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For more info see http://www.lyx.org/ \lyxformat 474 \begin_document \begin_header \textclass article \begin_preamble \usepackage{unicode-math} % Toto je trik, jimž se z fontspec získá familyname pro následující \ExplSyntaxOn \DeclareExpandableDocumentCommand{\getfamilyname}{m} { \use:c { g__fontspec_ \cs_to_str:N #1 _family } } \ExplSyntaxOff % definujeme novou rodinu, jež se volá pomocí \MyCyr pro běžné použití, avšak pro účely \DeclareSymbolFont je nutno získat název pomocí getfamilyname definovaného výše \newfontfamily\MyCyr{CMU Serif} \DeclareSymbolFont{cyritletters}{EU1}{\getfamilyname\MyCyr}{m}{it} \newcommand{\makecyrmathletter}[1]{% \begingroup\lccode`a=#1\lowercase{\endgroup \Umathcode`a}="0 \csname symcyritletters\endcsname\space #1 } \count255="409 \loop\ifnum\count255<"44F \advance\count255 by 1 \makecyrmathletter{\count255} \repeat \renewcommand{\lyxmathsym}[1]{#1} \end_preamble \use_default_options true \maintain_unincluded_children false \language english \language_package default \inputencoding auto \fontencoding global \font_roman TeX Gyre Pagella \font_sans default \font_typewriter default \font_math default \font_default_family default \use_non_tex_fonts true \font_sc false \font_osf true \font_sf_scale 100 \font_tt_scale 100 \graphics default \default_output_format pdf4 \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref true \pdf_title "Accelerating lattice mode calculations with T-matrix method" \pdf_author "Marek Nečada" \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder false \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard \begin_inset FormulaMacro \newcommand{\uoft}[1]{\mathfrak{F}#1} \end_inset \begin_inset FormulaMacro \newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1} \end_inset \begin_inset FormulaMacro \newcommand{\vect}[1]{\mathbf{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\ud}{\mathrm{d}} \end_inset \begin_inset FormulaMacro \newcommand{\basis}[1]{\mathfrak{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\dc}[1]{Ш_{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\rec}[1]{#1^{-1}} \end_inset \begin_inset FormulaMacro \newcommand{\recb}[1]{#1^{\widehat{-1}}} \end_inset \end_layout \begin_layout Title 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 \begin_layout Section Multidimensional Dirac comb \end_layout \begin_layout Subsection 1D \end_layout \begin_layout Standard This is all from Wikipedia \end_layout \begin_layout Subsubsection Definitions \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} Ш(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-k)\\ Ш_{T}(t) & \equiv & \sum_{k=-\infty}^{\infty}\delta(t-kT)=\frac{1}{T}Ш\left(\frac{t}{T}\right) \end{eqnarray*} \end_inset \end_layout \begin_layout Subsubsection Fourier series representation \end_layout \begin_layout Standard \begin_inset Formula \[ Ш_{T}(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T} \] \end_inset \end_layout \begin_layout Subsubsection Fourier transform \end_layout \begin_layout Standard With unitary ordinary frequency Ft., i.e. \end_layout \begin_layout Standard \begin_inset Formula \[ \uoft f(\vect{\xi})\equiv\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect{\xi}}\ud^{n}\vect x \] \end_inset we have \begin_inset Formula \[ \uoft{Ш_{T}}(f)=\frac{1}{T}Ш_{\frac{1}{T}}(f)=\sum_{n=-\infty}^{\infty}e^{-i2\pi fnT} \] \end_inset and with unitary angular frequency Ft., i.e. \begin_inset Formula \[ \uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect k}\ud^{n}\vect x \] \end_inset we have \begin_inset Formula \[ \uaft{Ш_{T}}(\omega)=\frac{\sqrt{2\pi}}{T}Ш_{\frac{2\pi}{T}}(\omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT} \] \end_inset \end_layout \begin_layout Subsection Dirac comb for multidimensional lattices \end_layout \begin_layout Subsubsection Definitions \end_layout \begin_layout Standard Let \begin_inset Formula $d$ \end_inset be the dimensionality of the real vector space in question, and let \begin_inset Formula $\basis u\equiv\left\{ \vect u_{i}\right\} _{i=1}^{d}$ \end_inset denote a basis for some lattice in that space. Let the corresponding lattice delta comb be \begin_inset Formula \[ \dc{\basis u}\left(\vect x\right)\equiv\sum_{n_{1}=-\infty}^{\infty}\ldots\sum_{n_{d}=-\infty}^{\infty}\delta\left(\vect x-\sum_{i=1}^{d}n_{i}\vect u_{i}\right). \] \end_inset \end_layout \begin_layout Standard Furthemore, let \begin_inset Formula $\rec{\basis u}\equiv\left\{ \rec{\vect u}_{i}\right\} _{i=1}^{d}$ \end_inset be the reciprocal lattice basis, that is the basis satisfying \begin_inset Formula $\vect u_{i}\cdot\rec{\vect u_{j}}=\delta_{ij}$ \end_inset . This slightly differs from the usual definition of a reciprocal basis, here denoted \begin_inset Formula $\recb{\basis u}\equiv\left\{ \recb{\vect u_{i}}\right\} _{i=1}^{d}$ \end_inset , which satisfies \begin_inset Formula $\vect u_{i}\cdot\recb{\vect u_{j}}=2\pi\delta_{ij}$ \end_inset instead. \end_layout \begin_layout Subsubsection Factorisation of a multidimensional lattice delta comb \end_layout \begin_layout Standard By simple drawing, it can be seen that \begin_inset Formula \[ \dc{\basis u}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right) \] \end_inset where \begin_inset Formula $c_{\basis u}$ \end_inset is some numerical volume factor. In order to determine \begin_inset Formula $c_{\basis u}$ \end_inset , let us consider only the \begin_inset Quotes eld \end_inset zero tooth \begin_inset Quotes erd \end_inset of the comb, leading to \begin_inset Formula \[ \delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\delta\left(\vect x\cdot\rec{\vect u_{i}}\right). \] \end_inset From the scaling property of delta function, \begin_inset Formula $\delta(ax)=\left|a\right|^{-1}\delta(x)$ \end_inset , we get \begin_inset Formula \[ \delta^{d}(\vect x)=c_{\basis u}\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert ^{-1}\delta\left(\vect x\cdot\frac{\rec{\vect u_{i}}}{\left\Vert \rec{\vect u_{i}}\right\Vert }\right). \] \end_inset Applying both sides to a test function that is one at the origin, we get \begin_inset Formula $c_{\basis u}=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert $ \end_inset , and hence \begin_inset Formula \[ \dc{\basis u}(\vect x)=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right). \] \end_inset \end_layout \begin_layout Subsubsection Fourier series representation \end_layout \begin_layout Subsubsection Fourier transform \end_layout \end_body \end_document