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{\Large\bf Computer Codes ``Separation of Variables Method''}

\medskip

{\Large\bf and ``TMM-Method''}


\medskip

\vspace{0.2cm}

{\large Final Report}

\vspace{0.2cm}

({\it September 30, 1998})
\ec

\bigskip
\bigskip

\vspace{0.6cm}

The computer codes which
treat the light scattering by spheroidal particles
basing on
the separation of variables method and a
new advanced method for axisymmetric particles were created.
The complete computer codes with extensive documentation and test
cases are on an  attached diskette in computer readable form.

\medskip
\medskip
\medskip
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noi {\sf I. Separation of Variables Method}

\medskip
\medskip

\noi New versions of numerical codes were developed
to calculate the optical properties
(scattering matrix and different cross-sections)
of homogeneous and core-mantle spheroidal particles.
They are based on the solutions to the light scattering problem of
Farafonov$^{1-3}$.

\medskip
\medskip

%---------------------------------------------------------
\noi {\bf 1$^0$.}
{\underline{\it Different methods of calculation of\ the spheroidal wave
functions\ were included into the com-}} \\ {\underline{\it puter codes.}}
Expansions in terms of either the Legendre functions or the Bessel ones
and a solution of the corresponding differential equation
were used in the calculations.
The method is chosen automatically
in the dependence on the value of calculated Wronskian and
available recommendations.
A comparison with the results obtained by previous versions
of the code as well as a thorough testing have been done.
The new approach allows to enlarge the codes' application area,
in particular as concern the size parameter values.

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 2$^0$.}
{\underline{\it A method of computations of the optical properties of highly
absorbing spheroidal particles}}\\ {\underline{\it was developed.}}
The method is based on a new approach to the determination of the eigenvalues
$\lambda_{m,n}(c)$ for complex values of the parameter $c$.
It is assumed that near to the branch points the eigenvalues
are changing non-monotonously with a growth of $n$.
A comparison of the efficiency factors for absorbing particles
of soot, graphite, iron, etc. calculated
with the numerical codes of Martin$^4$ and Mishchenko$^5$ as well as
using the quasistatic and Rayleigh approximations (see Section~4)
has been performed.  The validity of the
approach developed was confirmed in all cases.
The results of the work are presented in the paper submitted
to the Journal of Quantitative Spectroscopy \& Radiative Transfer$^6$.
Some cross-section values obtained with high accuracy (7 significant digits)
by different methods
are given in the paper in tabular form to serve as benchmarks
(see Chapter~III).

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 3$^0$.}
{\underline{\it A method of computations of scattering matrices for
spheroids in the frame related to the sca-}}
{\underline{\it ttering plane was worked out.}}
This method was applied to the case of a fixed orientation of the
particles. It may be considered as a first step to applications
of non-spherical particles in single and multiple scattering calculations
in complex objects.
Some relations between the elements of the scattering matrices
were checked and found to be in accordance with the analytical
expressions given in Section~4.

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 4$^0$.}
{\underline{\it Quasistatic approximation for spheroidal particles.}
The results presented in this Section are based on the paper of
Farafonov$^7$.
We give the expressions for efficiency factors and
scattering matrix (in two frames) in the quasistatic approximation
and show the relation to the Rayleigh and Rayleigh-Hans approximations.

\noi {\it Definitions:}

$\lambda$ -- wavelength of incident radiation

$k = 2 \pi /\lambda$ -- wavenumber

$d$ --  focal distance of spheroid

$c =  2 \pi/\lambda \cdot d/2$ -- size parameter

$a$ -- large semiaxis

$b$ -- small semiaxis

$a/b$ -- aspect ratio

$V$ -- volume

$r_V$ -- radius of equivolume sphere

$\alpha$ -- angle between rotation axis and wave-vector of
incident radiation

$m$ -- refractive index ($\varepsilon = m^2$;
       $\varepsilon_1$ for core, $\varepsilon_2$ for mantle)

$Q_{\rm ext}$ -- efficiency factor for extinction

$Q_{\rm sca}$ -- efficiency factor for scattering

$Q_{\rm abs}$ -- efficiency factor for absorption

$\tilde{\alpha}_j$ -- polarizability

$E^{(0)}_{||} = E^{(0)}_{\rm TM}, E^{(0)}_{\bot} = E^{(0)}_{\rm TE}$ --
                incident radiation field

$E^{(1)}_{||} = E^{(1)}_{\rm TM}, E^{(1)}_{\bot} = E^{(1)}_{\rm TE}$ --
                scattered radiation field

$\Re\{.\}$ -- real part

$\Im\{.\}$ -- imaginary part

$L_j$ -- geometrical factors ($L^{(1)}_j$ for core, $L^{(2)}_j$ for mantle)

$\theta, \varphi$ -- angles of scattered radiation in the reference system related to
         spheroid's rotation axis (particle frame)

$\Theta, \Phi$ -- angles of scattered radiation in the reference system related to
         scattering plane (laboratory frame)

$f=1$ for prolate spheroids

$f=-1$ for oblate spheroids

$\tilde{f} = V_{\rm core} /V_{\rm total}$ -- ratio of volumes for core-mantle
particle

\bea
& \mbox{\rm prolate} &
\xi_0 = \left (\frac{a}{b} \right )^{2}
\left [ \left ( \frac{a}{b} \right )^{2} - 1 \right ]^{-1/2},
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\frac{2\pi a}{\lambda} = c \xi_0,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
r_V^3  = a b^2
\eea

\bea
& \mbox{\rm oblate} &
\xi_0 = \left [ \left ( \frac{a}{b} \right )^{2} - 1 \right ]^{-1/2},
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\frac{2\pi a}{\lambda} = \frac{a}{b} c \xi_0,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
r_V^3  = a^2 b
\eea

\be
Q^{\rm TM, TE}_{\rm ext}\left(m, c, \frac{a}{b}, \alpha \right) =
Q^{\rm TM, TE}_{\rm abs}\left(m, c, \frac{a}{b}, \alpha \right) +
Q^{\rm TM, TE}_{\rm sca}\left(m, c, \frac{a}{b}, \alpha \right)
\ee

\bigskip

\be
Q^{\rm TE}_{\rm abs} = \left( Q^{\rm TE}_{\rm abs} \right)_{\rm Rayleigh} =
\frac{4}{3} \, c \xi_0 \left(\frac{\xi_0^2-f}{\xi_0^2-f \cos^2 \alpha}\right)^{1/2} \,
\frac{\Im \{ \varepsilon \}}{|L_1(\varepsilon-1)+1|^2}
\ee

\bigskip

\be
Q^{\rm TM}_{\rm abs} = \left( Q^{\rm TM}_{\rm abs} \right)_{\rm Rayleigh} =
\frac{4}{3} \, c \xi_0 \left(\frac{\xi_0^2-f}{\xi_0^2-f \cos^2 \alpha}\right)^{1/2} \,
\Im \{ \varepsilon \} \, \left(\frac{\sin^2 \alpha}{|L_3(\varepsilon-1)+1|^2} +
\frac{\cos^2 \alpha}{|L_1(\varepsilon-1)+1|^2} \right)
\ee
\bigskip
\be
Q^{\rm TE}_{\rm sca} =
\frac{c^4 \xi_0^2 (\xi_0^2-f)^{3/2}}
      {9 \pi (\xi_0^2-f \cos^2 \alpha)^{1/2}} \, |\tilde{\alpha}_1|^2 \,
      \int_0^{2 \pi} \int_0^{\pi}
      (\cos^2 \varphi + \cos^2 \theta \sin^2 \varphi) \,
      G^2(u) \sin \theta {\rm d}\theta {\rm d}\varphi
\ee
\bigskip
\bea
Q^{\rm TM}_{\rm sca} =
\frac{c^4 \xi_0^2 (\xi_0^2-f)^{3/2}}
      {9 \pi (\xi_0^2-f \cos^2 \alpha)^{1/2}} \, \times
      \nonumber \\
      \left [ |\tilde{\alpha}_1|^2 \, \cos^2 \alpha
      \int_0^{2 \pi} \int_0^{\pi}
      (\sin^2 \varphi + \cos^2 \theta \cos^2 \varphi) \,
      G^2(u) \sin \theta {\rm d}\theta {\rm d}\varphi \, +
      \right. \nonumber \\ \left.
      |\tilde{\alpha}_3|^2 \, \sin^2 \alpha
      \int_0^{2 \pi} \int_0^{\pi}
      G^2(u) \sin^3 \theta {\rm d}\theta {\rm d}\varphi \, +
      \right. \nonumber \\ \left.
      2 \Re \{\tilde{\alpha}_3 \tilde{\alpha}_1^* \} \sin \alpha \cos \alpha
      \int_0^{2 \pi} \int_0^{\pi}
      G^2(u) \cos \theta \sin^2 \theta \cos \varphi
      {\rm d}\theta {\rm d}\varphi \right ]
\eea
\bigskip
\be
G(u) = \frac{3}{u^3}(\sin u - u \cos u)
\ee
\bigskip
\be
u = c \xi_0 \sqrt{(\cos \theta - \cos \alpha)^2 +
    \left ( \frac{a}{b} \right )^{-2f}(\sin^2 \theta + \sin^2 \alpha
      -2 \sin \theta \sin \alpha \cos \varphi)}
\ee

homogeneous particles:
\be
\tilde{\alpha}_j = \frac{\varepsilon-1}{L_j(\varepsilon-1)+1},
\,\,\,\,\,\,\,\,\,\,
L_1 = \frac{1-L_3}{2}
\ee

\bea
& \mbox{\rm prolate} &
L_3 = \frac{1-e^2}{e^2} \left( \frac{1}{2e} \ln \frac{1+e}{1-e}-1 \right),
\,\,\,\,\,\,\,\,\,\, e = \sqrt{1-b^2/a^2}
\eea

\bea
& \mbox{\rm oblate} &
L_3 = \frac{1+\tilde{e}^2}{\tilde{e}^2} \left(1- \frac{1}{\tilde{e}^2} \arctan{\tilde{e}} \right),
\,\,\,\,\,\,\,\,\,\, \tilde{e} = \sqrt{a^2/b^2-1}
\eea

core-mantle particles:
\be
\tilde{\alpha}_j = \frac{(\varepsilon_2-1)
  [\varepsilon_2 + (\varepsilon_1 - \varepsilon_2)(L_j^{(1)} - \tilde{f}L_j^{(2)})]
 + \tilde{f}\varepsilon_2 (\varepsilon_1 - \varepsilon_2)}
 {[\varepsilon_2 + (\varepsilon_1 - \varepsilon_2)(L_j^{(1)} - \tilde{f}L_j^{(2)})]
 [1 + (\varepsilon_2 - 1)L_j^{(2)})] +
 \tilde{f}L_j^{(2)}\varepsilon_2 (\varepsilon_1 - \varepsilon_2)}
\ee

%\newpage

amplitude matrix (particle frame):
\begin{equation} \left( \begin{array}{c}
E^{(1)}_{||}\\ E^{(1)}_{\bot}  \end{array} \right)_{\rm part.} = \frac{1}{- i k r}
\exp \left[ {i (k r - \vec{k} \vec{r} )} \right]
\left( \begin{array}{cc} T_{22}  &  T_{12} \\
                         T_{21}  &  T_{11} \end{array} \right)_{\rm part.}
\left( \begin{array}{c} E^{(0)}_{||} \\ E^{(0)}_{\bot}
\end{array} \right)_{\rm part.},
\end{equation}
\be
\left\{
\begin{array}{ll}
 T_{11} =  {\cal{K}} \tilde{\alpha}_1 \cos \varphi G({u}) & \\
  & \\
 T_{12} =   {\cal{K}}  \tilde{\alpha}_1 \cos \theta \sin \varphi G({u}) & \\
  & \\
 T_{21} = -  {\cal{K}}  \tilde{\alpha}_1 \cos \alpha \sin \varphi G({u}) & \\
  & \\
 T_{22} =  {\cal{K}} (\tilde{\alpha}_3 \sin \alpha \sin \theta +
          \tilde{\alpha}_1 \cos \alpha \cos \theta \cos \varphi)
           G({u}) &
\end{array}
          \right.
\ee

\be
{\cal{K}} = - \frac{i k^3}{4 \pi} V =
            - \frac{i}{3} (c \xi_0)^3 \left ( \frac{a}{b} \right )^{-2f}
\ee

\be
{\rm prolate} \,\,\,\,\,\,\,\,\,\,  V = \frac{4 \pi}{3} a b^2,
 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
{\rm oblate} \,\,\,\,\,\,\,\,\,\,\, V = \frac{4 \pi}{3} a^2 b
\ee

\be
\left\{
\begin{array}{rll}
{\rm incident \,\,\, radiation} &  \vec{i}^{(0)}_{\bot} = - \vec{i}_{\rm y}, &
 \vec{i}^{(0)}_{||} =  \vec{i}_{\rm x} \cos \alpha - \vec{i}_{\rm z} \sin \alpha, \\
{\rm scattered \,\,\, radiation} & \vec{i}^{(1)}_{\bot} = - \vec{i}_{\varphi}, &
 \vec{i}^{(1)}_{||} =  \vec{i}_{\theta}
\end{array}
          \right.
\ee

\bigskip

amplitude matrix (laboratory frame):
\begin{equation} \left( \begin{array}{c}
E^{(1)}_{||}\\ E^{(1)}_{\bot}  \end{array} \right)_{\rm lab.} = \frac{1}{- i k r}
\exp \left[ {i (k r - \vec{k} \vec{r} )} \right]
\left( \begin{array}{cc} \tilde{T}_{22}  &  \tilde{T}_{12} \\
                         \tilde{T}_{21}  &  \tilde{T}_{11} \end{array} \right)_{\rm lab.}
\left( \begin{array}{c} E^{(0)}_{||} \\ E^{(0)}_{\bot}
\end{array} \right)_{\rm lab.},
\end{equation}

