650 lines
14 KiB
Plaintext
650 lines
14 KiB
Plaintext
#LyX 2.1 created this file. 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 10
|
||
\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 a5paper
|
||
\use_geometry true
|
||
\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
|
||
\leftmargin 2cm
|
||
\topmargin 2cm
|
||
\rightmargin 2cm
|
||
\bottommargin 2cm
|
||
\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
|
||
|
||
and
|
||
\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
|
||
\end_inset
|
||
|
||
.
|
||
In order to find lattice modes, we search for solutions with zero RHS
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
and we assume periodic solution
|
||
\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
|
||
\end_inset
|
||
|
||
, yielding
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\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,\\
|
||
\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,\\
|
||
\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,\\
|
||
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.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
Therefore, in order to solve the modes, we need to compute the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
lattice Fourier transform
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
of the translation operator,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
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}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Computing the Fourier sum of the translation operator
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The problem evaluating
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:W definition"
|
||
|
||
\end_inset
|
||
|
||
is the asymptotic behaviour of the translation operator,
|
||
\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
|
||
\end_inset
|
||
|
||
that makes the convergence of the sum quite problematic for any
|
||
\begin_inset Formula $d>1$
|
||
\end_inset
|
||
|
||
-dimensional lattice.
|
||
In electrostatics, one can solve this with problem with Ewald summation.
|
||
Its basic idea is that if what asymptoticaly decays poorly in the direct
|
||
space, will perhaps decay fast in the Fourier space.
|
||
I use the same idea here, but things will be somehow harder than in electrostat
|
||
ics.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
(Appendix) 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
|
||
\begin{equation}
|
||
Ш_{T}(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T}\label{eq:1D Dirac comb Fourier series}
|
||
\end{equation}
|
||
|
||
\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
|
||
\begin{equation}
|
||
\uoft{Ш_{T}}(f)=\frac{1}{T}Ш_{\frac{1}{T}}(f)=\sum_{n=-\infty}^{\infty}e^{-i2\pi fnT}\label{eq:1D Dirac comb Ft ordinary freq}
|
||
\end{equation}
|
||
|
||
\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
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the book:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\dc A(\vect x)=\frac{1}{\left|\det A\right|}\dc{}^{(d)}\left(A^{-1}\vect x\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
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
|
||
|
||
SRSLY?, and hence
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\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).\label{eq:Dirac comb factorisation}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier series representation
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Utilising the Fourier series for 1D Dirac comb
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:1D Dirac comb Fourier series"
|
||
|
||
\end_inset
|
||
|
||
and the factorisation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb factorisation"
|
||
|
||
\end_inset
|
||
|
||
, we get
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\dc{\basis u}(\vect x) & = & \prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \sum_{n_{j}=-\infty}^{\infty}e^{2\pi in_{i}\vect x\cdot\rec{\vect u_{i}}}\\
|
||
& = & \left(\prod_{j=1}^{d}\left\Vert \rec{\vect u_{j}}\right\Vert \right)\sum_{\vect n\in\mathbb{Z}^{d}}e^{2\pi i\vect x\cdot\sum_{k=1}^{d}n_{k}\rec{\vect u_{k}}}.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Fourier transform
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
From the book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft{\dc A}\left(\vect{\xi}\right)=\dc{}^{(d)}\left(A^{T}\vect{\xi}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
So, from the stretch theorem
|
||
\begin_inset Formula $\uoft{(f(A\vect x))}=\frac{1}{\left|\det A\right|}\uoft{f\left(A^{-T}\vect{\xi}\right)}=\left|\det A^{-T}\right|\uoft{f\left(A^{-T}\vect{\xi}\right)}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
From
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:Dirac comb factorisation"
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:1D Dirac comb Ft ordinary freq"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\uoft{\dc{\basis u}}(\vect{\xi})=\prod_{i=1}^{d}\left\Vert \rec{\vect u_{i}}\right\Vert \dc{}\left(\vect x\cdot\rec{\vect u_{i}}\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|