Numerical Aperture Formula
Calculate the light-gathering ability of an optical fiber or microscope objective.
Key to fiber optic and imaging design.
The Formula
NA = n × sin(θ)
Numerical aperture describes how much light an optical system can collect. Higher NA means better resolution and brighter images, but shallower depth of field.
Variables
| Symbol | Meaning |
|---|---|
| NA | Numerical aperture (unitless, typically between 0 and 1.5) |
| n | Refractive index of the medium between the lens and specimen |
| θ | Half-angle of the maximum cone of light entering the lens |
For optical fibers: NA = √(n₁² - n₂²), where n₁ is the core index and n₂ is the cladding index.
Example 1
A microscope objective in air has a half-angle of 45°. Find the NA.
n = 1.00 (air), θ = 45°
NA = 1.00 × sin(45°)
NA = 0.707
Example 2
An optical fiber has core index n₁ = 1.50 and cladding index n₂ = 1.46
NA = √(1.50² - 1.46²) = √(2.25 - 2.1316)
NA = √0.1184
NA ≈ 0.344
When to Use It
Use the numerical aperture formula when:
- Choosing microscope objectives for the resolution you need
- Designing optical fiber systems and calculating acceptance angles
- Comparing the light-gathering power of different optical systems
- Estimating the resolution limit of an imaging system