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%TCIDATA{Created=Wed Oct 09 16:25:51 1996}
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\markright{{\sf Scalar Diffraction Theory}}
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\section{9.8\quad Scalar Diffraction Theory.}

\subsection{9.8.1}

\quad \thinspace \textbf{a. }The word ``diffraction'' is applied to describe
the scattering from objects large compared to the wavelength. It is thus in
a way equivalent to ``semiclassical scattering''. The leading corrections to
geometrical optics are sought. The theory is rather subtle because there is
not a simple expansion in a parameter such as $\lambda /d,$ where $d$ is a
typical size of the system. Rather, one must seek approximations that have
an asymptotic character, for $\lambda /d\rightarrow 0.$ Further, WKB theory
cannot be used in the many cases where sharp edges occur.

\smallskip\ 

\textbf{b}. Much of the theory is based on Kirchhoff's integral and its
vector generalizations (next section). We work out the theory for a scalar
field first, although this applies to electromagnetic scattering only in
some special cases.

\smallskip\ 

\textbf{c.} Kirchhoff's integral is really a version of Green's theorem 
\begin{equation}
\int_V\left( G\nabla ^2\psi -\psi \nabla ^2G\right) d^3x=\int_S\left( G\frac{%
\partial \psi }{\partial n}-\psi \frac{\partial G}{\partial n}\right) da
\label{1}
\end{equation}
applied to two functions $\psi $ and $G,$ that satisfy 
\begin{equation}
\left( \nabla ^2+k^2\right) \psi (\mathbf{x)}=0,  \label{2}
\end{equation}
\begin{equation}
\left( \nabla ^2+k^2\right) G(\mathbf{x,x}^{\prime })=-\delta (\mathbf{x-x}%
^{\prime }).  \label{3}
\end{equation}
Using (\ref{2}) and (\ref{3}) in the l.h.s. of (\ref{1}) we find, \textsl{%
for }$\mathbf{x}^{\prime }$\textsl{\ in }$V,$ 
\begin{equation}
\psi (\mathbf{x}^{\prime }\mathbf{)=}\int_S\left[ \psi (\mathbf{x)n\cdot
\nabla }G(\mathbf{x,x}^{\prime })-G(\mathbf{x,x}^{\prime })\mathbf{n\cdot
\nabla }\psi (\mathbf{x)}\right] da  \label{4}
\end{equation}
where $\mathbf{n}$ is the normal directed \textsl{into} the volume $V$
surrounded by $S.$ For$\mathbf{\ x}^{\prime }$ outside $V,$ the r.h.s. of (%
\ref{4}) gives zero.

\newpage

\subsection{9.8.2}

\quad \quad \textbf{a. }For the applications of (\ref{4}), $S$ need not
consist of only one piece. This can be used to exclude the sources of $\psi
, $ if any, from the volume under consideration (see sketch).

\smallskip\ 

\textbf{b. }An explicit solution of (\ref{3}), valid over all space, is the
outgoing-wave Green function: 
\begin{equation}
\left( R=\left| \mathbf{x-x}^{\prime }\right| \right) \quad \quad \quad
\quad \quad \quad G(\mathbf{x,x}^{\prime })=G(\mathbf{x}^{\prime }\mathbf{,x)%
}=\frac{e^{ikR}}{4\pi R}  \label{5}
\end{equation}
\textbf{\ }

\textbf{c. }From (\ref{4}) and (\ref{5}) it follows that the solution of (%
\ref{2}) is known throughout $V,$ if $\psi $ and $\partial \psi /\partial n$
are given on the boundary $S$ (but both these boundary values cannot be
assigned arbitrarily, since either one is in fact sufficient to determine $%
\psi $ throughout $V).$

\smallskip\ 

\textbf{d. }If the sources of $\psi $ are all in the finite volume enclosed
by $S_1$ (see sketch), the solution of (\ref{4}), with $G$ given by (\ref{5}%
), is such that, at large distances $r$ from $S_1,$%
\begin{equation}
\psi \rightarrow f(\theta ,\varphi )\frac{e^{ikr}}r\quad \quad \quad \quad
\quad \quad \frac{\partial \psi }{\partial r}\rightarrow \left( ik-\frac
1r\right) \psi .  \label{6}
\end{equation}
It is easy to see that $G\left( \partial \psi /\partial r\right) -\psi
\left( \partial G/\partial r\right) $ vanishes faster than $r^{-2}$ for $%
R\approx r\rightarrow \infty .$ Therefore $S_2$ can be taken to infinity,
leaving only the integral over $S_1$ on the r.h.s. of (\ref{1}) and (\ref{4}%
).

