What is numerical aperture?

Numerical aperture (NA) quantifies the range of angles over which an optical system can accept or emit light. It equals n × sin(θ), where n is the refractive index of the medium and θ is the half-angle of the maximum light cone.

What refractive index should I use?

Use the index of the medium surrounding the objective or fiber tip: 1.0 for air, ~1.333 for water immersion, ~1.515 for oil immersion objectives. Optical fiber NA calculations typically use the core and cladding indices separately.

How is f-number related to NA?

For objectives working in air (n = 1), f/# ≈ 1 / (2 × NA). This approximation holds well for moderate NA values. High-NA objectives deviate slightly because the paraxial approximation breaks down.

What is the acceptance angle?

The acceptance angle (also called numerical aperture angle) is the full cone angle 2θ within which light entering the fiber or objective is captured. Light arriving outside this cone is lost.

Can NA exceed the medium refractive index?

No. The maximum NA equals n (when θ = 90°, sin θ = 1). In air the theoretical max is 1.0; in oil immersion it can reach ~1.4–1.5. Values above n are physically impossible.

Numerical Aperture Calculator

Calculate numerical aperture from refractive index and half-angle, or reverse-solve for angle or index — plus acceptance cone and f-number.

Inputs

Refractive index (n) — air = 1.0, water ≈ 1.333, oil ≈ 1.515

Half-acceptance angle (θ) — degrees, 0–90

Quick presets

Results

Enter values and click Calculate

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Summary

The numerical aperture (NA) of an optical system determines how much light it can collect and its resolving power. This calculator applies NA = n × sin(θ), where n is the refractive index of the surrounding medium and θ is the half-angle of the acceptance cone. It also derives the full acceptance angle and the equivalent f-number, and supports reverse calculation when NA or angle is the known quantity.

How it works

Enter the medium refractive index (n) and the half-acceptance angle (θ) in degrees.
The calculator applies NA = n × sin(θ) to compute the numerical aperture.
Acceptance angle (full cone) is shown as 2θ.
f-number is derived as f/# = 1 / (2 × NA) — valid for objectives in air.
Use Reverse Mode to solve for the unknown when NA or angle is known.

Use cases

Determine the light-gathering capability of a microscope objective.
Calculate the acceptance cone of a multimode optical fiber.
Convert between NA and f-number for camera or telescope optics.
Verify fiber coupling efficiency by comparing source NA to fiber NA.
Design optical systems where maximum angle of incidence matters.
Cross-check manufacturer NA specs against measured acceptance angles.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu