856 lines
19 KiB
Plaintext
856 lines
19 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\begin_layout Standard
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ush}[2]{Y_{#1,#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svwfr}[3]{\mathbf{u}_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svwfs}[3]{\mathbf{v}_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffs}{a}
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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}{p}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffri}[3]{p_{#1,#2}^{#3}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffrip}[4]{p_{#1}^{#2,#3,#4}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\coeffripext}[4]{p_{\mathrm{ext}(#1)}^{#2,#3,#4}}
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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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\end_layout
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\begin_layout Section
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\lang english
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:T-matrix-simulations"
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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 order to get more detailed insight into the mode structure of the lattice
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around the lasing
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\begin_inset Formula $\Kp$
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\end_inset
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-point – most importantly, how much do the mode frequencies at the
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\begin_inset Formula $\Kp$
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\end_inset
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-points differ from the empty lattice model – we performed multiple-scattering
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\begin_inset Formula $T$
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\end_inset
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-matrix simulations
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\begin_inset CommandInset citation
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LatexCommand cite
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key "mackowski_analysis_1991"
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\end_inset
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for an infinite lattice based on our systems' geometry.
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We give a brief overview of this method in the subsections
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sub:The-multiple-scattering-problem"
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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 "sub:Periodic-systems"
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\end_inset
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below.
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\lang finnish
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The top advantage of the multiple-scattering
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\begin_inset Formula $T$
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\end_inset
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-matrix approach is its computational efficiency for large finite systems
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of nanoparticles.
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In the lattice mode analysis in this work, however, we use it here for
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another reason, specifically the relative ease of describing symmetries
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\begin_inset CommandInset citation
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LatexCommand cite
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key "schulz_point-group_1999"
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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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The
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\begin_inset Formula $T$
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\end_inset
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-matrix of a single nanoparticle was computed using the scuff-tmatrix applicatio
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n from the SCUFF-EM suite~
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\lang finnish
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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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\lang english
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and the system was solved up to the
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\begin_inset Formula $l_{\mathrm{max}}=3$
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\end_inset
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(octupolar) degree of electric and magnetic spherical multipole.
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\end_layout
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\begin_layout Standard
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\lang english
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We did not find any deviation from the empty lattice diffracted orders exceeding
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the numerical precision of the computation (about 2 meV).
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This is most likely due to the frequencies in our experiment being far
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below the resonances of the nanoparticles, with the largest elements of
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the
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\begin_inset Formula $T$
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\end_inset
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-matrix being of the order of
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\begin_inset Formula $10^{-3}$
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\end_inset
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(for power-normalised waves).
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The nanoparticles are therefore almost transparent, but still suffice to
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provide feedback for lasing.
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\end_layout
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\begin_layout Subsection
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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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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 are first
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computed in terms of vector sperical wavefunctions (VSWFs)—the field incident
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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\svwfr 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 $\svwfr lmt$
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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 can be found e.g.
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in [REF]
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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few words about different conventions?
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\end_layout
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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 $\svwfs lmt$
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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\svwfs 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 polarisation amplitudes
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of the nanoparticle.
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\end_layout
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||
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\begin_layout Standard
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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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.
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\end_layout
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\begin_layout Standard
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The singular SVWFs 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 SVWFs,
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\begin_inset Formula
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\begin{equation}
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\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n'}-\vect R_{n}\right)\svwfr{l'}{m'}{t'}\left(\vect r_{n'}\right),\qquad\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.\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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||
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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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||
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\begin_layout Standard
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||
If we write the field incident onto
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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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,
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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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||
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\end_inset
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||
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and use eqs.
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||
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\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:E_inc"
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||
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\end_inset
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||
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–
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||
\begin_inset CommandInset ref
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||
LatexCommand eqref
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||
reference "eq:translation op def"
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||
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||
\end_inset
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||
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, we obtain a set of linear equations for the electromagnetic response (multiple
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scattering) of the whole set of nanoparticles,
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||
\end_layout
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||
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||
\begin_layout Standard
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||
\begin_inset Note Note
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||
status open
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||
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||
\begin_layout Plain Layout
|
||
\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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||
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||
\end_inset
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||
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||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
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||
\[
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||
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\coeffsip{n'}lmt\svwfs lmt\left(\vect r_{n'}\right)
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||
\]
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||
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||
\end_inset
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||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
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||
\[
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||
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\coeffsip{n'}lmt\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n}-\vect R_{n'}\right)\svwfr{l'}{m'}{t'}\left(\vect r_{n}\right)
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||
\]
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||
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||
\end_inset
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||
|
||
|
||
\begin_inset Formula
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||
\[
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||
\sum_{l,m,t}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right)+\sum_{n'\ne n}\sum_{l,m,t}\sum_{l',m',t'}\coeffsip{n'}{l'}{m'}{t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\svwfr lmt\left(\vect r_{n}\right)
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||
\]
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||
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||
\end_inset
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||
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||
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||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
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||
\[
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||
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\coeffsip{n'}{l'}{m'}{t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)
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||
\]
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||
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||
\end_inset
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||
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||
(
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||
\begin_inset Formula $\coeffsip{n'}{l'}{m'}{t'}=\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}$
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||
\end_inset
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||
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||
)
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||
\begin_inset Formula
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||
\[
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||
\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}
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||
\]
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||
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||
\end_inset
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||
|
||
|
||
\end_layout
|
||
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||
\end_inset
|
||
|
||
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||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
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||
\begin{equation}
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||
\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)\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''},\label{eq:multiplescattering element-wise}
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||
\end{equation}
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||
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||
\end_inset
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||
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||
where
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||
\begin_inset Formula $\coeffripext nlmt$
|
||
\end_inset
|
||
|
||
are the expansion coefficients of the external field around the
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
-th particle,
|
||
\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\svwfr lmt\left(\vect r_{n}\right).$
|
||
\end_inset
|
||
|
||
It is practical to get rid of the SVWF indices, rewriting
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 eqref
|
||
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 eqref
|
||
reference "eq:E_scat"
|
||
|
||
\end_inset
|
||
|
||
from all particles.
