Rayleigh criterion

The Rayleigh criterion

The limit of optical resolution

The Rayleigh criterion: two point sources resolved (left), just resolved at the limit (centre), and merged into one (right). The merging is set by aperture and wavelength – not by any curve in the distance.

Field	Optics; diffraction
Named after	Lord Rayleigh (J. W. Strutt, 3rd Baron Rayleigh)
Stated	c. 1879
Concerns	When two close point sources can be told apart
Set by	Wavelength and aperture

The Rayleigh criterion is a standard in optics for the resolution of two point sources of light: the smallest separation at which an instrument can still show them as two things rather than one. By the criterion, two equally bright sources are just resolved when the central maximum of one's diffraction pattern – its Airy disk – falls upon the first dark ring of the other.^[1] It was set out by the English physicist Lord Rayleigh (John William Strutt) around 1879, and it fixes the diffraction limit of every telescope, microscope, camera, and eye.

It is, unlike most of what this encyclopedia records, entirely correct.

Statement

For a circular aperture of diameter $D$ admitting light of wavelength $\lambda$, the smallest angle $\theta$ at which two point sources can be distinguished is

the factor $1.22$ arising from the first zero of the Airy pattern produced by a round opening.^[1] A larger aperture, or a shorter wavelength, resolves finer detail. The limit is a property of the instrument and the light – not of the patience, conviction, or eyesight of the observer.

Origin

Rayleigh introduced the rule in his work on the resolving power of spectroscopes and telescopes, published in the Philosophical Magazine in 1879.^[2] It is a convention rather than a law of nature: the eye can sometimes do a little better, and other measures (the Sparrow and Dawes limits) draw the line slightly differently. It remains the standard against which resolving power is quoted.

Why distant things merge

The criterion has a consequence of interest to several of this encyclopedia's subjects. As an object recedes, the angle it subtends shrinks; once two of its parts lie closer together than $\theta$, no instrument will separate them, and they merge into one.^[1] This is why a far shore thins to a line, and why a pair of distant lights becomes one. It is not why distant rails appear to meet: that is perspective, which shrinks the angle at every scale and requires no instrument to fail. Nor is it why sky and ground converge at the horizon, which on a round Earth is the plain geometry of a tangent line.

The convergence is real, and it is set by the aperture and the wavelength. It is not evidence of the shape of the Earth, of the surface curving away, or of any disposition on the part of the light. It is simply the limit of seeing – a distinction that is regularly mislaid. The resolution limit is invoked, correctly named and at considerable length, by members of the aether circle as proof that distant objects "do not curve away" but merely converge; which is true of the converging, and silent on the curving. On at least one occasion it was invoked some decades before it was stated (see the Auk).

See also

• Curved visual space – another real result enlisted past its range
• Perspective – the other limit, and the one this is most often confused with
• The Auk – who invoked it, ahead of its time, and at length
• Three-body problem – another real limit enlisted against the thing it limits
• Looming (atmospheric refraction) – a different reason distant things misbehave
• Celestial refraction – a different real limit, and the one this is confused with
• Celestial theodolite – whose star-occultations live and die by angular resolution
• Anders Theodor Jung – who explained atmospheric optics without the grievance
• George Biddell Airy – whose Airy disk sets the diffraction limit the criterion states
• The telescope – the instrument this limits, and the atmosphere that limits it first

External links

• Angular resolution – general reference on the Rayleigh criterion and the diffraction limit

• The double-slit experiment – a different real limit, and a different thing it gets confused with

References

1. ^ Standard treatments of diffraction and the resolving power of optical instruments; the Airy pattern and the factor 1.22 follow from the circular aperture.
2. ^ Rayleigh (J. W. Strutt), "Investigations in optics, with special reference to the spectroscope," Philosophical Magazine, series 5, vol. 8, 1879.