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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\begin_body

\begin_layout Subsection
The multiple-scattering problem
\begin_inset CommandInset label
LatexCommand label
name "subsec:The-multiple-scattering-problem"

\end_inset


\end_layout

\begin_layout Standard
In the 
\begin_inset Formula $T$
\end_inset

-matrix approach, scattering properties of single nanoparticles in a homogeneous
 medium are first computed in terms of vector sperical wavefunctions (VSWFs)—the
 field incident onto the 
\begin_inset Formula $n$
\end_inset

-th nanoparticle from external sources can be expanded as 
\begin_inset Formula 
\begin{equation}
\vect E_{n}^{\mathrm{inc}}(\vect r)=\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{t=\mathrm{E},\mathrm{M}}\coeffrip nlmt\vswfr lmt\left(\vect r_{n}\right)\label{eq:E_inc}
\end{equation}

\end_inset

where 
\begin_inset Formula $\vect r_{n}=\vect r-\vect R_{n}$
\end_inset

, 
\begin_inset Formula $\vect R_{n}$
\end_inset

 being the position of the centre of 
\begin_inset Formula $n$
\end_inset

-th nanoparticle and 
\begin_inset Formula $\vswfr lmt$
\end_inset

 are the regular VSWFs which can be expressed in terms of regular spherical
 Bessel functions of 
\begin_inset Formula $j_{k}\left(\left|\vect r_{n}\right|\right)$
\end_inset

 and spherical harmonics 
\begin_inset Formula $\ush km\left(\hat{\vect r}_{n}\right)$
\end_inset

; the expressions, together with a proof that the VSWFs span all the solutions
 of vector Helmholtz equation around the particle, justifying the expansion,
 can be found e.g.
 in 
\begin_inset CommandInset citation
LatexCommand cite
after "chapter 7"
key "kristensson_scattering_2016"
literal "true"

\end_inset

 (care must be taken because of varying normalisation and phase conventions).
 On the other hand, the field scattered by the particle can be (outside
 the particle's circumscribing sphere) expanded in terms of singular VSWFs
 
\begin_inset Formula $\vswfs lmt$
\end_inset

 which differ from the regular ones by regular spherical Bessel functions
 being replaced with spherical Hankel functions 
\begin_inset Formula $h_{k}^{(1)}\left(\left|\vect r_{n}\right|\right)$
\end_inset

, 
\begin_inset Formula 
\begin{equation}
\vect E_{n}^{\mathrm{scat}}\left(\vect r\right)=\sum_{l,m,t}\coeffsip nlmt\vswfs lmt\left(\vect r_{n}\right).\label{eq:E_scat}
\end{equation}

\end_inset

The expansion coefficients 
\begin_inset Formula $\coeffsip nlmt$
\end_inset

, 
\begin_inset Formula $t=\mathrm{E},\mathrm{M}$
\end_inset

 are related to the electric and magnetic multipole polarization amplitudes
 of the nanoparticle.
\end_layout

\begin_layout Standard
At a given frequency, assuming the system is linear, the relation between
 the expansion coefficients in the VSWF bases is given by the so-called
 
\begin_inset Formula $T$
\end_inset

-matrix, 
\begin_inset Formula 
\begin{equation}
\coeffsip nlmt=\sum_{l',m',t'}T_{n}^{lmt;l'm't'}\coeffrip n{l'}{m'}{t'}.\label{eq:Tmatrix definition}
\end{equation}

\end_inset

The 
\begin_inset Formula $T$
\end_inset

-matrix is given by the shape and composition of the particle and fully
 describes its scattering properties.
 In theory it is infinite-dimensional, but in practice (at least for subwaveleng
th nanoparticles) its elements drop very quickly to negligible values with
 growing degree indices 
\begin_inset Formula $l,l'$
\end_inset

, enabling to take into account only the elements up to some finite degree,
 
\begin_inset Formula $l,l'\le l_{\mathrm{max}}$
\end_inset

.
 The 
\begin_inset Formula $T$
\end_inset

-matrix can be calculated numerically using various methods; here we used
 the scuff-tmatrix tool from the SCUFF-EM suite 
\begin_inset CommandInset citation
LatexCommand cite
key "SCUFF2,reid_efficient_2015"
literal "true"

