The VRTE can be written in the form
where 
is the Stokes vector for the polarized radiation
field, 
is the propagation matrix, 
is the scalar source function, and 
is the formal vector
. The propagation
matrix is then a 
matrix, whose explicit form will be considered
later on.
The optical depth, 
, is defined such that it increases in the direction
opposite to that of the observed radiation. In general, the optical depth is
a function of the frequency as well, but in the application of Eq. (1) to the
calculation of line profiles it is customary to choose a frequency independent
representation of 
. In particular, if we are considering the case of
emission lines in the absence of continuum contribution, then we may define
, where 
is the
line absorption coefficient integrated in frequency, and s is the geometrical
depth.
In the following, we will adopt such definition of the optical depth. However,
since we are parameterizing the solution of Eq. (1) in terms of the optical
depth of the medium (instead of the geometrical depth), the explicit
dependence of 
on s, and therefore of 
on
s, is not needed.
Also, in order to keep the complexity of notation at a minimum, we will drop
the subscript ``
'' from the above defined quantities, implicitly
assuming their frequency (or wavelength) dependence.
From the general theory of linear ordinary differential equations (see,
for instance, Hochstadt 1964), it is known that the formal solution
of Eq. (1) can be written in the form
where U is the solution matrix of the adjoint equation of Eq. (1),
Incidentally, we note that U coincides with the evolution operator
which was introduced by Landi Degl'Innocenti & Landi Degl'Innocenti
(1985).
As we anticipated in the Introduction, we are interested in determining
the solution of Eq. (1) in the case of an isolated slab of emitting
hydrogen plasma of total optical depth 
. In the absence of
background sources of radiation 
, so that Eq. (2)
becomes
Since we have assumed that the propagation matrix, 
, is
independent of 
, the formal solution of Eq. (3) is simply given by
apart from an unnecessary multiplicative constant. Thus Eq. (2') becomes
which is the starting point for all the following calculations.
If we let 
(matching the case of uniform temperature and
pressure within the slab), we have, for any optical depth 
in the
interval 
,
where 
is the 
unit matrix.
We note that, in the limit of optically-thin (emission) lines, the above solution would become
where in the last equivalence we have introduced the emission vector
.
In general, we may want to allow the source function to vary with 
,
in order to mimic temperature variations through the slab, and to account
for possible departures from LTE. However, we keep
the approximation of a constant propagation matrix through the whole slab,
so that Eq. (4) is still valid.
Since we are mainly interested in determining the emergent polarized radiation from solar atmospheric structures such as prominences or post-flare loops, we make the additional assumption that the slab has planar symmetry with respect to its median plane. This requirement yields the following constraint on the source function,
which, in particular, gives the two-points boundary condition for
the source function
The next form of the source function that we consider, after the simplest
case of a constant source function which yielded Eq. (5), is the parabolic
approximation,
By imposing the two-points boundary condition, Eq. (6b), we easily find
Since the source function can never attain negative values, the
parameter 
must satisfy the constraint
where we also limited 
to a positive range of values, since we
want to describe the realistic physical situation of a slab temperature
increasing outward, which is more realistic for the application
to prominences and post-flare loops.
Substituting Eq. (7) into Eq. (4) yields, after some tedious calculation,
In the parabolic approximation of the source function, the observed
polarization profiles are then given by Eq. (9) for 
:
We next consider a source function with a Gaussian behavior,
where again we limit ourselves to considering only positive values of the
parameter 
, with the additional constraint 
. Substituting
Eq. (11) into Eq. (4), we then obtain
The above integral can be expressed through the error function
(e.g., Abramowitz & Stegun 1965), and therefore must
in general be calculated through some numerical method.
However, since we are finally interested in determining the outcoming
Stokes vector, 
, we will concentrate on
so to extend the integration limits to the boundary of the slab.
If we then make the assumption that 
, we can further
extend the integration limits to infinity. Such assumption corresponds
to imposing the boundary condition 
or, in
other words, it corresponds to the reasonable assumption that the
exponential contribution of the source function given by Eq. (11) be
negligibly small at the boundary of the slab.
In fact, letting 
, we note that the
exponential contribution to the source function at the boundary of the
slab is only about a fraction 
of the contribution at the center
of the slab, if we take 
. So we will consider the extension
of the integration limits to infinity a valid approximation, whenever
.
We then observe that we can write, through some simple algebraic
transformation,
The last integral can now be evaluated through the formula (see Gradshteyn
& Ryzhik 1980)
In fact, it can be shown that the same formula also applies when the
parameter q is replaced by any (real) square matrix 
(like
the propagation matrix 
). Thus letting 
and
, we easily obtain
so that
