# Scattering-matrix method

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{{Disputed|date=October 2009}}
In [computational electromagnetics](/source/computational_electromagnetics), the '''scattering-matrix method''' ('''SMM''') is a [numerical method](/source/numerical_method) used to solve [Maxwell's equations](/source/Maxwell's_equations),<ref>
{{cite book
 | title = Reciprocity, spatial mapping and time reversal in electromagnetics
 | author = C. Altman and K. Suchy
 | publisher = Springer
 | year = 1991
 | isbn = 978-0-7923-1339-7
 | page = 39
 | url = https://books.google.com/books?id=bQmQil-dMBUC&dq=scattering-matrix-method&pg=PA39
 }}</ref> related to the [transfer-matrix method](/source/Transfer-matrix_method_(optics)).

==Principles==

SMM can, for example, use cylinders to model [dielectric](/source/dielectric)/[metal](/source/metal) objects in the domain.<ref>
{{cite book
 | title = Electromagnetic theory and applications for photonic crystals
 | author = Kiyotoshi Yasumoto
 | publisher = CRC Press
 | year = 2006
 | isbn = 978-0-8493-3677-5
 | page = 3
 | url = https://books.google.com/books?id=fF6Gd5Wkb78C&dq=scattering-matrix-method&pg=PA3
 }}</ref>
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:
:<math>E_{tot} = E_{inc} + E_{scatt} \ </math>

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](/source/Bessel_function) and [Hankel function](/source/Hankel_function) solutions to the cylindrical [Helmholtz equation](/source/Helmholtz_equation). 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 2''M''+1 harmonic terms, SMM requires to solve a ''N''(2''M'' + 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 [Finite-difference time-domain (FDTD) method](/source/Finite-difference_time-domain_method).

==See also==
*[Eigenmode expansion](/source/Eigenmode_expansion)
*[Finite-difference time-domain method](/source/Finite-difference_time-domain_method)
*[Finite element method](/source/Finite_element_method)
*[Maxwell's equations](/source/Maxwell's_equations)
*[Method of Lines](/source/Method_of_lines)

==References==

{{reflist}}

Category:Scattering, absorption and radiative transfer (optics)
Category:Computational electromagnetics

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Adapted from the Wikipedia article [Scattering-matrix method](https://en.wikipedia.org/wiki/Scattering-matrix_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Scattering-matrix_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.