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

[math]\displaystyle{ \left(Z+\frac{\lambda}{4}\right)^2 =Z^2 + \left(\frac{D}{2} \right)^2 }[/math].

Rearranging, and simplifying

[math]\displaystyle{ Z = \frac{D^2}{2\lambda}-\frac{\lambda}{8} }[/math]

The constant term[math]\displaystyle{ \frac{\lambda}{8} }[/math] can be neglected.

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

1. ↑ Kraus, J (2002). Antennas for all applications. McGraw Hill. pp. 832. ISBN 0-07-232103-2.
2. ↑ "Radar Tutorial". http://www.radartutorial.eu/06.antennas/an62.en.html.
3. ↑ On Pinhole Photography

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