---
title: "Acceptance angle and NA of step-index fibre"
subject: "Propagation and Antenna"
chapter: 6
key: "semester_6/propagation_and_antenna:acceptance-angle-na-step-index"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/acceptance-angle-na-step-index
site: BCT Engineering
---

# Acceptance angle and NA of step-index fibre

Propagation and Antenna · Chapter 6: Optical Fibres

- Asked in: 2081 Chaitra, 2+5 marks; 2078 Chaitra, 3+5 marks (asked 2 times)
- 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/acceptance-angle-na-step-index
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-6

## Question

What is an acceptance angle? Derive the expression for numerical aperture (NA) of a stepped index optical fiber with neat diagram.

## Short answer

Acceptance angle is the maximum incident angle for guided rays. For a step-index fibre, NA = n0 sin θa = √(n1² - n2²) ≈ n1√(2Δ). Example: n1=1.48, n2=1.46 gives NA=0.2425, θa=14°.

## Answer

#### Acceptance angle

The **acceptance angle** θa is the maximum angle, measured from the fibre axis, at which a ray can enter the fibre core and still be guided by total internal reflection. Rays inside the cone of half-angle θa (the **acceptance cone**) are trapped; rays outside it are refracted into the cladding and lost.

#### Derivation of NA for a step-index fibre

```text
          n0 (air)  |  n2 cladding
                    |-----------------------------
        θa  ray     |   φ  \  /  φ  (φ ≥ φc: TIR)
   ---------------->*-------\/----------------->
     (acceptance)   |θ1  n1 core   axis
                    |-----------------------------
                    |      n2 cladding
```

Let a ray enter the core from a medium of index n0 at angle θa to the axis, refract to angle θ1 inside the core, and meet the core–cladding boundary at angle φ (measured from the normal), where φ = 90° − θ1.

1. Snell's law at the input face:

```text
n0 sin θa = n1 sin θ1
```

2. For total internal reflection at the core–cladding boundary, φ must be at least the critical angle φc, where

```text
sin φc = n2 / n1
```

3. The limiting (maximum) entry angle θa corresponds to φ = φc, i.e. θ1 = 90° − φc:

```text
n0 sin θa = n1 sin(90° − φc) = n1 cos φc
          = n1 √(1 − sin²φc)
          = n1 √(1 − n2²/n1²)
          = √(n1² − n2²)
```

4. The numerical aperture is defined as NA = n0 sin θa, so

```text
NA = n0 sin θa = √(n1² − n2²)
```

5. With the relative index difference Δ = (n1² − n2²)/(2n1²) ≈ (n1 − n2)/n1:

```text
NA = n1 √(2Δ)
```

For launching from air (n0 = 1): θa = sin⁻¹(NA), and the full acceptance cone angle is 2θa.

**Example:** n1 = 1.48, n2 = 1.46 gives NA = √(1.48² − 1.46²) = 0.2425 and θa = 14.0°.

## Check yourself

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

1. The acceptance angle is:
   A. The critical angle
   B. The bend angle
   C. The cladding angle
   D. The maximum angle from the axis at which light can enter and be guided
   Answer: D. Rays within the acceptance cone undergo TIR. (Key point: Acceptance angle)

2. The limiting entry angle corresponds to:
   A. φ = φc
   B. φ = 0
   C. φ = 90°
   D. θ1 = 0
   Answer: A. Boundary angle equals critical angle. (Key point: Derivation limit)

3. The numerical aperture is:
   A. n1 + n2
   B. sqrt(n1^2 − n2^2)
   C. n1 − n2
   D. n1^2 − n2^2
   Answer: B. n1 cos φc = sqrt(n1^2 − n2^2). (Key point: NA expression)

4. NA in terms of Δ is:
   A. sqrt(Δ)
   B. 2n1Δ
   C. n1 sqrt(2Δ)
   D. n1·Δ
   Answer: C. With Δ ≈ (n1−n2)/n1. (Key point: NA with Δ)

## Related in Propagation and Antenna

- [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
- [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
- [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
- [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 derivation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aacceptance-angle-derivation) — 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×
- [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