From d5b6f8f5d426f26de2e3eeb70b1f3b1cafb69b2d Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Wed, 31 Jul 2019 06:40:25 +0300
Subject: [PATCH] Infinite lattices WIP

Former-commit-id: 39facbe563492f4fce54104fd4cf16d9cf1b951b
---
 lepaper/infinite.lyx | 33 ++++++++++++++++++++++++++++++---
 1 file changed, 30 insertions(+), 3 deletions(-)

diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx
index df709e3..141050b 100644
--- a/lepaper/infinite.lyx
+++ b/lepaper/infinite.lyx
@@ -320,6 +320,33 @@ noprefix "false"
 .
 \end_layout
 
+\begin_layout Standard
+As in the case of a finite system, eq.
+ can be written in a shorter block-matrix form,
+\begin_inset Formula 
+\[
+\left(I-WT\right)\outcoeffp{\vect 0}\left(\vect k\right)=\rcoeffincp{\vect 0}\left(\vect k\right)
+\]
+
+\end_inset
+
+ Eq.
+ 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:Multiple-scattering problem unit cell"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ can be used to calculate electromagnetic response of the structure to external
+ quasiperiodic driving field – most notably a plane wave.
+ However, if one sets the right the right-hand side to zero, it can also
+ be used to find electromagnetic lattice modes
+\end_layout
+
 \begin_layout Subsection
 Computing the Fourier sum of the translation operator
 \begin_inset CommandInset label
@@ -590,8 +617,8 @@ reference "eq:W sum in reciprocal space"
 \begin_inset Formula 
 \begin{eqnarray}
 W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
-W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
-W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
+W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
 \end{eqnarray}
 
 \end_inset
@@ -629,7 +656,7 @@ CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
 
 \begin_inset Formula 
 \begin{equation}
-\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
+\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums}
 \end{equation}
 
 \end_inset