# Rayleigh distance

Rayleigh distance in optics is the axial distance from a radiating aperture to a point at which the path difference between the axial ray and an edge ray is λ / 4. An approximation of the Rayleigh Distance is $Z={\frac {D^{2}}{2\lambda }}$ , in which Z is the Rayleigh distance, D is the aperture of radiation, λ the wavelength. This approximation can be derived as follows. Consider a right angled triangle with sides adjacent $Z$ , opposite ${\frac {D}{2}}$ and hypotenuse $Z+{\frac {\lambda }{4}}$ . According to Pythagorean theorem,

${\big (}Z+{\frac {\lambda }{4}}{\big )}^{2}=Z^{2}+{\big (}{\frac {D}{2}}{\big )}^{2}$ .

Rearranging, and simplifying

$Z={\frac {D^{2}}{2\lambda }}-{\frac {\lambda }{8}}$ The constant term${\frac {\lambda }{8}}$ can be neglected.

In antenna applications, the Rayleigh distance is often given as four times this value, i.e. $Z={\frac {2D^{2}}{\lambda }}$ which corresponds to the border between the Fresnel and Fraunhofer regions and denotes the distance at which the beam radiated by a reflector antenna is fully formed (although sometimes the Rayleigh distance it is still given as per the optical convention e.g.).

Actually, Rayleigh distance is also a distance beyond which the distribution of the diffracted light energy no longer changes according to the distance Z from the aperture. It is the reduced Fraunhofer diffraction limitation.

Lord Rayleigh's paper on the subject was published in 1891.

1. Kraus, J (2002). Antennas for all applications. McGraw Hill. p. 832. ISBN 0-07-232103-2.
2. "Radar Tutorial".
3. On Pinhole Photography
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