### Gaussian beam propagation

Propagation of Gaussian beam in air is characterized by the change of its radius $$w$$ (at $$1/\mathrm{e}^2$$ ) and curvature radius $$R$$ dependence on coordinate $$z$$: $$w(z) = w_0\sqrt{1+{z^2}/{z_\mathrm{R}^2}},$$ $$R(z) = z \left( 1 + {z^2}/{z_\mathrm{R}^2} \right).$$ $$w_0$$ is beam radius at waist ( $$z=0$$ ), Rayleigh length $$z_\mathrm{R} = \pi w_0^2/\lambda$$. Gouy phase shift $$\phi_\mathrm{G} = -\arctan(z/z_\mathrm{R})$$ is difference between phase of propagating Gaussian beam and plane wave of the same frequency,

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##### Beam diameter at waist [mm]
at e⁻²:
at FWHM:

Beam waist diameter at focus: $$w_{0}'=\frac{w_{0}}{\sqrt{\left(1+s/f\right)^{2}+\left(z_{\mathrm{R}}/f\right)^{2}}}.$$ Distance to focus: $$s'=\frac{f\left[s(s-f)+z_{\mathrm{R}}^{2}\right]}{(s-f)^{2}+z_{\mathrm{R}}^{2}}.$$

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