Rayleigh criterion

A lens with a circular aperture images each point of an object as an Airy disk, a bright core surrounded by faint rings. When two points are close, their disks overlap, and at some separation the pair can no longer be told apart from a single blurred spot. Lord Rayleigh's criterion puts that limit where the peak of one pattern lies on the first zero of the other. For an aperture of diameter $D$ the angular separation is

the factor 1.22 being the first zero of the Bessel function $J_1$ divided by $\pi$. In the image plane of a lens at f-number $N$ the separation is $1.22\lambda N$, and for a microscope objective it is written in object space as $d = 0.61\lambda/\mathrm{NA}$.

At that separation the summed intensity, for two equally bright incoherent points, dips to 73.5% of the peaks midway between them, a dip of about 26%, which a human observer can see.

Typical values

An oil-immersion objective with an NA of 1.4 resolves 240 nm at 550 nm. A human eye with a 3 mm pupil has a Rayleigh limit of 46 arcseconds at 550 nm, close to its measured acuity of about one arcminute. The Hubble Space Telescope, with its 2.4 m mirror, reaches 0.052 arcseconds at 500 nm; a ground-based telescope of any size is limited by the atmosphere unless it uses adaptive optics.

Other criteria

The Rayleigh criterion is a convention for visual resolution, and other conventions give somewhat different numbers. The Sparrow criterion takes the separation at which the dip just vanishes, about $0.47\lambda/\mathrm{NA}$, 185 nm in the example above, and the Abbe limit for periodic structures, $0.5\lambda/\mathrm{NA}$, gives 196 nm. With a good signal-to-noise ratio and a known point spread function, sources closer than any of these can be distinguished by fitting, and the separation of two emitters can be measured to far better than the criterion; localization microscopy relies on this. The modulation transfer function gives the fuller picture, since contrast falls gradually rather than disappearing at one separation.

Measurement

Resolution is tested by imaging pairs of point sources, closely spaced pinholes or a resolution target such as the USAF 1951 bar chart, and finding the finest element still separated. In microscopy, sub-resolution fluorescent beads give the point spread function, from which the resolution follows.

References: Lord Rayleigh, Phil. Mag. 8, 261 (1879); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 8; C. M. Sparrow, Astrophys. J. 44, 76 (1916).