Angular Resolution Calculator

Use this Angular Resolution Calculator to find the smallest angle your optical system can separate using the Rayleigh diffraction limit, with optional Dawes and Sparrow telescope rules and a distance-to-target size estimate.

How to use our Angular Resolution Calculator

1. Enter the aperture diameter (mm) for your lens or telescope mirror.
2. Choose a result unit (arcseconds, degrees, or radians) for your main readout.
3. Open Advanced options and pick a resolution model: Rayleigh (physics-based) or Dawes/Sparrow (telescope rules of thumb).
4. If you picked Rayleigh, enter the wavelength (nm), like 550 nm for green light.
5. If you picked Dawes, choose the double-star assumption to match how the Dawes rule is usually described.
6. (Optional) Enter a distance to target and its unit to convert the angle into a minimum separable spacing at that range.
7. Click Calculate.
8. Read the angle in radians, degrees, and arcseconds to cross-check units, then use the linear spacing result (if provided) to answer "what size detail can I see?"

Definitions

Angular resolution: The smallest angle between two points that you can tell apart. Smaller angular resolution means sharper detail.

Diffraction: The spreading of light waves when they pass through an opening, which sets a fundamental sharpness limit for any optics.[1]

Wavelength (nm): The "size" of the light wave. Shorter wavelengths (bluer light) can, in theory, resolve finer detail.

Aperture diameter (mm): The width of the opening that collects light (lens diameter or mirror diameter). Bigger aperture usually improves resolution.

Rayleigh criterion: A common diffraction-based rule that defines when two point sources are just resolvable for a circular aperture.[2]

Dawes limit: A telescope rule of thumb for double stars that gives a resolution in arcseconds based on aperture size.

Sparrow limit: Another telescope rule of thumb that is slightly more optimistic than Dawes for the same aperture.

Arcsecond (arcsec): An angle unit equal to 1/3600 of a degree. Astronomers often use arcseconds for small angles.

Small-angle approximation: For small angles, the linear spacing is about angle (in radians) times distance.

Methodology

What this calculator computes

It computes the angular resolution angle (smallest resolvable angle) from your chosen model, then converts that same angle into radians, degrees, and arcseconds. If you provide a target distance, it also estimates the smallest separable spacing at that distance using the small-angle approximation.

Inputs and unit conversions used

D_m = D_mm / 1000

lambda_m = lambda_nm * 1e-9

deg_per_rad = 180 / pi

arcsec_per_deg = 3600

Angular resolution models

Rayleigh (diffraction limit for a circular aperture): Uses the first minimum of the diffraction pattern for a circular opening.[1][2]

theta_rad = 1.22 * lambda_m / D_m

Dawes (rule of thumb): Uses an aperture-only relation in arcseconds.

theta_arcsec = 116 / D_mm

Sparrow (rule of thumb): Similar form with a different constant.

theta_arcsec = 108 / D_mm

Angle unit conversions (always shown)

theta_deg = theta_rad * (180 / pi)

theta_arcsec = theta_deg * 3600

If you selected Dawes or Sparrow (which start in arcseconds), the calculator converts back to radians and degrees using the same relationships.

Linear detail at a target distance (optional)

If you enter a distance to target L, the calculator estimates the minimum separable spacing s at that range using the small-angle approximation. This is most accurate when the angle is small.

s = theta_rad * L_m

L_m = {distance converted into meters from your selected unit}

Validation and edge cases

Aperture diameter must be greater than 0 for all models. For Rayleigh, wavelength must be greater than 0. Distance is optional, but if provided it must be greater than 0; otherwise the linear spacing output shows N/A. If the computed angle is not small, the notes remind you that the small-angle spacing estimate can become less accurate.

Sources