**
QUASI-SEPARABLE T-SCATTERING OPERATOR
APPROACH TO LOCAL FIELD DIRECT CALCULATIONS IN MULTIPLE SCATTERING PROBLEMS**

**Yu. N. Barabanenkov **^{1}, M. Yu. Barabanenkov ^{2}

** **

^{1} V.A. Kotelnikov Institute
of Radioengineering and Electronics of RAS, Moscow

^{2} Institute of
Microelectronics Technology of RAS, Chernogolovka, Moscow Region

Received April 10, 2013

**
Abstract**. We present analytic solution to
fundamental in wave multiple scattering theory Lippmann-Schwinger (LS) integral
equation for electric field quantum mechanical type tensor T- scattering
operator by nonmagnetic arbitrary shaped particle with given scalar dielectric
permittivity and specific conductivity in free space. The solution is obtained with
the aid of a vector expansion functions’ basis and Galerkin method and written
as sum of separable scattering operators weighted by inverse of a generating
matrix, which is expressed through matrix describing wave coupling between the
particle elements. Similar quasi-separable (QS) form is obtained for T-scattering
operator of coupled particles’ ensemble, when generating matrix is related
with matrix describing wave coupling between particles; an equations’ system
for self consistent currents excited inside coupled particles is derived on
this way also. Having given directly the current
excited inside particle, T-scattering operator should be closed connected with
wave spatial dispersion effect in homogenized electromagnetic crystal
structure. Really, we show the rigorously defined a periodic structure effective
dielectric permittivity tensor is exactly expressed by unit cell QS T-
scattering operator, with generating matrix related to matrix of wave coupling
between unit cell particles directly and via crystal. In order to test and
apply the QS T-scattering operator approach, some different choosing the vector
expansion functions are considered. In the case of vector spherical wave
functions’ basis the QS T-scattering operator gives the Mie solution for
incident plane wave scattering from and transmitted into a spherical particle.
The another basis vector expansion functions defined on finite elements of
particle volume is consistent with QS approximation of particle scattering
potential operator, for which the LS equation is resolved exactly. Next, an asymptotic
formula is obtained for contribution of spatially resonant coupling between two
small spherical plasmonic particles inside unit cell of electromagnetic crystal
into the structure effective magnetic permeability. We study at last some
simple low dimensional ordered periodic arrays of particles, with particles’
coupling matrix obeying a stochastic property for the case of specifically
linearly polarized wave electric field, and find corresponding stochastic and overtone
eigenmodes and method of their excitation. Exact and asymptotic formulas are
found also for standing and propagating wave transfer of currents’ exciting
along a strait linear chain of particles with Jacobi’s coupling matrix.

**
Keywords: **
electromagnetic wave field,
arbitrary shaped nonmagnetic coupled particles, multiple scattering,
T-scattering operator, Lippmann-Schwinger integral equation, analytic solution,
currents excited inside particles, electromagnetic crystal structures, low
dimension arrays of particles.