From 74655c0210c2018f66ff49bc58710eb95cf91a86 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Thu, 18 Jul 2019 22:50:01 +0300
Subject: [PATCH] WIP finite systems + copypasta from hexlaser SM

Former-commit-id: 7fa2ae8308b4aa62fdb0986ac0ed669fad29d2a1
---
 lepaper/arrayscat.lyx    |  91 +++++++-
 lepaper/finite-old.lyx   | 378 +++++++++++++++++++++++++++++++
 lepaper/finite.lyx       | 152 ++++++++++++-
 lepaper/infinite-old.lyx | 476 +++++++++++++++++++++++++++++++++++++++
 4 files changed, 1088 insertions(+), 9 deletions(-)
 create mode 100644 lepaper/finite-old.lyx
 create mode 100644 lepaper/infinite-old.lyx

diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx
index bf9767a..0806f93 100644
--- a/lepaper/arrayscat.lyx
+++ b/lepaper/arrayscat.lyx
@@ -41,11 +41,11 @@
 \papersize default
 \use_geometry false
 \use_package amsmath 2
-\use_package amssymb 1
+\use_package amssymb 2
 \use_package cancel 1
 \use_package esint 1
 \use_package mathdots 1
-\use_package mathtools 1
+\use_package mathtools 2
 \use_package mhchem 1
 \use_package stackrel 1
 \use_package stmaryrd 1
@@ -158,7 +158,12 @@
 
 
 \begin_inset FormulaMacro
-\newcommand{\ush}[2]{Y_{#1,#2}}
+\newcommand{\spharm}[2]{Y_{#1,#2}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\ush}[2]{\spharm{#1}{#2}}
 \end_inset
 
 
@@ -232,6 +237,66 @@
 \end_inset
 
 
+\begin_inset FormulaMacro
+\newcommand{\transop}{S}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\vswfr}[3]{\vect{\vect v}_{#1#2#3}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\vswfs}[3]{\vect{\vect u}_{#1#2#3}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\vspharm}[3]{\vect A_{#1#2#3}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\uvec}[1]{\vect{\hat{#1}}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffs}{f}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffsi}[3]{\coeffs_{#1#2}^{#3}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffsip}[4]{\coeffs_{#1}^{#2,#3,#4}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffr}{a}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffri}[3]{\coeffr_{#1#2}^{#3}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffrip}[4]{\coeffr_{#1}^{#2,#3,#4}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\coeffripext}[4]{\coeffr_{\mathrm{ext}#1}^{#2,#3,#4}}
+\end_inset
+
+
 \end_layout
 
