---
title: "Short note: NA for meridional rays"
subject: "Propagation and Antenna"
chapter: 6
key: "semester_6/propagation_and_antenna:na-meridional-rays-short-note"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/na-meridional-rays-short-note
site: BCT Engineering
---

# Short note: NA for meridional rays

Propagation and Antenna · Chapter 6: Optical Fibres

- Asked in: 2072 Asoj, 3 marks
- Video: [Meridional Ray & Skew Ray I L 8 I Optical Communication](https://www.youtube.com/watch?v=P-33batWSb0) — Dopamine
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/na-meridional-rays-short-note
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-6

## Question

Write a short note on NA for meridional rays in optical fiber.

## Short answer

For meridional rays, which pass through the fibre axis, NA = n0 sin θa = sqrt(n1² − n2²) ≈ n1√(2Δ). Only rays inside the acceptance cone of half-angle θa are guided; NA depends only on core and cladding indices. Example: n1=1.48, n2=1.46 gives NA=0.2425, θa=14.03°.

## Answer

**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.

```text
        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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are meridional rays?
   A. Rays avoiding the axis
   B. Rays in the cladding
   C. Rays at the critical angle only
   D. Rays passing through the fibre axis, staying in one plane
   Answer: D. Skew rays do not cross the axis. (Key point: Meridional rays)

2. How is NA derived for meridional rays?
   A. Snell's law at the input face with the critical angle condition: NA = n1 cos(phi_c)
   B. From V-number
   C. From core diameter
   D. From attenuation
   Answer: A. NA = sqrt(n1^2 - n2^2). (Key point: NA derivation)

3. How do skew rays affect the NA?
   A. They reduce NA
   B. They can be accepted at slightly larger angles, so the meridional NA is conservative
   C. They have no acceptance
   D. They double NA
   Answer: B. Meridional NA is the standard value. (Key point: Skew rays)

## Related in Propagation and Antenna

- [Short note: Optical fiber loss](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aoptical-fiber-loss-short-note) — Propagation and Antenna, chapter 6
- [Acceptance angle and NA of step-index fibre](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aacceptance-angle-na-step-index) — Propagation and Antenna, chapter 6, asked 2×
- [OFC vs cable and radio; NA](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aofc-vs-cable-radio-na) — Propagation and Antenna, chapter 6
- [Light propagation; define NA and acceptance angle](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Alight-propagation-define-na-acceptance) — Propagation and Antenna, chapter 6
- [Multimode vs single mode; NA derivation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Amultimode-vs-single-na-derivation) — Propagation and Antenna, chapter 6
- [Types of optical fibre: construction and propagation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aoptical-fiber-types-construction) — Propagation and Antenna, chapter 6, asked 2×

## Related in other subjects

- [Optical fiber, NA, single vs multimode](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aoptical-fiber-na-acceptance-mode-comparison) — Engineering Physics, chapter 6
- [Derive numerical aperture of optical fiber](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ana-derivation-optical-fiber) — Engineering Physics, chapter 6, asked 4×
- [Optical fiber and its advantages](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aoptical-fiber-advantages) — Basic Electronics Engineering, chapter 5, asked 5×
- [Advantages of optical fiber over copper](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Aoptical-fiber-advantages) — Instrumentation II, chapter 5, asked 3×