Calculator
Example data table
| Scenario | n₀ | n₁ | n₂ | NA | θ (deg) | 2θ (deg) |
|---|---|---|---|---|---|---|
| Given NA in air | 1.000 | — | — | 0.220 | 12.70 | 25.40 |
| Step-index fiber (air) | 1.000 | 1.480 | 1.460 | 0.242 | 14.01 | 28.02 |
| Step-index fiber (water) | 1.333 | 1.480 | 1.460 | 0.242 | 10.47 | 20.94 |
Formula used
- NA = n₀ · sin(θ) where θ is the acceptance half‑angle.
- θ = asin(NA / n₀) (valid only when NA ≤ n₀).
- For a step‑index fiber (small-angle model), NA ≈ √(n₁² − n₂²).
- Full acceptance cone angle: 2θ.
How to use this calculator
- Select a calculation mode (angle from NA, NA from angle, or index‑based).
- Enter the required fields (unused fields can be left as-is).
- Choose the angle unit for θ input (degrees or radians).
- Press Calculate to show results above the form.
- Use Download CSV or Download PDF to export.
Numerical Aperture Angle Guide
1) Understanding numerical aperture (NA)
Numerical aperture is a unitless measure of light‑gathering strength. Higher NA usually means a wider acceptance cone. Many multimode fibers are around NA ≈ 0.20–0.30, while many single‑mode designs are lower, often near NA ≈ 0.10–0.14. In imaging, objectives can reach NA near 0.8–1.4 with immersion.
2) Acceptance half‑angle and full cone
This calculator treats θ as the half‑angle of the acceptance cone. The core relation is NA = n₀ · sin(θ). The full cone is 2θ. Example: NA = 0.22 in air (n₀ = 1.00) gives θ ≈ 12.70° and 2θ ≈ 25.40°.
3) Typical NA ranges in practice
NA below 0.10 can feel “tight,” often needing better alignment and cleaner optics. NA around 0.20–0.30 is common for coupling LEDs or broad sources into multimode fibers. As NA / n₀ approaches 1, the cone becomes very wide and the surrounding medium strongly affects θ.
4) Effect of surrounding medium (n₀)
The same NA can map to different angles in different media. If n₀ increases, θ decreases because sin(θ) = NA / n₀. With NA ≈ 0.242, θ is about 14.01° in air (n₀ = 1.00) but about 10.47° in water (n₀ ≈ 1.333). This matters for underwater sensors and immersion optics.
5) Step‑index calculation from n₁ and n₂
For step‑index fibers, a common model is NA ≈ √(n₁² − n₂²). With n₁ = 1.480 and n₂ = 1.460, NA ≈ 0.242. The tool flags when NA > n₀, because asin(NA/n₀) would be invalid for a real angle.
6) Degrees, radians, and precision
Specs are often shown in degrees, while simulations may use radians. You can input θ in either unit and the calculator outputs both. Use 2–4 decimals for quick notes, or 6+ decimals for close design comparisons. Consistent rounding helps when you compare multiple fibers. Small NA changes can shift θ noticeably at high NA, so precision can matter in tight designs.
7) Exporting results for reporting
After calculating, export CSV for spreadsheets or a simple PDF for sharing. Exports include mode, n₀, optional n₁/n₂, NA, θ, and 2θ—handy for documenting coupling tests, lens selection, and media changes. Keep exports with the same decimals across experiments.
FAQs
1) What does NA physically represent?
NA summarizes the maximum input cone a fiber or optic can accept. Larger NA means it accepts light from wider angles, which can improve coupling tolerance but may affect modal behavior in fibers.
2) Is θ the half‑angle or full angle?
θ is the half‑angle. The full acceptance cone is 2θ. This is why the calculator displays both θ and 2θ side by side for clarity.
3) Why can’t NA be greater than n₀?
Because the model uses sin(θ) = NA/n₀ and sin(θ) cannot exceed 1. If NA > n₀, no real acceptance angle exists in that surrounding medium.
4) Which mode should I use for fiber work?
If you know core and cladding indices, use the index‑based mode. If you already have NA from a datasheet, use “angle from NA” to get θ and 2θ in your chosen medium.
5) Do n₁ and n₂ change with wavelength?
Yes. Refractive index depends on wavelength and temperature. For precise work, use indices at your operating wavelength and conditions, then recompute NA and acceptance angle accordingly.
6) What decimals should I choose?
Use 2–4 decimals for routine lab notes. Use 5–8 decimals when comparing near‑identical designs or when downstream calculations are sensitive to small changes in NA or angle.