The 2D case is identified here as the waveguide with cylindrical symmetry discussed in
so the coordinates , and stand for horizontal distance,
depth and ray slope related to the horizontal,
The Gaussian beam expression for this case follows readily from the expression for the 2.5 case,
by taking and ,
which provides the expression
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
Those derivatives can be identified as the components of the polarization vector
which in the 2D case correspond to:
Polarization vectors for the 2D case.
The beam width and curvature, and , respectively[1,9],
can be calculated from and by comparing the following expressions:
which yields that
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