\smallskip\ 

\textbf{e.} Other typical conditions where $\int_{S_2}\rightarrow 0:$

\quad - $S_1$ is a plane, but $\psi \neq 0$ over only a finite part (shaded)
of $S_1$;

\quad - $S_1$ has periodic properties; then the two pieces $S_2^{\prime }$
give canceling contributions and $S_2^{\prime \prime }$ can be taken to $%
\infty .$

\newpage

\subsection{9.8.3}

\quad \thinspace \textbf{a.} \textsl{Kirchhoff's integral} (\ref{4}) can be
used to ``solve'' the problem exactly, in two steps:

\quad (i) coupled integral equations for $\psi $ and $\partial \psi
/\partial n$ \textsl{on }$S$\textsl{\ alone} are given by (\ref{4}) and its
gradient;

\quad (ii) once $\psi $ and $\partial \psi /\partial n$ are known on $S$, (%
\ref{4}) gives $\psi $ throughout $V.$

Unfortunately step (i) gives integral equations that can only be solved
numerically, or approximately. The main advantage is that \textsl{one can
choose} $S$ so that a great deal can be said about $\psi $ and $\partial
\psi /\partial n$ on $S,$ without numerical work or doubtful approximations.

\smallskip\ 

\textbf{b. }As a typical example, consider radiation incident from the left
on a flat screen with a hole in it. We take $S=S_1+S_2$, with $S_1$ a plane
just to the right of the screen (dotted line) and $S_2\rightarrow \infty .$
We know that $\psi $ decays rapidly to zero on the part of $S_1$ that is in
the shadow. Thus all we really need is $\psi $ and $\partial \psi /\partial
n $ across the opening.

\smallskip\ 

\textbf{c. }\textsl{Kirchhoff's approximation} is that, in the example given,

\quad (i) $\psi $ and $\partial \psi /\partial n$ vanish on $S_1,$ except in
the opening;

\quad (ii) in the opening, they can be replaced by the incident field and
its normal derivative.

\smallskip\ 

\textbf{d.} One must clearly distinguish Kirchhoff's approximation (KA) from
the integral formulation (\ref{4}), which is exact in principle.

\newpage

\subsection{9.8.4}

\quad \thinspace \textbf{a.} One obvious flaw of the KA is that it is
inconsistent, since $\psi $ and $\partial \psi /\partial n$ on the r.h.s. of
(\ref{4}) cannot vanish in the ``shadow'' region of the screen, unless $\psi
=0$ everywhere.

\smallskip\ 

\textbf{b. }The boundary conditions on the screen will typically be $\psi =0$
(Dirichlet) or $\partial \psi /\partial n=0$ (Neumann). If $\psi =0$
(strictly), then $\partial \psi /\partial n$ decays rapidly toward zero in
the shadow, but it is \textsl{not} zero, and vice versa in the Neumann case.

\smallskip\ 

\textbf{c. }The Dirichlet problem would be solved exactly if one knew the
function $G$ that obeys (\ref{3}) and satisfies $G=0$ for $\mathbf{x}$ or $%
\mathbf{x}^{\prime }$ on the screen, as in sketch (a). What one does know
(by the method of images) is a $G_D$ that vanishes over one side of the
screen and the opening (dotted line in sketch (b)).

\smallskip\ 

\textbf{d. }Since $G_D$ satisfies (\ref{3}) in the region to the right of
the screen, it can be used in (\ref{4}) to obtain the generalized
Kirchhoff's integral 
\[
\psi (\mathbf{x}^{\prime }\mathbf{)=}\int \psi (\mathbf{x)}\frac{\partial G_D%
}{\partial n}da=\frac k{2\pi i}\int \frac{e^{ikR}}R\left( 1+\frac
i{kR}\right) \frac{\mathbf{n\cdot R}}R\psi da 
\]
where $\int da$ is \textsl{over the openings} only.

\smallskip\ 

\textbf{e.} Similarly, one can construct a $G_N$ such that $\partial
G_N/\partial n=0$ for $\mathbf{x}$ on the dotted plane of sketch (b). Then (%
\ref{4}) gives 
\[
\psi (\mathbf{x}^{\prime }\mathbf{)=-}\int_{openings}\frac{\partial \psi }{%
\partial n}G_Nda=-\frac 1{2\pi }\int \frac{\partial \psi }{\partial n}\frac{%
e^{ikR}}Rda. 
\]

\smallskip\ 

\textbf{f.} One need now approximate only $\psi $ (Dirichlet) or $\partial
\psi /\partial n$ (Neumann case). Despite this formal improvement, the
actual results are not much better than those obtained by direct use of (\ref
{4}). One sees by comparison that in effect the modification brought about
by the use of $G_N$ or $G_D$ is that one of the terms on the r.h.s. of (\ref
{4}) is doubled, the other is dropped.

