386 lines
9.2 KiB
Plaintext
386 lines
9.2 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\end_header
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\begin_body
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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 "subsec: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 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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literal "true"
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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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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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literal "true"
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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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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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literal "true"
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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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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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