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.