A short, straight, large-core fibre used to pipe light into a machine tool has core…

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Beams, resonators and guided-wave photonics · Beams, resonators and guided-wave photonics

Optics and Photonics, Ch Optical fibres in communication and endoscopy

Problem

A short, straight, large-core fibre used to pipe light into a machine tool has core index n1=1.62n_1 = 1.62 and cladding index n2=1.58n_2 = 1.58, both uniform. Light in the core travels at c/n1c/n_1 with c=3.00×108c = 3.00\times10^{8} m s−1^{-1}. Consider meridional rays only, and measure the ray direction inside the core by the angle θ\theta it makes with the fibre axis. (a) Show that a ray at angle θ\theta to the axis needs a geometrical path L/cos⁡θL/\cos\theta to advance an axial distance LL, that the steepest guided ray has cos⁡θmax⁡=n2/n1\cos\theta_{\max} = n_2/n_1, and hence that Δt/L=(n1/c)(1/cos⁡θmax⁡−1)\Delta t/L = (n_1/c)(1/\cos\theta_{\max} - 1). (b) Evaluate θmax⁡\theta_{\max} in degrees and the exact modal spread in ns km−1^{-1}. (c) Expand for small θmax⁡\theta_{\max} to show Δt/L≈n1θmax⁡2/(2c)\Delta t/L \approx n_1\theta_{\max}^2/(2c) with θmax⁡\theta_{\max} in radians, and, using θmax⁡2≈2(n1−n2)/n1\theta_{\max}^2 \approx 2(n_1-n_2)/n_1, that this is (n1−n2)/c(n_1-n_2)/c to first order in the index step. Evaluate both approximate values in ns km−1^{-1}.

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