\be
\left\{
\begin{array}{ll}
  \tilde{T}_{11} =   {\cal{K}} [\tilde{\alpha}_1 +
          (\tilde{\alpha}_3 - \tilde{\alpha}_1)
          \sin^2 \alpha \sin^2 (\Phi - \Phi_0)] G(\tilde{u})  & \\
  & \\
  \tilde{T}_{12} =  {\cal{K}}
          (\tilde{\alpha}_3 - \tilde{\alpha}_1) \sin \alpha
          [\cos \Theta \sin \alpha \cos (\Phi - \Phi_0) -
          \sin \Theta \cos \alpha] \sin (\Phi - \Phi_0) G(\tilde{u})  &\\
  & \\
  \tilde{T}_{21} =  {\cal{K}}
          (\tilde{\alpha}_3 - \tilde{\alpha}_1) \sin^2 \alpha
          \cos (\Phi - \Phi_0) \sin (\Phi - \Phi_0) G(\tilde{u}) & \\
  & \\
  \tilde{T}_{22} =  {\cal{K}}
          \{\tilde{\alpha}_1 \cos \Theta +
          (\tilde{\alpha}_3 - \tilde{\alpha}_1) \sin \alpha
          [\cos \Theta \sin \alpha \cos (\Phi - \Phi_0) -
          \sin \Theta \cos \alpha] \times & \\
  \phantom{\tilde{T}_{22} =  {\cal{K}}\{}
         \cos (\Phi - \Phi_0) \} G(\tilde{u}) &
\end{array}
          \right.
\ee

non-rotating particle: $\Phi_0 = 0$


\be
\tilde{u} = 2 c \xi_0 \sin \frac{\Theta}{2}
        \sqrt{\cos^2 \beta +
    \left ( \frac{a}{b} \right )^{-2f} \sin^2 \beta}
\ee

\be
\cos \beta = -  \cos \alpha \sin \frac{\Theta}{2} +
        \sin \alpha \cos \frac{\Theta}{2} \cos (\Phi_0 - \Phi)
\ee

%\newpage

transformation: particle frame $\longrightarrow$ laboratory frame

(see Fig.1, Asano \& Sato, Appl. Opt. {\bf 19}, 962, 1980)

\be
\left( \begin{array}{cc} \tilde{T}_{22}  &  \tilde{T}_{12} \\
                         \tilde{T}_{21}  &  \tilde{T}_{11} \end{array} \right)_{\rm lab.}  =
\left( \begin{array}{rr} \cos \gamma  & -  \sin \gamma  \\
                         \sin \gamma  &  \cos \gamma  \end{array} \right)
\left( \begin{array}{cc} {T}_{22}  &  {T}_{12} \\
                         {T}_{21}  &  {T}_{11} \end{array} \right)_{\rm part.}
\left( \begin{array}{rr} \cos (\Phi - \Phi_0)  &  \sin (\Phi - \Phi_0)  \\
                        - \sin (\Phi - \Phi_0)  &  \cos (\Phi - \Phi_0)  \end{array} \right)
\ee

\be
\left\{
\begin{array}{ll}
 \sin \gamma \sin \theta =  \sin \alpha \sin (\Phi_0 - \Phi) & \\
  & \\
 \cos \gamma \sin \theta =  \cos \alpha \sin \Theta - \sin \alpha \cos \Theta
                         \cos (\Phi_0 - \Phi) &
\end{array}
          \right.
\ee

\bigskip

Rayleigh approximation: $G(u) = 1$

\bigskip

Rayleigh-Hans approximation: $\tilde{\alpha}_1 = \tilde{\alpha}_3 = |m-1|^2$

\bigskip

sphere: $\tilde{\alpha}_1 = \tilde{\alpha}_3, {a}/{b} = 1$

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 5$^0$.}
{\underline{\it Numerical code for calculations of extinction,
scattering and absorption efficiency factors by} \\
{\underline{\it homogeneous spheroidal particles.}
The directory  {\it SVM\_HOM/VER4\_3Q} contains the following
files:

{\tt readme} -- file `readme';

{\tt h4\_3q2a.for} -- program;

{\tt homn.dat} -- input file;

{\tt hom.res} -- output file (test results);

{\tt h.dat} -- output file (test results).

{\footnotesize
%{\tiny
\begin{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%   README: program  hom_c (version 4.3q2a)
%%%   -------
%%%
%%%   Light Scattering by Spheroidal Particles
%%%   ========================================
%%%
%%%     Calculations of the extinction, scattering and absorption
%%%     efficiency factors for prolate or oblate homogeneous spheroids.
%%%
%%%
%%%     Parameters:
%%%     -----------
%%%
%%%         nterms: number of terms in Eqs. for efficiency factors,
%%%        nalphas: number of angles for incident radiation.
%%%
%%%     The values of the parameters may be changed in file h4_3q2a.for.
%%%
%%%
%%%     Restrictions:
%%%     -------------
%%%
%%%         nterms:
%%%          c2 = ri*(2*pi/wvl)*(d/2) <= 135  for prolate spheroids,
%%%                                   <= 100  for oblate spheroids,
%%%          (d - focus distance of spheroid),
%%%
%%%          nterms ~= 1.2 * cdabs(c2) + 8 (+ ~30 if ksi0 > 2).
%%%
%%%        nalphas: <=  91
%%%
%%%
%%%     Recommended values:
%%%     -------------------
%%%                       
%%%        refractive index: k <= 1/2*n,
%%%                     eps: 1d-15 for prolate spheroids,
%%%                          1d-15 for oblate spheroids,
%%%                    eps1: 1d-3 - 1d-6
%%%
%%% 
%%%     Publications:
%%%     -------------
%%% 
%%%     Voshchinnikov N.V., Farafonov  V.G.
%%%     Optical properties of spheroidal particles,
%%%     Astrophys. Space Sci. 1993. V. 204. P. 19-86.
%%% 
%%%     Voshchinnikov N.V.
%%%     Electromagnetic scattering by homogeneous and coated spheroids:
%%%     calculations using the separation of variables method,
%%%     J. Quant. Spectrosc. Radiat. Transfer 1996. V. 55. P. 627-636.
%%% 
%%%     Farafonov V.G., Voshchinnikov N.V.,  Somsikov V.V.
%%%     Light scattering by a core-mantle spheroidal particle,
%%%     Applied Optics 1996. V. 35. P. 5412-5426.
%%% 
%%% 
%%%     Questions, more information, new versions:
%%%     -------------------------------------------
%%%
%%%     Dr. Nikolai Voshchinnikov
%%%     Astronomical Institute
%%%     St. Petersburg University
%%%     St. Petersburg-Peterhof
%%%     198904 Russia
%%%     e-mail: nvv@aispbu.spb.su
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%     The development of the new approach and the code was financially
%%%     supported by a grant of the Volkswagen Foundation (Germany).
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{verbatim}

\begin{verbatim}
----------------------------   homn.dat
   1                                ! K
1.5     0.00                        ! RI
1.      D-15                        ! EPS
1.      D-15                        ! EPS1
1.      D-05                        ! EPS2
0.5                                 ! WVL
   2                                ! icx:1-Rv, 2-Xv, 3-c, 4-cksi0, 5-Xc
   1                                ! NRv
3.0                                 ! aRv0
0.5                                 ! dRv
  10                                ! NALF
   0                                ! IAL0
  10                                ! IDAL
2.0                                 ! a/b
4.0                                 ! a/b
\end{verbatim}