 \begin_layout Title
@@ -440,6 +505,16 @@ filename "intro.lyx"
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand include
+filename "finite-old.lyx"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -460,6 +535,16 @@ filename "infinite.lyx"
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand include
+filename "infinite-old.lyx"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
diff --git a/lepaper/finite-old.lyx b/lepaper/finite-old.lyx
new file mode 100644
index 0000000..84eb30e
--- /dev/null
+++ b/lepaper/finite-old.lyx
@@ -0,0 +1,378 @@
+#LyX 2.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 474
+\begin_document
+\begin_header
+\textclass article
+\use_default_options true
+\maintain_unincluded_children false
+\language finnish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman TeX Gyre Pagella
+\font_sans default
+\font_typewriter default
+\font_math auto
+\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 "Sähköpajan päiväkirja"
+\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 swedish
+\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 Subsection
+
+\lang english
+The multiple-scattering problem
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:The-multiple-scattering-problem"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+In the 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix approach, scattering properties of single nanoparticles in a homogeneous
+ medium are first computed in terms of vector sperical wavefunctions (VSWFs)—the
+ field incident onto the 
+\begin_inset Formula $n$
+\end_inset
+
+-th nanoparticle from external sources can be expanded as 
+\begin_inset Formula 
+\begin{equation}
+\vect E_{n}^{\mathrm{inc}}(\vect r)=\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{t=\mathrm{E},\mathrm{M}}\coeffrip nlmt\vswfr lmt\left(\vect r_{n}\right)\label{eq:E_inc}
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\vect r_{n}=\vect r-\vect R_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $\vect R_{n}$
+\end_inset
+
+ being the position of the centre of 
+\begin_inset Formula $n$
+\end_inset
+
+-th nanoparticle and 
+\begin_inset Formula $\vswfr lmt$
+\end_inset
+
+ are the regular VSWFs which can be expressed in terms of regular spherical
+ Bessel functions of 
+\begin_inset Formula $j_{k}\left(\left|\vect r_{n}\right|\right)$
+\end_inset
+
+ and spherical harmonics 
+\begin_inset Formula $\ush km\left(\hat{\vect r}_{n}\right)$
+\end_inset
+
+; the expressions, together with a proof that the VSWFs span all the solutions
+ of vector Helmholtz equation around the particle, justifying the expansion,
+ can be found e.g.
+ in 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "chapter 7"
+key "kristensson_scattering_2016"
+
+\end_inset
+
+ (care must be taken because of varying normalisation and phase conventions).
+ On the other hand, the field scattered by the particle can be (outside
+ the particle's circumscribing sphere) expanded in terms of singular VSWFs
+ 
+\begin_inset Formula $\vswfs lmt$
+\end_inset
+
+ which differ from the regular ones by regular spherical Bessel functions
+ being replaced with spherical Hankel functions 
+\begin_inset Formula $h_{k}^{(1)}\left(\left|\vect r_{n}\right|\right)$
+\end_inset
+
+, 
+\begin_inset Formula 
+\begin{equation}
+\vect E_{n}^{\mathrm{scat}}\left(\vect r\right)=\sum_{l,m,t}\coeffsip nlmt\vswfs lmt\left(\vect r_{n}\right).\label{eq:E_scat}
+\end{equation}
+
+\end_inset
+
+The expansion coefficients 
+\begin_inset Formula $\coeffsip nlmt$
+\end_inset
+
+, 
+\begin_inset Formula $t=\mathrm{E},\mathrm{M}$
+\end_inset
+
+ are related to the electric and magnetic multipole polarization amplitudes
+ of the nanoparticle.
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+At a given frequency, assuming the system is linear, the relation between
+ the expansion coefficients in the VSWF bases is given by the so-called
+ 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix, 
+\begin_inset Formula 
+\begin{equation}
+\coeffsip nlmt=\sum_{l',m',t'}T_{n}^{lmt;l'm't'}\coeffrip n{l'}{m'}{t'}.\label{eq:Tmatrix definition}
+\end{equation}
+
+\end_inset
+
+The 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix is given by the shape and composition of the particle and fully
+ describes its scattering properties.
+ In theory it is infinite-dimensional, but in practice (at least for subwaveleng
+th nanoparticles) its elements drop very quickly to negligible values with
+ growing degree indices 
+\begin_inset Formula $l,l'$
+\end_inset
+
+, enabling to take into account only the elements up to some finite degree,
+ 
+\begin_inset Formula $l,l'\le l_{\mathrm{max}}$
+\end_inset
+
+.
+ The 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix can be calculated numerically using various methods; here we used
+ the scuff-tmatrix tool from the SCUFF-EM suite 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "SCUFF2,reid_efficient_2015"
+
+\end_inset
+
+, which implements the boundary element method (BEM).
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+The singular VSWFs originating at 
+\begin_inset Formula $\vect R_{n}$
+\end_inset
+
+ can be then re-expanded around another origin (nanoparticle location) 
+\begin_inset Formula $\vect R_{n'}$
+\end_inset
+
+ in terms of regular VSWFs, 
+\begin_inset Formula 
+\begin{equation}
+\begin{split}\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n'}-\vect R_{n}\right)\vswfr{l'}{m'}{t'}\left(\vect r_{n'}\right),\\
+\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.
+\end{split}
+\label{eq:translation op def}
+\end{equation}
+
+\end_inset
+
+Analytical expressions for the translation operator 
+\begin_inset Formula $\transop^{lmt;l'm't'}\left(\vect R_{n'}-\vect R_{n}\right)$
+\end_inset
+
+ can be found in 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "xu_efficient_1998"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+If we write the field incident onto the 
+\begin_inset Formula $n$
+\end_inset
+
+-th nanoparticle as the sum of fields scattered from all the other nanoparticles
+ and an external field 
+\begin_inset Formula $\vect E_{0}$
+\end_inset
+
+ (which we also expand around each nanoparticle, 
+\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\vswfr lmt\left(\vect r_{n}\right)$
+\end_inset
+
+), 
+\begin_inset Formula 
+\[
+\vect E_{n}^{\mathrm{inc}}\left(\vect r\right)=\vect E_{0}\left(\vect r\right)+\sum_{n'\ne n}\vect E_{n'}^{\mathrm{scat}}\left(\vect r\right)
+\]
+
+\end_inset
+
+and use eqs.
+ (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:E_inc"
+
+\end_inset
+
+)–(
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:translation op def"
+
+\end_inset
+
+), we obtain a set of linear equations for the electromagnetic response
+ (multiple scattering) of the whole set of nanoparticles, 
+\begin_inset Formula 
+\begin{equation}
+\begin{split}\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\\
+\times\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}.
+\end{split}
+\label{eq:multiplescattering element-wise}
+\end{equation}
+
+\end_inset
+
+It is practical to get rid of the VSWF indices, rewriting (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:multiplescattering element-wise"
+
+\end_inset
+
+) in a per-particle matrix form 
+\begin_inset Formula 
+\begin{equation}
+\coeffr_{n}=\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}T_{n'}p_{n'}\label{eq:multiple scattering per particle p}
+\end{equation}
+
+\end_inset
+
+and to reformulate the problem using (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:Tmatrix definition"
+
+\end_inset
+
+) in terms of the 
+\begin_inset Formula $\coeffs$
+\end_inset
+
+-coefficients which describe the multipole excitations of the particles
+ 
+\begin_inset Formula 
+\begin{equation}
+\coeffs_{n}-T_{n}\sum_{n'\ne n}S_{n,n'}\coeffs_{n'}=T_{n}\coeffr_{\mathrm{ext}(n)}.\label{eq:multiple scattering per particle a}
+\end{equation}
+
+\end_inset
+
+Knowing 
+\begin_inset Formula $T_{n},S_{n,n'},\coeffr_{\mathrm{ext}(n)}$
+\end_inset
+
+, the nanoparticle excitations 
+\begin_inset Formula $a_{n}$
+\end_inset
+
+ can be solved by standard linear algebra methods.
+ The total scattered field anywhere outside the particles' circumscribing
+ spheres is then obtained by summing the contributions (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:E_scat"
+
+\end_inset
+
+) from all particles.
+\end_layout
+
+\end_body
+\end_document
diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx
index c03ae8f..e7f7729 100644
--- a/lepaper/finite.lyx
+++ b/lepaper/finite.lyx
@@ -166,6 +166,10 @@ ity
 
