qpms/lepaper/intro.lyx

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\pdf_author "Marek Nečada"
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\begin_layout Section
Introduction
\begin_inset CommandInset label
LatexCommand label
name "sec:Introduction"

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\end_layout

\begin_layout Standard
The problem of electromagnetic response of a system consisting of many relativel
y small, 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
 are often unsuitable for simulating systems with larger number of scatterers
 due to their computational complexity: differential methods such as the
 finite difference time domain (FDTD) method or the finite element method
 (FEM) include the field degrees of freedom (DoF) of the background medium
 (which can have very large volumes), whereas integral approaches such as
 the boundary element method (BEM) need much less DoF but require working
 with dense matrices containing couplings between each pair of DoF.
 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 drastic reduction of the number of DoF
 is achieved by approximating individual scatterers to electric dipoles
 (characterised by a polarisability tensor) coupled to each other through
 Green's functions.
 
\end_layout

\begin_layout Standard
CDA is easy to implement and demands relatively little computational resources
 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, which results
 to quantitative errors.
 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 
\begin_inset CommandInset citation
LatexCommand cite
key "guo_lasing_2019,pourjamal_lasing_2019,hakala_lasing_2017,yang_real-time_2015,hakala_bose-einstein_2017"
literal "false"

\end_inset

 – 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), a.k.a.
 
\emph on
superposition 
\begin_inset Formula $T$
\end_inset

-matrix method
\emph default
 (TODO a.k.a something; refs??), 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

 
\end_layout

\begin_layout Standard
Due to the limitations of the existing available codes, we have been developing
 our own implementation of MSTMM, which has been used in several previous
 works studying various physical phenomena in plasmonic nanoarrays 
\begin_inset CommandInset citation
LatexCommand cite
key "pourjamal_lasing_2019,guo_lasing_2019,hakala_lasing_2017"
literal "false"

\end_inset

.
 During the process, it became apparent that although the size of the arrays
 we were able to simulate with MSTMM was far larger than with other methods,
 sometimes we were unable to match the full size of our physical arrays
 (typically consisting of tens of thousands of metallic nanoparticles) due
 to memory constraints.
 Moreover, to distinguish the effects attributable to the finite size of
 the arrays, it became desirable to simulate also 
\emph on
infinite periodic systems
\emph default
 with the same method, as choosing a completely different method could introduce
 differences stemming from the method choice itself.
 Unlike in differential methods where this can be achieved straightforwardly
 using periodic boundary conditions, this is not trivial in MSTMM where
 one has to deal with badly behaving infinite lattice sums.
\end_layout

\begin_layout Standard
Here we address both challenges: we extend the method on infinite periodic
 systems using Ewald-type summation techniques, and we exploit symmetries
 of the system to decompose the problem into several substantially smaller
 ones, which 1) reduces the demands on computational resources, hence speeds
 up the computations and allows for simulations of larger systems, and 2)
 provides better understanding of modes in periodic systems.
\end_layout

\begin_layout Standard
We hereby release our MSTMM implementation, the 
\emph on
QPMS Photonic Multiple Scattering
\emph default
 suite, as free 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, it includes the improvements covered in this paper, enabling
 to simulate even larger systems and also infinite structures with periodicity
 in one, two or three dimensions, which can be e.g.
 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.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
, such as TODO.
\end_layout

\end_inset

 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

 provides a review of 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.
 In section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Symmetries"
plural "false"
caps "false"
noprefix "false"

\end_inset

 we apply group theory on MSTMM to utilise the symmetries of the simulated
 system.
 Finally, section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Applications"

\end_inset

 shows some practical results that can be obtained using QPMS and benchmarks
 with BEM.
\begin_inset Note Note
status open

\begin_layout Plain Layout
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_inset


\end_layout

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