\begin{verbatim}
----------------------------   hom.res
 SPHEROID: Prolate
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv=  3.000
 a/b =  2.000    Ksi0= 1.155
 ========================================================================
 C1= 4.124  C2= 6.186 0.000  Cksi= 4.762  Nmax= 14  Nmxe= 18
 ------------------------------------------------------------------------
 alpha   m Cext/(Pi*Rv**2) Csca/(Pi*Rv**2) Cabs/(Pi*Rv**2)
 ------------------------------------------------------------------------
   0     1  4.689130E+00   4.689130E+00  -1.006689E-10  TM
  10     3  4.581254E+00   4.581254E+00   2.982751E-10  TM
  20     4  4.292827E+00   4.292827E+00   4.206238E-10  TM
  30     4  3.921359E+00   3.921359E+00   1.230543E-10  TM
  40     4  3.590461E+00   3.590461E+00  -2.961867E-10  TM
  50     5  3.382928E+00   3.382928E+00  -3.026627E-10  TM
  60     5  3.295448E+00   3.295448E+00   2.561503E-11  TM
  70     5  3.270856E+00   3.270856E+00   7.502259E-11  TM
  80     5  3.264661E+00   3.264661E+00  -1.369500E-10  TM
  90     5  3.263132E+00   3.263132E+00  -2.261804E-10  TM
 ........................................................................
  10     3  4.546672E+00   4.546672E+00  -2.895719E-08  TE
  20     4  4.159671E+00   4.159671E+00  -1.377982E-08  TE
  30     4  3.650252E+00   3.650252E+00   1.979189E-09  TE
  40     4  3.196037E+00   3.196037E+00   6.971372E-09  TE
  50     5  2.933618E+00   2.933618E+00   8.083837E-10  TE
  60     5  2.842884E+00   2.842884E+00  -7.045059E-10  TE
  70     5  2.774782E+00   2.774782E+00   6.612780E-09  TE
  80     5  2.655350E+00   2.655350E+00   5.420859E-09  TE
  90     5  2.586365E+00   2.586365E+00  -1.363656E-11  TE
 ========================================================================
 SPHEROID: Prolate
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv=  3.000
 a/b =  4.000    Ksi0= 1.033
 ========================================================================
 C1= 7.319  C2=10.979 0.000  Cksi= 7.560  Nmax= 20  Nmxe= 24
 ------------------------------------------------------------------------
 alpha   m Cext/(Pi*Rv**2) Csca/(Pi*Rv**2) Cabs/(Pi*Rv**2)
 ------------------------------------------------------------------------
   0     1  6.097449E+00   6.097449E+00   1.218605E-09  TM
  10     3  5.801787E+00   5.801787E+00   5.196346E-10  TM
  20     3  4.994845E+00   4.994845E+00  -5.858236E-10  TM
  30     4  3.987717E+00   3.987717E+00  -5.082343E-10  TM
  40     4  3.268992E+00   3.268992E+00   3.613986E-10  TM
  50     4  3.052207E+00   3.052207E+00   3.212005E-10  TM
  60     4  3.085555E+00   3.085555E+00  -5.236880E-12  TM
  70     4  3.135157E+00   3.135157E+00   1.627460E-10  TM
  80     4  3.105479E+00   3.105479E+00   7.040120E-11  TM
  90     4  3.048810E+00   3.048810E+00   5.340331E-12  TM
 ........................................................................
  10     3  5.752954E+00   5.752963E+00  -9.801123E-06  TE
  20     3  4.760930E+00   4.760933E+00  -2.606479E-06  TE
  30     4  3.405776E+00   3.405773E+00   2.357485E-06  TE
  40     4  2.366909E+00   2.366908E+00   8.576168E-07  TE
  50     4  2.161966E+00   2.161967E+00  -8.565377E-07  TE
  60     4  2.323285E+00   2.323284E+00   1.150746E-06  TE
  70     4  2.176374E+00   2.176374E+00   7.193061E-07  TE
  80     4  2.012766E+00   2.012766E+00   2.535630E-07  TE
  90     4  2.030483E+00   2.030482E+00   9.016812E-07  TE
 ========================================================================
 SPHEROID: Oblate
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv=  3.000
 a/b =  2.000    Ksi0= 0.577
 ========================================================================
 C1= 3.273  C2= 4.910 0.000  Cksi= 3.780  Nmax= 12  Nmxe= 26
 ------------------------------------------------------------------------
 alpha   m Cext/(Pi*Rv**2) Csca/(Pi*Rv**2) Cabs/(Pi*Rv**2)
 ------------------------------------------------------------------------
   0     1  2.618970E+00   2.618970E+00   2.183117E-08  TM
  10     4  2.590011E+00   2.590011E+00   1.445189E-08  TM
  20     5  2.538354E+00   2.538354E+00   1.204643E-09  TM
  30     5  2.537404E+00   2.537404E+00  -6.246299E-09  TM
  40     6  2.638029E+00   2.638029E+00  -6.320452E-09  TM
  50     6  2.836368E+00   2.836368E+00  -3.857735E-09  TM
  60     6  3.083531E+00   3.083531E+00  -1.965296E-09  TM
  70     6  3.314939E+00   3.314939E+00  -8.561860E-10  TM
  80     6  3.475843E+00   3.475843E+00  -3.476014E-10  TM
  90     6  3.533040E+00   3.533040E+00  -2.573503E-10  TM
 ........................................................................
  10     4  2.598937E+00   2.598937E+00  -1.802960E-12  TE
  20     5  2.567452E+00   2.567452E+00  -4.102167E-12  TE
  30     5  2.598581E+00   2.598581E+00  -2.660233E-12  TE
  40     6  2.767982E+00   2.767982E+00   1.164220E-12  TE
  50     6  3.092626E+00   3.092626E+00   4.294206E-12  TE
  60     6  3.508484E+00   3.508484E+00   6.138098E-12  TE
  70     6  3.906439E+00   3.906439E+00   7.503683E-12  TE
  80     6  4.186173E+00   4.186173E+00   8.580478E-12  TE
  90     6  4.285986E+00   4.285986E+00   9.022625E-12  TE
 ========================================================================
 SPHEROID: Oblate
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv=  3.000
 a/b =  4.000    Ksi0= 0.258
 ========================================================================
 C1= 4.611  C2= 6.916 0.000  Cksi= 4.762  Nmax= 16  Nmxe= 20
 ------------------------------------------------------------------------
 alpha   m Cext/(Pi*Rv**2) Csca/(Pi*Rv**2) Cabs/(Pi*Rv**2)
 ------------------------------------------------------------------------
   0     1  1.848859E+00   1.848859E+00   2.465295E-08  TM
  10     5  1.830933E+00   1.830933E+00   1.551204E-08  TM
  20     5  1.803608E+00   1.803608E+00   5.977330E-11  TM
  30     6  1.828019E+00   1.828019E+00  -6.098922E-09  TM
  40     6  1.959446E+00   1.959446E+00  -3.993869E-09  TM
  50     7  2.202201E+00   2.202201E+00  -1.114774E-09  TM
  60     7  2.494754E+00   2.494754E+00  -4.430255E-11  TM
  70     7  2.751899E+00   2.751899E+00   5.039459E-11  TM
  80     7  2.917280E+00   2.917280E+00  -2.881382E-11  TM
  90     7  2.972924E+00   2.972924E+00  -6.437863E-11  TM
 ........................................................................
  10     5  1.808093E+00   1.807343E+00   7.496233E-04  TE
  20     5  1.674922E+00   1.674773E+00   1.483554E-04  TE
  30     6  1.585682E+00   1.585848E+00  -1.664838E-04  TE
  40     6  1.857019E+00   1.857149E+00  -1.293722E-04  TE
  50     7  2.611194E+00   2.611204E+00  -9.847564E-06  TE
  60     7  3.608773E+00   3.608743E+00   2.987607E-05  TE
  70     7  4.498896E+00   4.498890E+00   5.743841E-06  TE
  80     7  5.068786E+00   5.068811E+00  -2.583103E-05  TE
  90     7  5.259634E+00   5.259672E+00  -3.801147E-05  TE
 ========================================================================
\end{verbatim}
\begin{verbatim}
----------------------------   h.dat
 SPHEROID: Prolate
 Refractive index:  1.5000  0.0000 i     Lambda= 0.500 mkm     a/b =  2.000
 ================================================================================================
    c    x_V 2*pi*a/w alpha  Qext(TM)    Qsca(TM)    Qabs(TM)    Qext(TE)    Qsca(TE)    Qabs(TE)
 ================================================================================================
   4.12  3.00  4.76     0  7.4435E+00  7.4435E+00 -1.5980E-10
   4.12  3.00  4.76    10  6.9641E+00  6.9641E+00  4.5342E-10  6.9115E+00  6.9115E+00 -4.4019E-08
   4.12  3.00  4.76    20  5.8629E+00  5.8629E+00  5.7446E-10  5.6811E+00  5.6811E+00 -1.8820E-08
   4.12  3.00  4.76    30  4.7055E+00  4.7055E+00  1.4766E-10  4.3802E+00  4.3802E+00  2.3750E-09
   4.12  3.00  4.76    40  3.8085E+00  3.8085E+00 -3.1418E-10  3.3902E+00  3.3902E+00  7.3948E-09
   4.12  3.00  4.76    50  3.2321E+00  3.2321E+00 -2.8917E-10  2.8028E+00  2.8028E+00  7.7235E-10
   4.12  3.00  4.76    60  2.9017E+00  2.9017E+00  2.2555E-11  2.5032E+00  2.5032E+00 -6.2034E-10
   4.12  3.00  4.76    70  2.7180E+00  2.7180E+00  6.2343E-11  2.3058E+00  2.3058E+00  5.4952E-09
   4.12  3.00  4.76    80  2.6210E+00  2.6210E+00 -1.0995E-10  2.1318E+00  2.1318E+00  4.3520E-09
   4.12  3.00  4.76    90  2.5899E+00  2.5899E+00 -1.7952E-10  2.0528E+00  2.0528E+00 -1.0823E-11
 ================================================================================================
 SPHEROID: Prolate
 Refractive index:  1.5000  0.0000 i     Lambda= 0.500 mkm     a/b =  4.000
 ================================================================================================
    c    x_V 2*pi*a/w alpha  Qext(TM)    Qsca(TM)    Qabs(TM)    Qext(TE)    Qsca(TE)    Qabs(TE)
 ================================================================================================
   7.32  3.00  7.56     0  1.5365E+01  1.5365E+01  3.0707E-09
   7.32  3.00  7.56    10  1.2131E+01  1.2131E+01  1.0865E-09  1.2029E+01  1.2029E+01 -2.0494E-05
   7.32  3.00  7.56    20  7.5833E+00  7.5833E+00 -8.8942E-10  7.2282E+00  7.2282E+00 -3.9572E-06
   7.32  3.00  7.56    30  4.6105E+00  4.6105E+00 -5.8761E-10  3.9377E+00  3.9377E+00  2.7257E-06
   7.32  3.00  7.56    40  3.0704E+00  3.0704E+00  3.3944E-10  2.2231E+00  2.2231E+00  8.0551E-07
   7.32  3.00  7.56    50  2.4565E+00  2.4565E+00  2.5851E-10  1.7400E+00  1.7400E+00 -6.8937E-07
   7.32  3.00  7.56    60  2.2215E+00  2.2215E+00 -3.7703E-12  1.6727E+00  1.6727E+00  8.2849E-07
   7.32  3.00  7.56    70  2.0931E+00  2.0931E+00  1.0865E-10  1.4530E+00  1.4530E+00  4.8023E-07
   7.32  3.00  7.56    80  1.9846E+00  1.9846E+00  4.4990E-11  1.2863E+00  1.2863E+00  1.6204E-07
   7.32  3.00  7.56    90  1.9206E+00  1.9206E+00  3.3642E-12  1.2791E+00  1.2791E+00  5.6802E-07
 ================================================================================================
 SPHEROID: Oblate
 Refractive index:  1.5000  0.0000 i     Lambda= 0.500 mkm     a/b =  2.000
 ================================================================================================
    c    x_V 2*pi*a/w alpha  Qext(TM)    Qsca(TM)    Qabs(TM)    Qext(TE)    Qsca(TE)    Qabs(TE)
 ================================================================================================
   3.27  3.00  3.78     0  1.6498E+00  1.6498E+00  1.3753E-08
   3.27  3.00  3.78    10  1.6504E+00  1.6504E+00  9.2088E-09  1.6561E+00  1.6561E+00 -1.1489E-12
   3.27  3.00  3.78    20  1.6742E+00  1.6742E+00  7.9453E-10  1.6934E+00  1.6934E+00 -2.7056E-12
   3.27  3.00  3.78    30  1.7733E+00  1.7733E+00 -4.3654E-09  1.8161E+00  1.8161E+00 -1.8592E-12
   3.27  3.00  3.78    40  2.0005E+00  2.0005E+00 -4.7929E-09  2.0990E+00  2.0990E+00  8.8285E-13
   3.27  3.00  3.78    50  2.3880E+00  2.3880E+00 -3.2479E-09  2.6037E+00  2.6037E+00  3.6153E-12
   3.27  3.00  3.78    60  2.9368E+00  2.9368E+00 -1.8718E-09  3.3415E+00  3.3415E+00  5.8460E-12
   3.27  3.00  3.78    70  3.5934E+00  3.5934E+00 -9.2810E-10  4.2346E+00  4.2346E+00  8.1339E-12
   3.27  3.00  3.78    80  4.1937E+00  4.1937E+00 -4.1939E-10  5.0507E+00  5.0507E+00  1.0353E-11
   3.27  3.00  3.78    90  4.4514E+00  4.4514E+00 -3.2424E-10  5.4000E+00  5.4000E+00  1.1368E-11
 ================================================================================================
 SPHEROID: Oblate
 Refractive index:  1.5000  0.0000 i     Lambda= 0.500 mkm     a/b =  4.000
 ================================================================================================
    c    x_V 2*pi*a/w alpha  Qext(TM)    Qsca(TM)    Qabs(TM)    Qext(TE)    Qsca(TE)    Qabs(TE)
 ================================================================================================
   4.61  3.00  4.76     0  7.3372E-01  7.3372E-01  9.7835E-09
   4.61  3.00  4.76    10  7.3710E-01  7.3710E-01  6.2449E-09  7.2790E-01  7.2760E-01  3.0178E-04
   4.61  3.00  4.76    20  7.5856E-01  7.5856E-01  2.5140E-11  7.0444E-01  7.0438E-01  6.2396E-05
   4.61  3.00  4.76    30  8.2909E-01  8.2909E-01 -2.7661E-09  7.1918E-01  7.1925E-01 -7.5508E-05
   4.61  3.00  4.76    40  9.9347E-01  9.9347E-01 -2.0250E-09  9.4154E-01  9.4160E-01 -6.5594E-05
   4.61  3.00  4.76    50  1.3030E+00  1.3030E+00 -6.5960E-10  1.5450E+00  1.5450E+00 -5.8267E-06
   4.61  3.00  4.76    60  1.8171E+00  1.8171E+00 -3.2268E-11  2.6284E+00  2.6284E+00  2.1760E-05
   4.61  3.00  4.76    70  2.6320E+00  2.6320E+00  4.8199E-11  4.3029E+00  4.3029E+00  5.4936E-06
   4.61  3.00  4.76    80  3.8427E+00  3.8427E+00 -3.7954E-11  6.6767E+00  6.6767E+00 -3.4025E-05
   4.61  3.00  4.76    90  4.7192E+00  4.7192E+00 -1.0219E-10  8.3491E+00  8.3492E+00 -6.0339E-05
 ================================================================================================
\end{verbatim}
}

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 6$^0$.}
{\underline{\it Numerical code for calculations of scattering matrices
by homogeneous spheroidal particles.}
The directory  {\it SVM\_HOM/VER4\_1I} contains the following
files:

{\tt readme} -- file `readme';

{\tt h4\_1i3.for} -- program;

{\tt s15n3.dat} -- input file;

{\tt s15.i} -- output file (test results);

{\footnotesize
\begin{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%   READ.ME: program  ver4_1i3 (version 4_1i3)
%%%   --------
%%%
%%%
%%%   Light Scattering by Spheroidal Particles
%%%   ========================================
%%%
%%%     Calculations of the extinction and absorption efficiency factors,
%%%     albedo and scattering matrix for prolate or oblate
%%%     homogeneous spheroids.
%%%
%%%     The calculations may be done in two coordinate systems:
%%%          1) particle's frame (related to the rotation axis of a spheroid)
%%%          2) laboratory frame (related to the scattering plane).
%%%
%%%
%%%     Parameters:
%%%     -----------
%%%
%%%         nterms: number of terms in Eqs. for efficiency factors,
%%%        nalphas: number of angles for incident radiation,
%%%        nthetas: number of angles for scattered radiation.
%%%
%%%     The values of the parameters may be changed in file ver4_1i.for.
%%%
%%%     !!!  The angles 'theta' and 'phi' are in the coordinate system
%%%          related to the rotation axis of !a! spheroid.
%%%
%%%     !!!  The angles 'THETA' and 'PHI' are in the coordinate system
%%%          related to the scattering plane.
%%%
%%%     Restrictions:
%%%     -------------
%%%
%%%         nterms: <~= 100 - 120
%%%
%%%        nalphas: <=  91
%%%        nthetas: <= 181
%%%
%%%     Recommended values:
%%%     -------------------
%%%                       
%%%                     eps: 1d-15
%%%                    eps1: 1d-15
%%%                    eps2: 1d-3 - 1d-6
%%% 
%%%     Publications:
%%%     -------------
%%% 
%%%     Voshchinnikov N.V., Farafonov  V.G.
%%%     Optical properties of spheroidal particles,
%%%     Astrophys. Space Sci. 1993. V. 204. P. 19-86.
%%% 
%%%     Voshchinnikov N.V.
%%%     Electromagnetic scattering by homogeneous and coated spheroids:
%%%     calculations using the separation of variables method,
%%%     J. Quant. Spectrosc. Radiat. Transfer 1996. V. 55. P. 627-636.
%%% 
%%%     Farafonov V.G., Voshchinnikov N.V.,  Somsikov V.V.
%%%     Light scattering by a core-mantle spheroidal particle,
%%%     Applied Optics 1996. V. 35. P. 5412-5426.
%%% 
%%% 
%%%     Questions, more information, new versions:
%%%     -------------------------------------------
%%%
%%%     Dr. Nikolai Voshchinnikov
%%%     Astronomical Institute
%%%     St. Petersburg University
%%%     St. Petersburg-Petrodvoretz
%%%     198904 Russia
%%%     e-mail: nvv@aispbu.spb.su
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%     The development of the new approach and the code was financially
%%%     supported by a grant of the Volkswagen Foundation (Germany).
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{verbatim}

\begin{verbatim}
----------------------------   s15n3.dat
   0                                ! k
1.5     0.0                         ! ri
1.0     D-15                        ! eps
1.      D-15                        ! EPS1
1.      D-06                        ! EPS2
0.5                                 ! wvl
   2                                ! icx:1-Rv, 2-Xv, 3-c, 4-cksi0, 5-Xc
   1                                ! NRv
3.0                                 ! aRv0
4.9                                 ! dRv
   2                                ! nalf
   0                                ! ial0
  50                                ! idal
   0                                ! kt0
  30                                ! kh
 180                                ! kt1
   2                                ! N/fi
   0                                ! Ifi0
  30                                ! Idfi
   1                                ! Isys: 0-part.frame, 1-lab.
2.0                                 ! a/b
\end{verbatim}

\begin{verbatim}
----------------------------   s15.i
 SPHEROID: Prolate /  Particle frame
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv= 3.000
 a/b =  2.000    Ksi0= 1.155
 ******************************************************************************
         C   2*pi*a/wvl Nmax alpha     Qext          Qabs        Albedo
 ------------------------------------------------------------------------------
 M= 1   4.124   4.762    22    0    7.443529E+00  1.261213E-13  1.000000E+00 TM
 ==============================================================================
 THETA       F11           F12           F33           F34          P
 ------------------------------------------------------------------------------
   0    1.175E+02  0.000E+00  1.175E+02  0.000E+00     0.00
  30    5.830E+01  8.810E-01  5.813E+01  4.418E+00    -1.51
  60    3.974E+00  2.180E+00  3.311E+00 -2.794E-01   -54.86
  90    1.186E+00  7.915E-02  1.183E+00 -1.509E-02    -6.67
 120    6.166E-01 -4.469E-02  1.914E-01 -5.845E-01     7.25
 150    6.575E-01  1.569E-01 -5.692E-01  2.895E-01   -23.86
 180    2.290E+00  0.000E+00 -2.290E+00  0.000E+00     0.00
 ==============================================================================
 SPHEROID: Prolate /  Particle frame
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv= 3.000
 a/b =  2.000    Ksi0= 1.155
 ******************************************************************************
         C   2*pi*a/wvl Nmax alpha     Qext          Qabs        Albedo
 ------------------------------------------------------------------------------
 M= 5   4.124   4.762    22   50    3.232122E+00 -1.909584E-14  1.000000E+00 TM
 M= 5   4.124   4.762    26   50    2.802842E+00  1.749711E-13  1.000000E+00 TE
 ==============================================================================================
                    alpha =  50 deg.     phi =   0 deg.

