Rayleigh Criterion
Calculate the minimum angular resolution of telescopes, microscopes, and cameras using the Rayleigh criterion.
The Formula
The Rayleigh criterion defines the minimum angular separation at which two point sources of light can be distinguished as separate objects through a circular aperture. Named after Lord Rayleigh (John William Strutt), who formalized it in 1879, this formula sets a fundamental physical limit on the resolution of any optical instrument, from the human eye to the largest space telescopes.
The limitation comes from diffraction — the bending of light waves as they pass through an opening. Even a perfect lens with no aberrations produces a diffraction pattern (called an Airy disk) instead of a perfect point image. The Airy disk consists of a central bright spot surrounded by concentric rings of decreasing brightness. The Rayleigh criterion states that two sources are just barely resolvable when the central maximum of one Airy pattern falls on the first minimum (dark ring) of the other.
The factor 1.22 comes from the first zero of the Bessel function J1, which describes the diffraction pattern of a circular aperture. For a slit (rectangular aperture), the factor would be 1.0 instead. This specific value is what makes circular apertures slightly less sharp than their diameter alone might suggest.
The formula reveals two ways to improve resolution: use shorter wavelengths or use larger apertures. This is why radio telescopes (which observe long wavelengths) must be enormous — sometimes kilometers across — to achieve the same angular resolution as a small optical telescope. It is also why electron microscopes, which use the extremely short de Broglie wavelength of electrons, can resolve individual atoms while optical microscopes cannot.
In photography, the Rayleigh criterion explains why larger camera lenses produce sharper images (up to a point) and why stopping down too far (using very small f-numbers) actually reduces sharpness due to diffraction. Most camera lenses reach their sharpest performance two to three stops from wide open, balancing aberrations against diffraction.
The angle θ is measured in radians. To convert to arcseconds (commonly used in astronomy), multiply by 206,265 (the number of arcseconds in one radian).
Variables
| Symbol | Meaning |
|---|---|
| θ | Minimum resolvable angle (radians) |
| λ | Wavelength of light (meters) |
| D | Diameter of the circular aperture (meters) |
| 1.22 | Constant from the first zero of the Bessel function J1 |
Example 1
What is the angular resolution of a 20 cm (0.2 m) telescope observing green light (550 nm)?
Convert wavelength: λ = 550 nm = 550 × 10−9 m
θ = 1.22 × 550 × 10−9 / 0.2
θ = 671 × 10−9 / 0.2 = 3.355 × 10−6 radians
Convert to arcseconds: 3.355 × 10−6 × 206,265 ≈ 0.69"
θ ≈ 0.69 arcseconds — enough to split close double stars
Example 2
The human eye has a pupil diameter of about 5 mm in daylight. What is its resolution limit?
D = 5 mm = 0.005 m, λ = 550 × 10−9 m (peak sensitivity)
θ = 1.22 × 550 × 10−9 / 0.005
θ = 1.342 × 10−4 radians
Convert to arcminutes: 1.342 × 10−4 × (180/π) × 60 ≈ 0.46 arcminutes
θ ≈ 0.46 arcminutes — close to the commonly quoted 1 arcminute limit of human vision
When to Use It
The Rayleigh criterion is essential in any field that depends on resolving fine detail through optical systems.
- Choosing telescope aperture size for astronomical observations
- Determining the resolution limit of microscopes in biology and materials science
- Evaluating camera lens sharpness and diffraction-limited performance
- Designing satellite imaging systems for Earth observation
- Calculating the resolving power of radar and radio telescope arrays