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. For a circular aperture, this gives angular resolution θ = 1.22λ/D radians, where λ is wavelength and D is aperture diameter.

What limits angular resolution in practice?

Diffraction sets the theoretical limit, but atmospheric turbulence (seeing), optical aberrations, and detector pixel size often limit practical angular resolution. Adaptive optics can partially correct atmospheric effects; space telescopes (above the atmosphere) come closest to the diffraction limit.

Why does aperture matter so much for telescopes?

Two reasons: more area collects more light (sensitivity scales with D²), and more aperture gives finer diffraction-limited resolution (scales with 1/D). Doubling diameter quadruples light gathering AND doubles angular resolution. This is why telescope diameter is the most important spec.

What is the diffraction limit in microscopy?

For light microscopy: d_min ≈ λ/(2×NA), known as the Abbe limit. With visible light and high-NA oil-immersion objectives (NA = 1.4), this is ~200 nm. Super-resolution techniques (STED, PALM, STORM) break this limit through clever fluorophore management, achieving 20–50 nm resolution.

Why does my camera lens get soft at small apertures?

Smaller aperture = larger Airy disk relative to pixel size. At f/22 on a full-frame camera, Airy disk ≈ 29 µm, much larger than typical 5-6 µm pixels. The image is uniformly soft across the frame. Most lenses have a "sweet spot" around f/5.6–f/8 where aberrations are minimized but diffraction isn't yet dominant.

What is the angular resolution of the human eye?

Diffraction-limited (8mm pupil at 550 nm): about 17 arcsec. Real-world acuity is limited by cone spacing in the fovea: about 1 arcmin = 60 arcsec ("20/20 vision"). At reading distance of 35 cm, this corresponds to features ~100 µm — small text is at this limit.

Can adaptive optics achieve diffraction-limited imaging?

For bright point sources (laser guide stars or natural reference stars), yes — modern AO systems on 8–10 m telescopes achieve close to diffraction-limited imaging at near-IR wavelengths. For visible wavelengths it's harder; for faint extended objects (galaxies), AO improvement is less dramatic. JWST and Hubble (space-based) sidestep the issue entirely.

Angular Resolution Calculator

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.

Inputs

Wavelength (nm)

Aperture Diameter (mm)

Distance to Target (m)

Distance to calculate minimum resolvable feature size

Results

Resolution

1.38 arcsec

Min Distance

6.71 mm

Resolution (mrad)

0.0067

Angular Resolution Results

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

Last updated: May 29, 2026

Formula

**Rayleigh criterion (angular resolution):** θ = 1.22 × λ / D Where: - **θ**: minimum resolvable angular separation (radians) - **λ**: wavelength of light - **D**: aperture diameter - **1.22**: numerical coefficient from the first zero of the Airy function To convert to arcseconds: θ (arcsec) = θ (radians) × 206,265 **Minimum resolvable distance at range R:** d_min = R × θ ≈ 1.22 × λ × R / D **Worked example: Hubble Space Telescope (D = 2.4 m, λ = 550 nm)** θ = 1.22 × 550 × 10⁻⁹ / 2.4 = 2.8 × 10⁻⁷ rad = 0.058 arcsec At Mars distance (~225 million km), Hubble can theoretically resolve features ~225 × 10⁹ × 2.8 × 10⁻⁷ = ~63 km. Real Hubble Mars images show resolved features ~50 km — close to the diffraction limit. **Telescope aperture vs resolution at visible wavelength (550 nm):** | Aperture D | θ (arcsec) | Best feature at 1 km | |---|---|---| | 1 mm (pinhole) | 138 | 0.67 m | | 1 cm (small lens) | 13.8 | 6.7 cm | | 5 cm | 2.76 | 1.3 cm | | 10 cm (4-inch telescope) | 1.38 | 0.67 cm | | 25 cm (10-inch telescope) | 0.55 | 2.7 mm | | 50 cm (20-inch) | 0.28 | 1.4 mm | | 1 m (medium telescope) | 0.138 | 0.7 mm | | 2.4 m (Hubble) | 0.058 | 0.28 mm | | 8 m (Gemini/Keck) | 0.017 | 84 µm | **Microscope resolution (numerical aperture form):** d_min ≈ 0.61 × λ / NA (Rayleigh, transverse) d_min ≈ λ / (2 × NA) (Abbe limit, more common) Where NA = n × sin(half-angle), and n is the immersion medium index. | NA | Min feature (λ=550 nm) | |---|---| | 0.25 | 1.10 µm | | 0.65 | 423 nm | | 0.95 (air) | 290 nm | | 1.40 (oil immersion) | 196 nm | Below ~200 nm in the visible is the "Abbe limit." Super-resolution microscopy (STED, PALM, STORM) breaks this limit by clever fluorescence tricks. **Camera lens diffraction (f-number = focal_length / D):** For a lens at f-number N, the Airy disk diameter on the sensor is: d_Airy = 2.44 × λ × N | f-number | Airy disk (visible) | |---|---| | f/2.8 | 3.8 µm | | f/5.6 | 7.5 µm | | f/8 | 10.7 µm | | f/16 | 21.4 µm | | f/22 | 29.4 µm | When Airy disk exceeds 2× pixel size, diffraction softens the image. On a typical full-frame 24-megapixel camera (pixel pitch 5.9 µm), stopping below f/8 starts hurting sharpness; f/22 is significantly soft. **Atmospheric "seeing":** - Best mountain-top observatory: 0.5–1 arcsec under good conditions. - Typical good location: 1.5–2 arcsec. - Adaptive optics correction: down to ~0.1 arcsec for bright stars. - Space (no atmosphere): diffraction-limited (matches the Rayleigh formula). **Human eye angular resolution:** - Pupil 2–8 mm; at λ = 550 nm, diffraction-limited θ ≈ 80–340 µrad = 16–70 arcsec. - Foveal cone spacing limits real-world acuity to ~30 arcsec (1 minute of arc). - 20/20 vision = 1 minute of arc resolution (~290 µrad).

How to use this calculator

Enter the wavelength (550 nm for green visible, 1064 nm for Nd:YAG laser, 21 cm for radio H I line).
Enter the aperture diameter in mm (1 inch = 25.4 mm; 1 ft = 305 mm).
For telescope work, distance is usually "infinity" — focus on angular resolution.
For nearby targets, enter the distance to see the minimum resolvable feature size at that range.
For camera diffraction questions, use Airy disk size = 2.44 × λ × f-number.
Real-world resolution is often worse than the diffraction limit due to atmosphere, aberrations, or detector limits.

Worked examples

Why amateur telescopes max out around 200×

**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.

Smartphone camera diffraction limit

**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).

Spy satellite ground resolution

**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.

When to use this calculator

**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)

Common mistakes to avoid

Confusing magnification with resolution. Magnification just makes images bigger; resolution determines what details you can see.
Forgetting wavelength dependence. UV gives much better resolution than IR at same aperture (shorter λ).
Treating diffraction as a "soft" limit. It's a hard physical limit set by the wave nature of light.
Ignoring sensor pixel size. Resolution is limited by the worse of optical (diffraction + aberrations) and sampling (pixel size, sensor density).
Computing resolution for circular apertures using rectangular formulas. The 1.22 coefficient is specifically for circular apertures.
Forgetting the Earth's atmosphere. Even ground-based 8m telescopes are seeing-limited, not diffraction-limited, without adaptive optics.
Trusting "claimed resolution" specs without understanding the underlying metric. Specs vary on what limits are included.

Frequently Asked Questions

Sources & further reading

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