Generally speaking,
the only way to ensure that the matrices
and
are complex,
being
a real matrix,
is by selecting a proper choice of complex initial conditions for both matrices.
As shown by the literature such proper choice is a matter of intense debate,
seemingly not yet solved.
There are however particular choices of real
and
,
which are relevant for the discussion of Gaussian beams.
Due to such relevance those choices are presented in this section for the case of a point source
and for the case of a linear source.

**Subsections**

Orlando Camargo Rodríguez
2012-06-21