in this expression and are related through the so-called dynamic equations:

The second-order derivative along the normal can be written in terms of derivatives along and as[1]

where

and

Those derivatives can be identified as the components of the polarization vector , which in the 2D case correspond to:

(see Fig.3.3).

The beam width and curvature, and , respectively[1,9],
can be calculated from and by comparing the following expressions:

which yields that

and

As shown by Eq.(3.11) the Gaussian beam approximation requires that . To this end it would be sufficient to select a non-zero real value for , plus the choice , being a real number, as small as possible. Orlando Camargo Rodríguez 2012-06-21