Asymptotic developments and scattering theory in terms of a vector combining the electric and magnetic fields

The vector combination
\overrightarrow{M}=(\frac{\mu}{\epsilon})^{1/4} \overrightarrow{H} + j (\frac {\epsilon}{\mu})^{1/4} \overrightarrow{E}
which was in principle introduced by Bateman and Silberstein in order to shorten Maxwell's equations for homogeneous media, also proves to be useful for the treatment of inhomogeneous media
\epsilon
and
\mu
not depending on the time). The vector
M\rightarrow
is to be considered together with its conjugated quantity
M^{x}\rightarrow
obtained by replacing the imaginary unit
j
by
-j
. In a source-free medium the Maxwell equations reduce to
curl \overrightarrow{M} + \frac{j}{c} (\epsilon \mu)^{1/2}\partial \overrightarrow{M}/ \partial^{f} = \frac{1}{4} grad \log \frac{\mu}{epsilon} \wedge \overrightarrow{M}^{x}
and to the equation obtained by taking the conjugated complex value. This relation shows how an interaction between
M\rightarrow
and
M^{x}\rightarrow
is produced only by the inhomogeneity of the medium. The theory of scattering by special volume elements, as well as that of partial reflections against layers with rapidly changing
\epsilon
and
\mu
, can be based on the single relation (1) while fully accounting for the vectorial character of the field, The introduction of
M\rightarrow
and
M^{x}\rightarrow
also enables one to put many results of Luneberg-Kline's theory concerning asymptotic developments in a very simple form. As an example we mention the equation:
grad S \wedge \overrightarrow{m}_{r} - i_r \mu)^{1/2}m_{r} = c curl \overrightarrow {m_{r-1}} - \frac{c}{\epsilon} grad \log \frac{\mu}{\epsilon} \wedge \overrightarrow {m_{r-1^{x}}}
, which fixes all recurrence relations between the consecutive terms of geometric-optical expansions; these expansions are defined by the asymptotic development
\overrightarrow{M} =e^{ik^{s}} \Sigma \min{r=0} \max{\infty}(\frac{i}{kc))^{r} \overrightarrow {m_{r}}
, for monochromatic solutions corresponding to some eiconal function
S
.