  and magnetic permeability 
 \begin_inset Formula $\mu(\vect r,\omega)=\mubg(\omega)$
+\end_inset
+
+ depending only on (angular) frequency 
+\begin_inset Formula $\omega$
 \end_inset
 
 , and that the whole system is linear, i.e.
@@ -176,7 +180,7 @@ ity
 \end_inset
 
  must satisfy the homogeneous vector Helmholtz equation 
-\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0$
+\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0$
 \end_inset
 
  
@@ -193,16 +197,44 @@ todo define
 
 \end_inset
 
- with 
-\begin_inset Formula $k=TODO$
+ with wavenumber 
+\begin_inset Formula $k=\omega\sqrt{\mu\epsilon}/c_{0}$
 \end_inset
 
- [TODO REF Jackson?].
- Its solutions (TODO under which conditions? What vector space do the SVWFs
+, and transversality condition 
+\begin_inset Formula $\nabla\cdot\vect{\Psi}\left(\vect r,\omega\right)=0$
+\end_inset
+
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "jackson_classical_1998"
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+ [TODO more specific REF Jackson?]
+\end_layout
+
+\end_inset
+
+.
+ 
+\lang english
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Its solutions (TODO under which conditions? What vector space do the SVWFs
  actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson)
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 
 \lang english
 Throughout this text, we will use the same normalisation conventions as
@@ -216,12 +248,120 @@ key "kristensson_scattering_2016"
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Subsubsection
 
