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%TCIDATA{Created=Fri Oct 11 17:37:08 1996}
%TCIDATA{LastRevised=Tue Oct 15 12:13:54 1996}
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\markright{{\sf Babinet's Principle of Complementarity}}
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\section{9.11\quad Babinet's Principle of Complementarity.}

\subsection{9.11.1}

\quad \thinspace \textbf{a. }It should be clear from the examples of Sect.
10 that the scattered field $\mathbf{B}^{\prime }$ from a screen and the
diffracted field $\mathbf{E}^{\prime \prime }$ from a hole are simply
related.

\quad \quad \quad $S_a$: diffracted screen with hole

\quad \quad \quad $S_b$: complementary screen

\quad \quad \quad $S_1$: plane of $S_a$ or of $S_b$

\smallskip\ 

\textbf{b}. In a scalar theory, the following exact statement can be made.
Let $\psi _b$ be the field scattered by $S_b$ with Dirichlet boundary
conditions on $S_b$; then $\partial \psi _b/\partial n=0$ in $S_1-S_b$ by
symmetry if the screen is flat. For the same incident field $\psi ^{(0)},$
let $\psi ^{(1)}$ be the scattered field with Neumann boundary conditions on
the whole plane $S_1$ which is trivially found (behind $S_1,$ we just have $%
\psi ^{(1)}=-\psi ^{(0)}).$ Then $\psi =\psi ^{(0)}+\psi ^{(1)}+\psi _b$
solves the scattering problem for the screen $S_a$ with Neumann boundary
conditions ($\partial \psi /\partial n=0$ on $S_a)$

\smallskip\ 

\textbf{c.} Similar statements hold if the roles of $S_a$ and $S_b,$ or of
Dirichlet and Neumann conditions, are interchanged.

\smallskip\ 

\textbf{d. }Actually, for a flat screen, the Neumann and Dirichlet problems
are simply related, since $\partial \psi /\partial z$ solves the Dirichlet
problem if $\psi $ solves the Neumann problem. Note that $\partial \psi
^{(0)}/\partial z$ is a plane wave if $\psi ^{(0)}$ is a plane wave.

\newpage

\subsection{9.11.2}

\quad \quad \textbf{a. }In practice, the usefulness of Babinet's principle
is that it remains approximately valid under much more general conditions
than those specified on previous page. It is intuitively true that the field
passing through a hole in an opaque screen depends much more sensitively on
the size and shape of the hole than on the nature of the screen. The
corresponding mathematical result is that a remarkably good description of
the field at large distances from the screen is given by the Kirchhoff
approximation, and that the improved or generalized formulae using $G_D$ or $%
G_N$, as appropriate, do not give very different answers, despite
appearances, near the forward or backward direction, where $\psi _s$ is
usually largest (see 8.4f and Jackson).

\smallskip\ 

\textbf{b. }Leaving aside all subtleties, we can formulate the principle as
follows. Let the diffracted field be defined by 
\[
\psi _{d}(\mathbf{x}^{\prime })=\int_{openings}\left( G\frac{\partial \psi
^{(0)}}{\partial n}-\psi ^{(0)}\frac{\partial G}{\partial n}\right) da 
\]
according to the traditional KA of 9.8\textbf{.}3c, with $\psi ^{(0)}$
coming (say) from the left of the screen and $\mathbf{n}$ pointing
downstream to the right. Then we have the scheme:

\begin{center}
\begin{tabular}{|ccccc|}
\hline
&  & $S_{a}$ &  & $S_{b}$ \\ 
&  &  &  &  \\ 
$\text{Right}$ &  & $\psi _{d}$ &  & $\psi ^{(0)}-\psi _{d}$ \\ 
&  &  &  &  \\ 
$\text{Left}$ &  & $\psi ^{(0)}+\psi ^{(1)}-\psi _{d}$ &  & $\psi
^{(0)}+\psi _{d}$ \\ 
&  &  &  &  \\ \hline
\end{tabular}
\end{center}

