qpms/notes/hexlaser-tmatrixtext.lyx

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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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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{\coeffsi}[3]{a_{#1,#2}^{#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffsip}[4]{a_{#1}^{#2,#3,#4}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffri}[3]{p_{#1,#2}^{#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\coeffrip}[4]{p_{#1}^{#2,#3,#4}}
\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
\[
\vect E_{n}^{\mathrm{inc}}(\vect r)=\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{t=\mathrm{E},\mathrm{M}}\coeffrip nlmt\svwfr lmt\left(\vect r_{n}\right)
\]
\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 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
\[
\vect E_{n}^{\mathrm{scat}}=\sum_{l,m,t}\coeffsip nlmt\svwfs lmt\left(\vect r_{n}\right).
\]
\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
\[
\coeffsip nlmt=\sum_{l,m,t}T_{n}^{l,m,t;l',m',t'}\coeffrip n{l'}{m'}{t'}.
\]
\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
.
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