 theta      F11        F12        F13        F14        F21        F22        F23        F24
 ----------------------------------------------------------------------------------------------
   0    1.180E+01  2.926E+00  0.000E+00  0.000E+00  2.926E+00  1.180E+01  0.000E+00  0.000E+00
  30    4.848E+01  3.885E+00  0.000E+00  0.000E+00  3.885E+00  4.848E+01  0.000E+00  0.000E+00
  60    6.772E+01  5.216E+00  0.000E+00  0.000E+00  5.216E+00  6.772E+01  0.000E+00  0.000E+00
  90    5.307E+00  3.966E-01  0.000E+00  0.000E+00  3.966E-01  5.307E+00  0.000E+00  0.000E+00
 120    4.259E-01 -1.215E-01  0.000E+00  0.000E+00 -1.215E-01  4.259E-01  0.000E+00  0.000E+00
 150    2.876E-01 -9.372E-02  0.000E+00  0.000E+00 -9.372E-02  2.876E-01  0.000E+00  0.000E+00
 180    3.058E-01 -7.977E-02  0.000E+00  0.000E+00 -7.977E-02  3.058E-01  0.000E+00  0.000E+00
 ----------------------------------------------------------------------------------------------
 theta      F31        F32        F33        F34        F41        F42        F43        F44
 ----------------------------------------------------------------------------------------------
   0    0.000E+00  0.000E+00  1.120E+01  2.300E+00  0.000E+00  0.000E+00 -2.300E+00  1.120E+01
  30    0.000E+00  0.000E+00  4.805E+01  5.185E+00  0.000E+00  0.000E+00 -5.185E+00  4.805E+01
  60    0.000E+00  0.000E+00  6.723E+01  6.329E+00  0.000E+00  0.000E+00 -6.329E+00  6.723E+01
  90    0.000E+00  0.000E+00  5.130E+00  1.303E+00  0.000E+00  0.000E+00 -1.303E+00  5.130E+00
 120    0.000E+00  0.000E+00  3.900E-01 -1.207E-01  0.000E+00  0.000E+00  1.207E-01  3.900E-01
 150    0.000E+00  0.000E+00  2.343E-01  1.380E-01  0.000E+00  0.000E+00 -1.380E-01  2.343E-01
 180    0.000E+00  0.000E+00  2.578E-01 -1.439E-01  0.000E+00  0.000E+00  1.439E-01  2.578E-01
 ==============================================================================================
                    alpha =  50 deg.     phi =  30 deg.

 theta      F11        F12        F13        F14        F21        F22        F23        F24
 ----------------------------------------------------------------------------------------------
   0    1.180E+01  2.926E+00 -2.949E-15 -1.476E-15  1.463E+00  5.900E+00  9.698E+00  1.992E+00
  30    3.899E+01  3.538E+00  1.696E+00 -7.674E-01  3.676E+00  2.904E+01  2.567E+01  3.892E+00
  60    4.706E+01  5.295E+00  5.787E-01  7.982E-01  4.794E+00  4.015E+01  2.440E+01 -1.351E+00
  90    3.006E+00  1.401E-01  3.942E-01  3.902E-01  2.333E-01  2.552E+00  1.401E+00 -5.348E-01
 120    4.429E-01 -5.963E-02 -2.729E-01  3.314E-02 -3.283E-02  3.415E-01 -1.715E-02  3.444E-02
 150    2.946E-01 -7.991E-02  2.749E-02  7.779E-02 -1.011E-01  2.578E-01 -1.034E-01 -8.147E-02
 180    3.058E-01 -7.977E-02  2.208E-17  6.914E-17 -3.989E-02  1.529E-01 -2.232E-01  1.246E-01
 ----------------------------------------------------------------------------------------------
 theta      F31        F32        F33        F34        F41        F42        F43        F44
 ----------------------------------------------------------------------------------------------
   0   -2.534E+00 -1.022E+01  5.599E+00  1.150E+00 -6.003E-16  9.090E-16 -2.300E+00  1.120E+01
  30   -1.122E+00 -2.587E+01  2.888E+01  1.524E+00 -1.102E+00 -1.979E+00 -3.765E+00  3.856E+01
  60   -2.203E+00 -2.438E+01  3.985E+01  2.929E+00  1.082E+00  2.784E+00 -1.785E+00  4.665E+01
  90    3.773E-01 -1.243E+00  2.604E+00  7.222E-01  3.613E-01  8.168E-01 -3.445E-01  2.837E+00
 120   -2.794E-01  5.430E-02  4.348E-01 -5.563E-02  7.889E-04 -3.176E-02  4.663E-02  3.375E-01
 150    1.483E-02  1.151E-01  2.274E-01  9.407E-02  5.244E-02  1.563E-02 -1.093E-01  2.533E-01
 180   -6.909E-02  2.648E-01  1.289E-01 -7.194E-02  7.920E-17 -4.030E-17  1.439E-01  2.578E-01
 ==============================================================================================
 SPHEROID: Prolate /  Laboratory frame
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv= 3.000
 a/b =  2.000    Ksi0= 1.155
 ******************************************************************************
         C   2*pi*a/wvl Nmax alpha     Qext          Qabs        Albedo
 ------------------------------------------------------------------------------
 M= 1   4.124   4.762    22    0    7.443529E+00  1.261213E-13  1.000000E+00 TM
 ==============================================================================
 THETA       F11           F12           F33           F34          P
 ------------------------------------------------------------------------------
   0    1.175E+02  0.000E+00  1.175E+02  0.000E+00     0.00
  30    5.830E+01  8.810E-01  5.813E+01  4.418E+00    -1.51
  60    3.974E+00  2.180E+00  3.311E+00 -2.794E-01   -54.86
  90    1.186E+00  7.915E-02  1.183E+00 -1.509E-02    -6.67
 120    6.166E-01 -4.469E-02  1.914E-01 -5.845E-01     7.25
 150    6.575E-01  1.569E-01 -5.692E-01  2.895E-01   -23.86
 180    2.290E+00  0.000E+00 -2.290E+00  0.000E+00     0.00
 ==============================================================================
 SPHEROID: Prolate /  Laboratory frame
 Refractive index:  1.5000  0.0000 I
 Rv= 0.239 mkm  Lambda= 0.500 mkm  Xv= 3.000
 a/b =  2.000    Ksi0= 1.155
 ******************************************************************************
         C   2*pi*a/wvl Nmax alpha     Qext          Qabs        Albedo
 ------------------------------------------------------------------------------
 M= 5   4.124   4.762    22   50    3.232122E+00 -1.909584E-14  1.000000E+00 TM
 M= 5   4.124   4.762    26   50    2.802842E+00  1.749711E-13  1.000000E+00 TE
 ==============================================================================================
                    alpha =  50 deg.     PHI =   0 deg.

 THETA      F11        F12        F13        F14        F21        F22        F23        F24
 ----------------------------------------------------------------------------------------------
   0    7.574E+01  6.333E+00  0.000E+00  0.000E+00  6.333E+00  7.574E+01  0.000E+00  0.000E+00
  30    3.102E+01  2.825E+00  0.000E+00  0.000E+00  2.825E+00  3.102E+01  0.000E+00  0.000E+00
  60    8.365E+00  3.123E+00 -9.175E-17 -1.745E-16  3.123E+00  8.365E+00 -1.320E-15 -1.069E-16
  90    4.928E+00  5.655E-01 -2.487E-16 -5.310E-16  5.655E-01  4.928E+00 -9.675E-16  2.025E-16
 120    2.227E+00 -2.017E+00  2.854E-16 -3.463E-16 -2.017E+00  2.227E+00 -4.989E-16  3.782E-16
 150    1.335E-01  1.609E-02  3.277E-18  2.361E-17  1.609E-02  1.335E-01 -1.260E-17  7.675E-17
 180    9.855E-01  9.010E-03  1.688E-17  3.229E-17  9.010E-03  9.855E-01 -1.695E-16  1.491E-16
 ----------------------------------------------------------------------------------------------
 THETA      F31        F32        F33        F34        F41        F42        F43        F44
 ----------------------------------------------------------------------------------------------
   0    0.000E+00  0.000E+00  7.522E+01  6.262E+00  0.000E+00  0.000E+00 -6.262E+00  7.522E+01
  30    0.000E+00  0.000E+00  3.056E+01  4.520E+00  0.000E+00  0.000E+00 -4.520E+00  3.056E+01
  60    4.154E-16  1.384E-15  7.720E+00  7.881E-01 -1.883E-16 -9.600E-17 -7.881E-01  7.720E+00
  90    2.373E-16  9.060E-16  3.840E+00 -3.036E+00 -5.237E-16  3.782E-16  3.036E+00  3.840E+00
 120   -4.229E-19  1.660E-16  2.256E-02 -9.455E-01 -3.919E-16  5.628E-16  9.455E-01  2.256E-02
 150   -2.371E-18 -2.524E-17 -1.307E-01  2.194E-02 -1.506E-17  7.126E-17 -2.194E-02 -1.307E-01
 180    1.592E-17 -2.172E-16 -5.362E-01  8.267E-01 -3.229E-17 -6.136E-17 -8.267E-01 -5.362E-01
 ==============================================================================================
                    alpha =  50 deg.     PHI =  30 deg.

 THETA      F11        F12        F13        F14        F21        F22        F23        F24
 ----------------------------------------------------------------------------------------------
   0    7.574E+01  3.167E+00  5.485E+00 -2.580E-16  3.167E+00  7.535E+01  2.275E-01 -5.423E+00
  30    3.410E+01  2.984E-01  3.485E+00 -9.268E-01  1.598E-02  3.365E+01 -3.376E+00 -2.451E+00
  60    7.763E+00  2.104E+00  9.856E-01 -2.507E+00  1.818E+00  6.902E+00 -1.968E+00 -6.105E-01
  90    3.832E+00  7.694E-01 -9.361E-01 -2.394E+00  7.540E-01  2.823E+00 -2.861E-02 -2.884E-01
 120    2.042E+00 -1.430E+00 -6.550E-01 -9.570E-01 -1.373E+00  1.401E+00 -2.927E-03  8.385E-01
 150    1.624E-01  1.153E-02 -5.208E-03  6.354E-02  6.238E-02  6.720E-02 -1.595E-02  1.459E-01
 180    9.855E-01  4.505E-03  7.803E-03  1.076E-17  4.505E-03  6.485E-01  1.945E-01  7.160E-01
 ----------------------------------------------------------------------------------------------
 THETA      F31        F32        F33        F34        F41        F42        F43        F44
 ----------------------------------------------------------------------------------------------
   0    5.485E+00  2.275E-01  7.561E+01  3.131E+00 -1.969E-16  5.423E+00 -3.131E+00  7.522E+01
  30    3.387E+00  3.628E+00  3.371E+01  3.350E+00 -1.275E+00  2.081E+00 -3.697E+00  3.366E+01
  60    8.901E-01  2.222E+00  6.459E+00  1.648E+00 -2.755E+00 -6.564E-01 -1.989E+00  7.195E+00
  90    9.663E-01  4.694E-02  1.833E+00 -2.249E+00 -2.386E+00 -3.244E-01  2.236E+00  2.842E+00
 120    4.902E-01  8.747E-02 -3.315E-01 -9.500E-01 -1.125E+00  9.253E-01  1.048E+00  2.999E-01
 150   -5.595E-03  2.865E-02 -1.429E-01 -3.120E-02 -1.658E-02  1.303E-01  3.919E-02 -6.280E-02
 180   -7.803E-03 -1.945E-01 -8.731E-01  4.134E-01 -3.155E-17  7.160E-01 -4.134E-01 -5.362E-01
 ==============================================================================================
\end{verbatim}
}

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 7$^0$.}
{\underline{\it Numerical code for calculations of extinction,
scattering and absorption efficiency factors by} \\
{\underline{\it core-mantle spheroidal particles}
The directory  {\it SVM\_COAT/VER3\_1Q} contains the following
files:

{\tt readme} -- file `readme';

{\tt cm3\_1q.for} -- main program and subroutines;

{\tt cmn.dat} -- input file;

{\tt cm.res} -- output file (test results);

{\tt cm.nrm} -- output file (test results).

