An optical fibre core has a wavelength-dependent refractive index given by the Cauchy…

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

Computational Physics, Ch Where a function vanishes, and what lies between the points

Problem

An optical fibre core has a wavelength-dependent refractive index given by the Cauchy formula n(λ)=1.4580+0.00354λ2n(\lambda) = 1.4580 + \dfrac{0.00354}{\lambda^2}, with λ\lambda in micrometres (μm\mu\mathrm{m}). It is surrounded by cladding of fixed index nclad=1.4500n_{\rm clad} = 1.4500. Total internal reflection at the core-cladding boundary occurs above the critical angle θc\theta_c, where sin⁡θc=nclad/n(λ)\sin\theta_c = n_{\rm clad}/n(\lambda). (a) Find the wavelength λ\lambda at which the critical angle is exactly θc=79.0∘\theta_c = 79.0^\circ, by writing the condition as f(λ)=0f(\lambda) = 0 with f(λ)=nclad−sin⁡θc n(λ)f(\lambda) = n_{\rm clad} - \sin\theta_c\, n(\lambda). (b) Visible light runs from λ=0.40 μm\lambda = 0.40\,\mu\mathrm{m} to λ=0.70 μm\lambda = 0.70\,\mu\mathrm{m}. Evaluate ff at both ends of this range and state whether it brackets the root.

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