Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

198 3 Simulation Results

3.2.4 CST Microwave Studio Program

CST Microwave Studio is a commercial software package which can be used for analyzing the electromagnetic scattering by particles with complex geometries and optical properties. Essentially, the electromagnetic simulator is based on the finite integration technique (FIT) developed by Weiland [260]. The finite integration technique is simple in concept and provides a discrete reformulation of Maxwell’s equations in their integral from. Following the analysis of Clemens and Weiland [38] we recall the basic properties of the discrete representation of Maxwell’s equation by the finite integration technique.

The first discretization step of the finite integration method consists in the restriction of the electromagnetic field problem, which represent an open boundary problem, to a simply connected and bounded space domain Ω containing the region of interest. The next step consists in the decomposition of the computational domain into a finite number of volume elements (cells). This decomposition yields the volume element complex G, which serves as computational grid. Assuming that Ω is brick-shaped, we have the volume element complex

G= {Dijk /Dijk = [xi, xi+1] × [yj , yj+1] × [zk , zk+1]

i = 1, 2, . . . , Nx − 1, j = 1, 2, . . . , Ny − 1, k = 1, 2, . . . , Nz − 1} ,

and note that each edge and facet of the volume elements are associated with a direction.

Next, we consider the integral form of the Maxwell equations, i.e., for Faraday’s induction law and Maxwell–Ampere law we integrate the di erential equations on an open surface and apply the Stokes theorem, while for Gauss’ electric and magnetic field laws we integrate the di erential equations on a bounded domain and use the Gauss theorem.

The Faraday law in integral form

with S being an arbitrary open surface contained in Ω, can be rewritten for the facet Sz,ijk of the volume element Dijk as the ordinary di erential equation

d

ex,ijk + ey,i+1jk − ex,ij+1k − ey,ijk = −dt bz,ijk ,

where

(xi+1,yj ,zk )

is the electric voltage along the edge Lx,ijk of the facet Sz,ijk , and

200 3 Simulation Results

are discretized in an analogous manner. The di erential equations are formulated in terms of the magnetic grid voltages hα,ijk along the dual edges Lα,ijk , the dielectric fluxes dα,ijk and the conductive currents jα,ijk through the dual facets Sα,ijk , α = x, y, z, and the electric charge qijk in the dual

Collecting the equations of all element surfaces of the complex pair {G, G} and introducing the integral voltage-vectors e and h, the flux state-vectors b and d, the conductive current-vector j, and the electric charge-vector q, we derive a set of discrete matrix equations, the so-called Maxwell grid equations

The transformation into frequency domain for the Maxwell grid equations with e(t) = Re{ee−jωt}, yields

Ce = jωb , Ch = −jωd + j .

The discrete curl-matrices C and C, and the discrete divergence matrices S and S are defined on the grids G and G, respectively, and depend only on the grid topology. The integral voltageand flux state-variables allocated on the two di erent volume element complexes are related to each other by the discrete material matrix relations

d = M εe + p , j = M κe , h = M ν b − m ,

where M ε is the permittivity matrix, M κ is the matrix of conductivities, M ν is the matrix of reluctivities, and p and m are the electric and magnetic polarization vectors, respectively. The discrete grid topology matrices have the same e ect as the vector operators curl and div. For instance, the discrete analog of the equation · × = 0 (div curl = 0) is SC = 0 and SC =

0. Transposition of these equations together with the relation between the

T

discrete curl-matrices C = C , yields the discrete equations CST = 0 and

T

CS = 0, both corresponding to the equation × = 0 (curl grad = 0). Basic algebraic properties of the Maxwell grid equations also allow to prove conservation properties with respect to energy and charges.

The finite integration technique su ers somewhat from a deficiency in being able to model very complicated cavities including curved boundaries with high precision, but the usage of the perfect boundary approximation eliminate this deficiency [123].