From 6092449db4c5623a11b47850e21acd6fd026d8a9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Tue, 16 Jun 2020 21:53:54 +0300 Subject: [PATCH] WIP Former-commit-id: 3ab2f0803f9399fe32c41325b9404f8e0b1ba18c --- lepaper/arrayscat.lyx | 38 +++++++++++++++++++------------- lepaper/infinite.lyx | 51 ++++++++++++++++++++++++++++++++++--------- 2 files changed, 64 insertions(+), 25 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 46b563c..40f5314 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -491,21 +491,12 @@ These are compatibility macros for the (...)-old files: \end_layout \begin_layout Title +Many-particle \begin_inset Formula $T$ \end_inset --matrix simulations in finite and infinite systems of electromagnetic scatterers - -\begin_inset Marginal -status open - -\begin_layout Plain Layout -(TODO better title) -\end_layout - -\end_inset - - +-matrix simulations for nanophotonics: symmetries, scattering and lattice + modes \end_layout \begin_layout Standard @@ -521,7 +512,7 @@ Multiple-scattering \end_layout \begin_layout Itemize -Multiple-scattering +Many-particle \begin_inset Formula $T$ \end_inset @@ -529,6 +520,13 @@ Multiple-scattering modes. \end_layout +\begin_layout Itemize +\begin_inset Formula $T$ +\end_inset + +-matrix simulations in finite and infinite systems of electromagnetic scatterers +\end_layout + \begin_layout Standard \begin_inset Note Note status open @@ -617,8 +615,13 @@ The T-matrix multiple scattering method (TMMSM) can be used to solve the \begin_layout Abstract Here we extend the method to infinite periodic structures using Ewald-type lattice summation, and we exploit the possible symmetries of the structure - to further improve its efficiency. + to further improve its efficiency, so that systems containing tens of thousands + of particles can be studied with relative ease. +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset Marginal status open @@ -629,6 +632,11 @@ Should I mention also the cross sections formulae in abstract / intro? \end_inset +\end_layout + +\end_inset + + \end_layout \begin_layout Abstract @@ -862,7 +870,7 @@ Given up \end_layout -\begin_layout Itemize +\begin_layout Standard \begin_inset Note Note status open diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 8d3ac86..f197291 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -370,7 +370,13 @@ noprefix "false" describes the \emph on -lattice modes. +lattice modes +\emph default +, i.e. + electromagnetic excitations that can sustain themselves for prolonged time + even without external driving +\emph on +. \emph default Non-trivial solutions to @@ -1700,7 +1706,32 @@ One pecularity of the two-dimensional case is the two-branchedness of the \begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$ \end_inset - appearing in the long-range part. + appearing in the long-range part (in the cases +\begin_inset Formula $d=1,3$ +\end_inset + + the function +\begin_inset Formula $\gamma\left(z\right)$ +\end_inset + + appears with even powers, and +\begin_inset Formula $\Gamma\left(-j,z\right)$ +\end_inset + + is meromorphic for integer +\begin_inset Formula $j$ +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +after "8.2.9" +key "NIST:DLMF" +literal "false" + +\end_inset + +). As a consequence, if we now explicitly label the dependence on the wavenumber, \begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\kappa,\vect k,\vect s\right)$ @@ -2213,8 +2244,8 @@ noprefix "false" translation operator: \begin_inset Formula \begin{align} -\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\nonumber \\ -\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves} +\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm21{m'}\left(0\right),\nonumber \\ +\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves} \end{align} \end_inset @@ -2241,7 +2272,7 @@ noprefix "false" \end_inset and the fact that all the other regular VSWFs except for -\begin_inset Formula $\vswfrtlm 21{m'}$ +\begin_inset Formula $\vswfrtlm21{m'}$ \end_inset vanish at origin. @@ -2314,10 +2345,10 @@ status open \begin_inset Formula \begin{align*} \vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\ - & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\ - & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right),\text{FIXME signs}\\ - & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\ - & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right) + & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm21{m'}\left(0\right)\\ + & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm21{m'}\left(0\right),\text{FIXME signs}\\ + & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm21{m'}\left(0\right)\\ + & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right) \end{align*} \end_inset @@ -2335,7 +2366,7 @@ TODO fix signs and exponential phase factors \begin_inset Formula \begin{align*} \vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\ - & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right). + & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right). \end{align*} \end_inset