326 lines
9.3 KiB
Plaintext
326 lines
9.3 KiB
Plaintext
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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 (TODO refs).
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The most commonly used general approaches used in computational electrodynamics
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are often unsuitable for simulating systems with larger number of scatterers
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due to their computational complexity: differential methods such as the
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finite difference time domain (FDTD) method or the finite element method
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(FEM) include the field degrees of freedom (DoF) of the background medium
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(which can have very large volumes), whereas integral approaches such as
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the boundary element method (BEM) need much less DoF but require working
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with dense matrices containing 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 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, regardless of how small
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the particles are, and 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_bose-einstein_2017"
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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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include higher multipoles into account.
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Instead of polarisability tensor, the scattering properties of an individual
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particle are then described a 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 in
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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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(TODO a.k.a something; refs??), and it has been implemented previously for
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a limited subset of problems (TODO refs and list the limitations of the
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available).
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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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TODO přestože blablaba, moc se to nepoužívalo, protože je težké udělat to
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správně.
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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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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) due
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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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\begin_layout Standard
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Here we address both challenges: we extend the method on infinite periodic
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systems using Ewald-type summation techniques, and we exploit symmetries
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of the system to decompose the problem into several substantially smaller
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ones, which 1) reduces the demands on computational resources, hence speeds
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up the computations and allows for simulations of larger systems, and 2)
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provides better understanding of modes in periodic systems.
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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, as free software under the GNU General Public License version 3.
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(TODO refs to the code repositories.) QPMS allows for linear optics simulations
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of arbitrary sets of compact 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 paper, 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 e.g.
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used for quickly evaluating dispersions of such structures, and also their
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topological invariants (TODO).
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The QPMS suite contains a core C library, Python bindings and several utilities
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for routine computations.
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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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, such as TODO.
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\end_layout
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\end_inset
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It includes extensive Doxygen documentation, together with description
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of the API, making extending and customising the code easy.
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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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\end_inset
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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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\end_inset
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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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\end_inset
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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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\end_inset
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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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\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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\end_inset
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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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\end_inset
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\end_layout
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\end_body
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\end_document
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