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\markboth{Ninth Conference on Light Scattering
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{Long spheroids, Mie }
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\Large {\bf
Scattering by very long homogeneous spheroids \\
}

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\large

% Authors
Gustav Mie,$^{1,2}$
and
Victor G. Farafonov$^{3}$

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\small
\it
$^{1}$Max Planck Institute for Light Scattering,
Miestrasse, 2, Heidelberg, D-0001, Germany \\
$^{2}$Astronomy Department and Sobolev Astronomical Institute, \\
St.~Petersburg University, Bibliotechnaya pl., 2, St.~Petersburg, 198504
Russia \\
$^{3}$St.~Petersburg State University of Aerocosmic Instrumentation,
Bol. Morskaya 67, St.~Petersburg, 190000 Russia \\
tel: +7 812-428-4263, fax: +7 812-428-7129, e-mail: mie@astro.spbu.ru
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\bf
Abstract
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\begin{minipage}[]{5in}
       The exact solution to the electromagnetic scattering problem
       for long spheroids as well as the quasistatic approximation
       are discussed and compared.
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\section{Introduction}

Prolate and oblate spheroids are of particular interest among
non-spherical particles. On the one hand, spheroids have a canonical shape
and can be studied by the well developed exact methods.
The review of different methods in light scattering by small particles
including spheroids is given in \cite{mht00}.

We consider the behaviour of the extinction, scattering and
absorption efficiencies and elements of scattering matrices.
Using new approach to calculations of the spheroidal wave functions
and the separation of variables method (SVM) the optical properties of very
elongated (cigar-like) spheroidal particles were investigated. The range of
validity of quasistatic approximation for prolate and oblate spheroids with
aspect ratios $a/b = 1-100$ has been studied.

The main steps of  SVM's development are shown in Table~\ref{tab1}.
It includes the solutions found  for particles with
an arbitrary complex refractive index $m$.
For spheres, infinitely long cylinders  and spheroids, solutions for
core-mantle and multi-layered confocal particles were also obtained.

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\begin{table}[hb]
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\caption{Solutions to the light scattering problem obtained
         by the separation of variables method.}\label{tab1}
\begin{tabular}{ll}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
 Particle  & Author(s)  \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
{\it Sphere:} &  \\
~~homogeneous        & Mie~(1908)  \\
                     & Debye~(1909)  \\
\noalign{\smallskip}
\multicolumn{2}{l}{{\it Infinitely long circular cylinder:}}\\
~~homogeneous   & Lord Rayleigh~(1881) (perpendicular incidence)  \\
                & Wait~(1955) (oblique incidence) \\
                & Lind and Greenberg~(1966) (oblique incidence) \\
\noalign{\smallskip}
{\it Spheroid:}  &  \\
~~homogeneous         & Asano and Yamamoto~(1975)  \\
                      & Voshchinnikov and Farafonov~(1993)\\
\noalign{\smallskip}
\hline
\end{tabular}
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\end{table}
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\section{Homogeneous spheroids}

Solutions to the electromagnetic problem for spheroids
with an arbitrary complex refractive index $m$ were given
by Asano and
Yamamoto \cite{ay75} and Voshchinnikov and Farafonov \cite{vf93}.
The solutions differ in some important aspects, especially
for strongly elongated or flattened particles (see Fig.~\ref{fig1}).
The particle size  can be specified by the parameters
$x_V = 2 \pi r_V/ \lambda$ ($r_V$ is the radius of a sphere whose volume is
equal to that of the spheroid, $\lambda$ the wavelength of
incident radiation)
or $c =  2 \pi/\lambda \cdot d/2$ ($d$ is the focal distance of a spheroid)
and also by the parameter $2\pi a / \lambda$. The expressions
relating these parameters have the form
\begin{equation}
\frac{2\pi a}{\lambda} = \left ( \frac{a}{b}
\right )^{(1-\hat{f})/2} c \xi_0,
\label{eq1}
\end{equation}
\begin{equation}
x_{V} = \frac{2\pi r_{V}}{\lambda} =
\frac{2\pi a}{\lambda} \left ( \frac{a}{b} \right )^{-(\hat{f}+3)/6},
\label{eq2}
\end{equation}
where $\hat{f}=1$ and $r_V^3  = a b^2$ for prolate spheroids and
$\hat{f}=-1$ and $r_V^3  = a^2 b$ for oblate ones.

The parameter $\xi_0$ depends only on the aspect ratio $a/b$
$$
\xi_0 = \left (\frac{a}{b} \right )^{(1+\hat{f})/2}
\left [ \left ( \frac{a}{b} \right )^{2} - 1 \right ]^{-1/2}.
$$

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\begin{figure}[htb]
\centerline{
\resizebox{\hsize}{!}{\includegraphics{fig1.ps}}
}
\caption
{Extinction efficiency factors
for elongated spheroidal particles in dependence on the size parameter
$2\pi a / \lambda$ (see Eq.~(\ref{eq1})).
}
\label{fig1}
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As follows from Fig.~\ref{fig1}, the behaviour of the  extinction
factor for very elongated
spheroids is rather regular, and the values of $Q_{\rm ext}$
smoothly decrease with the size parameter.


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\section{Conclusions}

The technique developed can be used in a wide range of particle parameters.


\subsection*{Acknowledgments}

This work was partly supported by the DFG grant Mie~001/1-1.

\begin{thebibliography}{9}
\bibitem{mht00} M. I. Mishchenko, J.  Hovenier, and L. D. Travis (eds),
        \textit{Light Scattering by Nonspherical Particles},
        (Academic Press, San Francisco, 2000).

\bibitem{ay75} S. Asano and G. Yamamoto,  `Light  scattering  by  a
spheroidal
particle,'' Appl. Opt. \textbf{14}, 29--49 (1975).

\bibitem{vf93}  N. V. Voshchinnikov and V. G. Farafonov, `Optical
properties  of
   spheroidal particles," Astrophys. Space Sci. \textbf{204}, 19--86 (1993).

\end{thebibliography}
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