Basics on the Gaussian beams

The radiation emitted by a laser source can be considered as Gaussian. Why should we consider Gaussian waves instead of spherical or plane waves? This is because the considered waves exhibit non-negligible diffraction phenomena due to their finite extension transversally to their propagation axis.

A Gaussian wave, just like a spherical or plane wave, is a solution of the wave equation. The electric field has the following expression :

where A(z) is the complex amplitude of the field along the propagation axis z. In this expression, the variations of the field E with respect to the radial coordinate r are expressed through the exponential terms.

• The term contains the Gaussian character of the beam : at a distance z from the focus, the field amplitude decays transversally by a factor 1/e² at the distance r = w(z) from the optical axis. The radius, or waist, w(z) thus depicts the beam radial extension.

• Similarly to the paraxial spherical wave, the Gaussian beam is characterized by the curvature radius R(z) of its wave fronts.

On the Fig. 8, at z = 0, one can define the waist at the focus w0, which is the minimal value of the radius w(z) along the propagation axis. From this primordial parameter, one can express the following characteristic quantities :

The Rayleigh length zR depicts the characteristic divergence length of the beam. Indeed, the smaller this value, the more divergent the beam. This divergence, as can be seen on Fig. 8, can also be measured using the asymptotic evolution of the radius w(z) :

When z >> zR , w(z) tends to :

and the associated angle can be written as follows :