Paper intro

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Marek Nečada 2019-07-17 23:06:56 +03:00
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\end_inset \end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand include
filename "comparison.lyx"
\end_inset
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard

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#LyX 2.1 created this file. For more info see http://www.lyx.org/
\lyxformat 474
\begin_document
\begin_header
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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\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

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

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

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@ -81,6 +81,13 @@
\begin_layout Section \begin_layout Section
Infinite periodic systems Infinite periodic systems
\begin_inset CommandInset label
LatexCommand label
name "sec:Infinite"
\end_inset
\end_layout \end_layout
\begin_layout Subsection \begin_layout Subsection
@ -190,9 +197,9 @@ and we assume periodic solution
\begin_inset Formula \begin_inset Formula
\begin{eqnarray*} \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{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{eqnarray*}
\end_inset \end_inset
@ -208,7 +215,7 @@ lattice Fourier transform
of the translation operator, of the translation operator,
\begin_inset Formula \begin_inset Formula
\begin{equation} \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{equation}
\end_inset \end_inset
@ -233,7 +240,7 @@ reference "eq:W definition"
\end_inset \end_inset
is the asymptotic behaviour of the translation operator, 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 \end_inset
that makes the convergence of the sum quite problematic for any that makes the convergence of the sum quite problematic for any
@ -330,7 +337,7 @@ translation operator for spherical waves originating in
\end_inset \end_inset
is in fact a function of a single 3d argument, 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 \end_inset
. .
@ -344,7 +351,7 @@ reference "eq:W integral"
can be rewritten as can be rewritten as
\begin_inset Formula \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 \end_inset
@ -369,10 +376,10 @@ reference "eq:Dirac comb uaFt"
for the Fourier transform of Dirac comb) for the Fourier transform of Dirac comb)
\begin_inset Formula \begin_inset Formula
\begin{eqnarray} \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{eqnarray}
\end_inset \end_inset
@ -478,8 +485,8 @@ reference "eq:W sum in reciprocal space"
\begin_inset Formula \begin_inset Formula
\begin{eqnarray} \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}\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{eqnarray}
\end_inset \end_inset

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@ -81,6 +81,159 @@
\begin_layout Section \begin_layout Section
Introduction 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_layout
\end_body \end_body