---
title: "Derive numerical aperture of optical fiber"
subject: "Engineering Physics"
chapter: 6
key: "semester_1/engineering_physics:na-derivation-optical-fiber"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/na-derivation-optical-fiber
site: BCT Engineering
---

# Derive numerical aperture of optical fiber

Engineering Physics · Chapter 6: Laser and Fiber Optics

- Asked in: 2072 Kartik, 5 marks; 2072 Chaitra, 5 marks; 2068 Shrawan, 5 marks; 2067 Asar, 5 marks (asked 4 times)
- Video: [Derivation for #Numerical aperture and #Acceptance angle of an #optical fibre (exam point of view)](https://www.youtube.com/watch?v=qYgz4tIoztY) — PHYSICS FREAK
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/na-derivation-optical-fiber
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-6

## Question

What is an optical fiber? Define acceptance angle and numerical aperture, and show that NA = μ_core√(2Δ), where the symbols have their usual meanings.

## Short answer

An optical fiber is a thin glass strand guiding light by total internal reflection. Acceptance angle θ0 is the maximum entrance angle for guided rays; numerical aperture NA = sin θ0. Using Snell's law and critical angle condition, NA = √(n1^2 - n2^2). With fractional index difference Δ = (n1 - n2)/n1, for small Δ, NA ≈ n1 √(2Δ).

## Answer

#### Optical fiber
An optical fiber is a thin, flexible strand of transparent glass or plastic that guides light by **total internal reflection**. It has a central **core** (refractive index n₁ = μ_core) surrounded by a **cladding** of lower refractive index n₂ (n₁ > n₂), covered by a protective jacket.

#### Acceptance angle and numerical aperture
- **Acceptance angle (θ₀):** the maximum angle, with respect to the fiber axis, at which light from outside can enter the core and still be guided by total internal reflection. The cone of half-angle θ₀ is the acceptance cone.
- **Numerical aperture (NA):** n₀ sin θ₀, a measure of the light-gathering ability of the fiber. For air, n₀ = 1, so NA = sin θ₀.

#### Derivation of NA = n₁ √(2Δ)
```text
          cladding n2
 ------------------------------
        \ phi   /|
 n0  ----\-----/ |  core n1
  theta0  \theta1
 ======axis=========================
```
1. A ray enters the face of the fiber at angle θ₀ to the axis from a medium of index n₀ (air, 1). It refracts at angle θ₁:
 n₀ sin θ₀ = n₁ sin θ₁. (1)
2. The ray strikes the core-cladding boundary at angle of incidence φ = 90° − θ₁.
3. For the limiting ray, φ equals the critical angle φ_c:
 sin φ_c = n₂/n₁.
4. Then sin θ₁ = cos φ_c = √(1 − sin² φ_c) = √(1 − n₂²/n₁²) = √(n₁² − n₂²)/n₁.
5. Substitute in (1):
 n₀ sin θ₀ = n₁ × √(n₁² − n₂²)/n₁ = √(n₁² − n₂²).
 For air: **NA = sin θ₀ = √(n₁² − n₂²)**.
6. Define the fractional index difference Δ = (n₁ − n₂)/n₁, so n₂ = n₁(1 − Δ). Then
 n₁² − n₂² = n₁² [1 − (1 − Δ)²] = n₁² (2Δ − Δ²) ≈ 2Δ n₁² (Δ ≪ 1).
7. Therefore

 **NA = n₁ √(2Δ) = μ_core √(2Δ)**

Any ray entering within the cone of full angle 2θ₀ is guided; rays outside it leak into the cladding.

## Check yourself

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

1. Why does an optical fiber guide light?
   A. Total internal reflection, n1 > n2
   B. Absorption
   C. Reflection from a metal coating
   D. Refraction in cladding
   Answer: A. The core has a higher index than the cladding. (Key point: Optical fiber)

2. For the limiting ray, the angle of incidence at the core-cladding boundary is:
   A. 90°
   B. 0°
   C. The critical angle φc with sin φc = n2/n1
   D. Brewster angle
   Answer: C. Beyond acceptance the ray leaks. (Key point: Limiting ray)

3. What is NA for air?
   A. n1² − n2²
   B. √(n1 n2)
   C. √(n1² − n2²)
   D. n1 − n2
   Answer: C. sin θ0 = n1 cos φc = √(n1² − n2²). (Key point: NA in terms of n)

4. With Δ = (n1 − n2)/n1 and Δ << 1, NA =
   A. 2n1Δ
   B. n1√(2Δ)
   C. n1/√(2Δ)
   D. n1 Δ
   Answer: B. n1² − n2² = n1²(2Δ − Δ²) ≈ 2Δn1². (Key point: Delta form)

## Related in Engineering Physics

- [Acceptance angle of fiber in a liquid](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Afiber-acceptance-angle-liquid-index-change) — Engineering Physics, chapter 6
- [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
- [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
- [Acceptance angle of fiber in water](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Afiber-acceptance-angle-water) — Engineering Physics, chapter 6
- [Physical significance of numerical aperture](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Anumerical-aperture-significance-dependence) — 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×
- [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
- [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
- [Numerical aperture and dispersion](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Ana-and-dispersion) — Propagation and Antenna, chapter 6