Rayleigh Length

Gaussian beam width [math]\displaystyle{ w(z) }[/math] as a function of the axial distance [math]\displaystyle{ z }[/math]. [math]\displaystyle{ w_0 }[/math]: beam waist; [math]\displaystyle{ b }[/math]: confocal parameter; [math]\displaystyle{ z_\mathrm{R} }[/math]: Rayleigh length; [math]\displaystyle{ \Theta }[/math]: total angular spread

In optics and especially laser science, the Rayleigh length or Rayleigh range, [math]\displaystyle{ z_\mathrm{R} }[/math], 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.^[1] A related parameter is the confocal parameter, b, which is twice the Rayleigh length.^[2] The Rayleigh length is particularly important when beams are modeled as Gaussian beams.

Explanation

For a Gaussian beam propagating in free space along the [math]\displaystyle{ \hat{z} }[/math] axis with wave number [math]\displaystyle{ k = 2\pi/\lambda }[/math], the Rayleigh length is given by^[2]

[math]\displaystyle{ z_\mathrm{R} = \frac{\pi w_0^2}{\lambda} = \frac{1}{2} k w_0^2 }[/math]

where [math]\displaystyle{ \lambda }[/math] is the wavelength (the vacuum wavelength divided by [math]\displaystyle{ n }[/math], the index of refraction) and [math]\displaystyle{ w_0 }[/math] 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; [math]\displaystyle{ w_0 \ge 2\lambda/\pi }[/math].^[3]

The radius of the beam at a distance [math]\displaystyle{ z }[/math] from the waist is^[4]

[math]\displaystyle{ w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 } . }[/math]

The minimum value of [math]\displaystyle{ w(z) }[/math] occurs at [math]\displaystyle{ w(0) = w_0 }[/math], by definition. At distance [math]\displaystyle{ z_\mathrm{R} }[/math] from the beam waist, the beam radius is increased by a factor [math]\displaystyle{ \sqrt{2} }[/math] and the cross sectional area by 2.

Related quantities

The total angular spread of a Gaussian beam in radians is related to the Rayleigh length by^[1]

[math]\displaystyle{ \Theta_{\mathrm{div}} \simeq 2\frac{w_0}{z_R}. }[/math]

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

[math]\displaystyle{ D = 2\,w_0 \simeq \frac{4\lambda}{\pi\, \Theta_{\mathrm{div}}} }[/math].

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

↑ ^1.0 ^1.1 Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0-935702-11-3. https://archive.org/details/lasers0000sieg.
↑ ^2.0 ^2.1 Damask, Jay N. (2004). Polarization Optics in Telecommunications. Springer. pp. 221–223. ISBN 0-387-22493-9. https://archive.org/details/polarizationopti00dama.
↑ Siegman (1986) p. 630.
↑ Meschede, Dieter (2007). Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics. Wiley-VCH. pp. 46–48. ISBN 978-3-527-40628-9. https://archive.org/details/opticslightlaser00mesc_263.

Rayleigh length RP Photonics Encyclopedia of Optics

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[Siegman1986-1] 1.0 ^1.1 Siegman, A. E. (1986). Lasers. University Science Books. pp. 664–669. ISBN 0-935702-11-3. https://archive.org/details/lasers0000sieg.

[Damask-2] 2.0 ^2.1 Damask, Jay N. (2004). Polarization Optics in Telecommunications. Springer. pp. 221–223. ISBN 0-387-22493-9. https://archive.org/details/polarizationopti00dama.

[3] Siegman (1986) p. 630.

[4] Meschede, Dieter (2007). Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics. Wiley-VCH. pp. 46–48. ISBN 978-3-527-40628-9. https://archive.org/details/opticslightlaser00mesc_263.

Rayleigh Length

Contents

Explanation

Related quantities

See also

References