Minimising modal dispersion with numerical aperture

Numerical aperture - Wikipedia

This note is about why, to minimise modal dispersion in fibre optics, it’s best to have a small difference between the refractive indices of the core and the cladding.

It can be derived from the geometry of optic fibres and Snell’s law that the maximum angle of incident light that the fibre will accept, $\theta$, is given by the following equation:

$$n\sin ({\theta }_{\text{max}})=\sqrt{n_{\text{core}}^2-n_{\text{clad}}^2}$$

This implies that, in order to minimise modal dispersion, it is better to have a smaller difference in the refractive indices of the core and the cladding, as this reduces the range of angles that will experience total internal reflection in the fibre, and thus reduces the range of paths that the light can take.