Rayleigh Criterion Calculator
Calculate the diffraction-limited angular resolution θ = 1.22λ/D from aperture diameter and wavelength.
Outputs radians, arcseconds, and arcminutes for any optical system.
The Rayleigh criterion is the fundamental physical limit on how sharply any optical instrument can resolve detail. It comes straight from the wave nature of light: pass a wave through a circular opening and it diffracts. No matter how good your lens or mirror is, two point sources closer together than this angle can’t be told apart:
θ = 1.22 × λ / D
θ is in radians, λ is the wavelength of light, D is the diameter of the aperture (telescope mirror, lens, microscope objective, camera pupil). Bigger aperture or shorter wavelength → finer resolution. There is no way around this with passive optics.
Where the 1.22 comes from
It’s the first zero of the first-order Bessel function J₁, which describes the Airy pattern (the diffraction image of a point source through a round aperture). The pattern is a bright central disk surrounded by faint rings. The Rayleigh criterion says: two sources are “just resolvable” when the central peak of one pattern sits at the first dark ring of the other. The factor 1.22 specifically applies to circular apertures. A slit (rectangular aperture) would use 1.0 instead.
Two ways to improve resolution
The formula gives you exactly two knobs:
- Larger D: doubling aperture halves the angle. Why telescopes have gotten huge: Keck (10 m), the Extremely Large Telescope (39 m), the James Webb (6.5 m). Each is hunting more resolution.
- Shorter λ: switching from visible (550 nm) to UV (200 nm) gains 2.75× resolution. Going to X-ray (1 nm) gains 550×. This is why electron microscopes (de Broglie wavelength ~0.005 nm for 100 keV electrons) routinely resolve individual atoms while light microscopes top out around 200 nm.
Quick reference for common instruments
Assuming green light (λ = 550 nm):
| Instrument | Aperture D | θ (radians) | θ (arcseconds) | Practical meaning |
|---|---|---|---|---|
| Human eye (day pupil) | 5 mm | 1.34 × 10⁻⁴ | 28 | About 1 arcminute (3 feet at 1 km) |
| Binoculars 50 mm | 50 mm | 1.34 × 10⁻⁵ | 2.8 | Splits binary stars 5"+ apart |
| Backyard telescope 200 mm | 200 mm | 3.4 × 10⁻⁶ | 0.69 | Resolves Jupiter’s cloud bands |
| Hubble Space Telescope | 2.4 m | 2.8 × 10⁻⁷ | 0.058 | Reads a license plate at 5,000 km (in theory) |
| ELT | 39 m | 1.7 × 10⁻⁸ | 0.0036 | Resolves planets around nearby stars |
| Very Large Array (radio, λ = 21 cm) | 36 km baseline | 7.1 × 10⁻⁶ | 1.5 | Matches a 60 cm optical telescope, but at radio wavelengths |
For radio telescopes the wavelength is so long that aperture has to be kilometers to match what a 1-meter optical telescope does. That’s why radio astronomy uses interferometry: a network of dishes acts like one effectively-huge aperture.
Two unit conversions worth memorizing
- 1 radian = 206,265 arcseconds (very useful for astronomy)
- 1 radian = 3,438 arcminutes (more for survey/optics)
- 1 arcsecond ≈ 4.85 × 10⁻⁶ radians
The “1 arcminute” rule of thumb for human vision (Snellen 20/20) corresponds to about 2.9 × 10⁻⁴ radians.
Worked example: can you split this binary star?
The double star Albireo (β Cygni) has components separated by 34 arcseconds. Through a backyard 200 mm telescope at 550 nm:
θ_limit = 1.22 × 550×10⁻⁹ / 0.2 = 3.4 × 10⁻⁶ rad = 0.69 arcseconds
Albireo’s 34" separation is about 50× wider than the limit, so it splits cleanly even in a small telescope.
But Sirius A and B (the Pup) sit only ~10" apart and the B component is 10,000× fainter. That’s a separate problem from pure angular resolution — it’s about contrast, not just angle. Rayleigh’s formula gives the angle; whether you actually see the fainter object depends on the optical system’s stray-light performance.
Photography: where the diffraction limit hits
For camera lenses, the relevant aperture isn’t the lens housing but the f-stop aperture. A 50 mm lens at f/16 has effective D ≈ 50/16 = 3.125 mm. At 550 nm:
θ = 1.22 × 550×10⁻⁹ / 0.003125 ≈ 2.15 × 10⁻⁴ rad
At a 36 mm sensor distance, that’s a spot size of: 36 × 10⁻³ × 2.15 × 10⁻⁴ ≈ 7.7 μm
Modern full-frame sensors have pixels around 4 to 6 μm. So at f/16 the diffraction spot is larger than a single pixel, and stopping down further only blurs the image. This is the “diffraction limit” photographers talk about. For sharpest results, most lenses peak around f/5.6 to f/8, where aberrations are minimized but diffraction hasn’t yet dominated.
Atmospheric seeing: the real ground-based limit
For ground-based optical telescopes, atmospheric turbulence dominates over Rayleigh resolution at apertures above about 15 to 20 cm. The “seeing” — atmospheric blur — typically limits resolution to 1 to 2 arcseconds even on excellent observing nights, regardless of telescope size. This is why large telescopes use adaptive optics (mirrors that flex in real time to cancel turbulence) or get launched into space.
Why this calculator matters
- Telescope buyers checking what their aperture can resolve
- Astrophotographers picking the right f-number to balance diffraction vs aberrations
- Microscope users matching objective NA to required resolution
- Anyone wondering why electron microscopes need vacuum and high voltages (the de Broglie wavelength shortens with electron energy)