{\footnotesize
\begin{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%   README: program  C-M_SPH (version 3.1qa)
%%%   -------
%%%
%%%   Light Scattering by Core-Mantle Spheroidal Particles
%%%   ====================================================
%%%
%%%     Calculations of the extinction, scattering and absorption
%%%     efficiency factors for prolate or oblate coated spheroids.
%%%
%%%     readme            -- this file
%%%     cm3_1qa.for       -- main program and subroutines
%%%     cm.dat            -- input parameters
%%%     cm.res & cm.nrm   -- results
%%%
%%%     Parameters:
%%%     -----------
%%%
%%%         nterms: number of terms in Eqs. for efficiency factors,
%%%        nalphas: number of angles for incident radiation.
%%%
%%%     The values of the parameters may be changed in file c-m3_1qa.for.
%%%
%%%
%%%     Restrictions:
%%%     -------------
%%%
%%%         nterms:
%%%          c2 = ri*(2*pi/wvl)*(d/2) <= 60 - 65 for prolate spheroids,
%%%                                   <= 55 - 60 for oblate spheroids,
%%%          (d - focus distance of spheroid),
%%%
%%%          nterms ~= 1.2 * cdabs(c2) + 8 (+ ~30 if ksi0 > 2).
%%%
%%%
%%%        nalphas: <=  91
%%%
%%%
%%%     for coated spheroids:
%%%        1) in all cases: (a/b)_core > (a/b)_mantle (!)
%%%        2) Vcore/Vtotal ~> 0.05 -- 0.1
%%%
%%%     Recommended values:
%%%     -------------------
%%%                       
%%%        refractive index: k <= 1/2*n,
%%%                     eps: 1d-15 for prolate spheroids,
%%%                          1d-10 for oblate spheroids,
%%%                    eps1: 1d-3 - 1d-6
%%%
%%% 
%%%     Publications:
%%%     -------------
%%%
%%%     Voshchinnikov N.V., Farafonov  V.G.
%%%     Optical properties of spheroidal particles,
%%%     Astrophys. Space Sci. 1993. V. 204. P. 19-86.
%%% 
%%%     Voshchinnikov N.V.
%%%     Electromagnetic scattering by homogeneous and coated spheroids:
%%%     calculations using the separation of variables method,
%%%     J. Quant. Spectrosc. Radiat. Transfer 1996. V. 55. P. 627-636.
%%% 
%%%     Farafonov V.G., Voshchinnikov N.V.,  Somsikov V.V.
%%%     Light scattering by a core-mantle spheroidal particle,
%%%     Applied Optics 1996. V. 35. P. 5412-5426.
%%% 
%%% 
%%%     Questions, more information, new versions:
%%%     -------------------------------------------
%%%
%%%     Dr. Nikolai Voshchinnikov
%%%     Astronomical Institute
%%%     St. Petersburg University
%%%     St. Petersburg-Petrodvoretz
%%%     198904 Russia
%%%     e-mail: nvv@aispbu.spb.su
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%     The development of the new approach and the code was financially
%%%     supported by a grant of the Volkswagen Foundation (Germany).
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{verbatim}

\begin{verbatim}
----------------------------   cmn.dat
   0             ! K: 0-prolate, 1-oblate
2.               ! a/b of mantle
1.5     0.0      ! RII - Refraction Index of core
1.3     0.0      ! RI  - Refraction Index of mantle
1.0     D-15     ! EPS
1.0     D-12     ! EPS1
1.0     D-06     ! EPS2
0.50             ! WVL - Wavelength of incident radiation
   2             ! icx:1-Rv of total particle, 2-Xv, 3-C, 4-Cksi0, 5-Xc
   1             ! NRV - number of sizes to be calculated
3.00             ! ARV0 - initial size
0.1              ! DRV  - step over size
  10             ! NALPH - number of angles to be calculated
   0             ! IAL0 - initial angle
  10             ! IDAL - step over angle
   0             ! iswitch: 0-a/b of core, 1-Vol_rat, 2-Xv_core, 3-Rv_core
3.0              ! a/b of core OR Vol_rat OR Xv_core OR Rv_core
   1             ! iswitch: 0-a/b of core, 1-Vol_rat, 2-Xv_core, 3-Rv_core
0.5              ! a/b of core OR Vol_rat OR Xv_core OR Rv_core
\end{verbatim}

\begin{verbatim}
----------------------------   cm.res
 CORE / MANTLE SPHEROID: Prolate
 Refractive index:  1.5000  0.0000i /  1.3000  0.0000i
 Lambda= 0.500  mkm,  Rv=   0.167 / 0.239 mkm,  Xv= 2.103 / 3.000,
 a/b=  3.00 /  2.00,  ksi = 1.061 / 1.155,   Volume: core / total= 0.34
 ..............................................................................
 C1= 4.12 C2= 5.36 0.00 C3= 6.19 0.00 Cksi= 4.76 Cksi2= 4.37 Nmax= 16 Nmxe= 22
 ------------------------------------------------------------------------------
 alpha   m      Qext           Pext           Qsca           Qabs
 ------------------------------------------------------------------------------
   0     1  5.275404E+00   0.000000E+00   5.275404E+00   3.018030E-12  TM
  10     4  4.958071E+00  -3.018782E+00   4.958071E+00   2.087219E-13  TM
  20     4  4.217058E+00  -2.668218E+00   4.217058E+00  -6.776801E-13  TM
  30     4  3.403554E+00  -2.332867E+00   3.403554E+00  -6.430412E-13  TM
  40     5  2.716999E+00  -2.109584E+00   2.716999E+00  -2.176037E-14  TM
  50     5  2.215417E+00  -1.992413E+00   2.215417E+00   2.970957E-13  TM
  60     5  1.892421E+00  -1.936324E+00   1.892421E+00   8.104628E-14  TM
  70     5  1.720430E+00  -1.904627E+00   1.720430E+00  -1.161293E-13  TM
  80     5  1.656420E+00  -1.884458E+00   1.656420E+00  -2.153833E-14  TM
  90     5  1.644703E+00  -1.877065E+00   1.644703E+00   8.970602E-14  TM
 ..............................................................................
  10     4  4.924347E+00  -2.991359E+00   4.924349E+00  -2.090003E-06  TE
  20     4  4.092549E+00  -2.586201E+00   4.092551E+00  -1.160048E-06  TE
  30     4  3.159727E+00  -2.223494E+00   3.159728E+00  -2.354428E-07  TE
  40     5  2.370791E+00  -2.037196E+00   2.370791E+00   2.854183E-07  TE
  50     5  1.830056E+00  -2.011510E+00   1.830056E+00   3.104289E-07  TE
  60     5  1.535325E+00  -2.040250E+00   1.535325E+00   9.637599E-08  TE
  70     5  1.405987E+00  -2.023418E+00   1.405987E+00   1.186606E-08  TE
  80     5  1.352381E+00  -1.956170E+00   1.352381E+00   1.411194E-07  TE
  90     5  1.335657E+00  -1.917230E+00   1.335657E+00   2.405806E-07  TE
 ==============================================================================
 CORE / MANTLE SPHEROID: Prolate
 Refractive index:  1.5000  0.0000i /  1.3000  0.0000i
 Lambda= 0.500  mkm,  Rv=   0.189 / 0.239 mkm,  Xv= 2.381 / 3.000,
 a/b=  2.58 /  2.00,  ksi = 1.085 / 1.155,   Volume: core / total= 0.50
 ..............................................................................
 C1= 4.12 C2= 5.36 0.00 C3= 6.19 0.00 Cksi= 4.76 Cksi2= 4.48 Nmax= 16 Nmxe= 22
 ------------------------------------------------------------------------------
 alpha   m      Qext           Pext           Qsca           Qabs
 ------------------------------------------------------------------------------
   0     1  6.003559E+00   0.000000E+00   6.003559E+00   3.555378E-12  TM
  10     4  5.629155E+00  -2.695999E+00   5.629155E+00   5.391243E-13  TM
  20     4  4.758362E+00  -2.415907E+00   4.758362E+00  -4.138911E-13  TM
  30     4  3.812910E+00  -2.169355E+00   3.812910E+00  -4.827250E-13  TM
  40     5  3.032224E+00  -2.033902E+00   3.032224E+00   6.927792E-14  TM
  50     5  2.480242E+00  -1.987861E+00   2.480242E+00   3.246292E-13  TM
  60     5  2.137945E+00  -1.973556E+00   2.137945E+00   4.884981E-14  TM
  70     5  1.962485E+00  -1.950555E+00   1.962485E+00  -1.447731E-13  TM
  80     5  1.900686E+00  -1.918531E+00   1.900686E+00   2.353673E-14  TM
  90     5  1.890583E+00  -1.902848E+00   1.890583E+00   1.891820E-13  TM
 ..............................................................................
  10     4  5.587304E+00  -2.669802E+00   5.587304E+00  -3.480401E-07  TE
  20     4  4.607631E+00  -2.340267E+00   4.607631E+00  -1.831900E-07  TE
  30     4  3.528676E+00  -2.076330E+00   3.528676E+00  -2.638694E-08  TE
  40     5  2.648158E+00  -1.987971E+00   2.648158E+00   5.211858E-08  TE
  50     5  2.077304E+00  -2.035265E+00   2.077304E+00   4.654729E-08  TE
  60     5  1.781854E+00  -2.097483E+00   1.781854E+00   1.153340E-08  TE
  70     5  1.639866E+00  -2.080984E+00   1.639866E+00   3.316533E-09  TE
  80     5  1.554533E+00  -2.002097E+00   1.554533E+00   2.281848E-08  TE
  90     5  1.519706E+00  -1.957345E+00   1.519706E+00   3.533187E-08  TE
 ==============================================================================
\end{verbatim}

\begin{verbatim}
----------------------------   cm.nrm
 CORE / MANTLE SPHEROID: Prolate
 Refractive index:  1.5000  0.0000i /  1.3000  0.0000i
 Lambda= 0.500  mkm,  Rv=   0.167 / 0.239 mkm,  Xv= 2.103 / 3.000,
 a/b=  3.00 /  2.00,  ksi = 1.061 / 1.155,   Volume: core / total= 0.34
 ..............................................................................
 C1= 4.12 C2= 5.36 0.00 C3= 6.19 0.00 Cksi= 4.76 Cksi2= 4.37 Nmax= 16 Nmxe= 22
 ------------------------------------------------------------------------------
 alpha   m Cext/(Pi*Rv**2) Csca/(Pi*Rv**2) Cabs/(Pi*Rv**2)
 ..............................................................................
   0     1  3.323296E+00    3.323296E+00    1.901240E-12   TM
  10     4  3.261604E+00    3.261604E+00    1.374135E-13   TM
  20     4  3.087734E+00    3.087734E+00   -4.961851E-13   TM
  30     4  2.836384E+00    2.836384E+00   -5.356917E-13   TM
  40     5  2.561421E+00    2.561421E+00   -2.056298E-14   TM
  50     5  2.318785E+00    2.318785E+00    3.108067E-13   TM
  60     5  2.149180E+00    2.149180E+00    9.196355E-14   TM
  70     5  2.070338E+00    2.070338E+00   -1.398042E-13   TM
  80     5  2.063224E+00    2.063224E+00   -2.667463E-14   TM
  90     5  2.072196E+00    2.072196E+00    1.130157E-13   TM
 ..............................................................................
  10     4  3.239419E+00    3.239420E+00   -1.374882E-06   TE
  20     4  2.996569E+00    2.996569E+00   -8.493881E-07   TE
  30     4  2.633189E+00    2.633189E+00   -1.962085E-07   TE
  40     5  2.235038E+00    2.235037E+00    2.690750E-07   TE
  50     5  1.915444E+00    1.915444E+00    3.249131E-07   TE
  60     5  1.743634E+00    1.743634E+00    1.094520E-07   TE
  70     5  1.691942E+00    1.691942E+00    1.427943E-08   TE
  80     5  1.684517E+00    1.684517E+00    1.757773E-07   TE
  90     5  1.682822E+00    1.682822E+00    3.031125E-07   TE
 ==============================================================================
 CORE / MANTLE SPHEROID: Prolate
 Refractive index:  1.5000  0.0000i /  1.3000  0.0000i
 Lambda= 0.500  mkm,  Rv=   0.189 / 0.239 mkm,  Xv= 2.381 / 3.000,
 a/b=  2.58 /  2.00,  ksi = 1.085 / 1.155,   Volume: core / total= 0.50
 ..............................................................................
 C1= 4.12 C2= 5.36 0.00 C3= 6.19 0.00 Cksi= 4.76 Cksi2= 4.48 Nmax= 16 Nmxe= 22
 ------------------------------------------------------------------------------
 alpha   m Cext/(Pi*Rv**2) Csca/(Pi*Rv**2) Cabs/(Pi*Rv**2)
 ..............................................................................
   0     1  3.782005E+00    3.782005E+00    2.239748E-12   TM
  10     4  3.703068E+00    3.703068E+00    3.546664E-13   TM
  20     4  3.484077E+00    3.484077E+00   -3.028920E-13   TM
  30     4  3.177524E+00    3.177524E+00   -4.023940E-13   TM
  40     5  2.858597E+00    2.858597E+00    6.509355E-14   TM
  50     5  2.595967E+00    2.595967E+00    3.398657E-13   TM
  60     5  2.428016E+00    2.428016E+00    5.543829E-14   TM
  70     5  2.361624E+00    2.361624E+00   -1.741881E-13   TM
  80     5  2.367481E+00    2.367481E+00    2.952456E-14   TM
  90     5  2.381985E+00    2.381985E+00    2.381871E-13   TM
 ..............................................................................
  10     4  3.675536E+00    3.675536E+00   -2.289538E-07   TE
  20     4  3.373712E+00    3.373712E+00   -1.341319E-07   TE
  30     4  2.940656E+00    2.940656E+00   -2.198981E-08   TE
  40     5  2.496522E+00    2.496522E+00    4.913423E-08   TE
  50     5  2.174228E+00    2.174228E+00    4.871912E-08   TE
  60     5  2.023611E+00    2.023611E+00    1.309823E-08   TE
  70     5  1.973388E+00    1.973388E+00    3.991063E-09   TE
  80     5  1.936316E+00    1.936316E+00    2.842254E-08   TE
  90     5  1.914709E+00    1.914709E+00    4.451537E-08   TE
 ==============================================================================
\end{verbatim}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\noi {\sf II. A new method for axisymmetric particles}

\medskip
\medskip
 An advanced method for axisymmetric particles was suggested by
Farafonov$^8$. It  combines
the principles of the Separation of Variables
and T-matrix Methods.

