While is a real vector both matrices and are complex. Therefore, the complex part of the product induces a Gaussian decay of ray amplitude along the normal (see Fig.3.1), while the real part introduces phase corrections to the travel time along . Additionally, a proper choice of initial conditions for will ensure that det , which frees the Gaussian beam approximation of singularities.
Both components of and can be considered as being dependent of a particular set of local ray parameters, let's say, ray arclength , plus angles and . At any point of the ray one can introduce a set of three orthogonal unit vectors, known as the polarization vectors; naturally, the first polarization vector is ; the other two polarization vectors, which are going to be represented as and , are within the plane perpendicular to (see Fig.3.2).
define the possible orientations of
coordinate of the ray:
the Gaussian beam approximation involves two more matrices,
all four matrices are related through the following relationships:
Orlando Camargo Rodríguez 2012-06-21