Rayleigh distance

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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 ${\displaystyle Z={\frac {D^{2}}{2\lambda }}}$, 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 ${\displaystyle Z}$, opposite ${\displaystyle {\frac {D}{2}}}$and hypotenuse ${\displaystyle Z+{\frac {\lambda }{4}}}$. According to Pythagorean theorem,

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

Rearranging, and simplifying

${\displaystyle Z={\frac {D^{2}}{2\lambda }}-{\frac {\lambda }{8}}}$

The constant term${\displaystyle {\frac {\lambda }{8}}}$ can be neglected.

In antenna applications, the Rayleigh distance is often given as four times this value, i.e. ${\displaystyle Z={\frac {2D^{2}}{\lambda }}}$^[1] 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.^[2]).

The Rayleigh distance is also the 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.^[3]