A straight step-index fibre has uniform core index and cladding index

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 straight step-index fibre has uniform core index n1=1.500n_1 = 1.500 and cladding index n2=1.465n_2 = 1.465. 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. A launching lens is arranged so that no ray inside the core makes an angle greater than θmax⁡=4.0∘\theta_{\max} = 4.0^\circ with the fibre axis (this is smaller than the cut-off angle, so all such rays are guided). (a) Show that with the ray directions inside the core restricted to 0≤θ≤θmax⁡0 \le \theta \le \theta_{\max} the spread in arrival time over an axial length LL is Δt/L=(n1/c)(1/cos⁡θmax⁡−1)\Delta t/L = (n_1/c)(1/\cos\theta_{\max} - 1), and that for small θmax⁡\theta_{\max} (in radians) this is n1θmax⁡2/(2c)n_1\theta_{\max}^2/(2c). (b) Evaluate the restricted spread in ns km−1^{-1}, exactly and by the small-angle form. (c) Evaluate the spread when the full guided cone is filled, and give the factor by which the restricted launch reduces it. Confirm your full-cone value against the first-order-in-index-step estimate (n1−n2)/c(n_1 - n_2)/c.

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