Angular Resolution Calculator (Rayleigh θ = 1.22 λ/D, smallest detail at a distance)

Understanding the angular resolution calculator — concept and principle

Angular resolution is the smallest angle between two points that an optical system can still separate. This calculator applies the Rayleigh criterion to find the diffraction limit of a telescope, camera lens, eye or radio dish, and the smallest detail visible at a given distance.

Diffraction blurs every point into a small disk; with 550 nm light a 200 mm aperture separates points 0.692″ apart.

How it is calculated

θ = 1.22 λ ÷ D (radians), where λ is the wavelength and D the aperture diameter.
Arcseconds = θ × 206,265; degrees = θ × 180 ÷ π.
Smallest separable detail at distance r: s = r × θ.

The principle

Light passing through a circular opening spreads into a diffraction pattern, the Airy disk, with a bright centre and faint rings. Its first dark ring lies at 1.22 λ ÷ D from the centre. Lord Rayleigh proposed that two point sources are just resolved when the peak of one falls on the first dark ring of the other. A bigger aperture or shorter wavelength makes the disk smaller and the resolution finer. That is why radio telescopes, working at long wavelengths, must be enormous or linked together.

Worked example

Default inputs: λ = 550 nm (green light), D = 200 mm, distance 10 km.

θ = 1.22 × 550 × 10⁻⁹ ÷ 0.200 ≈ 3.355 × 10⁻⁶ rad.
In arcseconds: 3.355 × 10⁻⁶ × 206,265 ≈ 0.692″; in degrees ≈ 0.0001922°.
At 10 km: s = 10,000 × 3.355 × 10⁻⁶ ≈ 0.03355 m (about 3.4 cm).
Aperture-to-wavelength ratio D ÷ λ ≈ 363,600.

Each point is the calculator re-run with only “Aperture diameter D” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Aperture diameter D	Angular resolution (arcseconds)
50	2.768
100	1.384
150	0.9227
200	0.692
250	0.5536
300	0.4613
400	0.346

How to read the result

This is a theoretical best. Ground-based telescopes are usually limited by atmospheric turbulence ("seeing") to around an arcsecond or worse, so a 200 mm telescope rarely reaches 0.69″ in practice. Space telescopes and adaptive optics get closer to the limit. For comparison, the Hubble Space Telescope's 2.4 m mirror at 550 nm has a limit of about 0.058″.

Common mistakes

Mixing nanometres and millimetres.
Forgetting the 1.22 factor for circular apertures.
Expecting the diffraction limit through turbulent air.
Confusing angular resolution with magnification.

Keep in mind: The result is an ideal diffraction limit. Real performance depends on optical quality, alignment, detector sampling and atmosphere.

Frequently asked questions

What is the Rayleigh criterion?

Two point sources are just resolved when one lies on the first dark ring of the other's diffraction pattern, θ = 1.22 λ ÷ D.

Does a larger telescope always see sharper?

In theory yes, but on the ground atmospheric seeing often limits resolution first.

What is the resolution of the human eye?

With a pupil of a few millimetres the diffraction limit is roughly an arcminute or below, and in practice normal visual acuity is about one arcminute.

Sources: Hecht, E., Optics, 5th ed. Pearson (diffraction, Rayleigh criterion) · Born, M. & Wolf, E., Principles of Optics. Cambridge University Press

Checked 9 Nov 2026. OpenStax, University Physics Volume 3 (2016), §4.5 “Circular Apertures and Resolution”: Rayleigh criterion θ = 1.22λ/D (Eq. 4.5), s = rθ, Hubble example (2.40 m, 550 nm, 2.80 × 10⁻⁷ rad).

Your numbers