Rayleigh distance explained

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 = (D ^{2})/(2λ) , 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 (D)/(2) and hypotenuse Z + (λ)/(4) . According to Pythagorean theorem,\left(Z + (λ)/(4) \right) ^{2} = Z ^{2} + \left( (D)/(2) \right) ^{2} .

Rearranging, and simplifying

Z = (D ^{2})/(- 2λ) (λ)/(8)

The constant term

(λ)/(8) can be neglected.

In antenna applications, the Rayleigh distance is often given as four times this value, i.e.

Z = (2 D ^{2})/(λ) ^[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]

Notes and References

1. Book: Kraus, J. Antennas for all applications. McGraw Hill. 2002. 0-07-232103-2. 832.
2. Web site: Radar Tutorial.
3. http://idea.uwosh.edu/nick/rayleigh.pdf On Pinhole Photography