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,


Maximal distance
m
Divergence angle
mrad
Rayleigh length
mm
Wavelength
nm
Beam quality factor \(M^2\)
Beam parameter product (BPP)
mm mrad
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: $$s'=\frac{f\left[s(s-f)+z_{\mathrm{R}}^{2}\right]}{(s-f)^{2}+z_{\mathrm{R}}^{2}}.$$

Lens focal length \( f \)
mm
Distance to lens \( s \)
mm
Beam diameter at focus \( 2w_0^\prime \) (at e⁻²)
mm
Distance to focus \( s' \)
mm
Beam diameter at lens \( d \)
mm
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