---
title: "Short note: Numerical aperture"
subject: "Propagation and Antenna"
chapter: 6
key: "semester_6/propagation_and_antenna:numerical-aperture-short-note"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/numerical-aperture-short-note
site: BCT Engineering
---

# Short note: Numerical aperture

Propagation and Antenna · Chapter 6: Optical Fibres

- Asked in: 2074 Bhadra, 4 marks
- Video: [Acceptance Angle & Numerical Aperture NA (Basics, Definition, Derivation & Formula) Explained](https://www.youtube.com/watch?v=q37CpocWZyw) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/numerical-aperture-short-note
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-6

## Question

Write a short note on numerical aperture (NA).

## Short answer

Numerical aperture is the light-gathering ability of a fibre, defined as NA = n0 sin θa = sqrt(n1² − n2²) ≈ n1√(2Δ). It depends only on core and cladding refractive indices, not fibre size; larger NA eases coupling but increases modal dispersion. For n1=1.48, n2=1.46, NA=0.2425 and acceptance angle θa=14.0°.

## Answer

**Numerical aperture (NA)** is a measure of the light-gathering (acceptance) ability of an optical fibre. It is defined as the sine of the acceptance angle multiplied by the refractive index of the medium from which light enters.

```text
NA = n0 sin θa
```

#### Expression

From Snell's law at the input face, n0 sin θa = n1 cos φc, and from the TIR condition sin φc = n2/n1:

```text
NA = √(n1² − n2²) ≈ n1 √(2Δ),   Δ = (n1 − n2)/n1
θa = sin⁻¹(NA)        (launch from air, n0 = 1)
```

```text
          \  θa
   air     \       core n1, cladding n2
   ---------*===========================
           /   acceptance cone 2θa
```

#### Points to note

- NA depends only on n1 and n2, not on the fibre size.
- Large NA: easy coupling of light from LEDs, but more modes and more modal dispersion, so less bandwidth.
- Typical values: 0.1–0.15 (single mode), 0.2–0.3 (multimode).
- NA also enters the V-number V = (2πa/λ)·NA, which fixes the number of modes (single mode if V < 2.405).

**Example:** n1 = 1.48, n2 = 1.46: NA = √(2.1904 − 2.1316) = 0.2425, θa = 14.0°.

## Check yourself

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

1. What does NA = n0 sin(theta_a) measure?
   A. The fibre's attenuation
   B. Core diameter
   C. Number of splices
   D. The light-gathering ability of the fibre
   Answer: D. A larger NA means a wider acceptance cone. (Key point: NA definition)

2. On what does NA depend?
   A. Only on n1 and n2, not on fibre size
   B. Core diameter
   C. Fibre length
   D. Wavelength only
   Answer: A. NA = sqrt(n1^2 - n2^2). (Key point: NA dependence)

3. What is the trade-off of a large NA?
   A. Lower coupling but higher bandwidth
   B. Easier coupling from LEDs but more modes and modal dispersion, so less bandwidth
   C. No effect on bandwidth
   D. More attenuation only
   Answer: B. More light is accepted into more modes. (Key point: Large NA trade-off)

4. Single-mode operation requires which condition?
   A. V > 2.405
   B. V = 1
   C. V < 2.405, where V = (2 pi a/lambda) NA
   D. V > 100
   Answer: C. V sets the number of modes. (Key point: V-number)

## Related in Propagation and Antenna

- [Fibre types and numerical aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afiber-types-define-na) — Propagation and Antenna, chapter 6
- [Numerical aperture and dispersion](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Ana-and-dispersion) — 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; NA and acceptance angle](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Alight-propagation-na-acceptance-relation) — 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

## Related in other subjects

- [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×
- [Physical significance of numerical aperture](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Anumerical-aperture-significance-dependence) — Engineering Physics, chapter 6
- [Glass-clad fiber numerical aperture](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Afiber-na-glass-clad-numerical) — Engineering Physics, chapter 6
- [Aperture effect, DPCM and DM slope overload numerical](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-effect-dpcm-dm-slope) — Communication Systems, chapter 4