More or less finished T-matrix description for hexlaser peper

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Marek Nečada 2018-09-25 14:01:46 +03:00
parent 33a603db38
commit 6a07f6a212
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@ -605,7 +605,7 @@ reference "eq:Tmatrix definition"
-coefficients which describe the multipole excitations of the particles -coefficients which describe the multipole excitations of the particles
\begin_inset Formula \begin_inset Formula
\begin{equation} \begin{equation}
\coeffs_{n}=T_{n}\left(\coeffr_{\mathrm{ext}(n)}+\sum_{n'\ne n}S_{n,n'}\coeffs_{n'}\right).\label{eq:multiple scattering per particle a} \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{equation}
\end_inset \end_inset
@ -641,6 +641,205 @@ name "sub:Periodic-systems"
\end_layout \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_layout Standard
\begin_inset CommandInset bibtex \begin_inset CommandInset bibtex
LatexCommand bibtex LatexCommand bibtex