 \lang english
 Spherical waves
 \end_layout
 
+\begin_layout Standard
+Inside a ball 
+\begin_inset Formula $B_{R}\left(\vect{r'}\right)\subset\medium$
+\end_inset
+
+ with radius 
+\begin_inset Formula $R$
+\end_inset
+
+ centered at 
+\begin_inset Formula $\vect{r'}$
+\end_inset
+
+, the transversal solutions of the vector Helmholtz equation can be expressed
+ in the basis of the regular transversal 
+\emph on
+vector spherical wavefunctions
+\emph default
+ (VSWFs) 
+\begin_inset Formula $\vswfr{\tau}lm\left(k\left(\vect r-\vect{r'}\right)\right)$
+\end_inset
+
+, which are found by separation of variables in spherical coordinates.
+ There is a large variety of VSWF normalisation and phase conventions in
+ the literature (and existing software), which can lead to great confusion
+ using them.
+ Throughout this text, we use the following convention, adopted from [Kristensso
+n 2014]:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{eqnarray}
+\vswfr 1lm\left(k\vect r\right) & = & j_{l}\left(kr\right)\vspharm 1lm\left(\uvec r\right),\nonumber \\
+\vswfr 2lm\left(k\vect r\right) & = & \frac{1}{kr}\frac{\ud\left(kr\, j_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vspharm 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vspharm 3lm\left(\uvec r\right),\label{eq:regular vswf}\\
+ &  & \qquad l=1,2,\dots;\, m=-l,-l+1,\dots,l;\nonumber 
+\end{eqnarray}
+
+\end_inset
+
+where we separated the position variable into its magnitude 
+\begin_inset Formula $r$
+\end_inset
+
+ and a unit vector 
+\begin_inset Formula $\uvec r$
+\end_inset
+
+, 
+\begin_inset Formula $\vect r=r\uvec r$
+\end_inset
+
+, the 
+\emph on
+vector spherical harmonics
+\emph default
+ 
+\begin_inset Formula $\vspharm{\sigma}lm$
+\end_inset
+
+ are defined as
+\begin_inset Formula 
+\begin{eqnarray}
+\vspharm 1lm\left(\uvec r\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\nabla\spharm lm\left(\uvec r\right)\times\vect r,\nonumber \\
+\vspharm 2lm\left(\uvec r\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\spharm lm\left(\uvec r\right),\label{eq:vspharm}\\
+\vspharm 2lm\left(\uvec r\right) & = & \uvec r\spharm lm\left(\uvec r\right),\nonumber 
+\end{eqnarray}
+
+\end_inset
+
+and for the scalar spherical harmonics 
+\begin_inset Formula $\spharm lm$
+\end_inset
+
+ we use the convention from [REF DLMF 14.30.1],
+\begin_inset Formula 
+\begin{equation}
+\spharm lm\left(\uvec r\right)=\spharm lm\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\mathsf{P}_{l}^{m}\left(\cos\theta\right),\label{eq:spharm}
+\end{equation}
+
+\end_inset
+
+where the Condon-Shortley phase factor 
+\begin_inset Formula $\left(-1\right)^{m}$
+\end_inset
+
+ is already included in the definition of Ferrers function 
+\begin_inset Formula $\mathsf{P}_{l}^{m}\left(\cos\theta\right)$
+\end_inset
+
+ [as in DLMF 14].
+ The main reason for this choice of VSWF 
+\emph on
+normalisation
+\emph default
+ is that it leads to simple formulae for power transport and scattering
+ cross sections without additional 
+\begin_inset Formula $l,m$
+\end_inset
+
+-dependent factors, see below.
+\end_layout
+
 \begin_layout Standard
 
