Lasers usually emit beams with a Gaussian profile. A Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions.
For a Gaussian beam, the amplitude of the complex electric field is given by
r – radial distance from the centre axis of the beam
z – axial distance from the beam’s narrowest point
i – imaginary unit (for which i2 = − 1)
k – wave number (in radians per meter).
w(z) – radius at which the field amplitude drops to 1/e and field intensity to 1/e2 of their axial values, respectively.
w(0) – waist size.
E0 = |E( 0,0) |
R(z) – radius of curvature of the beam’s wavefronts
ζ(z) – Gouy phase shift. It is an extra contribution to the phase that is seen in beams which obey Gaussian profiles.
The corresponding time-averaged intensity (or irradiance) distribution is
where I0 = I(0,0) is the intensity at the center of the beam at its waist. The constant is defined as the characteristic impedance of the medium through which the beam is propagating.