WIP finite systems + copypasta from hexlaser SM
Former-commit-id: 7fa2ae8308b4aa62fdb0986ac0ed669fad29d2a1
This commit is contained in:
Marek Nečada
parent
e30eb45a36
commit
74655c0210
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@ -41,11 +41,11 @@
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\papersize default
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\use_geometry false
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\use_package amsmath 2
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\use_package amssymb 1
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\use_package amssymb 2
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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 mathtools 2
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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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@ -158,7 +158,12 @@
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{Y_{#1,#2}}
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\newcommand{\spharm}[2]{Y_{#1,#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{\spharm{#1}{#2}}
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\end_inset
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@ -232,6 +237,66 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\transop}{S}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vswfr}[3]{\vect{\vect v}_{#1#2#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vswfs}[3]{\vect{\vect u}_{#1#2#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vspharm}[3]{\vect A_{#1#2#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\uvec}[1]{\vect{\hat{#1}}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffs}{f}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffsi}[3]{\coeffs_{#1#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffsip}[4]{\coeffs_{#1}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffr}{a}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffri}[3]{\coeffr_{#1#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffrip}[4]{\coeffr_{#1}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffripext}[4]{\coeffr_{\mathrm{ext}#1}^{#2,#3,#4}}
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\end_inset
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\end_layout
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\begin_layout Title
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@ -440,6 +505,16 @@ filename "intro.lyx"
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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filename "finite-old.lyx"
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -460,6 +535,16 @@ filename "infinite.lyx"
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand include
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filename "infinite-old.lyx"
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -0,0 +1,378 @@
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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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\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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\cite_engine basic
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\justification true
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\use_refstyle 1
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\index Index
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\shortcut idx
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\color #008000
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\secnumdepth 3
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\end_header
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\begin_body
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\begin_layout Subsection
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\lang english
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The multiple-scattering problem
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\begin_inset CommandInset label
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LatexCommand label
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name "sub:The-multiple-scattering-problem"
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\end_inset
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\end_layout
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\begin_layout Standard
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\lang english
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In the
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\begin_inset Formula $T$
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\end_inset
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-matrix approach, scattering properties of single nanoparticles in a homogeneous
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medium are first computed in terms of vector sperical wavefunctions (VSWFs)—the
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field incident onto the
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\begin_inset Formula $n$
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\end_inset
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-th nanoparticle from external sources can be expanded as
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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where
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\begin_inset Formula $\vect r_{n}=\vect r-\vect R_{n}$
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\end_inset
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,
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\begin_inset Formula $\vect R_{n}$
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\end_inset
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being the position of the centre of
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\begin_inset Formula $n$
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\end_inset
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-th nanoparticle and
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\begin_inset Formula $\vswfr lmt$
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\end_inset
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are the regular VSWFs which can be expressed in terms of regular spherical
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Bessel functions of
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\begin_inset Formula $j_{k}\left(\left|\vect r_{n}\right|\right)$
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\end_inset
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and spherical harmonics
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\begin_inset Formula $\ush km\left(\hat{\vect r}_{n}\right)$
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\end_inset
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; the expressions, together with a proof that the VSWFs span all the solutions
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of vector Helmholtz equation around the particle, justifying the expansion,
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can be found e.g.
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in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "chapter 7"
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key "kristensson_scattering_2016"
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\end_inset
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(care must be taken because of varying normalisation and phase conventions).
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On the other hand, the field scattered by the particle can be (outside
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the particle's circumscribing sphere) expanded in terms of singular VSWFs
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\begin_inset Formula $\vswfs lmt$
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\end_inset
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which differ from the regular ones by regular spherical Bessel functions
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being replaced with spherical Hankel functions
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\begin_inset Formula $h_{k}^{(1)}\left(\left|\vect r_{n}\right|\right)$
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\end_inset
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,
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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The expansion coefficients
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\begin_inset Formula $\coeffsip nlmt$
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\end_inset
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,
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\begin_inset Formula $t=\mathrm{E},\mathrm{M}$
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\end_inset
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are related to the electric and magnetic multipole polarization amplitudes
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of the nanoparticle.
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\end_layout
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\begin_layout Standard
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\lang english
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At a given frequency, assuming the system is linear, the relation between
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the expansion coefficients in the VSWF bases is given by the so-called
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\begin_inset Formula $T$
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\end_inset
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-matrix,
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\begin_inset Formula
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\begin{equation}
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\coeffsip nlmt=\sum_{l',m',t'}T_{n}^{lmt;l'm't'}\coeffrip n{l'}{m'}{t'}.\label{eq:Tmatrix definition}
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\end{equation}
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\end_inset
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The
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\begin_inset Formula $T$
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\end_inset
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-matrix is given by the shape and composition of the particle and fully
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describes its scattering properties.
