551 lines
14 KiB
Plaintext
551 lines
14 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\pdf_author "Marek Nečada"
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\end_header
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\begin_body
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\begin_layout Section
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Introduction
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Introduction"
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\end_inset
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\end_layout
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\begin_layout Standard
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The problem of electromagnetic response of a system consisting of many relativel
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y small, compact scatterers in various geometries, and its numerical solution,
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is relevant to many branches of nanophotonics.
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In practice, the scatterers often form some ordered structure, such as
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metalic or dielectric nanoparticle arrays
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\begin_inset CommandInset citation
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LatexCommand cite
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key "zou_silver_2004,garcia_de_abajo_colloquium:_2007,wang_rich_2018,kravets_plasmonic_2018"
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literal "false"
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\end_inset
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that offer many degrees of tunability, with applications including structural
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color, ultra-thin lenses
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\begin_inset CommandInset citation
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LatexCommand cite
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key "khorasaninejad_metalenses_2017"
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literal "false"
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\end_inset
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, strong coupling between light and quantum emitters
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\begin_inset CommandInset citation
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LatexCommand cite
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key "vakevainen_plasmonic_2014,ramezani_strong_2019,torma_strong_2015"
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literal "false"
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\end_inset
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, weak and strong coupling lasing and Bose-Einstein condensation
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\begin_inset CommandInset citation
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LatexCommand cite
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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"
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literal "false"
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\end_inset
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, magneto-optical effects
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kataja_surface_2015"
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literal "false"
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\end_inset
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, or sensing
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kuttner_plasmonics_2018"
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literal "false"
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\end_inset
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.
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The number of scatterers tends to be rather large; unfortunately, the most
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common general approaches used in computational electrodynamics are often
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unsuitable for simulating systems with larger number of scatterers due
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to their computational complexity: differential methods such as the finite
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difference time domain (FDTD,
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\begin_inset CommandInset citation
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LatexCommand cite
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key "sullivan_electromagnetic_2013"
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literal "false"
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\end_inset
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) method or the finite element method (FEM,
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\begin_inset CommandInset citation
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LatexCommand cite
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key "raiyan_kabir_finite_2017"
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literal "false"
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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zkontroluj reference, přidej referenci na frequency domain fem
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\end_layout
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\end_inset
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) include the field degrees of freedom (DoF) of the background medium (which
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can have very large volumes), whereas integral approaches such as the boundary
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element method (BEM, a.k.a the method of moments, MOM,
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\begin_inset CommandInset citation
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LatexCommand cite
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key "medgyesi-mitschang_generalized_1994,reid_efficient_2015"
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literal "false"
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\end_inset
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) need much less DoF but require working with dense matrices containing
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couplings between each pair of DoF.
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Therefore, a common (frequency-domain) approach to get an approximate solution
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of the scattering problem for many small particles has been the coupled
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dipole approximation (CDA) where a drastic reduction of the number of DoF
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is achieved by approximating individual scatterers to electric dipoles
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(characterised by a polarisability tensor) coupled to each other through
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Green's functions.
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\end_layout
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\begin_layout Standard
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CDA is easy to implement and demands relatively little computational resources
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but suffers from at least two fundamental drawbacks.
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The obvious one is that the dipole approximation is too rough for particles
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with diameter larger than a small fraction of the wavelength, which results
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to quantitative errors.
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The other one, more subtle, manifests itself in photonic crystal-like structure
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s used in nanophotonics: there are modes in which the particles' electric
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dipole moments completely vanish due to symmetry, and regardless of how
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small the particles are, the excitations have quadrupolar or higher-degree
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multipolar character.
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These modes typically appear at the band edges where interesting phenomena
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such as lasing or Bose-Einstein condensation have been observed
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\begin_inset CommandInset citation
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LatexCommand cite
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key "guo_lasing_2019,pourjamal_lasing_2019,hakala_lasing_2017,yang_real-time_2015,hakala_boseeinstein_2018"
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literal "false"
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\end_inset
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– and CDA by definition fails to capture such modes.