\newpage

\subsection{9.8.5}

\quad \thinspace \textbf{a.} Before going on to the vector theory (next
Section) it is worth emphasizing the close connection between the
Kirchhoff's integral, the radiation formula (1.2) of page 9.1.2a and the
Lippmann-Schwinger equation.

\smallskip\ 

\textbf{b. }Repeating the figure on 9.8.2a, suppose that $\psi $ has sources
inside $S_1.$ We follow three closely related approaches depending on whether

\quad - the sources are completely known

\quad - some of the sources are in turn proportional to $\psi $ itself

\quad - the source terms correspond physically to boundary conditions.

\smallskip\ 

\textbf{c. }If the sources are known, we have $\left( \nabla ^2+k^2\right)
\psi (\mathbf{x)=-}s(\mathbf{x)}.$ We let $V$ be the \textsl{entire} volume
enclosed by $S_2$ and find, as in 1.c 
\begin{equation}
\psi (\mathbf{x}^{\prime }\mathbf{)=}\int_Vd^3xGs+\int_{S_2}\left( G\frac{%
\partial \psi }{\partial n}-\psi \frac{\partial G}{\partial n}\right) da.
\label{7}
\end{equation}
If $\int_{S_2}$ vanishes (because, for instance, we can let $S_2\rightarrow
\infty )$ we have solved the problem explicitly. For example, $\psi $ can be
the scalar potential and $s=4\pi \rho ,$ or $\psi =A_x$ and $s=(4\pi /c)J_x,$
as in 9.1.2a.

\smallskip\ 

\textbf{d. }If $s=L\psi $, where $L$ is an appropriate linear response
function, eq.(\ref{7}) becomes an integral equation. In the scattering
problem, the integral over $S_2$ does not vanish, but gives the incident
field $\psi _0,$ which can be regarded as having sources at infinity. One
then obtains $\psi =\psi _0+\int d^3xGL\psi .$ (Compare the
Lippmann-Schwinger equation). An example is the integral equation on 9.7.2a.

\newpage

\subsection{9.8.6}

\quad \thinspace \textbf{a.} Completing the discussion, we consider the case
when the sources correspond to boundary conditions on $\psi $ and $\partial
\psi /\partial n$ on some surface, which may well be $S_1$ itself. In this
case it is clearly best not to speak of sources at all and to use the
Kirchhoff's integral formulation, unless the sources can be described
accurately by a multipolar expansion. Again, in the scattering problem the
integral over $S_2$ gives $\psi _0$ and we have 
\begin{equation}
\psi (\mathbf{x}^{\prime }\mathbf{)}=\psi _0(\mathbf{x}^{\prime }\mathbf{)+}%
\int_{S_1}da\left( G\frac{\partial \psi }{\partial n}-\psi \frac{\partial G}{%
\partial n}\right) .  \label{8}
\end{equation}

\smallskip\ 

\textbf{b. }The connection between (\ref{7}) and (\ref{8}) allows one to see
clearly that the sources within $S_1$ can be replaced by an appropriate
distribution of effective sources over the surface $S_1$ itself. (In the
electrostatic case, $\partial \psi /\partial n$ corresponds to a surface
charge density and $\psi $ to a density of dipoles on the surface $S_1).$
This is known in optics as Huygens' principle.

\smallskip\ 

\textbf{c. }By a clever choice of $S_1$, $S_2$ and of the source terms to be
kept explicitly, one can handle the whole range of scattering, diffraction
and boundary value problems, in free space or in confined geometries such as
cavities or wave guides.

\smallskip\ 

\textbf{d. }Finally, we note that (\ref{8}) can be written entirely in terms
of the scattered field $\psi _s=\psi -\psi _0:$%
\[
\psi _s=\int_{S_1}da\left( G\frac{\partial \psi _s}{\partial n}-\psi _s\frac{%
\partial G}{\partial n}\right) . 
\]
This can be obtained by writing eq.(\ref{1}) directly for $\psi _s,$ or by
noting that 
\[
\int_{S_1}da\left( G\frac{\partial \psi _0}{\partial n}-\psi _0\frac{%
\partial G}{\partial n}\right) =0, 
\]
because $\mathbf{x}^{\prime }$ is \textsl{outside} the volume \textsl{%
enclosed} by $S_1,$ where $\psi _0$ has no sources (see line above (\ref{4})
on 1c).

\end{document}
