---
title: "Physical significance of numerical aperture"
subject: "Engineering Physics"
chapter: 6
key: "semester_1/engineering_physics:numerical-aperture-significance-dependence"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/numerical-aperture-significance-dependence
site: BCT Engineering
---

# Physical significance of numerical aperture

Engineering Physics · Chapter 6: Laser and Fiber Optics

- Asked in: 2074 Asoj, 5 marks
- Video: [Numerical Aperture of an Optical Fibre](https://www.youtube.com/watch?v=jbwu6_cajoQ) — Physics Jessy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/numerical-aperture-significance-dependence
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-6

## Question

Discuss the physical significance of numerical aperture (NA). How does it depend on refractive index of core and cladding?

## Short answer

NA determines the light-gathering ability and acceptance cone of a fiber; larger NA couples more light but increases modal dispersion, reducing bandwidth. NA = √(n1^2 - n2^2), so it increases with higher core index n1 or lower cladding index n2. It is independent of core diameter.

## Answer

#### Physical significance of NA
Numerical aperture NA = sin θ₀ = √(n₁² − n₂²) describes how much light a fiber accepts and guides.

1. **Light-gathering power:** NA fixes the acceptance cone with half-angle θ₀. Larger NA means a wider cone, so more light from a source (LED, laser) can be launched, giving higher coupling efficiency.
2. **Acceptance angle:** θ₀ = sin⁻¹(NA). For NA = 0.2, θ₀ ≈ 11.5°.
3. **Number of modes:** the V-number is V = (2πa/λ) NA, and the number of modes in a multimode step-index fiber is about V²/2. Higher NA means more modes.
4. **Dispersion and bandwidth:** more modes lead to more intermodal dispersion, so a very high NA reduces the information capacity. A very low NA, on the other hand, makes the fiber more sensitive to bending loss, because the guiding is weak.
5. NA cannot exceed 1 for air.

Typical values: single-mode fibers 0.1 to 0.14; multimode fibers 0.2 to 0.5.

#### Dependence on the refractive indices
```text
NA = √(n₁² − n₂²) = n₁ √(2Δ),   Δ = (n₁ − n₂)/n₁
```
- NA **increases** when n₁ (core) increases or n₂ (cladding) decreases; it depends only on the difference of the squares n₁² − n₂².
- If n₁ = n₂, then NA = 0 and no light is guided.
- It is independent of the core diameter.
- **Example:** n₁ = 1.48, n₂ = 1.46 gives NA = √(2.1904 − 2.1316) = 0.242, so θ₀ ≈ 14°.

## Check yourself

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

1. A larger NA means:
   A. Lower coupling efficiency
   B. Wider acceptance cone, more light coupled
   C. Fewer modes
   D. Narrower cone
   Answer: B. NA = sin θ0 measures light gathering. (Key point: Significance)

2. NA increases when:
   A. n1 increases or n2 decreases
   B. Core diameter increases
   C. n2 increases
   D. Wavelength increases
   Answer: A. NA = √(n1² − n2²). (Key point: Dependence)

3. NA depends on core diameter?
   A. Yes, proportional
   B. Only for single-mode
   C. No, it is independent
   D. Yes, inversely
   Answer: C. It depends only on the indices. (Key point: Independence)

## Related in Engineering Physics

- [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
- [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×
- [Newton's rings, refractive index of liquid](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Anewton-rings-liquid-refractive-index) — Engineering Physics, chapter 4, asked 2×
- [Refractive index of liquid film by interference](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acoherent-sources-film-refractive-index-method) — Engineering Physics, chapter 4
- [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

## Related in other subjects

- [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×
- [Refractive index, critical frequency and MUF](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Arefractive-index-fc-muf) — Propagation and Antenna, chapter 4
- [Short note: Numerical aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Anumerical-aperture-short-note) — Propagation and Antenna, chapter 6
- [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