DDSCAT.5a internally computes the
scattering properties of the dipole array in terms of a complex
scattering matrix 
(Draine 1988),
where index l=1,2 denotes the incident polarization states,
m=1,2 denotes the scattered polarization state, and 
,
specify the scattering direction.
Normally DDSCAT is used with IORTH=2 in ddscat.par, so that
the scattering problem will be solved for both incident polarization states
(l=1 and 2); in this subsection it will be assumed that this is the case.
Incident polarization states l=1,2 correspond to polarization states
, 
; recall that polarization state
is user-specified, and
.
Scattered polarization state m=1 corresponds to linear polarization of the
scattered wave parallel to the scattering plane
(
)
and m=2 corresponds to linear polarization
perpendicular to the scattering plane (in the
direction).
The scattering matrix 
was defined (Draine 1988) so that
the scattered electric field 
is related to the incident
electric field 
at the origin (where the target is
assumed to be located) by
The 2
2 complex
amplitude scattering matrix (with elements 
, 
,
, and 
) is defined so that (see Bohren & Huffman 1983)
where
, 
are (real) unit vectors
for incident polarization
parallel and perpendicular to the scattering plane (with the
customary definition of 
).
Let
Note that since 
could be
complex (i.e., elliptical polarization),
the quantities a,b,c,d are complex.
Then
and eq. (27) can be written
The incident polarization states and
are related to 
, 
by
substituting (34) into (33) we
obtain
Eq. (35) must be true for all , so we
obtain an expression for the complex scattering amplitude matrix in terms
of the :
This provides the 4 equations used in subroutine GETMUELLER to
compute the amplitude scattering matrix elements:
It is convenient to describe the scattering properties in terms of
the Mueller matrix relating the Stokes parameters 
and
of
the incident and scattered radiation:
Once the amplitude scattering matrix elements are obtained, the Mueller
matrix elements can be computed (Bohren & Huffman 1983):
These matrix elements are computed in DDSCAT and passed to subroutine
WRITESCA which handles output of scattering properties.
As delivered, WRITESCA writes out 6 selected elements: 
, 
,
, 
(these 4 elements describe the intensity and polarization
state for scattering of unpolarized incident radiation), 
, and 
.
In addition, WRITESCA writes out the linear polarization P of the
scattered light for incident unpolarized light:
Of course, other elements 
may be of interest.
It is relatively straightforward for the user to modify subroutine
WRITESCA to write out whatever elements of the Mueller matrix (or the
scattering amplitude matrix) are desired.