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Rayleigh length

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Rayleigh length

In optics and especially laser science, the Rayleigh length or Rayleigh range is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. A related parameter is the confocal parameter, b, which is twice the Rayleigh length. The Rayleigh length is particularly important when beams are modeled as Gaussian beams.

Contents

Explanation

For a Gaussian beam propagating in free space along the z ^ axis, the Rayleigh length is given by

z R = π w 0 2 λ ,

where λ is the wavelength and w 0 is the beam waist, the radial size of the beam at its narrowest point. This equation and those that follow assume that the waist is not extraordinarily small; w 0 ≥ 2 λ / π .

The radius of the beam at a distance z from the waist is

w ( z ) = w 0 1 + ( z z R ) 2 .

The minimum value of w ( z ) occurs at w ( 0 ) = w 0 , by definition. At distance z R from the beam waist, the beam radius is increased by a factor 2 and the cross sectional area by 2.

The total angular spread of a Gaussian beam in radians is related to the Rayleigh length by

Θ d i v ≃ 2 w 0 z R .

The diameter of the beam at its waist (focus spot size) is given by

D = 2 w 0 ≃ 4 λ π Θ d i v .

These equations are valid within the limits of the paraxial approximation. For beams with much larger divergence the Gaussian beam model is no longer accurate and a physical optics analysis is required.

References

Rayleigh length Wikipedia
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