Home/Optics & Photonics/Geometric Optics/Numerical Aperture

Formula & Calculator

Numerical Aperture

Characterizes the range of angles over which an optical system can accept or emit light, key to resolution and light-gathering power.

Written by the Vorzep Team

OpticsGeometric OpticsMicroscopy

Numerical Aperture CalculatorOptics · Light Collection

NA = n · sin(θ)
NA = numerical aperture  ·  n = refractive index  ·  θ = acceptance half‑angle
⟹ Solve⟷NA, n, θ
dimensionless
dimensionless
°
Please fix the errors above.
Solve for:
Presets:
NA—
—
NA: —n: —θ: —
✓ Copied!
Numerical Aperture (NA) Gauge—
Low NA (< 0.3) Medium (0.3–0.7) High NA (> 0.7)
NA = n · sin(θ)  ·  n is the refractive index of the medium (air = 1.0, water = 1.33, glass = 1.5–1.9)

Variables

SymbolQuantityUnit
NANumerical aperture
nRefractive index of the medium
θHalf-angle of the maximum light conedegrees

What it means

Numerical aperture (NA) is a dimensionless number that characterises the range of angles over which an optical system can accept or emit light. It is a key parameter in microscopy: higher NA gives better resolution. In fiber optics, NA determines the acceptance angle of the fiber. In lithography, NA limits the minimum feature size. Understanding NA is essential for designing high‑resolution imaging systems and for coupling light into optical fibers.

Worked example

Numerical Aperture – Two Detailed Examples

Real‑World
Scenario: A microscope objective has a refractive index n = 1.0 (air) and half‑acceptance angle θ = 30°. The numerical aperture NA = n sinθ = 1.0 × sin30° = 0.5. A larger NA means higher resolution. The biologist uses this to estimate the resolving power of the objective, which is crucial for distinguishing fine cellular structures.
ParameterValue
n1.0
θ (°)30
1NA = 1.0 × sin30° = 0.5
Result 0.5 ✓ Numerical aperture
Scenario: An oil‑immersion objective uses oil with n = 1.5 and a cone half‑angle θ = 25°. The NA = 1.5 × sin25° ≈ 0.634. This high NA provides excellent resolution, enabling the researcher to see details at the nanoscale. The higher NA is a key advantage of immersion microscopy.
ParameterValue
n1.5
θ25
1NA = 1.5 × sin25° ≈ 1.5 × 0.4226 = 0.6339
Result 0.634 ✓ High NA
Insight: Numerical aperture quantifies the light‑gathering ability and resolving power of an optical system. It depends on the refractive index of the medium and the maximum acceptance angle. Higher NA gives higher resolution.

Common mistakes

  • Numerical aperture (NA): NA = n·sin(θ) – where n is the refractive index of the medium (usually air, n=1).
  • θ: The half‑angle of the cone of light accepted by the lens – measured from the optical axis.
  • Units: NA is dimensionless – sinθ is dimensionless.
  • Relation to f‑number: NA ≈ 1/(2N) for small angles – but not exact for high NA.
  • Resolution: Higher NA gives better resolution (Abbe diffraction limit).

Applications

Numerical aperture, NA = n·sinθ, characterises the light‑gathering ability of an optical system, such as a microscope objective or optical fibre. A higher NA allows finer resolution and more light collection. Engineers use NA to specify microscope objectives, to design fibre optic couplers, and to evaluate the performance of imaging systems. It is also used in photolithography for defining the resolution limit. By maximising NA, optical systems can achieve higher resolution and brighter images. Understanding NA is essential for designing high‑performance optical instruments and for selecting appropriate components for specific applications.

  • Microscope objective selection and resolution determination
  • Fibre optic system design and coupling efficiency
  • Photolithography and semiconductor manufacturing
  • Optical sensor and detector design
  • Educational understanding of light collection