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## Main Question or Discussion Point

**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)

*E*=

_{o}*E*at

*r*=0

*I, I*= irradiance (W/m

_{o}^{2})

*r*= transverse or radial coordinate (m)

*R(z)*= radius of curvature of wavefronts (m)

*w(z)*= spot size (m)

*w*= beam waist (m), or spot size at

_{o}*z*=0

*z*= longitudinal coordinate (m)

*z*= Rayleigh range

_{R}*λ*= wavelength (m)

*θ*= divergence half-angle

__Descriptive figure__

* This entry is from our old Library feature, and was originally created by Redbelly98.

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