\smallskip\ 

\textbf{c. }In the vector theory of diffraction, the above conclusion
remains true if the holes are large or if they are nearly circular. However,
for apertures such as a narrow slit one must pay closer attention to the
orientation of the incident fields $\mathbf{E}^{(0)}$ and $\mathbf{B}^{(0)}$
with respect to the aperture. In this case even the approximate relation
between the original and complementary problems must be according to the
scheme
\[
\begin{tabular}{cccc}
$\text{Original}$ & $S_{a}$ & $\mathbf{E}^{(0)}$ & $\mathbf{B}^{(0)}$ \\ 
&  &  &  \\ 
$\text{Complement}$ & $S_{b}$ & $\mathbf{E}_{c}^{(0)}=-\mathbf{B}^{(0)}$ & $%
\mathbf{B}_{c}^{(0)}=\mathbf{E}^{(0)}$ \\ 
&  &  & 
\end{tabular}
\]

\newpage

\subsection{9.11.3}

\quad \thinspace \textbf{a.} The exact statement of Babinet's principle for
a thin, flat metallic screen is based on the correspondence of the results
of 10.2d and of 10.4d when $\mathbf{B}^{(0)}$ for the complementary ($S_b)$
problem is numerically equal to $\mathbf{E}^{(0)}$ for the original ($S_a)$
problem. We have the scheme: 
\begin{eqnarray*}
&&\quad \quad \quad \quad \quad \quad \quad \quad \text{Original }\mathbf{%
E\quad \quad \quad \quad \quad }\text{Complementary }\mathbf{B} \\
&&\text{Right:\quad \quad \quad \quad \quad \quad \quad \quad }\mathbf{E}%
^{\prime \prime }\quad \quad \quad \quad \quad \quad \quad \quad \mathbf{E}%
^{(0)}-\mathbf{E}^{\prime \prime } \\
&&\text{Left: \quad \quad \quad \thinspace \quad \quad }\mathbf{E}^{(0)}+%
\mathbf{E}^{(1)}+\mathbf{E}^{\prime \prime }\quad \quad \quad \quad \,\text{%
\quad }\mathbf{E}^{(0)}+\mathbf{E}^{\prime \prime }
\end{eqnarray*}
where the diffracted field $\mathbf{E}^{\prime \prime }$ is defined
according to 9.10.3c 
\[
\mathbf{E}^{\prime \prime }(\mathbf{x}^{\prime })=\frac{\mathbf{\nabla
\times }}{2\pi }\int_{apertures\,\,of\,\,S_a}\left( \mathbf{n\times E}(%
\mathbf{x)}\right) \frac{e^{ikR}}Rdxdy. 
\]
and $\mathbf{n}$ points to the right.

\smallskip\ 

\textbf{b. }To be explicit about it, if $\mathbf{E}^{(0)}=E_0\mathbf{%
\epsilon }_0\exp (ik(k_xx+k_yy+k_zz))$, with $\mathbf{k\cdot \epsilon }_0=0,$
then, if $\mathbf{\epsilon }_0=\mathbf{\epsilon }_{0/\!/}+\mathbf{\epsilon }%
_{0\perp },$%
\[
\mathbf{E}^{(1)}=E_0(\mathbf{\epsilon }_{0\perp }-\mathbf{\epsilon }%
_{0/\!/})e^{i(k_xx+k_yy+k_zz)}. 
\]
\[
\]
\[
\]
\[
\]

\smallskip\ 

\textbf{c. }As an example, consider a narrow slit in the side of a cavity or
wave guide, with $b\ll a\ll A,B$ as in the sketch. If $\mathbf{B}$ is the
magnetic field in the slit, the radiated power will be the same as that of a
narrow strip (antenna) of length $b$ with an \textsl{electric} field that is
numerically equal to $\mathbf{B}$ in the slit. Of course, the polarization
of the antenna's radiation will be opposite (at 90$^{\circ })$ to that of
the slit. (i.e. to the lowest order the slit's fields are magnetic, not
electric, dipole).

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