, T. 2014 Predicting the drawing conditions for microstructured optical fiber fabrication. Opt. Mater. Express 4, 2940.
Kropf, E.2009 A Fornberg-like method for the numerical conformal mapping of bounded multiply connected domains, Master’s thesis, Wichita State University.
Langlois, W. 1964 Slow Viscous Flow. Macmillan.
Lyytikäinen, K. J.2004 Control of complex structural geometry in optical fibre drawing, PhD thesis, University of Sydney.
Pearson, J. R. A. & Matovich, M. A. 1969 Spinning a molten threadline; stability. Ind. Engng Chem. Fundam. 8, 605609.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flows. Cambridge University Press.
Pozrikidis, C. 2001 Expansion of a compressible gas bubble in Stokes flow. J. Fluid Mech. 442, 171189.
Pozrikidis, C. 2003 Computation of the pressure inside bubbles and pores in Stokes flow. J. Fluid Mech. 474, 319337.
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. Appl. Maths 3, 193207.
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, 155176.
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, 176203.
Tanveer, S. & Vasconcelos, G. L. 1995 Time-evolving bubbles in two-dimensional Stokes flow. J. Fluid Mech. 301, 325344.
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, 120.
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, 667689.
Voyce, C. J., Fitt, A. D. & Monro, T. M. 2004 Mathematical model of the spinning of microstructured fibres. Opt. Express 12 (23), 58105820.
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, 6987.
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, 853860.
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, 22452254.
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, 22552266.
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), 5562.
Yarin, A. L. 1995 Surface-tension-driven flows at low Reynolds number arising in optoelectronic technology. J. Fluid Mech. 286, 173200.
Yarin, A. L., Gospodinov, P. & Roussinov, V. I. 1994 Stability loss and sensitivity in hollow fiber drawing. Phys. Fluids 6, 14541463.
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