|
||
\end_layout
|
||
|
||
\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
|
||
\[
|
||
\coeffs_{i\alpha}=e^{i\vect k\cdot\vect R_{i}}\coeffs_{\alpha}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
(assuming the incident external field has the same periodicity,
|
||
\begin_inset Formula $\coeffr_{\mathrm{ext}(i\alpha)}=e^{i\vect k\cdot\vect R_{i}}p_{\mathrm{ext}\left(\alpha\right)}$
|
||
\end_inset
|
||
|
||
) where
|
||
\begin_inset Formula $\alpha$
|
||
\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 eqref
|
||
reference "eq:multiple scattering per particle a"
|
||
|
||
\end_inset
|
||
|
||
then takes the form
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\[
|
||
\coeffs_{i\alpha}=T_{\alpha}\left(\coeffr_{\mathrm{ext}(i\alpha)}+\sum_{(i',\alpha')\ne\left(i,\alpha\right)}S_{i\alpha,i'\alpha}\coeffs_{i'\alpha'}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\[
|
||
\coeffs_{i\alpha}-T_{\alpha}\sum_{(i',\alpha')\ne\left(i,\alpha\right)}S_{i\alpha,i'\alpha'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}\coeffs_{i\alpha'}=T_{\alpha}\coeffr_{\mathrm{ext}(i\alpha)}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
or, labeling
|
||
\begin_inset Formula $W_{\alpha\alpha'}=\sum_{i';(i',\alpha')\ne\left(i,\alpha\right)}S_{i\alpha,i'\alpha'}e^{i\vect k\cdot\left(\vect R_{i'}-\vect R_{i}\right)}=\sum_{i';(i',\alpha')\ne\left(0,\alpha\right)}S_{0\alpha,i'\alpha'}e^{i\vect k\cdot\vect R_{i'}}$
|
||
\end_inset
|
||
|
||
and using the quasiperiodicity,
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\sum_{\alpha'}\left(\delta_{\alpha\alpha'}\mathbb{I}-T_{\alpha}W_{\alpha\alpha'}\right)\coeffs_{\alpha'}=T_{\alpha}\coeffr_{\mathrm{ext}(\alpha)},\label{eq:multiple scattering per particle a periodic}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\coeffs_{\alpha}-T_{\alpha}\sum_{\alpha'}W_{\alpha\alpha'}\coeffs_{\alpha'}=T_{\alpha}\coeffr_{\mathrm{ext}(\alpha)},\label{eq:multiple scattering per particle a periodic-2}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
which reduces the linear problem
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
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 $W_{\alpha\alpha'}$
|
||
\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 $a_{\alpha}$
|
||
\end_inset
|
||
|
||
that satisfy eq.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:multiple scattering per particle a periodic-2"
|
||
|
||
\end_inset
|
||
|
||
with zero right-hand side.
|
||
That can happen if the block matrix
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)=\left\{ \delta_{\alpha\alpha'}\mathbb{I}-T_{\alpha}\left(\vect{\omega}\right)W_{\alpha\alpha'}\left(\omega,\vect k\right)\right\} _{\alpha\alpha'}$
|
||
\end_inset
|
||
|
||
from the left hand side of
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:multiple scattering per particle a periodic"
|
||
|
||
\end_inset
|
||
|
||
is singular (here we explicitely note the
|
||
\begin_inset Formula $\omega,\vect k$
|
||
\end_inset
|
||
|
||
depence).
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
In other words, the energy bands of the lattice are given by
|
||
\begin_inset Formula
|
||
\[
|
||
\det M\left(\omega,\vect k\right)=0.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\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 sld
|
||
\end_inset
|
||
|
||
almost zero
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
singular value.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
bibfiles "hexarray-theory"
|
||
options "plain"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|