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Optics

Angular Resolution Calculator

Calculate telescope and microscope resolving power using Rayleigh criterion, Dawes’ limit, and Abbe diffraction limit.

Interactive calculator

Angular Resolution Calculator

Calculate the angular resolution of telescopes and microscopes using Rayleigh criterion, Dawes' limit, and Abbe diffraction limit.

Try an example

Your result will appear here.

Choose a calculation mode, fill in the known values, and click Calculate.

Quick Guide

  • Choose Rayleigh, Dawes’, microscope, or linear resolution mode.
  • Enter wavelength and aperture diameter.
  • Results show resolution in arcsec, arcmin, degrees, and radians.

Table of Contents

Key Takeaways

  • Angular resolution measures the smallest angle between two distinguishable objects.
  • Rayleigh criterion: θ = 1.22λ/D — larger aperture = finer resolution.
  • Dawes’ limit (θ = 116/Dₘₘ) is an empirical rule for visual double-star splitting.
  • Microscope resolution uses the Abbe limit: d = 0.61λ/NA.
  • Shorter wavelengths improve resolution (UV > visible > IR).
  • Atmospheric seeing usually limits ground telescopes to ~1″.

What Is Angular Resolution?

Angular resolution is the ability of an optical system to distinguish between two closely spaced objects. It is limited by diffraction — light bending around the aperture edge creates an Airy pattern rather than a perfect point image.

Rayleigh Criterion

θ=1.22λD\theta = 1.22 \frac{\lambda}{D}

θ is the minimum resolvable angle (radians), λ is wavelength, and D is aperture diameter. Two point sources closer than this angle merge into a single blurred spot.

Dawes’ Limit

θDawes=116Dmm arcsec\theta_{Dawes} = \frac{116}{D_{mm}} \text{ arcsec}

An empirical formula developed by W. R. Dawes for splitting equal-brightness double stars visually. It gives slightly finer values than Rayleigh for green light.

Microscope Resolution (Abbe Limit)

d=0.61λNAd = \frac{0.61\lambda}{NA}

The Abbe diffraction limit gives the minimum resolvable distance d for a microscope with numerical aperture NA. Oil-immersion objectives (NA up to ~1.4) achieve ~200 nm resolution at visible wavelengths.

How to Use

  1. Select the appropriate mode for your instrument.
  2. Enter wavelength and aperture/NA values.
  3. Click Calculate.

Examples

60 mm refractor at 550 nm

θ = 1.22 × 550e-9 / 0.06 ≈ 11.18 μrad ≈ 2.31"

Hubble (2.4 m, 500 nm)

θ = 1.22 × 500e-9 / 2.4 ≈ 0.053"

FAQ

What is angular resolution?›

Angular resolution (or resolving power) is the smallest angular separation at which two point sources can be distinguished as separate objects through an optical system.

What is the Rayleigh criterion?›

The Rayleigh criterion states that two point sources are just resolved when the central maximum of one Airy pattern falls on the first minimum of the other: θ = 1.22λ/D.

How does aperture affect resolution?›

Larger apertures yield finer resolution. Doubling the aperture halves the minimum resolvable angle.

What is the Dawes’ limit?›

Dawes’ limit (θ = 116/Dₘₘ arcseconds) is an empirical formula for the closest double star a telescope can split visually. It applies to equal-brightness pairs at visible wavelengths.

Why is shorter wavelength better?›

Resolution is proportional to λ. X-ray and electron microscopes achieve far finer detail than optical instruments because their effective wavelengths are much shorter.

What limits ground telescope resolution?›

Atmospheric turbulence (‘seeing’) blurs images, typically limiting resolution to about 1″ for ground telescopes without adaptive optics.

Sources

  1. HyperPhysics – Rayleigh Criterion
  2. OpenStax – Resolution of Optical Instruments
Manish Kumar

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

Learn more about Manish