The incident, scattered and internal fields are divided in two parts:
axisymmetric and non-axisymmetric ones
\be
\vec{E} = \vec{E}_{1} + \vec{E}_{2}, \ \ \ \
\vec{H} = \vec{H}_{1} + \vec{H}_{2}.
\ee
It is possible to do so that the axisymmetric part does not depend
on the azimuthal angle ($\varphi$)
and the averaging of the non-axisymmetric part
over all the azimuthal angle values gives zero.$^2$
Then the scattering problem can be solved
for each of the parts separately.

The scattering problem is formulated in the integral form
like in the TMM (e.g., Ref.~3).
 The Abraham's potentials
\be
p = E_{1\varphi} \cos\varphi, \ \ \ \
q = H_{1\varphi} \cos\varphi
\ee
are introduced for the axisymmetric part.
 A superposition of the Debye and Hertz potentials
is used for the non-axisymmetric part:
\be
\vec{E}_{2} = \vec{\nabla} \times (U \vec{e}_{z} + V \vec{r}), \ \ \ \
\vec{H}_{2} = \frac{1}{i \mu k_0} \vec{\nabla} \times \vec{\nabla}
              \times (U \vec{e}_{z} + V \vec{r})
\ee
for TE mode and
\be
\vec{E}_{2} = -\frac{1}{i \varepsilon k_0} \vec{\nabla} \times \vec{\nabla}
              \times (U \vec{e}_{z} + V \vec{r}), \ \ \ \
\vec{H}_{2} = \vec{\nabla} \times (U \vec{e}_{z} + V \vec{r})
\ee
for TM mode.
 Here $\varepsilon$ is the complex permittivity,
$\mu$ the magnetic permeability,
$k_0$ the wavevector in vacuum,
$\vec{r}$ the radius-vector,
$\vec{e}_{z}$ the unit vector,
and $i$ the complex unity.

The scalar potentials are expanded in terms of the spherical wave functions.
The coefficients of the expansions are determined from solution
of the algebraic systems similar to those obtained in the TMM (e.g., Ref.~3).

 Thus, the new approach
combines the strong aspects of the basic methods:
the simple solution scheme typical of the TMM and
the ability to treat particles whose shape can be
very elongated or flattened as in the SVM.

As a first step the method was applied to homogeneous spheroids.
The calculations consisted of
several independent steps: computations of the spherical functions,
of integrals of these functions, and of the scalar potentials
as surface integrals, solution of the systems of algebraic
equations, etc.


\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 8$^0$.}
{\underline{\it A numerical code for calculations of the Bessel and Hankel
spherical functions and the}} \\
{\underline{\it Legendre functions with the size parameters and
orders up to 5000 was developed.}}
Some results for the Bessel functions of the first and third kinds are
presented in Table~1. It gives the time of calculations using the libraries
{\it Mathematica 3.0} and {\it MathLib}.
The number of correct digits in the results is 16 for {\it Mathematica 3.0}
and 77  for {\it MathLib}.
The quantity $\Delta W=|W_{\rm calc} - W_{\rm theor}|$ is the difference
between the computed Wronskian
\be
W_{\rm calc} = J_{n+1/2}(x) H_{n+1/2}^{(1)'}(x) -
J_{n+1/2}^{'}(x) H_{n+1/2}^{(1)}(x)
\ee
and the theoretical one
\be
W_{\rm theor} = \frac{i}{x^2}.
\ee
The numerical code was written using the programming language C++
The calculations were performed on PC's and a workstation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\bc
\caption{{\it Calculations of the spherical Bessel functions}}
{\small
\begin{tabular}{clllll}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{1}{c}{$Method$} &&
\multicolumn{1}{c}{$n = 100$} &
\multicolumn{1}{c}{500} &
\multicolumn{1}{c}{1000} &
\multicolumn{1}{c}{5000} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{6}{c}{$x = 100$} \\
\noalign{\smallskip}
Math3.0  & &$~~~~~t=0.06$~s &  0.06 & 0.96 & 2.88  \\
         & &$\Delta W=1.4 \cdot 10^{-18}$ & $1.6 \cdot 10^{-17}$ & $2.8 \cdot 10^{-17}$ & $1.2 \cdot 10^{-16}$  \\
\noalign{\smallskip}
MathLib  & &$~~~~~t=0.005$~s &  0.005 & 0.015 & 0.08  \\
         & &$\Delta W=1.7 \cdot 10^{-80}$ & $5.9 \cdot 10^{-81}$ & $6.0 \cdot 10^{-81}$ & $6.5 \cdot 10^{-81}$  \\
\noalign{\medskip}
\multicolumn{6}{c}{$x = 500$} \\
\noalign{\smallskip}
Math3.0  & &$~~~~~t=0.06$~s &  0.06 & 1.28 & 204.1  \\
         & &$\Delta W=0$ & $0$ & $0$ & $8.5 \cdot 10^{-17}$  \\
\noalign{\smallskip}
MathLib  & &$~~~~~t=0.005$~s &  0.005 & 0.015 & 0.08  \\
         & &$\Delta W=2.4 \cdot 10^{-81}$ & $3.7 \cdot 10^{-80}$ & $9.2 \cdot 10^{-81}$ & $7.6 \cdot 10^{-81}$  \\
\noalign{\smallskip}
\multicolumn{6}{c}{$x = 1000$} \\
\noalign{\smallskip}
Math3.0  & &$~~~~~t=0.06$~s &  0.06 & 1.02 & --    \\
         & &$\Delta W=1.4 \cdot 10^{-20}$ & $0$ & $4.2 \cdot 10^{-22}$ & -- \\
\noalign{\medskip}
MathLib  & &$~~~~~t=0.005$~s &  0.005 & 0.015 & 0.08  \\
         & &$\Delta W=6.0 \cdot 10^{-81}$ & $2.2 \cdot 10^{-81}$ & $4.1 \cdot 10^{-80}$ & $7.6 \cdot 10^{-81}$  \\
\noalign{\medskip}
\multicolumn{6}{c}{$x = 5000$} \\
\noalign{\smallskip}
Math3.0  & &$~~~~~t=0.06$~s &  0.06 & 0.13 & --    \\
         & &$\Delta W=6.4 \cdot 10^{-21}$ & $4.0 \cdot 10^{-23}$ & $6.6 \cdot 10^{-24}$ & --  \\
\noalign{\smallskip}
MathLib  & &$~~~~~t=0.005$~s &  0.005 & 0.015 & 0.08  \\
         & &$\Delta W=2.9 \cdot 10^{-83}$ & $1.2 \cdot 10^{-82}$ & $2.2 \cdot 10^{-82}$ & $6.6 \cdot 10^{-80}$  \\
\noalign{\smallskip}
\hline
\end{tabular}
}
\ec
\end{table}

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 9$^0$.}
{\underline{\it Numerical code was written for spheroids.}
A comparison of the results obtained by the code developed
and by the Separation of Variables Method as well as from
the optical theorem has been performed.
In the case of non-absorbing particles, it was found
that about 20 and 6 digits for
the axisymmetric and non-axisymmetric parts  coincide
in the efficiency extinction and scattering factors
calculated in different ways.

{\footnotesize
\begin{verbatim}
 Parameter values:
   k = 0   nr = 13   ngauss =120
   alpha =  0
   a/b = 2.0000   x =  1.000   m2 =  2.500+  0.000

 Results:
   part.case of alpha = 0 m=   1  Qext=  0.12999513E+01  Qsca=  0.12999513E+01
                          sphere  Qext=  0.18547441E+01  Qsca=  0.18547441E+01
--------------------------------------------------------------------------------
alpha=  0 xv= 1.0000  Qext= 0.129995E+01  Qsca= 0.129995E+01  Qabs= 0.167507E-09
--------------------------------------------------------------------------------

 Parameter values:
   k = 0   nr = 13   ngauss =120
   alpha = 90
   a/b = 2.0000   x =  1.000   m2 =  2.500+  0.000

 Results:
   the axisymmetric part  m=   0  Qext=  0.34385022E+01  Qsca=  0.34385022E+01
   non-axisymmetric part  m=   1  Qext=  0.29829570E+00  Qsca=  0.29829570E+00
   non-axisymmetric part  m=   2  Qext=  0.35792080E-04  Qsca=  0.35792080E-04
   non-axisymmetric part  m=   3  Qext=  0.34278691E-08  Qsca=  0.34278691E-08
   non-axisymmetric part  m=   4  Qext=  0.14614717E-12  Qsca=  0.14614719E-12
   non-axisymmetric part  m=   5  Qext=  0.30287336E-17  Qsca=  0.30287336E-17
   total  (both parts)    m=  **  Qext=  0.37368337E+01  Qsca=  0.37368337E+01

--------------------------------------------------------------------------------
alpha= 90 xv= 1.0000  Qext= 0.373683E+01  Qsca= 0.373683E+01  Qabs= 0.152271E-10
--------------------------------------------------------------------------------

 Parameter values:
   k = 1   nr = 13   ngauss =120
   alpha =  0
   a/b = 2.0000   x =  1.000   m2 =  2.500+  0.000

 Results:
   part.case of alpha = 0 m=   1  Qext=  0.24647598E+01  Qsca=  0.24647598E+01

--------------------------------------------------------------------------------
alpha=  0 xv= 1.0000  Qext= 0.246476E+01  Qsca= 0.246476E+01  Qabs= 0.993194E-09
--------------------------------------------------------------------------------

 Parameter values:
   k = 1   nr = 13   ngauss =120
   alpha = 90
   a/b = 2.0000   x =  1.000   m2 =  2.500+  0.000

 Results:
   the axisymmetric part  m=   0  Qext=  0.50877087E+00  Qsca=  0.50877087E+00
   non-axisymmetric part  m=   1  Qext=  0.15561469E+00  Qsca=  0.15561469E+00
   non-axisymmetric part  m=   2  Qext=  0.37744037E-03  Qsca=  0.37744037E-03
   non-axisymmetric part  m=   3  Qext=  0.30549577E-06  Qsca=  0.30549577E-06
   non-axisymmetric part  m=   4  Qext=  0.97772147E-10  Qsca=  0.97772147E-10
   non-axisymmetric part  m=   5  Qext=  0.14458332E-13  Qsca=  0.14458332E-13
   non-axisymmetric part  m=   6  Qext=  0.11159338E-17  Qsca=  0.11159332E-17
   total  (both parts)    m=  **  Qext=  0.66476331E+00  Qsca=  0.66476331E+00

--------------------------------------------------------------------------------
alpha= 90 xv= 1.0000  Qext= 0.664763E+00  Qsca= 0.664763E+00  Qabs=-0.138672E-12
--------------------------------------------------------------------------------
\end{verbatim}
}

The comparison with the Mie theory for spheres was also made.

{\footnotesize
\begin{verbatim}
 Parameter values:
   k = 1   nr = 12   ngauss = 60
   alpha =  0
   a/b = 1.0000   x =  3.000   m2 =  2.500+  1.500

 Results:
   part.case of alpha = 0 m=   1  Qext=  0.28400452E+01  Qsca=  0.15543512E+01
                          sphere  Qext=  0.28400452E+01  Qsca=  0.15543512E+01

--------------------------------------------------------------------------------
alpha=  0 xv= 3.0000  Qext= 0.284005E+01  Qsca= 0.155435E+01  Qabs= 0.128569E+01
--------------------------------------------------------------------------------

 Parameter values:
   k = 1   nr = 12   ngauss = 60
   alpha = 90
   a/b = 1.0000   x =  3.000   m2 =  2.500+  1.500

 Results:
   the axisymmetric part  m=   0  Qext=  0.41661175E+00  Qsca=  0.16129612E+00
   non-axisymmetric part  m=   1  Qext=  0.11902145E+01  Qsca=  0.76944212E+00
   non-axisymmetric part  m=   2  Qext=  0.92499123E+00  Qsca=  0.54696679E+00
   non-axisymmetric part  m=   3  Qext=  0.25095903E+00  Qsca=  0.73817462E-01
   non-axisymmetric part  m=   4  Qext=  0.50177267E-01  Qsca=  0.27983076E-02
   non-axisymmetric part  m=   5  Qext=  0.65887749E-02  Qsca=  0.30253222E-04
   non-axisymmetric part  m=   6  Qext=  0.47958353E-03  Qsca=  0.11661183E-06
   non-axisymmetric part  m=   7  Qext=  0.22300402E-04  Qsca=  0.20965276E-09
   non-axisymmetric part  m=   8  Qext=  0.74102233E-06  Qsca=  0.20734656E-12
   non-axisymmetric part  m=   9  Qext=  0.18877409E-07  Qsca=  0.12526335E-15
   non-axisymmetric part  m=  10  Qext=  0.38536847E-09  Qsca=  0.49566801E-19
   non-axisymmetric part  m=  11  Qext=  0.64755913E-11  Qsca=  0.13469564E-22
   non-axisymmetric part  m=  12  Qext=  0.99319320E-13  Qsca=  0.27284710E-26
   non-axisymmetric part  m=  13  Qext=  0.00000000E+00  Qsca=  0.00000000E+00
   total  (both parts)    m=  **  Qext=  0.28400452E+01  Qsca=  0.15543512E+01
                          sphere  Qext=  0.28400452E+01  Qsca=  0.15543512E+01

--------------------------------------------------------------------------------
alpha= 90 xv= 3.0000  Qext= 0.284005E+01  Qsca= 0.155435E+01  Qabs= 0.128569E+01
--------------------------------------------------------------------------------

\end{verbatim}
}


%The code was written using the programming language C++ and
The code was written using the programming language FORTRAN and
was executed on PC's.

