qpms/notes/hexlaser-tmatrixtext.lyx

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

\begin_layout Standard

\lang english
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ush}[2]{Y_{#1,#2}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\svwfr}[3]{\mathbf{u}_{#1,#2}^{#3}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\svwfs}[3]{\mathbf{v}_{#1,#2}^{#3}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\coeffs}{a}
\end_inset


\begin_inset FormulaMacro
\newcommand{\coeffsi}[3]{\coeffs_{#1,#2}^{#3}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\coeffsip}[4]{\coeffs_{#1}^{#2,#3,#4}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\coeffr}{p}
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\begin_inset FormulaMacro
\newcommand{\coeffri}[3]{p_{#1,#2}^{#3}}
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\begin_inset FormulaMacro
\newcommand{\coeffrip}[4]{p_{#1}^{#2,#3,#4}}
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\begin_inset FormulaMacro
\newcommand{\transop}{S}
\end_inset


\end_layout

\begin_layout Standard
In this approach, scattering properties of single nanoparticles 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\svwfr 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 $\svwfr lmt$
\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 can be found e.g.
 in REF 
\begin_inset Note Note
status open

\begin_layout Plain Layout
few words about different conventions?
\end_layout

\end_inset

.
 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 $\svwfs lmt$
\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}}=\sum_{l,m,t}\coeffsip nlmt\svwfs 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 polarisation 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}^{l,m,t;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 [REF].
\end_layout

\begin_layout Standard
The singular SVWFs 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 SVWFs,
\begin_inset Formula 
\[
\svwfs lmt\left(\vect r_{n}\right)=\sum_{l',m',t'}
\]

\end_inset


\end_layout

\end_body
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