Result

Rayleigh criterion

θ_{m i n} = 1.22 \frac{λ}{D}

Statement

The minimum angular separation two point sources can have and still be resolved through a circular aperture of diameter D.

Symbols and units

$θ_{m i n}$	minimum resolvable angular separation	rad
$λ$	wavelength	m
$D$	aperture diameter	m

When it applies

A circular aperture; diffraction-limited imaging (no other aberration dominating).

Don't use when

Assumes diffraction-limited imaging through a circular aperture with no other aberration dominating — a poorly-figured lens or mirror may be limited by aberration well before reaching this diffraction limit.

Derivation

Set by the position of the first dark ring in the diffraction pattern of a circular aperture — analogous to single-slit diffraction, with a geometric factor of 1.22 for a circular rather than rectangular opening.

θ_{m i n} = 1.22 \frac{λ}{D}

Watch out for

Resolution improves — θ_min shrinks — with a LARGER aperture and SHORTER wavelength, which is why telescopes are built as large as possible, and why electron microscopes achieve far finer resolution than light microscopes.

Connects to

Single-slit diffraction