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What is a Gaussian beam

  1. Jul 24, 2014 #1
    Definition/Summary

    A Gaussian beam is an electromagnetic wave, usually a laser beam, with a Gaussian cross-sectional irradiance pattern. The Gaussian irradiance profile results in minimal spreading due to diffraction effects.

    The spot size [itex]w[/itex] represents the radius or half-width at which the irradiance is a factor of [itex]1/e^2[/itex] less than the central-axis irradiance.

    Equations

    For a Gaussian laser beam propagating along the z-axis, the electric field strength is a Gaussian function of the transverse (or radial) coordinate r:

    [tex]E = E_0 \cdot e^{-r^2/w^2}[/tex]

    where Eo and w are both functions of z.

    It is common practice to work in terms of the irradiance, which is proportional to the square of the electric field, so that

    [tex]I = I_0(z) \cdot e^{- 2 r^2 / w(z)^2}[/tex]


    The various parameters of a Gaussian beam are related as follows:

    [tex] \begin{align*}

    \theta & = & & \frac{\lambda}{\pi \ w_o}
    & = & & \sqrt{\frac{\lambda}{\pi \ z_R}} \
    & = & & \ \frac{w_o}{z_R}
    \\ \\

    w_o & = & & \frac{\lambda}{\pi \ \theta}
    & = & & \sqrt{\frac{\lambda \ z_R}{\pi}} \\ \\

    z_R & = & & \frac{\pi \ w_o^2}{\lambda}
    & = & & \frac{\lambda}{\pi \ \theta^2} \\ \\

    b & = & & 2 \ z_R \\

    \end{align*} [/tex]


    Moreover,

    [tex] \begin{align*}

    w(z) & = & & w_o \sqrt{1 + \left(\frac{z}{z_R}\right)^2} \\ \\

    R(z) & = & & z + z_R^2/z
    \ = \ z \left[ 1 + \left( \frac{z_R}{z} \right) ^2 \right]

    \end{align*} [/tex]

    Extended explanation

    Definitions of terms

    (SI units for quantities are shown in parantheses)
    b = confocal parameter (m)
    E = electric field (V/m)
    Eo = E at r=0
    I, Io = irradiance (W/m2)
    r = transverse or radial coordinate (m)
    R(z) = radius of curvature of wavefronts (m)
    w(z) = spot size (m)
    wo = beam waist (m), or spot size at z=0
    z = longitudinal coordinate (m)
    zR = Rayleigh range
    λ = wavelength (m)
    θ = divergence half-angle​


    Descriptive figure

    GaussianBeam.gif

    * This entry is from our old Library feature, and was originally created by Redbelly98.
     
    Last edited by a moderator: Jul 27, 2014
  2. jcsd
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