\vspace{0.5cm}

%---------------------------------------------------------
\noi {\bf 10$^0$.}
The directory  {\it NEW\_TMAT} contains the following
files:

{\tt readme} -- file `readme';

{\tt axisym0.for} -- main program and subroutines;

{\tt in} -- input file;

{\tt out-test} -- output file (test results);

{\footnotesize
\begin{verbatim}
----------------------------   readme
            Light scattering by axisymmetric particles
           within a new approach to the T-matrix method

The code AXISYM0.FOR realizes the new approach to the solution of
the light scattering problem suggested by Farafonov (Opt. Spectr.,
1998, in preparation). Some features of the approach are described in
the first lines of the code. Below them one finds also the description
of the input and output parameters, and a list of some used subroutines.
The comments given through the code's text allow one to follow its
general structure.

The file IN contains the input parameter values for a test run. The
results of this run must be similar to those in the file OUT-TEST.
I have carefully checked the code by comparison with results derived
for spheres with the Mie theory and for spheroids of quite different
aspect ratios, sizes, compositions obtained with the T-matrix codes of
Barber & Hill and Mishchenko and with the SVM code of Voshchinnikov.
In general, the code created is at least as fast and correct as
the T-matrix codes mentioned. Though the code is a bit slower but
many parts of it can be easily speeded up.

The code is written in standard Fortran 77 and does not contain
any particular computational tools. However, please feel free to
contact me provided you meet problems.

September, 1998                      Vladimir Il'in.
                              (e-mail: vi2087@vi2087.spb.edu)

The development of the new approach and the code was financially
supported by a grant of the Volkswagen Foundation (Germany).

\end{verbatim}

\begin{verbatim}
----------------------------   in
   1           k
   8           nn
  60           ngauss
  90           ialpha
 2.000d 00     a/b
 0.100d 00     x_b
 2.500d 00     m2 Re
 1.500d 00        Im

\end{verbatim}

\begin{verbatim}
----------------------------   out-test
 Parameter values:
   k = 1   nr =  8   ngauss = 60
   alpha = 90
   a/b = 2.0000   x =  0.100   m2 =  2.500+  1.500

 Results:
   the axisymmetric part  m=   0  Qext=  0.45135634E-01  Qsca=  0.86857008E-04
   non-axisymmetric part  m=   1  Qext=  0.76912434E-03  Qsca=  0.13197826E-08
   non-axisymmetric part  m=   2  Qext=  0.66661053E-06  Qsca=  0.57653347E-15
   non-axisymmetric part  m=   3  Qext=  0.24386599E-09  Qsca=  0.57425020E-22
   non-axisymmetric part  m=   4  Qext=  0.50437166E-13  Qsca=  0.20147638E-29
   non-axisymmetric part  m=   5  Qext=  0.67699345E-17  Qsca=  0.31316529E-37
   non-axisymmetric part  m=   6  Qext=  0.64004639E-21  Qsca=  0.24913054E-45
   total  (both parts)    m=  **  Qext=  0.45905425E-01  Qsca=  0.86858328E-04
                          sphere  Qext=  0.99863353E-01  Qsca=  0.19023658E-03

--------------------------------------------------------------------------------
alpha= 90 xv= 0.1000  Qext= 0.459054E-01  Qsca= 0.868583E-04  Qabs= 0.458186E-01
--------------------------------------------------------------------------------
\end{verbatim}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%\newpage
\phantom{\noi {\sf IV. References}}
\vspace{3.5cm}
\noi {\sf III. Benchmark results}

In Tables~2~--~6,  the normalized extinction and scattering
cross-sections for prolate and oblate spheroids with
the refractive indices $m = 2.5+0.0i, 1.7+0.7i, 2.5 +1.5i$ and $3.0 +4.0i$
are presented.
 The results --- all 7 (or in a few cases 6) significant digits --- were obtained with
{\it at least two} of four codes used:

 {\tt SVM1} -- the solution of Asano \& Yamamoto$^9$ (numerical code of Peter
         Martin$^4$);

 {\tt SVM2} -- the solution of Farafonov (numerical code of Nikolai Voshchinnikov,
 see Section 5);

{\tt TMM1} -- the T-matrix code of Barber \& Hill$^10$;

{\tt TMM2} -- the T-matrix code of Mishchenko$^5$.

The numbers printed by italics in Table~1
were calculated using the SVM2 code only.
 As the extinction and scattering
cross-sections were the same (with 8 and more digits),
the values presented in the Table are expected to be correct.

\newpage
%{\small
{\footnotesize
\begin{table}[htb]
\begin{center}
\caption{Normalized scattering cross-sections $C_{\rm sca}/\pi r_{V}^{2}$
for spheroids with $m = 1.3 +0.0i$}
\medskip
\begin{tabular}{lrlllllll} \hline
\noalign{\smallskip}
&&\multicolumn{3}{c}{Prolate spheroid} &&
\multicolumn{3}{c}{Oblate spheroid}  \\
\noalign{\smallskip}
\cline{3-5} \cline{7-9}
\noalign{\smallskip}
$a/b$ &$x_V$ &{~~~~$\alpha = 0^{\circ}$} & {$\alpha = 90^{\circ}$(TM)} & {$\alpha = 90^{\circ}$(TE)}
&&{~~~~$\alpha = 0^{\circ}$} & {$\alpha = 90^{\circ}$(TM)} & {$\alpha = 90^{\circ}$(TE)} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
1.1 &  0.1 & 9.182080E-06 & 9.584544E-06 & 9.185689E-06 &
           & 9.453529E-06 & 9.049059E-06 & 9.450043E-06 \\
    &  0.3 & 7.353049E-04 & 7.711279E-04 & 7.378481E-04 &
           & 7.608544E-04 & 7.250718E-04 & 7.583891E-04 \\
    &  1.0 & 7.447337E-02 & 8.134987E-02 & 7.649733E-02 &
           & 8.037420E-02 & 7.362975E-02 & 7.832033E-02 \\
    &  3.0 & 1.540311E-00 & 1.437064E-00 & 1.395298E-00 &
           & 1.380468E-00 & 1.472662E-00 & 1.519971E-00 \\
    & 10.0 & 2.230447E-00 & 2.822134E-00 & 2.867127E-00 &
           & 3.034794E-00 & 2.416342E-00 & 2.441388E-00 \\
    & 30.0 & 2.006623E-00 & 2.572447E-00 &              &
           &              &              &              \\
\noalign{\smallskip}
2.0 &  0.1 & 8.497336E-06 & 1.125232E-05 & 8.529378E-06 &
           & 1.042893E-05 & 7.558188E-06 & 1.040419E-05 \\
    &  0.3 & 6.605367E-04 & 9.119126E-04 & 6.825033E-04 &
           & 8.450193E-04 & 5.950242E-04 & 8.275387E-04 \\
    &  1.0 & 5.447619E-02 & 1.025111E-01 & 6.691507E-02 &
           & 9.418041E-02 & 5.141485E-02 & 7.960716E-02 \\
    &  3.0 & 1.968352E-00 & 1.333052E-00 & 1.083852E-00 &
           & 1.024408E-00 & 1.427102E-00 & 1.865927E-00 \\
    & 10.0 & {\it 1.538734E-00} & {\it 4.353976E-00} & {\it 4.273941E-00} &
           & 5.229347E-00 & 1.270176E-00 & 1.336617E-00 \\
\noalign{\smallskip}
4.0 &  0.1 & 7.979755E-06 & 1.273937E-05 & 8.074529E-06 &
           & 1.160638E-05 & 6.345801E-06 & 1.155186E-05 \\
    &  0.3 & 5.722482E-04 & 1.026860E-03 & 6.318278E-04 &
           & 9.361833E-04 & 4.847639E-04 & 8.983728E-04 \\
    &  1.0 & {\it 3.647086E-02} & {\it 1.085455E-01} & {\it 4.947725E-02} &
           & {\it 9.776480E-02} & {\it 3.277807E-02} & {\it 7.254568E-02} \\
    &  3.0 & {\it 2.208284E-00} & {\it 1.168012E-00} & {\it 7.362909E-01} &
           & {\it 7.018565E-01} & {\it 1.055080E-00} & {\it 1.967821E-00} \\
    & 10.0 & {\it 1.812869E-00} & {\it 5.680530E-00} & {\it 5.416727E-00} &
           & {\it 5.183390E-00} & {\it 1.225550E-00} & {\it 1.425348E-00} \\
\noalign{\smallskip}
10.0&  0.1 & 7.436218E-06 & 1.363190E-05 & 7.751983E-06 &
           & 1.280422E-05 & 5.467201E-06 & 1.268771E-05 \\
    &  0.3 & 4.111163E-04 & 1.049295E-03 & 5.521820E-04 &
           & 1.010622E-03 & 3.922034E-04 & 9.341944E-04 \\
    &  1.0 & {\it 2.210801E-02} & {\it 8.074041E-02} & {\it 2.516490E-02} &
           & {\it 8.226629E-02} & {\it 1.862260E-02} & {\it 5.558987E-02} \\
    &  3.0 & {\it 1.704760E-00} & {\it 9.341732E-01} & {\it 4.068989E-01} &
           & {\it 5.908465E-01} & {\it 5.660661E-01} & {\it 1.610737E-00} \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\label{tab1}
\end{table}
}

\newpage
\vspace{1.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{\small
{\footnotesize
%\begin{table}[htb]
\begin{table}
\begin{center}
\caption{Normalized scattering cross-sections $C_{\rm sca}/\pi r_{V}^{2}$
for spheroids with $m = 2.5 +0.0i$}
\medskip
\begin{tabular}{lrlllllll} \hline
\noalign{\smallskip}
&&\multicolumn{3}{c}{Prolate spheroid} &&
\multicolumn{3}{c}{Oblate spheroid}  \\
\noalign{\smallskip}
\cline{3-5} \cline{7-9}
\noalign{\smallskip}
$a/b$ &$x_V$ &{~~~~$\alpha = 0^{\circ}$} & {$\alpha = 90^{\circ}$(TM)} & {$\alpha = 90^{\circ}$(TE)}
&&{~~~~$\alpha = 0^{\circ}$} & {$\alpha = 90^{\circ}$(TM)} & {$\alpha = 90^{\circ}$(TE)} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
~2 &  0.1 & 8.132300E-05 & 2.262901E-04 & 8.162806E-05 &
           & 1.641502E-04 & 5.756336E-05 & 1.637629E-04 \\
    &  0.3 & 6.610480E-03 & 2.018734E-02 & 6.819180E-03 &
           & 1.433111E-02 & 4.685330E-03 & 1.404947E-02 \\
    &  1.0 & 1.299951E-00 & 3.736834E-00 & 9.854978E-01 &
           & 2.464760E-00 & 6.647633E-01 & 3.680223E-00 \\
    &  3.0 & 2.756951E-00 & 2.423317E-00 & 1.456577E-00 &
           & 4.484763E-00 & 1.777398E-00 & 3.846601E-00 \\
\noalign{\smallskip}
~4 &  0.1 & 6.871662E-05 & 4.258699E-04 & 6.952952E-05 &
           & 2.612223E-04 & 3.688507E-05 & 2.600006E-04 \\
    &  0.3 & {\it 5.105309E-03} & {\it 3.919277E-02} & {\it 5.615168E-03} &
           & {\it 2.307084E-02} & {\it 2.874841E-03} & {\it 2.217663E-02} \\
    &  1.0 & {\it 7.055768E-01} & {\it 5.493155E-00} & {\it 5.426740E-01} &
           & {\it 3.362340E-00} & {\it 2.595873E-01} & {\it 5.360019E-00} \\
    &  3.0 & {\it 1.673046E-00} & {\it 5.755844E-00} & {\it 4.730299E-00} &
           & {\it 7.392485E-00} & {\it 2.580128E-00} & {\it 3.791884E-00} \\
\noalign{\smallskip}
10&  0.1 & {\it 6.058991E-05} & {\it 6.745045E-04} & {\it 6.315583E-05} &
           & {\it 4.423506E-04} & {\it 2.643817E-05} & {\it 4.383382E-04} \\
    &  0.3 & {\it 3.446015E-03} & {\it 5.950350E-02} & {\it 4.588377E-03} &
           & {\it 3.852378E-02} & {\it 1.918230E-03} & {\it 3.569470E-02} \\
    &  1.0 & {\it 3.088317E-01} & {\it 6.153316E-00} & {\it 2.283901E-01} &
           & {\it 3.605985E-00} & {\it 1.053865E-01} & {\it 5.126838E-00} \\
    &  3.0 & {\it 4.612520E-01} & {\it 9.781859E-00} & {\it 7.081355E-00} &
           & {\it 7.766600E-00} & {\it 7.497780E-00} & {\it 4.701476E-00} \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\label{tab2}
\end{table}
}

