Answer: \(NA=\sin\theta_a=\sqrt{\mu_1^2-\mu_2^2}\approx\mu_1\sqrt{2\Delta}\) (fibre in air). It measures the light-gathering ability of the fibre.
Figure (described): a step index fibre lies along the horizontal axis with core of index \(\mu_1\) and cladding of index \(\mu_2\) (\(\mu_1>\mu_2\)); outside is a medium of index \(\mu_0\) (air). A ray enters the flat end face at angle \(\theta_0\) to the axis, refracts into the core at angle \(\theta_r\), and meets the core–cladding interface at angle \(\phi=90^\circ-\theta_r\) to the normal.
At the end face (Snell's law): \(\mu_0\sin\theta_0=\mu_1\sin\theta_r\) ...(1)
At the core–cladding interface, total internal reflection needs \(\phi\ge\theta_c\), where \(\sin\theta_c=\mu_2/\mu_1\). The limiting ray (\(\theta_0=\theta_a\), the acceptance angle) has \(\phi=\theta_c\), so
\[\sin(90^\circ-\theta_r)=\cos\theta_r=\frac{\mu_2}{\mu_1}\ \Rightarrow\ \sin\theta_r=\sqrt{1-\frac{\mu_2^2}{\mu_1^2}}=\frac{\sqrt{\mu_1^2-\mu_2^2}}{\mu_1}\quad\ldots(2)\]
Put (2) in (1) with \(\theta_0=\theta_a\):
\[\mu_0\sin\theta_a=\sqrt{\mu_1^2-\mu_2^2}\]
The numerical aperture is defined as \(NA=\mu_0\sin\theta_a\). For a fibre in air (\(\mu_0=1\)):
\[\boxed{NA=\sin\theta_a=\sqrt{\mu_1^2-\mu_2^2}}\]
In terms of \(\Delta\): with \(\Delta=\dfrac{\mu_1-\mu_2}{\mu_1}\) and \(\mu_1+\mu_2\approx2\mu_1\), \(\mu_1^2-\mu_2^2=(\mu_1+\mu_2)(\mu_1-\mu_2)\approx2\mu_1^2\Delta\), so \(NA\approx\mu_1\sqrt{2\Delta}\).
Significance:
- NA measures the light-gathering (light-collecting) ability of the fibre: all rays within the cone of half-angle \(\theta_a\) (the acceptance cone) are guided.
- Larger NA means more light is coupled in, but also more modes and more intermodal dispersion; long-distance fibres use small NA.
- NA depends only on the refractive indices, not on the fibre dimensions.
