Start writing notes on dirac combs

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+\begin_document
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+\language english
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+\default_output_format pdf4
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+\bibtex_command default
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+\spacing single
+\use_hyperref true
+\pdf_title "Accelerating lattice mode calculations with T-matrix method"
+\pdf_author "Marek Nečada"
+\pdf_bookmarks true
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+\index Index
+\shortcut idx
+\color #008000
+\end_index
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+\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