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On applicability of T-matrix-like methods

Victor G. Farafonov^1, Vladimir B. Il'in^2,
and Andrey A. Loskutov^1

^1 St.Petersburg University of Aerocosmic Instrumentation
^2 Astronomical Institute, St.Petersburg University


Abstract

The T-matrix method and its modifications are very popular now
in applications of the light scattering theory (e.g., [1-3]).
The method is based on:
i) the integral representation of the scattering problem;
ii) expansion of the fields (or their potentials) in terms of
the spherical wavefunctions, and
iii) solution of infinite systems of algebraic equations
for expansion coefficients.

Analytical investigation of the systems
in the case of perfectly conducting particles
was recently made in [3].
It was shown that only for ``weakly non-convex scatterers''
the Fredholm alternative is valid for the systems and
their solution can be obtained by reduction.
However, the T-matrix method is not known to work when the scatterer
shape is essentially non-spherical. For instance, for spheroids
numerical instability can arise already for the aspect ratio
a/b > 4 [4],
although spheroids, being smooth convex bodies, belong to
the weakly non-convex ones.

In this paper we analytically investigate the algebraic systems
for dielectric axisymmetric particles in frame of a recently
suggested modification of the T-matrix method [2].
It is demonstrated that the Fredholm alternative is valid for
the same condition as in [3], and namely: s_1 < s_2,
where $s_1$ is the distance from the coordinate origin to the farthest
peculiarity of the scattered field,
s_2 that to the nearest peculiarity of the internal field.

The condition (1) is weaker than
that of validity of the Rayleigh hypothesis
(convergence of the expansions everywhere up to the border of scatterer)
r_{\rm min} > s_1, \ \ r_{\rm max} < s_2,
where r_{\rm min} and r_{\rm max} are the distances from
the coordinate origin to the nearest and farthest
points of the scatterer surface, respectively

Our computations indicate that for spheroids
the expansions of the scattered field in contrast to the internal one
do not converge already for a/b > 2,
although the tests for the scattered field in a distant zone
(the optical theorem, reciprocity relation, etc. [1])
are satisfied with a high accuracy.
For Chebyshev particles,
both scattered and internal fields have peculiarities and
the regions of convergence for the fields are found to be more close.

Similar results should occur for the classical T-matrix method as well.
Moreover we believe that any T-matrix-like method involving
the above mentioned features i)--iii)
can be considered as theoretically correct
when the Rayleigh hypothesis is satisfied.
If the condition of weak non-convexity is not valid
the method should not work at all.
In the intermediate region
one must take the results of calculations with a care,
in particular as concern the internal fields and the scattered fields
close to particle surface.
Note that in numerical realizations a reduction of the systems
must be done and
a problem that is different from the initial one is actually solved.
As a consequence we observe that
the conditions (1) and (2) rather approximately
define the region of convergence.

References

1. Mishchenko M.I. et al. (2000)
Light Scattering by Non-Spherical Particles, Acad. Press.

2. Farafonov V.G. et al. (1999)
JQSRT, v.63, p.205--215.

3. Kyurkchan A.G. (2000)
Radiotech. Elektron., v.45, p.1078--1083.

4. Voshchinnikov N.V. et al. (2000)
JQSRT, v.65, p.877--893.

5. Voshchinnikov N.V., Farafonov V.G. (1993)
Astrophys. Space Sci., v.204, p.19--87.

6. Farafonov V.G. (1983)
Diff. Equat. (Sov.), v.19, p.1765--1777.