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\end_layout
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\begin_layout Standard
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The natural way to overcome both limitations of CDA mentioned above is to
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take higher multipoles into account.
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Instead of a polarisability tensor, the scattering properties of an individual
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particle are then described with more general
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\begin_inset Formula $T$
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\end_inset
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-matrix, and different particles' multipole excitations are coupled together
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via translation operators, a generalisation of the Green's functions used
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in CDA.
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This is the idea behind the
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\emph on
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multiple-scattering
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\begin_inset Formula $T$
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\end_inset
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-matrix method
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\emph default
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(MSTMM), a.k.a.
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\emph on
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superposition
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\begin_inset Formula $T$
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\end_inset
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-matrix method
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\emph default
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\begin_inset CommandInset citation
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LatexCommand cite
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key "litvinov_rigorous_2008"
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literal "false"
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\emph on
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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a.k.a.
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something else?
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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, and it has been implemented many times in the context of electromagnetics
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\begin_inset CommandInset citation
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LatexCommand cite
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key "scattport_multiple_nodate"
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literal "false"
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\end_inset
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, but usually only as specific codes for limited subsets of problems, such
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as scattering by clusters of spheres, circular cylinders, or Chebyshev
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particles
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\begin_inset CommandInset citation
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LatexCommand cite
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key "mackowski_multiple_2011,mackowski_mstm_2013,xu_fortran_2003"
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literal "false"
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\end_inset
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.
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Perhaps the most general MSTMM software with respect to the system geometry
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has been FaSTMM
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\begin_inset CommandInset citation
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LatexCommand cite
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key "markkanen_fast_2017,markkanen_fastmm_2017"
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literal "false"
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\end_inset
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, which also a rare example is in this field of a code with a clear licence.
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\end_layout
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\begin_layout Standard
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However, the potential of MSTMM reaches far beyond its past implementations.
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Here we present several enhancements to the method, which are especially
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useful in metamaterial and nanophotonics simulations.
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We extend the method on infinite periodic systems using Ewald-type summation
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techniques.
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This enables, among other things, to use MSTMM for fast solving of the
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lattice modes of such periodic systems, and comparing them to their finite
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counterparts with respect to electromagnetic response, which is useful
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to isolate the bulk and finite-size phenomena of photonic arrays.
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Moreover, we exploit symmetries of the system to decompose the problem
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into several substantially smaller ones, which provides better understanding
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of modes, mainly in periodic systems, and substantially reduces the demands
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on computational resources, hence speeding up the computations and allowing
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for finite size simulations of systems with particle counts practically
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impossible to reliably simulate with any other method.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Due to the limitations of the existing available codes, we have been developing
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our own implementation of MSTMM, which has been used in several previous
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works studying various physical phenomena in plasmonic nanoarrays
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\begin_inset CommandInset citation
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LatexCommand cite
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key "pourjamal_lasing_2019,guo_lasing_2019,hakala_lasing_2017"
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literal "false"
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\end_inset
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.
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During the process, it became apparent that although the size of the arrays
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we were able to simulate with MSTMM was far larger than with other methods,
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sometimes we were unable to match the full size of our physical arrays
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(typically consisting of tens of thousands of metallic nanoparticles) mainly
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due to memory constraints.
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Moreover, to distinguish the effects attributable to the finite size of
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the arrays, it became desirable to simulate also
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\emph on
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infinite periodic systems
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\emph default
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with the same method, as choosing a completely different method could introduce
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differences stemming from the method choice itself.
