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%TCIDATA{Created=Tue Oct 15 11:32:49 1996}
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\markright{{\sf The vector Kirchhoff integral (VKI)}}
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\section{9.9\quad The Vector Kirchhoff Integral (VKI).}

\subsection{9.9.1}

\quad \thinspace \textbf{a. }Since each \textsl{cartesian }component of $%
\mathbf{E},$ $\mathbf{B}$ obeys the wave equation in source-free regions,
most of the results of the scalar theory could be applied rather directly to
the scattering and diffraction of electromagnetic waves. Basically all
results of interest are contained in the formula ($\partial _\beta =\partial
/\partial x_\beta )$%
\begin{equation}
E_\alpha (\mathbf{x}^{\prime })=\sum_\beta \int_S\left( E_\alpha n_\beta
\partial _\beta G-Gn_\beta \partial _\beta E_\alpha \right) da  \label{1}
\end{equation}
with the appropriate specification of $S$, and a similar formula for $%
B_\alpha $ (also for the potentials $A_\alpha ,\phi ).$

\smallskip\ \thinspace

\textbf{b}. Proceeding in this way, one must determine the twelve unknowns
quantities $E_\alpha ,\,B_\alpha ,\,\partial E_\alpha /\partial n,$ $%
\,\partial B_\alpha /\partial n$ on $S.$ Any method of cutting this number
down is of great help. We know that all fields can be derived from just 
\textsl{two} components, so that one should be able to write down just two
Kirchhoff-like integrals. In special cases the two integral equations that
arise are decoupled and can be solved separately to find what are in fact
the ``superpotentials'' of the problem. One drawback of working with
components is that one loses sight of the vector nature of the fields.

\smallskip\ \thinspace

\textbf{c.} An alternative way, favored by Jackson, is to convert (\ref{1})
in an equation involving only $\mathbf{E}$ and $\mathbf{B}$: 
\begin{equation}
\mathbf{E}(\mathbf{x}^{\prime })=\int_S\left[ ik(\mathbf{n\times B)}G+(%
\mathbf{n\times E)\times \nabla }G+(\mathbf{n\cdot E)\nabla }G\right] da.
\label{2}
\end{equation}
This and the analogous equation with $\mathbf{E}\rightarrow \mathbf{B,B}%
\rightarrow -\mathbf{E}$ are obviously vectorial in nature and involve only
four unknown quantities, $E_{/\!/,}$ $B_{/\!/}$ and $E_{\perp },$ $B_{\perp
} $ on $S.$ The derivation of (\ref{2}) is given by Jackson.

\newpage

\subsection{9.9.2}

\quad \quad \textbf{a. }If all the sources of $\mathbf{E},$ $\mathbf{B}$ are
within a finite surface $S_1$ and if an outgoing wave Green function is
used, it follows that the integration in (\ref{1}) or (\ref{2}) can be taken
over $S_1$ alone.

\smallskip\ 

\textbf{b. \ }For the scattering geometry, it is convenient to take $\mathbf{%
E},$ $\mathbf{B}$ to be just the \textsl{scattered} (rather than the total)
fields. Then, with 
\[
G=\frac{e^{ikR}}{4\pi R}\,, 
\]
the asymptotic $\mathbf{E}_s$ field is of the form 
\[
\frac{e^{ikr}}r\,\mathbf{f}E_0, 
\]
with 
\[
\mathbf{f}E_0=\frac i{4\pi }\int_{S_1}e^{-i\mathbf{k\cdot x}}\left[ k(%
\mathbf{n\times B}_s\mathbf{)+k\times }(\mathbf{n\times E}_s\mathbf{)-k\cdot 
}(\mathbf{n\times E}_s\mathbf{)}\right] da. 
\]

\smallskip\ 

\textbf{c. }For scattered polarization $\mathbf{\epsilon },$ the scalar
scattering amplitude is then 
\[
\mathbf{\epsilon }^{*}\cdot \mathbf{f}=\frac i{4\pi E_0}\int_{S_1}e^{-i%
\mathbf{k\cdot x}}\left[ k\mathbf{\epsilon }^{*}\cdot (\mathbf{n\times B}_s%
\mathbf{)+\epsilon ^{*}(k\times }(\mathbf{n\times E}_s\mathbf{))}\right] da. 
\]

\smallskip\ 

\textbf{d. }The differential cross section is of course $d\sigma /d\Omega $ =%
$\left| \mathbf{\epsilon }^{*}\cdot \mathbf{f}\right| ^2.$ The formal
analogy with the formulae of sections 9.6 and 9.7, for instance (\ref{2}) of
9.6.2b, is not accidental and can be usefully exploited. Of course, one
should keep in mind the small detail that $\mathbf{E}_s,$ $\mathbf{B}_s$ are
not known, so that the above formulae are mostly useful for formal
manipulations. (See 9.14).

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