Determine the minimum angular separation at which two point sources can be resolved by a circular aperture. Uses the Rayleigh criterion and calculates the minimum resolvable distance at a given range.
The angular resolution of any imaging system is fundamentally limited by diffraction. Even a perfect, aberration-free lens or mirror with a circular aperture of diameter D will spread a point source into a finite-sized "Airy disk" — a bright central spot surrounded by faint rings. The Rayleigh criterion says two point sources can just be resolved when the central maximum of one Airy pattern falls on the first dark ring of the other, giving the famous formula θ = 1.22λ/D radians.
This calculator returns angular resolution in arc seconds and the minimum resolvable feature size at a specified distance. Use it for telescope design (bigger aperture = sharper images), microscope objective specifications (NA = sin θ_max relates to resolution), camera lens diffraction limits (when stopping down past a certain f-stop, diffraction softens the image), and surveillance and machine vision (how small a feature can a system spot at a given distance).
Practical resolution is almost always worse than the diffraction limit. Atmospheric "seeing" limits ground-based telescopes to ~0.5–2 arc seconds typically (vs ~0.05 arc seconds diffraction-limited for a 2.5 m telescope at visible wavelengths). Adaptive optics partially corrects this but doesn't fully reach the diffraction limit. Space telescopes (Hubble, JWST) achieve close to the diffraction limit because they're above the atmosphere. Microscopes are limited by the wavelength of light and the numerical aperture — modern super-resolution techniques (PALM, STORM, STED) get below the classical "Abbe limit" by clever fluorophore tricks.
**Scenario:** A 10-inch (254 mm) amateur telescope viewing the moon at high magnification. What's its diffraction-limited resolution? Is 200× useful? **Calculation:** θ = 1.22 × 550 × 10⁻⁹ / 0.254 = 2.64 × 10⁻⁶ rad = 0.54 arcsec. At moon distance (~384,000 km): d_min = 384,000 × 10³ × 2.64 × 10⁻⁶ = 1000 m = 1 km. **Result:** A 10-inch scope can theoretically resolve 1 km features on the moon. Useful magnification rule of thumb: ~50× per inch of aperture = 500× for a 10-inch scope, but atmospheric seeing typically limits real performance to ~200×. For lunar craters and Saturn's rings, 200× is plenty; beyond that, more magnification just makes the same blur larger.
**Scenario:** Smartphone camera with 4mm effective focal length at f/1.8 (aperture diameter ≈ 2.2 mm). Pixel size ≈ 1.5 µm. Does diffraction limit image sharpness? **Calculation:** Airy disk = 2.44 × 550 × 10⁻⁹ × 1.8 m = 2.4 µm diameter. Compared to 1.5 µm pixel size, Airy disk is bigger than 1 pixel — borderline diffraction-limited even wide open. Stopping down to f/2.8 would give Airy disk = 3.8 µm = 2.5 pixels, clearly softer. **Result:** Most smartphone cameras are diffraction-limited or close to it. There's a hardware physics reason cameras can't just keep making smaller pixels: below a certain pixel size, diffraction blur exceeds the pixel size and adding more pixels doesn't help. The phone sensor war has therefore shifted toward bigger sensors with bigger pixels (Sony IMX989 in flagships, ~2.4 µm pitch).
**Scenario:** A reconnaissance satellite at 500 km altitude with a 2.4 m primary mirror images the ground at visible wavelength. What's the diffraction-limited ground resolution? **Calculation:** θ = 1.22 × 550 × 10⁻⁹ / 2.4 = 2.8 × 10⁻⁷ rad. d_min = 500 × 10³ × 2.8 × 10⁻⁷ = 0.14 m = 14 cm. **Result:** Diffraction-limited ground resolution ~14 cm — fine enough to identify vehicles, count people, read very large signs. Real reconnaissance satellites with 2.4m mirrors (KH-11 class) reportedly achieve ~10–20 cm resolution under good conditions, matching the theoretical limit. The atmosphere adds some blur but adaptive optics and good site selection minimize it.
**Use angular resolution math for:**
- **Telescope buying decisions**: aperture is the most important spec for resolution. - **Microscopy choice**: NA, immersion medium, and wavelength all set practical resolution. - **Camera lens optimization**: choosing aperture for sharpest image (small aperture = diffraction; large = aberrations; sweet spot in middle). - **Surveillance and aerial imaging**: estimating ground resolution from camera specs and altitude. - **Radar and antenna design**: same Rayleigh limit applies (with λ at radar/radio wavelengths). - **Laser system design**: beam divergence is similar to diffraction-limited angular spread. - **Optical communications**: pointing accuracy required to maintain link over distance. - **Space-based imaging**: planning instrument resolution for satellite missions.
**Camera diffraction sweet spot:**
For each lens, there's an optimal aperture where aberrations have decreased but diffraction hasn't taken over. Usually 2–4 f-stops below maximum aperture:
- Full-frame DSLR/mirrorless: sweet spot ~f/5.6 to f/8. - APS-C: similar f-stops but smaller effective Airy disk. - Smartphone: usually fixed aperture near diffraction limit anyway.
**Telescope buying guidance:**
- Aperture matters most. Doubling diameter doubles angular resolution and quadruples light-gathering area. - 6–10 inches is the practical amateur sweet spot for portability and price. - Mount stability matters more than slightly larger aperture beyond a certain point. - Light pollution is a bigger limit than diffraction for many users.
**Beyond Rayleigh:**
- **Sparrow criterion**: slightly less stringent than Rayleigh, two points "merged" when no central dip between them. - **Dawes' limit** (for binary stars, lower coefficient ~1.0 instead of 1.22): empirical. - **Super-resolution microscopy** (PALM, STORM, STED): clever fluorescence makes effective resolution 20–50 nm despite diffraction. - **Adaptive optics**: real-time correction of atmospheric distortion can approach diffraction-limited. - **Speckle interferometry, lucky imaging**: get diffraction-limited images from short exposures, post-process to combine.
**Common units for angular resolution:**
- arcseconds (1° = 3600 arcsec) - microradians (1 µrad ≈ 0.206 arcsec) - milliradians (1 mrad ≈ 206 arcsec ≈ 3.4 arcmin)
Calculate diffraction angles, angular dispersion, and resolving power using the grating equation mλ = d(sinα + sinβ).
Calculate Rayleigh range, divergence angle, beam diameter at distance, and depth of focus for a Gaussian beam.
Calculate the focused spot diameter, depth of focus, and peak intensity of a laser beam focused by a lens.
Calculate focal length from magnification, object/image distance, or field of view. Versatile focal length solver.
Calculate optical fiber numerical aperture, acceptance angle, and V-number. NA = sqrt(n_core² - n_clad²).
Calculate prism deviation angle, minimum deviation, and angular dispersion for optical prisms.
Distance to calculate minimum resolvable feature size
Resolution
1.38 arcsec
Min Distance
6.71 mm
Resolution (mrad)
0.0067
| Parameter | Value |
|---|---|
| Angular Resolution | 6.7100e-6 rad |
| Angular Resolution | 1.3840 arcsec |
| Angular Resolution | 0.006710 mrad |
| Min Resolvable Distance | 6.7100 mm at 1000 m |
| Wavelength | 550 nm |
| Aperture Diameter | 100 mm |
| Formula | θ = 1.22λ/D |