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#LyX 2.1 created this file. For more info see http://www.lyx.org/
\lyxformat 474
\begin_document
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% definujeme novou rodinu, jež se volá pomocí \MyCyr pro běžné použití, avšak pro účely \DeclareSymbolFont je nutno získat název pomocí getfamilyname definovaného výše
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\pdf_title "Accelerating lattice mode calculations with T-matrix method"
\pdf_author "Marek Nečada"
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\end_header
\begin_body
\begin_layout Title
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset
Accelerating lattice mode calculations with
\begin_inset Formula $T$
\end_inset
-matrix method
\end_layout
\begin_layout Author
Marek Nečada
\end_layout
\begin_layout Section
Formulation of the problem
\end_layout
\begin_layout Standard
Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
medium, and assume that the system, i.e.
both the medium and the scatterers, have linear response.
A scattering problem in such system can be written as
\begin_inset Formula
\[
A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
\]
\end_inset
where
\begin_inset Formula $T_{α}$
\end_inset
is the
\begin_inset Formula $T$
\end_inset
-matrix for scatterer α,
\begin_inset Formula $A_{α}$
\end_inset
is its vector of the scattered wave expansion coefficient (the multipole
indices are not explicitely indicated here) and
\begin_inset Formula $P_{α}$
\end_inset
is the local expansion of the incoming sources.
\begin_inset Formula $S_{α\leftarrowβ}$
\end_inset
is ...
and ...
is ...
\end_layout
\begin_layout Standard
...
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
\]
\end_inset
\end_layout
\begin_layout Standard
Now suppose that the scatterers constitute an infinite lattice
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
\]
\end_inset
Due to the periodicity, we can write
\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
\end_inset
.
In order to find lattice modes, we search for solutions with zero RHS
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
\]
\end_inset
and we assume periodic solution
\begin_inset Formula $A_{\vect b\alpha}(\vect k)=A_{\vect a\alpha}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
\end_inset
.
\end_layout
\end_body
\end_document