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%TCIDATA{Created=Tue Oct 15 12:16:47 1996}
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\markright{{\sf Circular aperture - Small apertures}}
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\begin{document}


\section{9.12\quad Circular Aperture - Small Apertures.}

\subsection{9.12.1}

\quad \thinspace \textbf{a. }As an example of Kirchhoff's vector formula, we
consider a circular hole in a conducting sheet. The field $\mathbf{E}%
^{\prime \prime }$ (which coincides with the total field ``on the right'',
according to the definition in 9.10.4d) is obtainable explicitly in terms of
the Bessel function $J_1(ka\xi ),$ where $k=\omega /c,$ $a$ is the radius
and $\xi $ is related to the incidence angle $\alpha $ and to the final
angles $\theta ,\varphi $ by (see Jackson) 
\[
\xi =\left[ \sin ^2\theta +\sin ^2\alpha -2\sin \theta \sin \alpha \cos
\varphi \right] ^{1/2}. 
\]

\smallskip\ 

\textbf{b}. The time-averaged power coming through the hole, for P-polarized
incidence, is 
\begin{equation}
\frac{dP}{d\Omega }=P_i\cos \alpha \frac{(ka)^2}{4\pi }\left( \cos ^2\theta
+\cos ^2\varphi \sin ^2\theta \right) \left| \frac{2J_1(ka\xi )}{ka\xi }%
\right| ^2  \label{1}
\end{equation}
where 
\[
P_i=\left( \frac{cE_0^2}{8\pi }\right) \pi a^2\cos \alpha 
\]
is the total power that would pass through the hole if there were no
diffraction.

\smallskip\ 

\textbf{c.} The polarization of the transmitted radiation can be described
as follows: the transmitted $\mathbf{B}$ field is proportional to $\mathbf{B}%
_{i\perp },$ the component of the incident $\mathbf{B}$ field perpendicular
to the final wave vector $\mathbf{k.}$ A view is given below. 
\begin{eqnarray*}
&& \\
&& \\
&&
\end{eqnarray*}

\smallskip\ 

\textbf{d. }The transmission coefficient T is obtained by integrating over
angles 
\[
\frac 1{P_1}\frac{dP}{d\Omega }. 
\]
It is found that T$\rightarrow \cos \alpha $, for $ka\gg $ $1,$ as expected;
for a small hole $(ka\ll 1)$ this theory gives T$\rightarrow (1/3)(ka)^2\cos
\alpha ,$ which is not right (see Problem 9.24).

\newpage 

\subsection{9.12.2}

\quad \quad \textbf{a. }The result (\ref{1}) should be compared with that of
the scalar Kirchhoff approximation 
\begin{equation}
\frac{dP}{d\Omega }=P_1\cos \alpha \frac{(ka)^2}{4\pi }\left( \frac{\cos
\alpha +\cos \theta }{2\cos \alpha }\right) ^2\left| \frac{2J_1(ka\xi )}{%
ka\xi }\right| ^2  \label{2}
\end{equation}

\smallskip\ 

\textbf{b.} We see that the results (\ref{1}) and (\ref{2}) both contain the 
\textsl{diffraction factor} $\left| 2J_1/ka\xi \right| ^2,$ which for $ka\gg 
$ $1$ gives strong peaking in the forward direction with a characteristic
pattern as sketched:

\begin{eqnarray*}
&& \\
&&
\end{eqnarray*}

\smallskip\ 

\textbf{c. }The different ``obliquity factors'' in (\ref{1}) and (\ref{2})
are due to the polarization properties of the electromagnetic field. Note in
particular that the scalar theory's obliquity factor is independent of the
azimuth $\varphi .$ The calculation of the obliquity factor for $S$%
-polarized incidence is left as a problem (Jackson's 9.17).

\smallskip\ 

\textbf{d.} The result quoted in 12.1d, that all Kirchhoff-like
approximations fail for small $ka,$ should not be surprising. In this limit
one can use a different approximation in the exact result (\ref{4}) or other
formulae: one simply expands $\exp (ikR)/R$ in spherical harmonics and
obtains a multipolar expansion. The leading terms (dipole and electric
quadrupole) can be obtained as is Sect. 9.1, without recourse to the full
machinery of Sect. 17.

\smallskip\ 

\textbf{e. }The correct expressions for $\mathbf{E}$ and $\mathbf{B}$ in an
opening, in the limit $ka\ll 1,$ are obtained by assuming that $\mathbf{E}%
^{(0)}$ and $\mathbf{B}^{(0)}$ are constant and time-independent. The
resulting static problem can be solved for a circular hole in a conducting
screen. For the results, see page 411 and problems 9.23 and 9.24 of Jackson.

\end{document}