\end_inset

, which implements the boundary element method (BEM).
\end_layout

\begin_layout Standard
The singular VSWFs originating at 
\begin_inset Formula $\vect R_{n}$
\end_inset

 can be then re-expanded around another origin (nanoparticle location) 
\begin_inset Formula $\vect R_{n'}$
\end_inset

 in terms of regular VSWFs, 
\begin_inset Formula 
\begin{equation}
\begin{split}\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}\transop^{l'm't';lmt}\left(\vect R_{n'}-\vect R_{n}\right)\vswfr{l'}{m'}{t'}\left(\vect r_{n'}\right),\\
\left|\vect r_{n'}\right|<\left|\vect R_{n'}-\vect R_{n}\right|.
\end{split}
\label{eq:translation op def}
\end{equation}

\end_inset

Analytical expressions for the translation operator 
\begin_inset Formula $\transop^{lmt;l'm't'}\left(\vect R_{n'}-\vect R_{n}\right)$
\end_inset

 can be found in 
\begin_inset CommandInset citation
LatexCommand cite
key "xu_efficient_1998"
literal "true"

\end_inset

.
\end_layout

\begin_layout Standard
If we write the field incident onto the 
\begin_inset Formula $n$
\end_inset

-th nanoparticle as the sum of fields scattered from all the other nanoparticles
 and an external field 
\begin_inset Formula $\vect E_{0}$
\end_inset

 (which we also expand around each nanoparticle, 
\begin_inset Formula $\vect E_{0}\left(\vect r\right)=\sum_{l,m,t}\coeffripext nlmt\vswfr lmt\left(\vect r_{n}\right)$
\end_inset

), 
\begin_inset Formula 
\[
\vect E_{n}^{\mathrm{inc}}\left(\vect r\right)=\vect E_{0}\left(\vect r\right)+\sum_{n'\ne n}\vect E_{n'}^{\mathrm{scat}}\left(\vect r\right)
\]

\end_inset

and use eqs.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:E_inc"

\end_inset

)–(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:translation op def"

\end_inset

), we obtain a set of linear equations for the electromagnetic response
 (multiple scattering) of the whole set of nanoparticles, 
\begin_inset Formula 
\begin{equation}
\begin{split}\coeffrip nlmt=\coeffripext nlmt+\sum_{n'\ne n}\sum_{l',m',t'}\transop^{lmt;l'm't'}\left(\vect R_{n}-\vect R_{n'}\right)\\
\times\sum_{l'',m'',t''}T_{n'}^{l'm't';l''m''t''}\coeffrip{n'}{l''}{m''}{t''}.
\end{split}
\label{eq:multiplescattering element-wise}
\end{equation}

\end_inset

It is practical to get rid of the VSWF indices, rewriting (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multiplescattering element-wise"

\end_inset

) in a per-particle matrix form 
\begin_inset Formula 
\begin{equation}
\coeffr_{n}=\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}T_{n'}p_{n'}\label{eq:multiple scattering per particle p}
\end{equation}

\end_inset

and to reformulate the problem using (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Tmatrix definition"

\end_inset

) in terms of the 
\begin_inset Formula $\coeffs$
\end_inset

-coefficients which describe the multipole excitations of the particles
 
\begin_inset Formula 
\begin{equation}
\coeffs_{n}-T_{n}\sum_{n'\ne n}S_{n,n'}\coeffs_{n'}=T_{n}\coeffr_{\mathrm{ext}(n)}.\label{eq:multiple scattering per particle a}
\end{equation}

\end_inset

Knowing 
\begin_inset Formula $T_{n},S_{n,n'},\coeffr_{\mathrm{ext}(n)}$
\end_inset

, the nanoparticle excitations 
\begin_inset Formula $a_{n}$
\end_inset

 can be solved by standard linear algebra methods.
 The total scattered field anywhere outside the particles' circumscribing
 spheres is then obtained by summing the contributions (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:E_scat"

\end_inset

) from all particles.
\end_layout

\end_body
\end_document
