Rayleigh criterion

A point source imaged through an optical system with a finite circular aperture does not form a perfect point. Diffraction produces a central bright spot surrounded by rings, known as an Airy pattern. Two close sources produce overlapping patterns, so their images may blend and become difficult to distinguish. This applies to systems such as telescopes, microscopes and cameras, although real performance is also affected by aberrations, detector properties, turbulence and data processing.

Lord Rayleigh proposed a practical convention for deciding when two point-source images are just resolved: the central maximum of one Airy pattern falls at the first dark minimum of the other. For an ideal circular aperture, the corresponding small-angle separation is θ ≈ 1.22 λ/D, where θ is in radians, λ is the wavelength, and D is the aperture diameter. The factor 1.22 comes from the first minimum of the circular-aperture diffraction pattern. The formula assumes diffraction-limited imaging and a circular pupil; other aperture shapes or imaging conditions have different point-spread patterns.

The expression shows that, all else equal, a larger aperture or shorter wavelength reduces the minimum angular separation under this criterion. In the small-angle regime, an object separation s at distance r corresponds approximately to θ = s/r. Thus a telescope's angular resolution can be translated into a minimum physical separation at a stated distance. The Rayleigh estimate describes angular separation, not a universal minimum feature size independent of distance and setup.

Rayleigh's rule is a useful and widely taught threshold for conventional imaging of two point sources, but the word 'just' matters: it is an operational criterion rather than a claim that no information is available below that separation. The detail that can be inferred depends on source brightness and coherence, noise, the point-spread function and the measurement procedure. Research on optical imaging has shown that appropriately designed measurements can estimate properties of some sub-diffraction source configurations. Such results refine how resolution is assessed; they do not make the ideal-aperture formula irrelevant for its intended classical benchmark.