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In theory it is infinite-dimensional, but in practice (at least for subwaveleng
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th nanoparticles) its elements drop very quickly to negligible values with
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growing degree indices
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\begin_inset Formula $l,l'$
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\end_inset
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, enabling to take into account only the elements up to some finite degree,
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\begin_inset Formula $l,l'\le l_{\mathrm{max}}$
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\end_inset
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.
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The
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\begin_inset Formula $T$
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\end_inset
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-matrix can be calculated numerically using various methods; here we used
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the scuff-tmatrix tool from the SCUFF-EM suite
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\begin_inset CommandInset citation
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LatexCommand cite
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key "SCUFF2,reid_efficient_2015"
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\end_inset
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, which implements the boundary element method (BEM).
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\end_layout
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\begin_layout Standard
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\lang english
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The singular VSWFs originating at
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\begin_inset Formula $\vect R_{n}$
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\end_inset
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can be then re-expanded around another origin (nanoparticle location)
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\begin_inset Formula $\vect R_{n'}$
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\end_inset
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in terms of regular VSWFs,
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\begin_inset Formula
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\begin{equation}
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\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),\\
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\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.
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\end{split}
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\label{eq:translation op def}
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\end{equation}
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\end_inset
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Analytical expressions for the translation operator
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\begin_inset Formula $\transop^{lmt;l'm't'}\left(\vect R_{n'}-\vect R_{n}\right)$
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\end_inset
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can be found in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "xu_efficient_1998"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\lang english
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If we write the field incident onto the
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\begin_inset Formula $n$
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\end_inset
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-th nanoparticle as the sum of fields scattered from all the other nanoparticles
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and an external field
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\begin_inset Formula $\vect E_{0}$
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\end_inset
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(which we also expand around each nanoparticle,
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\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\vswfr lmt\left(\vect r_{n}\right)$
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\end_inset
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),
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\begin_inset Formula
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\[
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\vect E_{n}^{\mathrm{inc}}\left(\vect r\right)=\vect E_{0}\left(\vect r\right)+\sum_{n'\ne n}\vect E_{n'}^{\mathrm{scat}}\left(\vect r\right)
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\]
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\end_inset
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and use eqs.
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(
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:E_inc"
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\end_inset
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)–(
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:translation op def"
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\end_inset
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), we obtain a set of linear equations for the electromagnetic response
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(multiple scattering) of the whole set of nanoparticles,
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\begin_inset Formula
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\begin{equation}
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\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)\\
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\times\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}.
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\end{split}
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\label{eq:multiplescattering element-wise}
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\end{equation}
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\end_inset
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It is practical to get rid of the VSWF indices, rewriting (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:multiplescattering element-wise"
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\end_inset
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) in a per-particle matrix form
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\begin_inset Formula
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\begin{equation}
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\coeffr_{n}=\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}T_{n'}p_{n'}\label{eq:multiple scattering per particle p}
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\end{equation}
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\end_inset
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and to reformulate the problem using (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Tmatrix definition"
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\end_inset
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) in terms of the
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\begin_inset Formula $\coeffs$
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\end_inset
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-coefficients which describe the multipole excitations of the particles
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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Knowing
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\begin_inset Formula $T_{n},S_{n,n'},\coeffr_{\mathrm{ext}(n)}$
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\end_inset
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, the nanoparticle excitations
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\begin_inset Formula $a_{n}$
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\end_inset
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can be solved by standard linear algebra methods.
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The total scattered field anywhere outside the particles' circumscribing
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spheres is then obtained by summing the contributions (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:E_scat"
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\end_inset
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) from all particles.
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\end_layout
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\end_body
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\end_document
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@ -166,6 +166,10 @@ ity
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and magnetic permeability
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\begin_inset Formula $\mu(\vect r,\omega)=\mubg(\omega)$
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\end_inset
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depending only on (angular) frequency
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\begin_inset Formula $\omega$
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\end_inset
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, and that the whole system is linear, i.e.
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@ -176,7 +180,7 @@ ity
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\end_inset
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must satisfy the homogeneous vector Helmholtz equation
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\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0$
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\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0$
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\end_inset
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@ -193,16 +197,44 @@ todo define
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\end_inset
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||||
|
||||
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
|
||||
|
|
|
|||
|
|
@ -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
|
||||
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|
||||
\font_sans default
|
||||
\font_typewriter default
|
||||
\font_math auto
|
||||
\font_default_family default
|
||||
\use_non_tex_fonts false
|
||||
\font_sc false
|
||||
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|
||||
\font_sf_scale 100
|
||||
\font_tt_scale 100
|
||||
\graphics default
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\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
|
||||
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|
||||
\cite_engine basic
|
||||
\cite_engine_type default
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\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
|
||||
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|
||||
\papercolumns 1
|
||||
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||||
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|
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\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
|
||||
Loading…
Reference in New Issue