Numerical Aperture of a Step-Index Fibre

A step-index fibre is a cylinder of glass with a slightly denser core running down the middle, and it guides light for exactly the reason the critical-angle page next door describes: a ray inside the core that strikes the boundary at a shallow enough angle is totally internally reflected and cannot escape. Work that condition back out through the flat end face with Snell's law and it becomes a cone of acceptance in the outside world, whose half-angle satisfies $n_0\sin\theta_a = \sqrt{n_1^2 - n_2^2}$. In air $n_0 = 1$, so the sine of the acceptance half-angle IS the numerical aperture.

Light arriving inside that cone is guided. Light arriving outside it enters the core, meets the boundary past the critical angle, and is gone into the cladding within a few centimetres. There is no partial credit, and this is why launching into fibre is a matter of matching cones rather than of aiming carefully.

Look at how small the index difference is. A typical silica single-mode fibre has $n_1 = 1.4682$ and $n_2 = 1.4629$ — a relative difference $\Delta$ of about 0.36%, and an NA of 0.125. That tininess is not incidental; it is the weakly guiding condition, and every simple fibre formula on this site, including the mode count, assumes it. The exact vector treatment is needed when it fails, as it does in plastic fibre or an air-clad design.

A large NA gathers more light, tolerates worse alignment, and is easier to couple an LED or a diode into. It pays for that in modal dispersion: the steep rays bounce along a measurably longer path than the axial ones, so a pulse launched sharp arrives spread out. That single trade explains most of the fibre catalogue — 0.12 for telecom single-mode, 0.20 to 0.29 for silica multimode, up to 0.5 for plastic fibre used over a few metres where nobody cares about bandwidth.

Two things that quietly break the acceptance angle in practice. The cone shrinks if the launch medium is not air: index-matching gel, water or cured epoxy all have $n_0 > 1$, so a pigtail potted into adhesive does not accept the cone the datasheet quotes. And the cladding is an OPTICAL part of the waveguide, not a coating — the guided field extends into it as an evanescent tail, a real fraction of the power travels there, and that is why the cladding glass must be as pure as the core, why a bend that strips the tail costs loss, and why touching bare stripped fibre with an index-matched fluid can attenuate a signal the same fibre carried perfectly in air.