Paper intro
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LatexCommand include
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filename "comparison.lyx"
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\begin_layout Standard
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\begin_layout Standard
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@ -0,0 +1,95 @@
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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\use_default_options true
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\spacing single
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\use_hyperref true
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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\use_package undertilde 1
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\end_header
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\begin_body
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\begin_layout Section
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Comparison to other methods
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Comparison"
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\end_document
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@ -81,6 +81,13 @@
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\begin_layout Section
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\begin_layout Section
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Applications
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Applications
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Applications"
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\end_body
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\end_body
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@ -81,6 +81,13 @@
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\begin_layout Section
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\begin_layout Section
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Finite systems
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Finite systems
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Finite"
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\begin_layout Itemize
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\begin_layout Itemize
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@ -81,6 +81,13 @@
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\begin_layout Section
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\begin_layout Section
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Infinite periodic systems
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Infinite periodic systems
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Infinite"
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\begin_layout Subsection
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\begin_layout Subsection
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@ -190,9 +197,9 @@ and we assume periodic solution
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\begin_inset Formula
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\begin_inset Formula
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\begin{eqnarray*}
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\begin{eqnarray*}
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
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\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
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\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
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\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect0α\leftarrow\vect bβ})A_{\vect0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
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\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
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\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect0\beta}\left(\vect k\right) & = & 0,\\
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A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
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A_{\vect0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect0\beta}\left(\vect k\right) & = & 0.
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\end{eqnarray*}
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\end{eqnarray*}
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\end_inset
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\end_inset
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@ -208,7 +215,7 @@ lattice Fourier transform
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of the translation operator,
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of the translation operator,
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\begin_inset Formula
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\begin_inset Formula
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\begin{equation}
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\begin{equation}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
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\end{equation}
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\end{equation}
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\end_inset
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\end_inset
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@ -233,7 +240,7 @@ reference "eq:W definition"
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\end_inset
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\end_inset
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is the asymptotic behaviour of the translation operator,
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is the asymptotic behaviour of the translation operator,
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\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
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\begin_inset Formula $S_{\vect0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
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\end_inset
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\end_inset
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that makes the convergence of the sum quite problematic for any
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that makes the convergence of the sum quite problematic for any
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@ -330,7 +337,7 @@ translation operator for spherical waves originating in
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\end_inset
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\end_inset
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is in fact a function of a single 3d argument,
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is in fact a function of a single 3d argument,
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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\end_inset
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\end_inset
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.
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.
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@ -344,7 +351,7 @@ reference "eq:W integral"
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can be rewritten as
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can be rewritten as
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\begin_inset Formula
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\begin_inset Formula
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\[
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\[
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W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
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W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)}
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\]
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\]
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\end_inset
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\end_inset
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@ -369,10 +376,10 @@ reference "eq:Dirac comb uaFt"
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for the Fourier transform of Dirac comb)
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for the Fourier transform of Dirac comb)
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\begin_inset Formula
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\begin_inset Formula
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\begin{eqnarray}
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\begin{eqnarray}
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W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
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W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)(\vect k)\nonumber \\
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& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
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& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)\left(\vect k\right)\nonumber \\
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\nonumber
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\end{eqnarray}
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\end{eqnarray}
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\end_inset
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\end_inset
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@ -478,8 +485,8 @@ reference "eq:W sum in reciprocal space"
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\begin_inset Formula
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\begin_inset Formula
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\begin{eqnarray}
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\begin{eqnarray}
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W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
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W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
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W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
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W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
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W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
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W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
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\end{eqnarray}
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\end{eqnarray}
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\end_inset
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\begin_layout Section
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\begin_layout Section
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Introduction
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Introduction
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Introduction"
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\end_inset
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\end_layout
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\begin_layout Standard
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The problem of electromagnetic response of a system consisting of many compact
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scatterers in various geometries, and its numerical solution, is relevant
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to many branches of nanophotonics (TODO refs).
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The most commonly used general approaches used in computational electrodynamics
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, such as the finite difference time domain (FDTD) method or the finite
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element method (FEM), are very often unsuitable for simulating systems
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with larger number of scatterers due to their computational complexity.
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Therefore, a common (frequency-domain) approach to get an approximate solution
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of the scattering problem for many small particles has been the coupled
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dipole approximation (CDA) where individual scatterers are reduced to electric
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dipoles (characterised by a polarisability tensor) and coupled to each
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other through Green's functions.
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\end_layout
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\begin_layout Standard
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CDA is easy to implement and has favorable computational complexity but
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suffers from at least two fundamental drawbacks.
