Numerical Aperture Calculator

Frequently Asked Questions

What does numerical aperture tell you?

Numerical aperture describes the cone of light a lens or fiber can accept or emit. A larger value means more light gathering and finer resolution.

How is numerical aperture related to f-number?

The f-number is approximately the reciprocal of twice the numerical aperture, so a small f-number corresponds to a large numerical aperture.

Why do microscope lenses use immersion oil?

Oil has a higher refractive index than air, which raises the numerical aperture above the limit set by air and improves resolution.

How is the numerical aperture of an optical fiber calculated?

A step-index fiber's numerical aperture is usually found from its two refractive indices rather than an angle: NA = √(ncore2 − nclad2). For example, a core of 1.48 and cladding of 1.46 give NA = √(1.482 − 1.462) ≈ 0.24. This is equivalent to the NA = n · sinθ definition used here, since the index difference sets the maximum acceptance half-angle; this calculator takes the angle directly, so compute the angle from your fiber's indices first if you only have ncore and nclad.

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How This Calculator Works

Numerical aperture describes the range of angles over which an optical system such as a lens, microscope objective, or optical fiber can accept or emit light. It is defined as NA = n · sinθ, where n is the refractive index of the medium in front of the lens and θ is the acceptance half-angle measured from the optical axis. A larger numerical aperture means the system gathers light from a wider cone, which improves light-collecting power and resolving ability. The Abbe diffraction limit ties NA directly to the smallest feature an optical system can resolve: d ≈ λ ÷ (2 · NA).

How To Use It

Enter the refractive index of the medium between the object and the lens. This is 1.0 for air, about 1.33 for water, and roughly 1.52 for immersion oil. Then enter the acceptance half-angle in degrees, which is half of the full cone angle. The calculator returns the numerical aperture and an equivalent f-number, a familiar way photographers describe the light-gathering speed of a lens.

Worked Example

Find the numerical aperture of a lens in air (n = 1.0) with an acceptance half-angle of θ = 30°.

Use NA = n · sinθ.

sin 30° = 0.5, so NA = 1.0 × 0.5 = 0.5.

The equivalent f-number is about 1 ÷ (2 · NA) = 1 ÷ (2 × 0.5) = f/1.0, a fast light-gathering lens.

The relation f/# = 1 ÷ (2 · NA) is an object-space, paraxial approximation valid for a lens working in air (n = 1); at large apertures the exact form is f/# = 1 ÷ (2 · tanθ), and for image-space or immersion systems the effective f-number is defined differently. Treat the f-number output as a convenient estimate, not an exact conversion.

Why It Matters

In microscopy a higher numerical aperture allows finer detail to be resolved because the diffraction-limited resolution improves as numerical aperture rises. Oil immersion objectives reach values above 1.0 precisely because oil raises the refractive index of the medium. In fiber optics the numerical aperture sets how easily light couples into the core.

Tips

Use the half-angle, not the full cone angle.

Immersion media raise numerical aperture above what air allows.

The f-number falls as numerical aperture rises.

Numerical Aperture Calculator

Frequently Asked Questions

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