Optical Fibre Numerical Aperture Calculator
Numerical aperture, acceptance angle, V number and single-mode cut-off of a step-index optical fibre from its core and cladding indices.
Results
What this tool does
Light stays inside a fibre by total internal reflection, and only rays that enter within a certain cone are trapped; the sine of that cone's half-angle is the numerical aperture. The two glasses differ by surprisingly little: in standard telecom fibre the core index is only about 0.35 % higher than the cladding, which gives an NA near 0.12 and an acceptance half-angle of about 7°. The V number decides how many paths, or modes, the light can take: below 2.405 only one survives, which is why the same 8 µm fibre is single-mode at 1310 and 1550 nm but not with visible light. The formulas are for step-index fibre; graded-index multimode fibre carries roughly half as many modes.
Formula
NA = √(n₁² − n₂²) · θ = asin(NA ÷ n₀) · V = π d NA ÷ λ
Variables
| Symbol | Meaning | Unit |
|---|---|---|
n1 | Refractive index of the core | — |
n2 | Refractive index of the cladding | — |
dc | Core diameter | µm |
lam | Wavelength | nm |
n0 | Index of the medium light comes from (air = 1) | — |
NA | Numerical aperture | — |
TH | Acceptance half-angle | ° |
V | Normalised frequency (V number) | — |
LC | Single-mode above this wavelength | nm |
MO | Mode regime | — |
NM | Approximate number of modes (large V) | — |
Worked example
- Refractive index of the core1.4504
- Refractive index of the cladding1.4454
- Core diameter8.2 µm
- Wavelength1310 nm
- Index of the medium light comes from (air = 1)1
- Numerical aperture0.1203
- Acceptance half-angle6.91 °
- Normalised frequency (V number)2.366
- Single-mode above this wavelength1289 nm
- Mode regimeSingle-mode at this wavelength
- Approximate number of modes (large V)3
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Coverage and consumption figures vary between products. Take the values from the manufacturer's datasheet for the product you are actually using.