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\pdf_author "Marek Nečada"
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\begin_body

\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 relativel
y small, compact scatterers in various geometries, and its numerical solution,
 is relevant to many branches of nanophotonics.
 In practice, the scatterers often form some ordered structure, such as
 metalic or dielectric nanoparticle arrays 
\begin_inset CommandInset citation
LatexCommand cite
key "zou_silver_2004,garcia_de_abajo_colloquium:_2007,wang_rich_2018,kravets_plasmonic_2018"
literal "false"

\end_inset

 that offer many degrees of tunability, with applications including structural
 color, ultra-thin lenses 
\begin_inset CommandInset citation
LatexCommand cite
key "khorasaninejad_metalenses_2017"
literal "false"

\end_inset

, strong coupling between light and quantum emitters 
\begin_inset CommandInset citation
LatexCommand cite
key "vakevainen_plasmonic_2014,ramezani_strong_2019,torma_strong_2015"
literal "false"

\end_inset

, weak and strong coupling lasing and Bose-Einstein condensation 
\begin_inset CommandInset citation
LatexCommand cite
key "zhou_lasing_2013,hakala_lasing_2017,guo_lasing_2019,hakala_boseeinstein_2018,yang_real-time_2015,ramezani_plasmon-exciton-polariton_2017,vakevainen_sub-picosecond_2020,wang_structural_2018"
literal "false"

\end_inset

, magneto-optical effects 
\begin_inset CommandInset citation
LatexCommand cite
key "kataja_surface_2015"
literal "false"

\end_inset

, or sensing 
\begin_inset CommandInset citation
LatexCommand cite
key "kuttner_plasmonics_2018"
literal "false"

\end_inset

.
 The number of scatterers tends to be rather large; unfortunately, the most
 common 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, 
\begin_inset CommandInset citation
LatexCommand cite
key "sullivan_electromagnetic_2013"
literal "false"

\end_inset

) method or the finite element method (FEM, 
\begin_inset CommandInset citation
LatexCommand cite
key "raiyan_kabir_finite_2017"
literal "false"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
zkontroluj reference, přidej referenci na frequency domain fem
\end_layout

\end_inset

) 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, a.k.a the method of moments, MOM 
\begin_inset CommandInset citation
LatexCommand cite
key "harrington_field_1993,medgyesi-mitschang_generalized_1994,reid_efficient_2015"
literal "false"

\end_inset

) 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 a 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, and regardless of how
 small the particles are, the excitations have quadrupolar or higher-degree
 multipolar character.
 These modes, belonging to a category that is sometimes called 
\emph on
optical bound states in the continuum (BIC)
\emph default
 
\begin_inset CommandInset citation
LatexCommand cite
key "hsu_bound_2016"
literal "false"

\end_inset

, 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_boseeinstein_2018"
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
 take higher multipoles into account.
 Instead of a polarisability tensor, the scattering properties of an individual
 particle are then described with more general 
\emph on
transition matrix
\emph default
 (commonly known as 
\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 used
 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
 
\begin_inset CommandInset citation
LatexCommand cite
key "litvinov_rigorous_2008"
literal "false"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout

\emph on
\begin_inset Marginal
status open

\begin_layout Plain Layout
a.k.a.
 something else?
\end_layout

\end_inset


\end_layout

\end_inset

, and it has been implemented many times in the context of electromagnetics
 
\begin_inset CommandInset citation
LatexCommand cite
key "scattport_multiple_nodate"
literal "false"

\end_inset

, but usually only as specific codes for limited subsets of problems, such
 as scattering by clusters of spheres, circular cylinders, or Chebyshev
 particles 
\begin_inset CommandInset citation
LatexCommand cite
key "mackowski_multiple_2011,mackowski_mstm_2013,xu_fortran_2003"
literal "false"

\end_inset

.
 Perhaps the most general MSTMM software with respect to the system geometry
 has been FaSTMM 
\begin_inset CommandInset citation
LatexCommand cite
key "markkanen_fast_2017,markkanen_fastmm_2017"
literal "false"

\end_inset

, which also a rare example is in this field of a code with a clear licence.
\end_layout

\begin_layout Standard
However, the potential of MSTMM reaches far beyond its past implementations.
 Here we present several enhancements to the method, which are especially
 useful in metamaterial and nanophotonics simulations.
 We extend the method on infinite periodic systems using Ewald-type summation
 techniques.
 This enables, among other things, to use MSTMM for fast solving of the
 lattice modes of such periodic systems, and comparing them to their finite
 counterparts with respect to electromagnetic response, which is useful
 to isolate the bulk and finite-size phenomena of photonic lattices.
 Moreover, we exploit symmetries of the system to decompose the problem
 into several substantially smaller ones, which provides better understanding
 of modes, mainly in periodic systems, and substantially reduces the demands
 on computational resources, hence speeding up the computations and allowing
 for finite size simulations of systems with particle numbers practically
 impossible to reliably simulate with any other method.
\begin_inset Note Note
status open

\begin_layout Plain Layout
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) mainly
 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

\end_inset


\end_layout

\begin_layout Standard
The power of the method has been already demonstrated by several works where
 we used it to explain experimental observations: the finite lattice size
 effects on dipole patterns and phase profiles of the nanoparticle lattice
 modes in 
\begin_inset CommandInset citation
LatexCommand cite
key "hakala_lasing_2017"
literal "false"

\end_inset

, symmetry and polarisation analysis of the modes at the 
\begin_inset Formula $K$
\end_inset

-point of a honeycomb nanopatricle lattice in 
\begin_inset CommandInset citation
LatexCommand cite
key "guo_lasing_2019"
literal "false"

\end_inset

, the structure of lasing modes in a Ni nanoparticle array 
\begin_inset CommandInset citation
LatexCommand cite
key "pourjamal_lasing_2019"
literal "false"

\end_inset

 and energy spacing between the 
\begin_inset Formula $\Gamma$
\end_inset

-point modes in a finite lattice 
\begin_inset CommandInset citation
LatexCommand cite
key "vakevainen_sub-picosecond_2020"
literal "false"

\end_inset

.
 
\end_layout

\begin_layout Standard
We hereby release our MSTMM implementation, the 
\emph on
QPMS Photonic Multiple Scattering
\emph default
 suite 
\begin_inset CommandInset citation
LatexCommand cite
key "necada_qpms_2019"
literal "false"

\end_inset

, as free software under the GNU General Public License version 3.
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset Marginal
status open

\begin_layout Plain Layout
(remember to clean / update the repos before submitting)
\end_layout

\end_inset


\end_layout

\end_inset

 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 article, enabling
 to simulate even larger systems and also infinite structures with periodicity
 in one, two or three dimensions, which can be used e.g.
 for evaluating dispersions of such structures.
 The QPMS suite contains a core C library, Python bindings and several utilities
 for routine computations, such as scattering cross sections under plane
 wave irradiation or lattice modes of two-dimensional periodic arrays.
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO před odesláním zkontrolovat, co všechno to v danou chvíli umí.
\end_layout

\end_inset

 It includes extensive Doxygen documentation, together with description
 of the API.
 It has been written with customisability and extendibility in mind, so
 that including e.g.
 alternative methods of 
\begin_inset Formula $T$
\end_inset

-matrix calculations of a single matrix are as easy as possible.
\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

\end_body
\end_document