From 6a07f6a212555d7a896ec0d55bd303a8258187af Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Tue, 25 Sep 2018 14:01:46 +0300 Subject: [PATCH] More or less finished T-matrix description for hexlaser peper Former-commit-id: c8e05c42d157909658061fbf6afce23d33be88ac --- notes/hexlaser-tmatrixtext.lyx | 201 ++++++++++++++++++++++++++++++++- 1 file changed, 200 insertions(+), 1 deletion(-) diff --git a/notes/hexlaser-tmatrixtext.lyx b/notes/hexlaser-tmatrixtext.lyx index 828e9ed..6c13282 100644 --- a/notes/hexlaser-tmatrixtext.lyx +++ b/notes/hexlaser-tmatrixtext.lyx @@ -605,7 +605,7 @@ reference "eq:Tmatrix definition" -coefficients which describe the multipole excitations of the particles \begin_inset Formula \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_inset @@ -641,6 +641,205 @@ name "sub:Periodic-systems" \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_inset CommandInset bibtex LatexCommand bibtex