Numerical Aperture Formula
Calculate numerical aperture (NA) for optical fibers and microscope objectives.
Covers acceptance angle, light-gathering power, and resolution limit formulas.
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
Key Notes
- Minimum resolvable feature (Abbe's diffraction limit): d = λ / (2 × NA) — doubling NA halves the resolution limit; this is why oil-immersion objectives (n ≈ 1.515) resolve ~50% finer detail than air objectives of the same design
- NA > 1.0 is achievable only in an immersion medium (water n=1.33, oil n=1.515) — dry air-based systems are fundamentally limited to NA < 1.0 by Snell's law
- For optical fibers, the acceptance angle (full cone) = 2 × arcsin(NA) — light entering outside this cone escapes into the cladding and is not guided along the fiber
- Depth of field is inversely proportional to NA² — high-NA objectives used in microscopy have extremely shallow depth of field, requiring precise focusing but enabling optical sectioning