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Unlike in differential methods where this can be achieved straightforwardly
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using periodic boundary conditions, this is not trivial in MSTMM where
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one has to deal with badly behaving infinite lattice sums.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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The power of the method has been already demonstrated by several works where
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we used it to explain experimental observations: the finite lattice size
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effects on dipole patterns and phase profiles of the nanoparticle lattice
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modes in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "hakala_lasing_2017"
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literal "false"
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\end_inset
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, symmetry and polarisation analysis of the modes at the
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\begin_inset Formula $K$
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\end_inset
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-point of a honeycomb nanopatricle lattice in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "guo_lasing_2019"
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literal "false"
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\end_inset
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, the structure of lasing modes in a Ni nanoparticle array
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\begin_inset CommandInset citation
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LatexCommand cite
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key "pourjamal_lasing_2019"
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literal "false"
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\end_inset
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and energy spacing between the
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\begin_inset Formula $\Gamma$
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\end_inset
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-point modes in a finite lattice
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\begin_inset CommandInset citation
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LatexCommand cite
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key "vakevainen_sub-picosecond_2020"
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literal "false"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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We hereby release our MSTMM implementation, the
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\emph on
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QPMS Photonic Multiple Scattering
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\emph default
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suite
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\begin_inset CommandInset citation
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LatexCommand cite
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key "necada_qpms_2019"
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literal "false"
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\end_inset
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, as free software under the GNU General Public License version 3.
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\begin_inset Note Note
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status open
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||
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\begin_layout Plain Layout
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||
\begin_inset Marginal
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||
status open
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||
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||
\begin_layout Plain Layout
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||
(remember to clean / update the repos before submitting)
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||
\end_layout
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||
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\end_inset
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||
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\end_layout
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\end_inset
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QPMS allows for linear optics simulations of arbitrary sets of compact
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scatterers in isotropic media.
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The features include computations of electromagnetic response to external
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driving, the related cross sections, and finding resonances of finite structure
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s.
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Moreover, it includes the improvements covered in this article, enabling
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to simulate even larger systems and also infinite structures with periodicity
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in one, two or three dimensions, which can be used e.g.
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for evaluating dispersions of such structures.
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The QPMS suite contains a core C library, Python bindings and several utilities
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for routine computations, such as scattering cross sections under plane
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wave irradiation or lattice modes of two-dimensional periodic arrays.
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\begin_inset Note Note
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||
status open
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||
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||
\begin_layout Plain Layout
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||
TODO před odesláním zkontrolovat, co všechno to v danou chvíli umí.
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||
\end_layout
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||
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||
\end_inset
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||
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||
It includes extensive Doxygen documentation, together with description
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||
of the API.
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||
It has been written with customisability and extendibility in mind, so
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||
that including e.g.
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||
alternative methods of
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\begin_inset Formula $T$
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||
\end_inset
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||
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-matrix calculations of a single matrix are as easy as possible.
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||
\end_layout
|
||
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||
\begin_layout Standard
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||
The current paper is organised as follows: Section
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "sec:Finite"
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||
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||
\end_inset
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||
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||
provides a review of MSTMM theory for finite systems.
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||
In Section
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\begin_inset CommandInset ref
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||
LatexCommand ref
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reference "sec:Infinite"
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||
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||
\end_inset
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||
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we develop the theory for infinite periodic structures.
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In Section
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "sec:Symmetries"
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plural "false"
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||
caps "false"
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||
noprefix "false"
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||
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||
\end_inset
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||
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we apply group theory on MSTMM to utilise the symmetries of the simulated
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system.
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||
Finally, Section
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "sec:Applications"
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||
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||
\end_inset
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||
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||
shows some practical results that can be obtained using QPMS and benchmarks
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||
with BEM.
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||
\begin_inset Note Note
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||
status open
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||
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||
\begin_layout Plain Layout
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||
Finally, in Section
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "sec:Comparison"
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||
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||
\end_inset
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||
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||
we comment on the computational complexity of MSTMM in comparison to other
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||
methods.
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||
\end_layout
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||
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||
\end_inset
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||
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||
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||
\end_layout
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||
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||
\end_body
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||
\end_document
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