What is the numerical aperture of an optical fibre and how is its formula derived?

In one line

Numerical aperture (NA) is the sine of the acceptance angle (times the index of the outside medium) and measures how much light a fibre can collect; \(NA = \sqrt{n_1^2 - n_2^2}\).

Explained simply

NA is the fibre's "light-gathering power". A wide funnel catches more rain than a narrow one; a fibre with a large NA accepts light from a wider range of angles and so collects more light from a source like an LED.

How it works

1. Snell's law at the entry face: \(n_0\sin\theta_a = n_1\sin\theta_r\).
2. At the core-cladding boundary, the limiting ray meets it at \(\theta_c\), with \(\theta_r = 90° - \theta_c\), so \(\sin\theta_r = \cos\theta_c\).
3. \(\sin\theta_c = n_2/n_1\), so \(\cos\theta_c = \sqrt{1 - n_2^2/n_1^2}\).
4. Therefore \(n_0\sin\theta_a = n_1\sqrt{1 - n_2^2/n_1^2} = \sqrt{n_1^2 - n_2^2}\).
5. By definition \(NA = n_0\sin\theta_a\), so \(NA = \sqrt{n_1^2 - n_2^2}\).
6. Using fractional index difference \(\Delta = (n_1 - n_2)/n_1\) and \(n_1 + n_2 \approx 2n_1\): \(NA \approx n_1\sqrt{2\Delta}\).

Key formula

\(NA = n_0\sin\theta_a = \sqrt{n_1^2 - n_2^2} \approx n_1\sqrt{2\Delta}\), \(\Delta = \dfrac{n_1 - n_2}{n_1}\). All quantities are dimensionless. \(NA\) is always less than 1 for a fibre in air. For step-index fibres, the V-number \(V = \dfrac{2\pi a}{\lambda}NA\) (\(a\) = core radius) decides how many modes travel.

Quick example

A fibre has \(n_1 = 1.48\) and \(\Delta = 0.01\). Then \(NA = 1.48\sqrt{0.02} = 1.48\times0.1414 \approx 0.209\), and \(\theta_a = \sin^{-1}(0.209) \approx 12.1°\). Then \(n_2 = n_1(1-\Delta) = 1.4652\).

Common mistakes

• Calling NA an angle; it is a number (sine of an angle).
• Writing \(NA = n_1 - n_2\) or \(\sqrt{n_1 - n_2}\); the squares are essential.
• Using \(\Delta = (n_1-n_2)/n_2\); the denominator is \(n_1\).

How MU asks it

One of the most frequently asked items: "Define NA and derive its expression" (5 marks), plus numericals asking for NA, acceptance angle, cone angle or \(n_2\) (3 to 5 marks). For full marks, draw the diagram, show every step of the derivation, and in numericals give NA, \(\theta_a\) and \(2\theta_a\) with units of degrees for angles.