Last updated: 2026-06-16

Rayleigh criterion

The Rayleigh criterion defines the minimum angular separation between two point sources that a circular aperture can resolve. When two equally bright points are separated by exactly this angle, the central maximum of one Airy pattern falls on the first dark ring of the other — the boundary often taken as the limit of resolvability.

Diffraction and the Airy disk

When light from a distant point source passes through a circular aperture of diameter $D$, diffraction spreads the beam into a pattern of concentric rings called an Airy pattern. The bright central spot — the Airy disk — contains about 84 % of the total energy. The angular radius of the first dark ring is

where $\lambda$ is the wavelength and $\theta$ is in radians. Two point sources separated by less than $\theta$ produce overlapping Airy disks that cannot be distinguished.

Origin of the 1.22 factor

The factor 1.22 is not arbitrary. The intensity of the Airy pattern along a radial direction is proportional to $[J_1(x)/x]^2$, where $J_1$ is the Bessel function of the first kind of order one. The first zero of $J_1(x)$ occurs at $x \approx 3.8317$, giving $x/\pi \approx 1.22$ once the geometry of a circular aperture is applied. For a rectangular slit the diffraction pattern is a sinc function whose first zero gives a factor of exactly unity, so the Rayleigh criterion for a slit is $\theta = \lambda / D$.

Formula summary

To convert from radians to arcseconds, multiply by 206,265. To convert to arcminutes, multiply by 3,438.

Worked example

A 100 mm refractor telescope observes a double star in yellow-green light at $\lambda = 550\,\text{nm}$. The minimum resolvable angle is

Two stars closer than 1.38 arcseconds would appear as a single point in this telescope under perfect conditions. Dawes' empirical limit for the human eye at the eyepiece gives a similar figure of 116 / D (mm) arcseconds, which equals 1.16 arcsec for 100 mm — close to the diffraction prediction.

Aperture and resolution

Angular resolution improves linearly with aperture. Doubling $D$ halves $\theta$, so the telescope can distinguish objects twice as close in angle. This relationship drives the design of large instruments across optics:

• Telescopes — the Hubble Space Telescope's 2.4 m mirror gives $\theta \approx 0.05\,\text{arcsec}$ at 550 nm, roughly ten times finer than a 200 mm amateur telescope.
• Microscopes — resolution is governed by the same physics via the numerical aperture $\text{NA} = n\sin\alpha$, where $n$ is the refractive index of the medium. Oil-immersion objectives achieve $\text{NA} > 1$, which reduces the effective wavelength and pushes resolution below 200 nm.
• Camera lenses — diffraction sets the sharpest aperture of a lens; stopping down too far (high f-number) increases $\lambda / D$ and softens fine detail even though depth of field grows.
• The human eye — the pupil diameter ranges from about 2 mm in bright light to 8 mm in the dark. At 550 nm and 2 mm the diffraction limit is about 1.4 arcmin, consistent with the accepted visual acuity limit of 1 arcmin.

Limits of the model

The Rayleigh criterion is the theoretical limit for a perfect, aberration-free circular aperture. Several factors degrade real-world performance:

• Atmospheric seeing — turbulence in the atmosphere blurs images, typically limiting ground-based telescopes to 0.5–2 arcsec regardless of aperture. Adaptive optics systems correct much of this in real time.
• Optical aberrations — spherical aberration, coma, and astigmatism in imperfect optics spread the point-spread function beyond the Airy disk.
• Obstruction — a central obstruction (as in a Cassegrain telescope) redistributes energy from the Airy disk into the rings, reducing peak contrast.
• Detector sampling — to record detail at the diffraction limit the pixel scale must be fine enough to satisfy the Nyquist criterion: at least two pixels per resolution element.