Short note: NA for meridional rays

Numerical aperture (NA) of a fibre measures its light-gathering ability. For meridional rays (rays that pass through the fibre axis and stay in one plane), it is the sine of the maximum acceptance angle multiplied by the index of the outside medium.

        n0       cladding n2
   \  ________________________
 th0\/  phi /\      /\
 ----*-----/--\----/--\---- axis  core n1
   ________________________

A ray entering at angle $\theta_0$ is refracted to $\theta_1$ (Snell's law $n_0\sin\theta_0 = n_1\sin\theta_1$) and meets the core–cladding boundary at $\phi = 90^\circ - \theta_1$. It is guided only if $\phi \ge \phi_c$ where $\sin\phi_c = n_2/n_1$. At the limit:

$$\begin{aligned} n_0\sin\theta_a &= n_1\cos\phi_c = n_1\sqrt{1-\frac{n_2^2}{n_1^2}} \\ \text{NA} = n_0\sin\theta_a &= \sqrt{n_1^2 - n_2^2} \approx n_1\sqrt{2\Delta} \end{aligned}$$

where $\Delta = (n_1 - n_2)/n_1$ is the relative index difference.

Key points

• NA depends only on $n_1$ and $n_2$, not on core size.
• Rays inside the cone of half-angle $\theta_a$ are guided; others leak into the cladding.
• Typical values: 0.1–0.15 (single mode), 0.2–0.3 (multimode).
• Example: $n_1 = 1.48$, $n_2 = 1.46$ gives NA $= 0.2425$ and $\theta_a = 14.03^\circ$ in air.
• Skew rays (not passing through the axis) can be accepted at slightly larger angles, so the meridional NA is the standard, conservative value.