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The obvious one is that the dipole approximation is too rough for particles
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with diameter larger than a small fraction of the wavelength.
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The other one, more subtle, manifests itself in photonic crystal-like structure
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s used in nanophotonics: there are modes in which the particles' electric
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dipole moments completely vanish due to symmetry, regardless of how small
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the particles are, and the excitations have quadrupolar or higher-degree
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multipolar character.
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These modes typically appear at the band edges where interesting phenomena
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such as lasing or Bose-Einstein condensation have been observed – and CDA
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by definition fails to capture such modes.
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\end_layout
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\begin_layout Standard
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The natural way to overcome both limitations of CDA mentioned above is to
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include higher multipoles into account.
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Instead of polarisability tensor, the scattering properties of an individual
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particle are then described a more general
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\begin_inset Formula $T$
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\end_inset
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-matrix, and different particles' multipole excitations are coupled together
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via translation operators, a generalisation of the Green's functions in
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CDA.
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This is the idea behind the
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\emph on
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multiple-scattering
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\begin_inset Formula $T$
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\end_inset
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-matrix method
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\emph default
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(MSTMM) (TODO a.k.a something??), and it has been implemented previously for
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a limited subset of problems (TODO refs and list the limitations of the
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available).
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO přestože blablaba, moc se to nepoužívalo, protože je težké udělat to
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správně.
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\end_layout
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||||||
|
|
||||||
|
\end_inset
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|
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|
Due to the limitations of the existing available codes, we have been developing
|
||||||
|
our own implementation of MSTMM, which we have used in several previous
|
||||||
|
works studying various physical phenomena in plasmonic nanoarrays (TODO
|
||||||
|
examples with refs).
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|
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||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Standard
|
||||||
|
Hereby we release our MSTMM implementation, the
|
||||||
|
\emph on
|
||||||
|
QPMS Photonic Multiple Scattering
|
||||||
|
\emph default
|
||||||
|
suite, as an open source software under the GNU General Public License
|
||||||
|
version 3.
|
||||||
|
(TODO refs to the code repositories.) QPMS allows for linear optics simulations
|
||||||
|
of arbitrary sets of compact scatterers in isotropic media.
|
||||||
|
The features include computations of electromagnetic response to external
|
||||||
|
driving, the related cross sections, and finding resonances of finite structure
|
||||||
|
s.
|
||||||
|
Moreover, in QPMS we extensively employ group theory to exploit the physical
|
||||||
|
symmetries of the system to further reduce the demands on computational
|
||||||
|
resources, enabling to simulate even larger systems.
|
||||||
|
|
||||||
|
\begin_inset Note Note
|
||||||
|
status open
|
||||||
|
|
||||||
|
\begin_layout Plain Layout
|
||||||
|
(TODO put a specific example here of how large system we are able to simulate?)
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
Although systems of large
|
||||||
|
\emph on
|
||||||
|
finite
|
||||||
|
\emph default
|
||||||
|
number of scatterers are the area where MSTMM excels the most—simply because
|
||||||
|
other methods fail due to their computational complexity—we also extended
|
||||||
|
the method onto infinite periodic systems (photonic crystals); this can
|
||||||
|
be used for quickly evaluating dispersions of such structures and also
|
||||||
|
their topological invariants (TODO).
|
||||||
|
The QPMS suite contains a core C library, Python bindings and several utilities
|
||||||
|
for routine computations, such as TODO.
|
||||||
|
It includes extensive Doxygen documentation, together with description
|
||||||
|
of the API, making extending and customising the code easy.
|
||||||
|
\end_layout
|
||||||
|
|
||||||
|
\begin_layout Standard
|
||||||
|
The current paper is organised as follows: Section
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand ref
|
||||||
|
reference "sec:Finite"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
is devoted to MSTMM theory for finite systems, in Section
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand ref
|
||||||
|
reference "sec:Infinite"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
we develop the theory for infinite periodic structures.
|
||||||
|
Section
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand ref
|
||||||
|
reference "sec:Applications"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
demonstrates some basic practical results that can be obtained using QPMS.
|
||||||
|
Finally, in Section
|
||||||
|
\begin_inset CommandInset ref
|
||||||
|
LatexCommand ref
|
||||||
|
reference "sec:Comparison"
|
||||||
|
|
||||||
|
\end_inset
|
||||||
|
|
||||||
|
we comment on the computational complexity of MSTMM in comparison to other
|
||||||
|
methods.
|
||||||
\end_layout
|
\end_layout
|
||||||
|
|
||||||
\end_body
|
\end_body
|
||||||
|
|
Loading…
Reference in New Issue