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Numerical Aperture Calculator

Reviewed by the CalculatorHub team, edited by James Graham. Formula per OpenStax University Physics and Optica (formerly OSA). Last verified 15 June 2026.

Numerical aperture (NA) is a key parameter for any optical system including microscope objectives, camera lenses, and optical fibers. It is defined as NA = n sin(theta), where n is the refractive index of the medium surrounding the lens and theta is the half-angle of the maximum cone of light. A higher NA means the system can gather more light and resolve finer detail. For an optical fiber, the formula becomes NA = sqrt(n_core^2 - n_cladding^2), which this calculator supports in fiber mode. The Rayleigh resolution limit of a circular aperture is approximately 0.61 lambda/NA. For a 550 nm wavelength and NA = 1.25, resolution is about 0.27 micrometers.

Air = 1.000, immersion oil = 1.515
Half the cone angle of light accepted by the lens
0.50
0.67 um

Numerical aperture formulas

Lens: NA = n sin(theta)
Fiber: NA = sqrt(ncore2 - nclad2)
Resolution: d = 0.61 lambda / NA (Rayleigh criterion)

Where n is the refractive index of the medium, theta is the half-angle of the acceptance cone, and lambda is the wavelength of light.

Typical numerical aperture values

  • Human eye (pupil 5 mm, focal length 17 mm in vitreous n=1.336): NA approximately 0.19.
  • Microscope objectives: 4x dry (NA 0.10) up to 100x oil immersion (NA 1.40).
  • Multimode fiber: NA 0.20 to 0.50, accepting a broad cone of light.
  • Single-mode fiber (e.g. SMF-28): NA approximately 0.12 to 0.14.

Numerical aperture: frequently asked questions

What is numerical aperture?

Numerical aperture (NA) is a dimensionless number that characterizes the range of angles over which an optical system can accept or emit light. For a lens in a medium of refractive index n, NA = n sin(theta), where theta is the half-angle of the maximum cone of light that can enter or exit the lens.

What is a good NA for a microscope objective?

Dry objectives (air, n=1) typically range from NA 0.1 to 0.95. Oil-immersion objectives (n=1.515) reach NA up to 1.45, improving resolution. The minimum resolvable distance is approximately 0.61 lambda / NA (Rayleigh criterion for a circular aperture).

How does NA relate to fiber optics?

For an optical fiber, NA = sqrt(n_core^2 - n_cladding^2). A higher NA means the fiber accepts light over a wider angle cone. Typical single-mode fibers have NA 0.12-0.14; multimode fibers have NA 0.20-0.50.

What is the relationship between NA and f-number?

For a lens working at infinity focus, the f-number N and NA are related by NA = 1/(2N) in air (n=1). So an f/2 lens has NA approximately 0.25. This relationship is approximate and assumes the paraxial regime.

Why does immersion oil increase NA?

Immersion oil (n ~1.515) fills the gap between the objective lens and the cover glass, replacing air (n=1) as the medium. Since NA = n sin(theta), multiplying n by 1.515 instead of 1 increases NA by up to 51.5%, directly improving resolving power.

Official sources

  • OpenStax University Physics Volume 3, Chapter 2: Geometric Optics. openstax.org.
  • Optica (formerly OSA) Fundamentals of Photonics. optica.org.

Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.