Paper intro

Former-commit-id: ccbb88524c7561b087d89267c6cd09bc0daa3f08

lepaper/arrayscat.lyx

@@ -470,6 +470,16 @@ filename "examples.lyx"
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand include
+filename "comparison.lyx"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard

lepaper/comparison.lyx

@@ -0,0 +1,95 @@
+#LyX 2.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 474
+\begin_document
+\begin_header
+\textclass article
+\use_default_options true
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+\language finnish
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+\spacing single
+\use_hyperref true
+\pdf_title "Sähköpajan päiväkirja"
+\pdf_author "Marek Nečada"
+\pdf_bookmarks true
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+\cite_engine basic
+\cite_engine_type default
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+\use_bibtopic false
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+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
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+\end_header
+
+\begin_body
+
+\begin_layout Section
+Comparison to other methods
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Comparison"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

lepaper/examples.lyx

@@ -81,6 +81,13 @@
 
 \begin_layout Section
 Applications
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Applications"
+
+\end_inset
+
+
 \end_layout
 
 \end_body

lepaper/finite.lyx

@@ -81,6 +81,13 @@
 
 \begin_layout Section
 Finite systems
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Finite"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Itemize

lepaper/infinite.lyx

@@ -81,6 +81,13 @@
 
 \begin_layout Section
 Infinite periodic systems
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Infinite"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Subsection
@@ -190,9 +197,9 @@ and we assume periodic solution
 \begin_inset Formula 
 \begin{eqnarray*}
 \sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
-\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
+\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect0α\leftarrow\vect bβ})A_{\vect0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
-\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
+\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect0\beta}\left(\vect k\right) & = & 0,\\
-A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
+A_{\vect0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect0\beta}\left(\vect k\right) & = & 0.
 \end{eqnarray*}
 
 \end_inset
@@ -208,7 +215,7 @@ lattice Fourier transform
  of the translation operator,
 \begin_inset Formula 
 \begin{equation}
-W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
+W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
 \end{equation}
 
 \end_inset
@@ -233,7 +240,7 @@ reference "eq:W definition"
 \end_inset
 
  is the asymptotic behaviour of the translation operator, 
-\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
+\begin_inset Formula $S_{\vect0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
 \end_inset
 
  that makes the convergence of the sum quite problematic for any 
@@ -330,7 +337,7 @@ translation operator for spherical waves originating in
 \end_inset
 
  is in fact a function of a single 3d argument, 
-\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
+\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
 \end_inset
 
 .
@@ -344,7 +351,7 @@ reference "eq:W integral"
  can be rewritten as
 \begin_inset Formula 
 \[
-W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
+W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)}
 \]
 
 \end_inset
@@ -369,10 +376,10 @@ reference "eq:Dirac comb uaFt"
  for the Fourier transform of Dirac comb)
 \begin_inset Formula 
 \begin{eqnarray}
-W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
+W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)(\vect k)\nonumber \\
- & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
+ & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)\left(\vect k\right)\nonumber \\
- & = & \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(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
+ & = & \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(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
- & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber 
+ & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\nonumber 
 \end{eqnarray}
 
 \end_inset
@@ -478,8 +485,8 @@ 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{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\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long 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

lepaper/intro.lyx

@@ -81,6 +81,159 @@
 
 \begin_layout Section
 Introduction
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:Introduction"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The problem of electromagnetic response of a system consisting of many compact
+ scatterers in various geometries, and its numerical solution, is relevant
+ to many branches of nanophotonics (TODO refs).
+ The most commonly used general approaches used in computational electrodynamics
+, such as the finite difference time domain (FDTD) method or the finite
+ element method (FEM), are very often unsuitable for simulating systems
+ with larger number of scatterers due to their computational complexity.
+ Therefore, a common (frequency-domain) approach to get an approximate solution
+ of the scattering problem for many small particles has been the coupled
+ dipole approximation (CDA) where individual scatterers are reduced to electric
+ dipoles (characterised by a polarisability tensor) and coupled to each
+ other through Green's functions.
+ 
+\end_layout
+
+\begin_layout Standard
+CDA is easy to implement and has favorable computational complexity but
+ suffers from at least two fundamental drawbacks.
+ The obvious one is that the dipole approximation is too rough for particles
+ with diameter larger than a small fraction of the wavelength.
+ The other one, more subtle, manifests itself in photonic crystal-like structure
+s used in nanophotonics: there are modes in which the particles' electric
+ dipole moments completely vanish due to symmetry, regardless of how small
+ the particles are, and the excitations have quadrupolar or higher-degree
+ multipolar character.
+ These modes typically appear at the band edges where interesting phenomena
+ such as lasing or Bose-Einstein condensation have been observed – and CDA
+ by definition fails to capture such modes.
+\end_layout
+
+\begin_layout Standard
+The natural way to overcome both limitations of CDA mentioned above is to
+ include higher multipoles into account.
+ Instead of polarisability tensor, the scattering properties of an individual
+ particle are then described a more general 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix, and different particles' multipole excitations are coupled together
+ via translation operators, a generalisation of the Green's functions in
+ CDA.
+ This is the idea behind the 
+\emph on
+multiple-scattering 
+\begin_inset Formula $T$
+\end_inset
+
+-matrix method 
+\emph default
+(MSTMM) (TODO a.k.a something??), and it has been implemented previously for
+ a limited subset of problems (TODO refs and list the limitations of the
+ available).
+ 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO přestože blablaba, moc se to nepoužívalo, protože je težké udělat to
+ správně.
+\end_layout
+
+\end_inset
+
+ Due to the limitations of the existing available codes, we have been developing
+ our own implementation of MSTMM, which we have used in several previous
+ works studying various physical phenomena in plasmonic nanoarrays (TODO
+ examples with refs).
+ 
+\end_layout
+
+\begin_layout Standard
+Hereby we release our MSTMM implementation, the 
+\emph on
+QPMS Photonic Multiple Scattering
+\emph default
+ suite, as an open source software under the GNU General Public License
+ version 3.
+ (TODO refs to the code repositories.) QPMS allows for linear optics simulations
+ of arbitrary sets of compact scatterers in isotropic media.
+ The features include computations of electromagnetic response to external
+ driving, the related cross sections, and finding resonances of finite structure
+s.
+ Moreover, in QPMS we extensively employ group theory to exploit the physical
+ symmetries of the system to further reduce the demands on computational
+ resources, enabling to simulate even larger systems.
+ 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+(TODO put a specific example here of how large system we are able to simulate?)
+\end_layout
+
+\end_inset
+
+ Although systems of large 
+\emph on
+finite
+\emph default
+ number of scatterers are the area where MSTMM excels the most—simply because
+ other methods fail due to their computational complexity—we also extended
+ the method onto infinite periodic systems (photonic crystals); this can
+ be used for quickly evaluating dispersions of such structures and also
+ their topological invariants (TODO).
+ The QPMS suite contains a core C library, Python bindings and several utilities
+ for routine computations, such as TODO.
+ It includes extensive Doxygen documentation, together with description
+ of the API, making extending and customising the code easy.
+\end_layout
+
+\begin_layout Standard
+The current paper is organised as follows: Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Finite"
+
+\end_inset
+
+ is devoted to MSTMM theory for finite systems, in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Infinite"
+
+\end_inset
+
+ we develop the theory for infinite periodic structures.
+ Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Applications"
+
+\end_inset
+
+ demonstrates some basic practical results that can be obtained using QPMS.
+ Finally, in Section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "sec:Comparison"
+
+\end_inset
+
+ we comment on the computational complexity of MSTMM in comparison to other
+ methods.
 \end_layout
 
 \end_body