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Numerical Aperture Calculator

Find a lens's numerical aperture.

How it is calculated

The tool computes NA = n × sin(θ), where n is the refractive index of the medium between the specimen and the lens (about 1.00 for air, 1.33 for water, about 1.5 for immersion oil) and θ is the half-angle of the cone of light in degrees. It then reports an equivalent f-number with N = 1 / (2 × NA), shown as f/…. The angle must be greater than 0 and at most 90°, and n must be positive; otherwise there is no result. NA is dimensionless, and it is shown to four decimals. The f-number conversion is the standard approximation and is applied the same way whatever n you enter, so for immersion lenses it is only a rough comparison with photographic lenses.

Worked example

An oil-immersion objective: n = 1.515, half-angle θ = 60°. sin(60°) = 0.8660. NA = 1.515 × 0.8660 = 1.3120. The equivalent f-number is 1 ÷ (2 × 1.3120) = 1 ÷ 2.624 = 0.381, shown as f/0.38. For comparison, a dry lens in air (n = 1) can never exceed NA 1.0, which is reached at a 90° half-angle and gives f/0.50.

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Common mistakes

  • Entering the full cone angle instead of the half-angle. A lens that accepts a 120° cone needs 60° here.
  • Leaving n at 1 for an oil or water immersion objective. The refractive index of the medium is the whole reason the NA can exceed 1.
  • Expecting an NA above 1 in air. With n = 1 the maximum is 1.0, at a 90° half-angle.
  • Entering the angle in radians. The field is in degrees.
  • Comparing the f-number with a camera lens rating at face value. The conversion is a rough approximation, mostly meaningful for lenses working in air.

About this calculator

This calculator finds a lens's numerical aperture, NA = n·sinθ, where n is the refractive index of the medium the lens is working in (air, oil, water) and θ is the half-angle of the cone of light entering or leaving the lens. NA determines two things at once: resolving power (a higher NA resolves finer detail, following the diffraction-limited resolution formula) and light-gathering ability, and it relates directly to the more familiar f-number used in photography via roughly N ≈ 1/(2·NA) — a higher NA corresponds to a lower f-number and a shallower depth of field.

Microscopists in biology and histology labs use NA ratings to choose objectives capable of resolving the fine structures they need to see; optical and fiber-optic engineers use it to design lenses and fibers that couple light efficiently at a required angle; and photographers use the underlying relationship to understand why wide-aperture (low f-number) lenses gather more light but produce a shallower depth of field.

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