 \lang english
diff --git a/lepaper/infinite-old.lyx b/lepaper/infinite-old.lyx
new file mode 100644
index 0000000..ffefe93
--- /dev/null
+++ b/lepaper/infinite-old.lyx
@@ -0,0 +1,476 @@
+#LyX 2.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 474
+\begin_document
+\begin_header
+\textclass article
+\use_default_options false
+\maintain_unincluded_children false
+\language english
+\language_package none
+\inputencoding auto
+\fontencoding default
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_math auto
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 0
+\use_package cancel 0
+\use_package esint 1
+\use_package mathdots 0
+\use_package mathtools 0
+\use_package mhchem 0
+\use_package stackrel 0
+\use_package stmaryrd 0
+\use_package undertilde 0
+\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 0
+\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 Subsection
+Periodic systems and mode analysis
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:Periodic-systems"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In an infinite periodic array of nanoparticles, the excitations of the nanoparti
+cles take the quasiperiodic Bloch-wave form 
+\begin_inset Formula 
+\[
+\coeffs_{i\nu}=e^{i\vect k\cdot\vect R_{i}}\coeffs_{\nu}
+\]
+
+\end_inset
+
+(assuming the incident external field has the same periodicity, 
+\begin_inset Formula $\coeffr_{\mathrm{ext}(i\nu)}=e^{i\vect k\cdot\vect R_{i}}p_{\mathrm{ext}\left(\nu\right)}$
+\end_inset
+
+) where 
+\begin_inset Formula $\nu$
+\end_inset
+
+ is the index of a particle inside one unit cell and 
+\begin_inset Formula $\vect R_{i},\vect R_{i'}\in\Lambda$
+\end_inset
+
+ are the lattice vectors corresponding to the sites (labeled by multiindices
+ 
+\begin_inset Formula $i,i'$
+\end_inset
+
+) of a Bravais lattice 
+\begin_inset Formula $\Lambda$
+\end_inset
+
+.
+ The multiple-scattering problem (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:multiple scattering per particle a"
+
+\end_inset
+
+) then takes the form
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\coeffs_{i\nu}-T_{\nu}\sum_{(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}\coeffs_{i\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(i\nu)}
+\]
+
+\end_inset
+
+or, labeling 
+\begin_inset Formula $W_{\nu\nu'}=\sum_{i';(i',\nu')\ne\left(i,\nu\right)}S_{i\nu,i'\nu'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}=\sum_{i';(i',\nu')\ne\left(0,\nu\right)}S_{0\nu,i'\nu'}e^{i\vect k\cdot\vect R_{i'}}$
+\end_inset
+
+ and using the quasiperiodicity, 
+\begin_inset Formula 
+\begin{equation}
+\sum_{\nu'}\left(\delta_{\nu\nu'}\mathbb{I}-T_{\nu}W_{\nu\nu'}\right)\coeffs_{\nu'}=T_{\nu}\coeffr_{\mathrm{ext}(\nu)},\label{eq:multiple scattering per particle a periodic}
+\end{equation}
+
+\end_inset
+
+which reduces the linear problem (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:multiple scattering per particle a"
+
+\end_inset
+
+) to interactions between particles inside single unit cell.
+ A problematic part is the evaluation of the translation operator lattice
+ sums 
+\begin_inset Formula $W_{\nu\nu'}$
+\end_inset
+
+; this is performed using exponentially convergent Ewald-type representations
+ 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "linton_lattice_2010"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+In an infinite periodic system, a nonlossy mode supports itself without
+ external driving, i.e.
+ such mode is described by excitation coefficients 
+\begin_inset Formula $a_{\nu}$
+\end_inset
+
+ that satisfy eq.
+ (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:multiple scattering per particle a periodic"
+
+\end_inset
+
+) with zero right-hand side.
+ That can happen if the block matrix 
+\begin_inset Formula 
+\begin{equation}
+M\left(\omega,\vect k\right)=\left\{ \delta_{\nu\nu'}\mathbb{I}-T_{\nu}\left(\omega\right)W_{\nu\nu'}\left(\omega,\vect k\right)\right\} _{\nu\nu'}\label{eq:M matrix definition}
+\end{equation}
+
+\end_inset
+
+from the left hand side of (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:multiple scattering per particle a periodic"
+
+\end_inset
+
+) is singular (here we explicitly note the 
+\begin_inset Formula $\omega,\vect k$
+\end_inset
+
+ depence).
+\end_layout
+
+\begin_layout Standard
+For lossy nanoparticles, however, perfect propagating modes will not exist
+ and 
+\begin_inset Formula $M\left(\omega,\vect k\right)$
+\end_inset
+
+ will never be perfectly singular.
+ Therefore in practice, we get the bands by scanning over 
+\begin_inset Formula $\omega,\vect k$
+\end_inset
+
+ to search for 
+\begin_inset Formula $M\left(\omega,\vect k\right)$
+\end_inset
+
+ which have an 
+\begin_inset Quotes erd
+\end_inset
+
+almost zero
+\begin_inset Quotes erd
+\end_inset
+
