Rayleigh length

{{more|Gaussian beam}}

For a Gaussian beam propagating in free space along the $\backslash hat\{z\}$ axis with wave number $k\; =\; 2\backslash pi/\backslash lambda$, the Rayleigh length is given by{{cite book | last = Damask | first = Jay N. | title = Polarization Optics in Telecommunications | url = https://archive.org/details/polarizationopti00dama | url-access = limited | publisher = Springer | date = 2004 | pages = [https://archive.org/details/polarizationopti00dama/page/n233 221]–223 | isbn = 0-387-22493-9 }}

:$z\_\backslash mathrm\{R\}\; =\; \backslash frac\{\backslash pi\; w\_0^2\}\{\backslash lambda\}\; =\; \backslash frac\{1\}\{2\}\; k\; w\_0^2$

where $\backslash lambda$ is the wavelength (the vacuum wavelength divided by $n$, the index of refraction) 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; $w\_0\; \backslash ge\; 2\backslash lambda/\backslash pi$.Siegman (1986) p. 630.

The radius of the beam at a distance $z$ from the waist is{{cite book | last = Meschede | first = Dieter | title = Optics, Light and Lasers: The Practical Approach to Modern Aspects of Photonics and Laser Physics | url = https://archive.org/details/opticslightlaser00mesc_263 | url-access = limited | publisher = Wiley-VCH | date = 2007 | pages = [https://archive.org/details/opticslightlaser00mesc_263/page/n57 46]–48 | isbn = 978-3-527-40628-9 }}

:$w(z)\; =\; w\_0\; \backslash ,\; \backslash sqrt\{\; 1+\; \{\backslash left(\; \backslash frac\{z\}\{z\_\backslash mathrm\{R\}\}\; \backslash right)\}^2\; \}\; .$

The minimum value of $w(z)$ occurs at $w(0)\; =\; w\_0$, by definition. At distance $z\_\backslash mathrm\{R\}$ from the beam waist, the beam radius is increased by a factor $\backslash sqrt\{2\}$ and the cross sectional area by 2.

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

:$\backslash Theta\_\{\backslash mathrm\{div\}\}\; \backslash simeq\; 2\backslash frac\{w\_0\}\{z\_R\}.$

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

:$D\; =\; 2\backslash ,w\_0\; \backslash simeq\; \backslash frac\{4\backslash lambda\}\{\backslash pi\backslash ,\; \backslash Theta\_\{\backslash mathrm\{div\}\}\}$.

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.

• Beam divergence
• Beam parameter product
• Gaussian function
• Electromagnetic wave equation
• John Strutt, 3rd Baron Rayleigh
• Robert Strutt, 4th Baron Rayleigh
• Depth of field

{{reflist}}

• [https://www.rp-photonics.com/rayleigh_length.html Rayleigh length] RP Photonics Encyclopedia of Optics

Category:Optical quantities

Category:Laser science