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Reciprocity relation in checking calculations
of the optical properties of non-spherical particles

V.G.Farafonov
St.Petersburg University of Aerocosmic Instrumentation,
St.Petersburg, 190000, Russia

and

V.B.Il'in
Astronomical Institute, St.Petersburg University
St.Petersburg, 198504, Russia


ABSTRACT

There are many methods to solve the problem of light scattering by
non-spherical particles, but only the separation of variables method (SVM)
for spheroids and the T-matrix method (TMM) appear to be capable to provide
very accurate results (5 and more correct digits) for particles
compared to or larger than the wavelength of incident radiation [1].

Though both the existence and uniqueness of solution to the problem
have been proved [2], a careful mathematical analysis of TMM
and of most versions of SVM is absent even for convex axisymmetric particles.
Moreover it is known (see, e.g., [3]) that
for particle shapes essentially deviating from the spherical one,
the classical versions of TMM [1] and SVM [4]
work well only for particle small in comparison with the wavelength.
Apparently, only for one version of SVM [5],
a relatively full analytic investigation of the solution
was made [6], and this approach looks to be more or less well mathematically
grounded for spheroids of any degree of asphericity (i.e. any aspect ratio).
In such a situation a careful check of calculations of
the optical properties of non-spherical particles is always required.

Up to now such checking is made mainly in two ways:
(i) basing on the optical theorem that states that for non-absorbing particles
the cross-sections of extinction and scattering must be equal, and
(ii) on the fact that the results should converge with an increase of the
number of terms taking into account in fields' expansions.
The second criterion can be used for absorbing particles and applied to
intensity of radiation scattered in different directions [7]. However, it
should be emphasized that the convergence does not guarantee that the results
obtained are correct.

In this paper we consider the light scattering by axisymmetric particles.
The problem is solved by a recently suggested method
that combines TMM and one of the version of SVM for spheroids [8].
To check computational results, we use the reciprocity relation
for the amplitude matrix (see [1] for more details)
\begin{equation}
{S_{11} (- \vec{n}^\prime, - \vec{n}) \ \ \
S_{12} (- \vec{n}^\prime, - \vec{n})
\choose
S_{21} (- \vec{n}^\prime, - \vec{n}) \ \ \
S_{22} (- \vec{n}^\prime, - \vec{n}) }
\ \ \ = \ \ \
{S_{11} ( \vec{n}, \vec{n}^\prime) \ \ \
- S_{12} ( \vec{n}, \vec{n}^\prime)
\choose
- S_{21} ( \vec{n}, \vec{n}^\prime) \ \ \
S_{22} ( \vec{n}, \vec{n}^\prime) },
\end{equation}
where $\vec{n}, \vec{n}^\prime$ are the unit vectors in the directions of
incident and scattered radiations, respectively.

Note that the criterion is applicable to absorbing particles and
is highly efficient due to the usage of the differential characteristics
of scattered radiation instead of the integral ones such as
the scattering cross-section.

The criterion is most efficient when the directions of incident and scattered
radiations and the symmetry axis of a particle are not in a plane. In this
case the amplitude matrix (1) is not diagonal, and the check involves the
waves of both polarizations (TE and TM modes).

We present some results obtained for particles of various
shapes (prolate and oblate spheroids, finite circular cylinders,
Chebyshev particles), sizes and refractive indices.
The behaviour of the criterion with variations of the scattering
angle and the particle parameters is discussed.
It is shown that for absorbing particles the criterion allows one to predict
the number of correct digits in the cross-sections, etc.

In particular useful should be the criterion for (SVM) solutions
where the spheroidal wave functions are used for fields' expansions,
since it provides a way to control calculations of the radial spheroidal
functions with a complex value of $c$ which often meet problems.

The work was partly supported by INTAS (grant 99/652).


References:

1. Mishchenko M.I., Hovenier J.W., Travis L.D. (2000) Light Scattering
by Non-Spherical Particles, Acad. Press.
2. Colton D., Kress R. (1983) Integral Equation Methods in Scattering
Theory, Wiley, New York.
3. Voshchinnikov N.V., Il'in V.B., Henning Th., Michel B., Farafonov V.G.
(2000) J. Quant. Spectr. Rad. Transf. 65, 877.
4. Asano S., Yamamoto G. (1975) Appl. Opt. 14, 29.
5. Voshchinnikov N.V., Farafonov V.G. (1993) Astrophys. Space Sci. 204, 19.
6. Farafonov V.G. (1983) Diff. Equat. (Sov.) 19, 1765.
7. Barber P.W., Hill S.C. (1990) Light Scattering by Particles: Computational
Methods, World Scientific, Singapore.
8. Farafonov V.G., Il'in V.B., Henning T. (1999) J. Quant. Spectr. Rad.
Transf. 63, 205.