+ singular value.
+\end_layout
+
+\begin_layout Section
+\begin_inset ERT
+status collapsed
+
+\begin_layout Plain Layout
+
+{
+\end_layout
+
+\end_inset
+
+Symmetries
+\begin_inset ERT
+status collapsed
+
+\begin_layout Plain Layout
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset label
+LatexCommand label
+name "sm:symmetries"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+A general overview of utilizing group theory to find lattice modes at high-symme
+try points of the Brillouin zone can be found e.g.
+ in 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "chapters 10–11"
+key "dresselhaus_group_2008"
+
+\end_inset
+
+; here we use the same notation.
+\end_layout
+
+\begin_layout Standard
+We analyse the symmetries of the system in the same VSWF representation
+ as used in the 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix formalism introduced above.
+ We are interested in the modes at the 
+\begin_inset Formula $\Kp$
+\end_inset
+
+-point of the hexagonal lattice, which has the 
+\begin_inset Formula $D_{3h}$
+\end_inset
+
+ point symmetry.
+ The six irreducible representations (irreps) of the 
+\begin_inset Formula $D_{3h}$
+\end_inset
+
+ group are known and are available in the literature in their explicit forms.
+ In order to find and classify the modes, we need to find a decomposition
+ of the lattice mode representation 
+\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
+\end_inset
+
+ into the irreps of 
+\begin_inset Formula $D_{3h}$
+\end_inset
+
+.
+ The equivalence representation 
+\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
+\end_inset
+
+ is the 
+\begin_inset Formula $E'$
+\end_inset
+
+ representation as can be deduced from 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "eq. (11.19)"
+key "dresselhaus_group_2008"
+
+\end_inset
+
+, eq.
+ (11.19) and the character table for 
+\begin_inset Formula $D_{3h}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
+\end_inset
+
+ operates on a space spanned by the VSWFs around each nanoparticle in the
+ unit cell (the effects of point group operations on VSWFs are described
+ in 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "schulz_point-group_1999"
+
+\end_inset
+
+).
+ This space can be then decomposed into invariant subspaces of the 
+\begin_inset Formula $D_{3h}$
+\end_inset
+
+ using the projectors 
+\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
+\end_inset
+
+ defined by 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "eq. (4.28)"
+key "dresselhaus_group_2008"
+
+\end_inset
+
+.
+ This way, we obtain a symmetry adapted basis 
+\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
+\end_inset
+
+ as linear combinations of VSWFs 
+\begin_inset Formula $\vswfs lm{p,t}$
+\end_inset
+
+ around the constituting nanoparticles (labeled 
+\begin_inset Formula $p$
+\end_inset
+
+), 
+\begin_inset Formula 
+\[
+\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
+\]
+
+\end_inset
+
+where 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+ stands for one of the six different irreps of 
+\begin_inset Formula $D_{3h}$
+\end_inset
+
+, 
+\begin_inset Formula $r$
+\end_inset
+
+ labels the different realisations of the same irrep, and the last index
+ 
+\begin_inset Formula $i$
+\end_inset
+
+ going from 1 to 
+\begin_inset Formula $d_{\Gamma}$
+\end_inset
+
+ (the dimensionality of 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+) labels the different partners of the same given irrep.
+ The number of how many times is each irrep contained in 
+\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
+\end_inset
+
+ (i.e.
+ the range of index 
+\begin_inset Formula $r$
+\end_inset
+
+ for given 
+\begin_inset Formula $\Gamma$
+\end_inset
+
+) depends on the multipole degree cutoff 
+\begin_inset Formula $l_{\mathrm{max}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Each mode at the 
+\begin_inset Formula $\Kp$
+\end_inset
+
+-point shall lie in the irreducible spaces of only one of the six possible
+ irreps and it can be shown via 
+\begin_inset CommandInset citation
+LatexCommand cite
+after "eq. (2.51)"
+key "dresselhaus_group_2008"
+
+\end_inset
+
+ that, at the 
+\begin_inset Formula $\Kp$
+\end_inset
+
+-point, the matrix 
+\begin_inset Formula $M\left(\omega,\vect k\right)$
+\end_inset
+
+ defined above takes a block-diagonal form in the symmetry-adapted basis,
+ 
+\begin_inset Formula 
+\[
+M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
+\]
+
+\end_inset
+
+This enables us to decompose the matrix according to the irreps and to solve
+ the singular value problem in each irrep separately, as done in Fig.
+ 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "smfig:dispersions"
+
+\end_inset
+
+(a).
+\end_layout
+
+\end_body
+\end_document