# Gaussian Beam

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

where,

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.

For vacuum,

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