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Understanding Gaussian Beams: Definition, Equations, and Parameters
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[QUOTE="Greg Bernhardt, post: 4804854, member: 1"] [SIZE="4"][U][B]Definition/Summary[/B][/U][/SIZE] 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. [SIZE="4"][U][B]Equations[/B][/U][/SIZE] 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] [SIZE="4"][U][B]Extended explanation[/B][/U][/SIZE] [u]Definitions of terms[/u] (SI units for quantities are shown in parantheses) [indent][i]b[/i] = confocal parameter (m) [i]E[/i] = electric field (V/m) [i]E[sub]o[/sub][/i] = [i]E[/i] at [i]r[/i]=0 [i]I, I[sub]o[/sub][/i] = irradiance (W/m[sup]2[/sup]) [i]r[/i] = transverse or radial coordinate (m) [i]R(z)[/i] = radius of curvature of wavefronts (m) [i]w(z)[/i] = spot size (m) [i]w[sub]o[/sub][/i] = beam waist (m), or spot size at [i]z[/i]=0 [i]z[/i] = longitudinal coordinate (m) [i]z[sub]R[/sub][/i] = Rayleigh range [i]λ[/i] = wavelength (m) [i]θ[/i] = divergence half-angle[/indent][u]Descriptive figure[/u] [indent][indent][ATTACH=full]171537[/ATTACH][/indent][/indent] * This entry is from our old Library feature, and was originally created by Redbelly98. [/QUOTE]
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