Correct ray-tracing analysis for interference microscopy of fibres

Abstract

A ray-tracing analysis for calculating, by means of interference microscope data, the radial distribution of refractive index, n ( r ), for fibres of round cross section has been formulated, solved, tested and used to determine n ( r ) for high-speed direct-spun polyethylene terephthalate (PET) fibres. The formulation was based on work by Kahl & Mylin (1965) originally performed to explore the radial density profiles in cylindrically symmetrical explosions. The equations were formulated with proper boundary conditions at the fibre surface, correcting a fundamental error that has caused problems for 20 years. This correct boundary condition unfortunately made it impossible to invert, as had been done in all previous work, the resulting integral equations, which consequently were solved by a nonlinear least-squares approximation. The analysis corrected a serious problem noted in the literature: namely, that the refractive index profile obtained for a particular fibre depended on the refractive index of the immersion liquid used to make the measurement. Refractive index profiles observed for PET fibres produced by high-speed direct spinning had concave-down curvature, the opposite of expectations based on previous work. This initial result suggests that when direct-spun, under crystallizing conditions, PET fibres develop a mainly radial density or quench profile, whereas fibres spun under conditions that give little crystallinity have a mainly radial orientation gradient. Experiments to test this suggestion further have not been done yet.