%\newpage
{\footnotesize
%\voffset= -1.5cm
%\begin{table}[htb]
\begin{table}
\begin{center}
\caption{Normalized extinction and scattering cross-sections
$C_{\rm ext, \, sca}/\pi r_{V}^{2}$ for spheroids with $m = 1.7 +0.7i$}
\medskip
\begin{tabular}{lrlllllll} \hline
\noalign{\smallskip}
&&\multicolumn{2}{c}{$\alpha = 0^{\circ}$}  &&
\multicolumn{4}{c}{$\alpha = 90^{\circ}$} \\
\noalign{\smallskip}
\cline{3-4} \cline{6-9}
\noalign{\smallskip}
$a/b$ &$x_V$ & $~~C_{\rm ext}/\pi r_{V}^{2}$          & $~~C_{\rm sca}/\pi r_{V}^{2}$ &
             & $~~C^{\rm TM}_{\rm ext}/\pi r_{V}^{2}$ & $~~C^{\rm TM}_{\rm sca}/\pi r_{V}^{2}$ &
               $~~C^{\rm TE}_{\rm ext}/\pi r_{V}^{2}$ & $~~C^{\rm TE}_{\rm sca}/\pi r_{V}^{2}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{9}{c}{Prolate spheroid} \\
\noalign{\smallskip}
 ~2 &  ~0.1 & 9.260996E-02 & 6.520100E-05 && 1.867292E-01 & 1.323250E-04 & 9.250492E-02 & 6.544660E-05 \\
     & ~0.3 & 3.001614E-01 & 5.177635E-03 && 6.027228E-01 & 1.099226E-02 & 2.974687E-01 & 5.348186E-03 \\
     & ~1.0 & 1.697352E-00 & 3.846683E-01 && 2.477888E-00 & 7.841288E-01 & 1.556310E-00 & 4.492447E-01 \\
     & ~3.0 & 2.481402E-00 & 1.039641E-00 && 3.195685E-00 & 1.600695E-00 & 3.021085E-00 & 1.360101E-00 \\
     & 10.0 & 1.804330E-00 & 9.737252E-01 && 2.874307E-00 & 1.607439E-00 & 2.908106E-00 & 1.453882E-00 \\
\noalign{\smallskip}
 ~4 &  0.1 & 8.155853E-02 & 5.681280E-05 && 2.550670E-01 & 1.807060E-04 & 8.14151~E-02 & 5.748690E-05 \\
     &  0.3 & 2.62113~E-01 & 4.141768E-03 && 8.11370~E-01 & 1.476105E-02 & 2.585912E-01 & 4.568657E-03 \\
     &  1.0 & 1.50146~E-00 & 2.669788E-01 && 2.713089E-00 & 8.506337E-01 & 1.24346~E-00 & 3.192207E-01 \\
     &  3.0 & 2.580295E-00 & 9.865859E-01 && 3.848184E-00 & 1.966160E-00 & 3.40958~E-00 & 1.515164E-00 \\
\noalign{\smallskip}
10 &  0.1 & 7.60280~E-02 & 5.084200E-05 && 3.01197~E-01 & 2.122690E-04 & 7.58943~E-02 & 5.299880E-05 \\
     &  0.3 & 2.39882~E-01 & 2.843924E-03 && 9.378379E-01 & 1.631801E-02 & 2.368506E-01 & 3.807946E-03 \\
\noalign{\smallskip}
\multicolumn{9}{c}{Oblate spheroid}  \\
\noalign{\smallskip}
 ~2 &  0.1 & 1.533476E-01 & 1.086960E-04 && 6.964599E-02 & 4.913110E-05 & 1.534705E-01 & 1.084390E-04 \\
     &  0.3 & 4.941085E-01 & 9.029149E-03 && 2.246184E-01 & 3.938983E-03 & 4.971749E-01 & 8.843852E-03 \\
     &  1.0 & 2.198062E-00 & 7.105949E-01 && 1.291186E-00 & 3.151001E-01 & 2.326983E-00 & 6.221156E-01 \\
     &  3.0 & 3.378955E-00 & 1.672422E-00 && 2.603296E-00 & 1.040019E-00 & 2.768031E-00 & 1.330862E-00 \\
\noalign{\smallskip}
 ~4 &  0.1 & 2.021662E-01 & 1.432990E-04 && 4.737106E-02 & 3.331120E-05 & 2.023435E-01 & 1.426270E-04 \\
     &  0.3 & 6.442415E-01 & 1.176912E-02 && 1.50949~E-01 & 2.574738E-03 & 6.483922E-01 & 1.129677E-02 \\
     &  1.0 & 2.394211E-00 & 7.756880E-01 && 8.254487E-01 & 1.707659E-01 & 2.658211E-00 & 6.204606E-01 \\
\noalign{\smallskip}
10 &  0.1 & 2.59201~E-01 & 1.833420E-04 && 3.51934~E-02 & 2.458170E-05 & 2.59355~E-01 & 1.816740E-04 \\
     &  0.3 & 8.101747E-01 & 1.452893E-02 && 1.102168E-01 & 1.774836E-03 & 8.13450~E-01 & 1.343542E-02 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\label{tab3}
\end{table}
}

%\newpage
{\footnotesize
%\begin{table}[htb]
\begin{table}
\begin{center}
\caption{Normalized extinction and scattering cross-sections
$C_{\rm ext, \, sca}/\pi r_{V}^{2}$ for spheroids with $m = 2.5 +1.5i$}
\medskip
\begin{tabular}{lrlllllll} \hline
\noalign{\smallskip}
&&\multicolumn{2}{c}{$\alpha = 0^{\circ}$}  &&
\multicolumn{4}{c}{$\alpha = 90^{\circ}$} \\
\noalign{\smallskip}
\cline{3-4} \cline{6-9}
\noalign{\smallskip}
$a/b$ &$x_V$ & $~~C_{\rm ext}/\pi r_{V}^{2}$          & $~~C_{\rm sca}/\pi r_{V}^{2}$ &
             & $~~C^{\rm TM}_{\rm ext}/\pi r_{V}^{2}$ & $~~C^{\rm TM}_{\rm sca}/\pi r_{V}^{2}$ &
               $~~C^{\rm TE}_{\rm ext}/\pi r_{V}^{2}$ & $~~C^{\rm TE}_{\rm sca}/\pi r_{V}^{2}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{9}{c}{Prolate spheroid} \\
\noalign{\smallskip}
 ~2 &  0.1 & 7.015863E-02 & 1.324916E-04 && 2.555773E-01 & 4.901860E-04 & 6.978007E-02 & 1.329890E-04 \\
     &  0.3 & 2.606615E-01 & 1.085718E-02 && 9.351659E-01 & 4.418671E-02 & 2.499507E-01 & 1.120452E-02 \\
     &  1.0 & 2.545641E-00 & 8.269798E-01 && 4.506818E-00 & 2.290117E-00 & 2.378954E-00 & 1.018659E-00 \\
     &  3.0 & 2.543020E-00 & 1.196810E-00 && 3.377612E-00 & 2.030276E-00 & 3.018778E-00 & 1.561478E-00 \\
\noalign{\smallskip}
 ~4 &  0.1 & 5.77412~E-02 & 1.079560E-04 && 5.61456~E-01 & 1.078514E-03 & 5.73428~E-02 & 1.092344E-04 \\
     &  0.3 & 2.09854~E-01 & 8.054995E-03 && 1.994433E-00 & 9.529425E-02 & 1.99005~E-01 & 8.869991E-03 \\
     &  1.0 & 2.23898~E-00 & 5.318469E-01 && 4.844781E-00 & 2.443652E-00 & 1.709826E-00 & 6.654597E-01 \\
     &  3.0 & 2.564187E-00 & 9.810827E-01 && 4.17736~E-00 & 2.576599E-00 & 3.444722E-00 & 1.733285E-00 \\
\noalign{\smallskip}
10 &  0.1 & 5.18757~E-02 & 9.344730E-05 && 8.83893~E-01 & 1.689619E-03 & 5.15928~E-02 & 9.740630E-05 \\
\noalign{\smallskip}
\multicolumn{9}{c}{Oblate spheroid}  \\
\noalign{\smallskip}
 ~2 &  0.1 & 1.683208E-01 & 3.228280E-04 && 4.590543E-02 & 8.685830E-05 & 1.690525E-01 & 3.220660E-04 \\
     &  0.3 & 6.046926E-01 & 2.854920E-02 && 1.682741E-01 & 7.112129E-03 & 6.24682~E-01 & 2.798368E-02 \\
     &  1.0 & 4.007697E-00 & 2.005398E-00 && 1.852325E-00 & 6.182319E-01 & 3.928646E-00 & 1.726163E-00 \\
     &  3.0 & 3.644115E-00 & 2.139153E-00 && 2.436512E-00 & 1.057701E-00 & 2.916411E-00 & 1.654727E-00 \\
\noalign{\smallskip}
 ~4 &  0.1 & 3.07216~E-01 & 5.903690E-04 && 2.714797E-02 & 5.130230E-05 & 3.085216E-01 & 5.876060E-04 \\
     &  0.3 & 1.089947E-00 & 5.221910E-02 && 9.612426E-02 & 4.009507E-03 & 1.123093E-00 & 5.017062E-02 \\
     &  1.0 & 4.550598E-00 & 2.313767E-00 && 1.04303~E-00 & 2.876447E-01 & 4.68404~E-00 & 1.847488E-00 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\label{tab4}
\end{table}
}

%\newpage
{\footnotesize
%\begin{table}[htb]
\begin{table}
\begin{center}
\caption{Normalized extinction and scattering cross-sections
$C_{\rm ext, \, sca}/\pi r_{V}^{2}$ for spheroids with $m = 3.0 +4.0i$}
\medskip
\begin{tabular}{lrlllllll} \hline
\noalign{\smallskip}
&&\multicolumn{2}{c}{$\alpha = 0^{\circ}$}  &&
\multicolumn{4}{c}{$\alpha = 90^{\circ}$} \\
\noalign{\smallskip}
\cline{3-4} \cline{6-9}
\noalign{\smallskip}
$a/b$ &$x_V$ & $~~C_{\rm ext}/\pi r_{V}^{2}$          & $~~C_{\rm sca}/\pi r_{V}^{2}$ &
             & $~~C^{\rm TM}_{\rm ext}/\pi r_{V}^{2}$ & $~~C^{\rm TM}_{\rm sca}/\pi r_{V}^{2}$ &
               $~~C^{\rm TE}_{\rm ext}/\pi r_{V}^{2}$ & $~~C^{\rm TE}_{\rm sca}/\pi r_{V}^{2}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{9}{c}{Prolate spheroid} \\
\noalign{\smallskip}
 ~2 &  0.1 & 3.459049E-02 & 1.836889E-04 && 1.929191E-01 & 1.115412E-03 & 3.339258E-02 & 1.843746E-04 \\
     &  0.3 & 1.944302E-01 & 1.537309E-02 && 9.098178E-01 & 1.129382E-02 & 1.661101E-01 & 1.585916E-02 \\
     &  1.0 & 1.894857E-00 & 9.206887E-01 && 5.495079E-00 & 3.970900E-00 & 2.283547E-00 & 1.438695E-00 \\
     &  3.0 & 2.388466E-00 & 1.478325E-00 && 3.453263E-00 & 2.702600E-00 & 2.762240E-00 & 1.941164E-00 \\
\noalign{\smallskip}
\multicolumn{9}{c}{Oblate spheroid}  \\
\noalign{\smallskip}
 ~2 &  0.1 & 1.017897E-01 & 5.864570E-04 && 2.106561E-02 & 1.116025E-04 & 1.046258E-01 & 5.850970E-04 \\
     &  0.3 & 4.583023E-01 & 5.527781E-02 && 1.197910E-01 & 9.258967E-03 & 5.167953E-01 & 5.426552E-02 \\
     &  1.0 & 5.236749E-00 & 3.731015E-00 && 1.404715E-00 & 6.821323E-01 & 4.215200E-00 & 2.800509E-00 \\
     &  3.0 & 3.647681E-00 & 2.814034E-00 && 2.009705E-00 & 1.173967E-00 & 3.040199E-00 & 2.239816E-00 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\label{tab5}
\end{table}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\phantom{\noi {\sf IV. References}}
\newpage
\phantom{\noi {\sf IV. References}}
\newpage
\noi {\sf IV. References}

\bigskip

\bitem{}~1. V.G. Farafonov, Differential Equations (Sov.)
         {\bf 19}, 1765 (1983).

\bitem{}~2. N.V. Voshchinnikov and V.G. Farafonov,
        Astrophys. Space Sci. {\bf 204,} 19 (1993).

\bitem{}~3. V.G. Farafonov, N.V. Voshchinnikov and V.V. Somsikov,
        Appl. Opt. {\bf 35,} 5412 (1996).

\bitem{}~4. C. Rogers and P.G. Martin,
          Astrophys. J, {\bf 228,} 450 (1979).

\bitem{}~5. M.I. Mishchenko, L.D. Travis and
      D.W. Mackowski,  JQSRT {\bf 55}, 535  (1996).

\bitem{}~6.  N.V.~Voshchinnikov, V.B.~Il'in, Th.~Henning, B.~Michel, V.G.~Farafonov,
      ``Extinction and Polarization of Radiation by Absorbing
      Spheroids: Shape/Size Effects and Some Benchmarks"
      submitted to the Journal of Quantitative Spectroscopy \&
      Radiative Transfer.


\bitem{}~7. V.G. Farafonov,  Optics Spectrosc., {\bf 77}, 402, 1994.


\bitem{}~8. V.G.~Farafonov,
       ``New method of calculations of light scattering by
       dielectric axisymmetric particles"
       prepared for publication in
       ``Optics and Spectroscopy''.

\bitem{}~9. S. Asano and G. Yamamoto,
      Applied Optics {\bf 14}, 29 (1975).
Asano

\bitem{}10. P.W. Barber and  S.C. Hill,  Light Scattering by Particles:
         Computational Methods, World Scientific, Singapore, 1990.


\vfill
\begin{tabular}{ll}
Prof. Dr. Nikolai Voshchinnikov & \hspace{3cm} AIU\\
Representative of the Software  & \hspace{3cm} Friedrich Schiller
University\\
Development Group & \hspace{3cm} \\
\end{tabular}

\end{document}
