Answer: \(NA = \sin\theta_a = \sqrt{n_1^{2} - n_2^{2}}\) (for a fibre in air), where \(n_1\) and \(n_2\) are the core and cladding refractive indices.
Consider a step-index fibre with core index \(n_1\), cladding index \(n_2\) (\(n_1 > n_2\)), placed in a medium of index \(n_0\) (air, \(n_0 = 1\)).
A ray enters the end face at angle \(\theta_i\) to the fibre axis and is refracted into the core at angle \(\theta_r\). It meets the core–cladding boundary at angle \(\phi = 90^\circ - \theta_r\) to the normal.
Snell's law at the entrance face:
\[n_0\sin\theta_i = n_1\sin\theta_r = n_1\sin(90^\circ - \phi) = n_1\cos\phi\]
For total internal reflection at the core–cladding boundary, \(\phi \ge \theta_c\), where
\[\sin\theta_c = \frac{n_2}{n_1}\]
The maximum entry angle, the acceptance angle \(\theta_a\), corresponds to \(\phi = \theta_c\):
\[n_0\sin\theta_a = n_1\cos\theta_c = n_1\sqrt{1 - \sin^{2}\theta_c}\]
\[= n_1\sqrt{1 - \frac{n_2^{2}}{n_1^{2}}} = \sqrt{n_1^{2} - n_2^{2}}\]
The numerical aperture is defined as \(NA = n_0\sin\theta_a\). For air (\(n_0 = 1\)):
\[NA = \sin\theta_a = \sqrt{n_1^{2} - n_2^{2}}\]
Writing \(n_1^{2} - n_2^{2} = (n_1+n_2)(n_1-n_2)\) and using the fractional index change \(\Delta = \dfrac{n_1 - n_2}{n_1}\) with \(n_1 + n_2 \approx 2n_1\):
\[NA \approx n_1\sqrt{2\Delta}\]
NA measures the light-gathering ability of the fibre: rays entering within the cone of half-angle \(\theta_a\) are guided; rays outside it leak into the cladding.