Numerical Aperture Angle Calculator

Calculator

This page uses one stacked layout for the content sections, while the input area below adapts to large, small, and mobile screens.

Calculation mode

Choose the quantity set you want to solve.

Core refractive index (n₁)

Needed when the core index is part of the solution.

Cladding refractive index (n₂)

Use the lower refractive index surrounding the core.

Numerical aperture (NA)

Enter NA directly when it is known.

Acceptance half-angle (degrees)

Use the half-angle, not the full cone angle.

External medium index (n₀)

Air is usually 1.0. Water is about 1.33.

Distance from fiber tip (cm)

Optional. Estimates cone diameter at that distance.

Core radius (µm)

Optional. Used to estimate normalized frequency.

Wavelength (nm)

Optional. Needed with radius for V-number and mode estimate.

Example data table

Case	n₁	n₂	n₀	NA	Half-Angle (°)	Full Cone (°)
Silica example A	1.48	1.46	1.00	0.2425	14.03	28.06
Silica example B	1.50	1.47	1.00	0.2985	17.37	34.74
Higher contrast fiber	1.62	1.58	1.00	0.3578	20.96	41.92
Known NA in water	—	—	1.33	0.2200	9.52	19.04

Formula used

NA = n₀ sin(θ_a)

NA = √(n₁² − n₂²)

θ_a = sin⁻¹(NA / n₀)

Acceptance cone angle = 2θ_a

Critical angle at the core-cladding boundary = sin⁻¹(n₂ / n₁)

Relative index difference, Δ = (n₁ − n₂) / n₁

V = (2πa / λ) × NA

Estimated guided modes for a step-index multimode fiber ≈ V² / 2

Here, n₁ is the core refractive index, n₂ is the cladding refractive index, n₀ is the surrounding medium index, and θ_a is the acceptance half-angle.

A real acceptance angle exists only when NA / n₀ is less than or equal to 1.

How to use this calculator

Select the calculation mode that matches your known values.
Enter the refractive indices, numerical aperture, or acceptance half-angle.
Set the external medium index. Use 1.0 for air unless your setup differs.
Add optional core radius and wavelength to estimate V-number and guided modes.
Add an optional propagation distance to estimate acceptance cone diameter.
Press Calculate Now to show results above the form.
Review the graph and recent history table for trend checking.
Export the results using the CSV or PDF buttons.

Frequently asked questions

1) What does numerical aperture measure?

Numerical aperture measures how strongly an optical fiber or lens system can accept incoming light. Larger NA means a wider acceptance angle and easier light coupling.

2) Is the acceptance angle the same as the cone angle?

No. The acceptance angle usually means the half-angle from the axis to the cone edge. The full acceptance cone angle is twice that half-angle.

3) Why does the surrounding medium matter?

The surrounding medium changes the relation between NA and the acceptance half-angle. A higher outside refractive index reduces the angle for the same NA.

4) Can NA be greater than 1?

It can, depending on the surrounding medium and system design. In air-coupled fiber work, practical values are often below 1, but the calculator still checks the angle condition correctly.

5) What happens when NA exceeds the medium index?

The inverse sine step becomes invalid, so no real acceptance angle exists in that external medium. The calculator flags this immediately.

6) Why is V-number useful here?

V-number helps estimate how many guided modes a step-index fiber can support. It combines radius, wavelength, and NA into one practical design metric.

7) Does this tool work for lenses too?

The core NA-angle relation is broadly useful, but the fiber-specific index formulas and mode estimate are intended mainly for optical fibers and waveguides.

8) Should I enter the full cone angle directly?

No. Enter the acceptance half-angle only. The tool automatically doubles it to report the full acceptance cone angle.