The graph updates after each calculation and shows how the full convergence angle changes with the active input relationship.
| Time | Method | Full Angle (deg) | Half Angle (deg) | Equivalent NA | Notes |
|---|---|---|---|---|---|
| No calculations yet. | |||||
| Method | Inputs | Half Angle (deg) | Full Angle (deg) | Equivalent NA |
|---|---|---|---|---|
| Diameter + Focal Length | D = 20 mm, f = 100 mm, n = 1.00 | 5.7106 | 11.4212 | 0.0995 |
| Diameter + Focal Length | D = 36 mm, f = 180 mm, n = 1.00 | 5.7106 | 11.4212 | 0.0995 |
| Radius + Focal Length | r = 12 mm, f = 60 mm, n = 1.00 | 11.3099 | 22.6199 | 0.1961 |
| NA + Refractive Index | NA = 0.25, n = 1.00 | 14.4775 | 28.955 | 0.25 |
These examples demonstrate common optical input combinations and their resulting half-angle, full-angle, and equivalent numerical aperture.
Half-angle: α = tan-1((D / 2) / f)
Full angle: θ = 2α = 2tan-1(D / 2f)
Half-angle: α = tan-1(r / f)
Full angle: θ = 2α
NA = n sin(α)
α = sin-1(NA / n)
θ = 2α
Beam slope = tan(α)
Solid angle: Ω = 2π(1 − cos α)
Equivalent f-number = 1 / (2tan α)
In this page, α is the half-angle of convergence and θ is the full included angle. For geometric inputs, keep diameter, radius, and focal length in the same unit.
- Select the calculation mode that matches your available optical data.
- Enter focal length with diameter or radius, or enter NA with refractive index.
- Choose an angle conversion mode if you already know a half-angle or full angle.
- Press the calculate button to show the result above the form.
- Review the detailed metrics, the graph, and the recent history table.
- Use the CSV or PDF buttons to export the current result and records.
1) What is the angle of convergence?
It is the angle formed by converging rays or beam edges as they move toward a focus. Many optics texts use a half-angle and a full included angle.
2) What is the difference between half-angle and full angle?
The half-angle is measured from the optical axis to one beam edge. The full angle is measured from one beam edge to the opposite edge, so it equals twice the half-angle.
3) When should I use diameter instead of radius?
Use diameter when you know the full beam width or aperture opening. Use radius when you know the distance from the axis to one beam edge.
4) Why is refractive index included?
Refractive index affects the link between numerical aperture and convergence half-angle. A higher index changes the same NA into a different physical angle.
5) Can I mix millimeters and centimeters?
Not inside the same geometric calculation. Diameter, radius, and focal length must use the same length unit, otherwise the computed angle becomes inconsistent.
6) What does equivalent f-number mean here?
It is a geometric optics interpretation of the converging cone. Smaller f-numbers correspond to steeper convergence and larger included angles.
7) Is numerical aperture always less than refractive index?
For this simple model, yes. Since NA = n sin(α), and sin(α) cannot exceed 1, numerical aperture cannot be larger than the refractive index.
8) Why does the graph change with the selected method?
Each mode has a different governing relationship. Geometry-based modes vary with aperture, NA mode varies with numerical aperture, and conversion mode shows full angle as twice the half-angle.