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 = D 2 2 λ , 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,

( Z + λ 4 ) 2 = Z 2 + ( D 2 ) 2 .

Rearranging, and simplifying

Z = D 2 2 λ − 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 which corresponds to the border between the Fresnel and Frauenhofer 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.