In
computational electromagnetics
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment.
It typically involves using computer ...
, the scattering-matrix method (SMM) is a
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
used to solve
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
.
Principles
SMM can, for example, use cylinders to model
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
/
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
objects in the domain.
[
]
The total-field/scattered-field (TF/SF) formalism where the total field is written as sum of incident and scattered at each point in the domain:
:
By assuming series solutions for the total field, the SMM method transforms the domain into a cylindrical problem. In this domain total field is written in terms of
Bessel and
Hankel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
solutions to the cylindrical
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
. SMM method formulation, finally helps compute these coefficients of the cylindrical harmonic functions within the cylinder and outside it, at the same time satisfying EM boundary conditions.
Finally, SMM accuracy can be increased by adding (removing) cylindrical harmonic terms used to model the scattered fields.
SMM, eventually leads to a matrix formalism, and the coefficients are calculated through matrix inversion. For N-cylinders, each scattered field modeled using 2M+1 harmonic terms, SMM requires to solve a N(2M + 1) system of equations.
Advantages
SMM, is a rigorous and accurate method deriving from first principles. Hence, it is guaranteed to be accurate within limits of model, and not show spurious effects of numerical dispersion arising in other techniques like
FDTD
Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to ...
.
See also
*
Eigenmode expansion
Eigenmode expansion (EME) is a computational electrodynamics modelling technique. It is also referred to as the mode matching technique or the bidirectional eigenmode propagation method (BEP method). Eigenmode expansion is a linear frequency-domain ...
*
Finite-difference time-domain method
Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to ...
*
Finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
*
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
*
Method of Lines
The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. By reducing a PDE to a single continuous dimension, the method of lines allows solutions to be ...
References
Scattering, absorption and radiative transfer (optics)
Computational electromagnetics
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