561 lines
12 KiB
Plaintext
561 lines
12 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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Applications
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Applications"
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\end_inset
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\end_layout
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\begin_layout Standard
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Finally, we present some results obtained with the QPMS suite as well as
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benchmarks with BEM.
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Scripts to reproduce these results are available under the
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\family typewriter
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examples
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\family default
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directory of the QPMS source repository.
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\end_layout
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\begin_layout Subsection
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Response of a rectangular nanoplasmonic array
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\end_layout
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\begin_layout Standard
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Our first example deals with a plasmonic array made of golden nanoparticles
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placed in a rectangular planar configuration.
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The nanoparticles have shape of right circular cylinder with radius 50
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nm and height 50 nm.
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The particles are placed with periodicities
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\begin_inset Formula $p_{x}=\SI{621}{nm}$
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\end_inset
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,
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\begin_inset Formula $p_{y}=\SI{571}{nm}$
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\end_inset
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into an isotropic medium with a constant refraction index
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\begin_inset Formula $n=1.52$
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\end_inset
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.
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For gold, we use the optical properties listed in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "johnson_optical_1972"
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literal "false"
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\end_inset
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interpolated with cubical splines.
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The particles' cylindrical shape is approximated with a triangular mesh
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with XXX boundary elements.
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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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Show the mesh as well?
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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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We consider finite arrays with
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\begin_inset Formula $N_{x}\times N_{y}=\ldots\times\ldots,\ldots\times\ldots,\ldots\times\ldots$
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\end_inset
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particles and also the corresponding infinite array, and simulate their
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absorption when irradiated by circularly polarised plane waves with energies
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from xx to yy and incidence direction lying in the
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\begin_inset Formula $xz$
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\end_inset
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plane.
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The results are shown in Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Example rectangular absorption"
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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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.
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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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Mention lMax.
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\end_layout
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\end_inset
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\begin_inset Float figure
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placement document
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alignment document
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Absorption of rectangular arrays of golden nanoparticles with periodicities
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\begin_inset Formula $p_{x}=\SI{621}{nm}$
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\end_inset
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,
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\begin_inset Formula $p_{y}=\SI{571}{nm}$
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\end_inset
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with a)
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\begin_inset Formula $\ldots\times\ldots$
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\end_inset
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, b)
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\begin_inset Formula $\ldots\times\ldots$
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\end_inset
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, c)
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\begin_inset Formula $\ldots\times\ldots$
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\end_inset
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and d) infinitely many particles, irradiated by circularly polarised plane
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waves.
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e) Absoption profile of a single nanoparticle.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:Example rectangular absorption"
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\end_inset
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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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We compared the
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\begin_inset Formula $\ldots\times\ldots$
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\end_inset
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case with a purely BEM-based solution obtained using the
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\family typewriter
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scuff-scatter
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\family default
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utility.
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TODO WHAT DO WE GET?
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\end_layout
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\begin_layout Standard
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In the infinite case, we benchmarked against a pseudorandom selection of
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\begin_inset Formula $\left(\vect k,\omega\right)$
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\end_inset
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pairs and the difference was TODO WHAT? We note that evaluating one
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\begin_inset Formula $\left(\vect k,\omega\right)$
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\end_inset
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pair took xxx miliseconds with MSTMM and truncation degree
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\begin_inset Formula $L=?$
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\end_inset
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, the same took xxx hours with BEM.
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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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TODO also details about the machines used.
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More info about time also at least for the largest case.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Subsection
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Modes of a rectangular nanoplasmonic array
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\end_layout
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\begin_layout Standard
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Next, we study the eigenmode problem of the same rectangular arrays.
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The system is lossy, therefore the eigenfrequencies are complex and we
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need to have a model of the material optical properties also for complex
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frequencies.
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So in this case we use the Drude-Lorentz model for gold with parameters
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as in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "rakic_optical_1998"
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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 Subsubsection
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Effects of multipole cutoff
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\end_layout
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\begin_layout Standard
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In order to demonstrate some of the consequences of multipole cutoff, we
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consider a square lattice with periodicity
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\begin_inset Formula $p_{x}=p_{y}=580\,\mathrm{nm}$
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\end_inset
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filled with spherical golden nanoparticles (with Drude-Lorentz model for
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permittivity; one sphere per unit cell) embedded in a medium with a constant
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refractive index
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\begin_inset Formula $n=1.52$
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\end_inset
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.
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We vary the multipole cutoff
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\begin_inset Formula $l_{\max}=1,\dots,5$
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\end_inset
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and the particle radius
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\begin_inset Formula $r=50\,\mathrm{nm},\dots,300\,\mathrm{nm}$
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\end_inset
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(note that right end of this interval is unphysical, as the spheres touch
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at
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\begin_inset Formula $r=290\,\mathrm{nm}$
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\end_inset
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) We look at the lattice modes at the
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\begin_inset Formula $\Gamma$
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\end_inset
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point right below the diffracted order crossing at 1.406 eV using Beyn's
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algorithm; the integration contour for Beyn's algorithm being a circle
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with centre at
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\begin_inset Formula $\omega=\left(1.335+0i\right)\mathrm{eV}/\hbar$
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\end_inset
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and radius
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\begin_inset Formula $70.3\,\mathrm{meV}/\hbar$
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\end_inset
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, and 410 sample points.
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We classify each of the found modes as one of the ten irreducible representatio
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ns of the corresponding little group at the
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\begin_inset Formula $\Gamma$
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\end_inset
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point,
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\begin_inset Formula $D_{4h}$
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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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The real and imaginary parts of the obtained mode frequencies are shown
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in Fig.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "square lattice var lMax, r at gamma point Au"
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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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.
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The most obvious (and expected) effect of the cutoff is the reduction of
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the number of modes found: the case
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\begin_inset Formula $l_{\max}=1$
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\end_inset
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(dipole-dipole approximation) contains only the modes with nontrivial dipole
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excitations (
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\begin_inset Formula $x,y$
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\end_inset
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dipoles in
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\begin_inset Formula $\mathrm{E}'$
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\end_inset
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and
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\begin_inset Formula $z$
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\end_inset
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dipole in
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\begin_inset Formula $\mathrm{A_{2}''})$
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\end_inset
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.
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For relatively small particle sizes, the main effect of increasing
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\begin_inset Formula $l_{\max}$
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\end_inset
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is making the higher multipolar modes accessible at all.
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As the particle radius increases, there start to appear more non-negligible
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elements in the
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\begin_inset Formula $T$
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\end_inset
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-matrix, and the cutoff then affects the mode frequencies as well.
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\end_layout
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\begin_layout Standard
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Another effect related to mode finding is, that increasing
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\begin_inset Formula $l_{\max}$
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\end_inset
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leads to overall decrease of the lowest singular values of the mode problem
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matrix
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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, so that they are very close to zero for a large frequency area, making
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it harder to determine the exact roots of the mode equation
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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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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, which might lead to some spurious results: Fig.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "square lattice var lMax, r at gamma point Au"
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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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shows modes with positive imaginary frequencies for
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\begin_inset Formula $l_{\max}\ge3$
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\end_inset
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, which is unphysical (positive imaginary frequency means effective losses
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of the medium, which, together with the lossy particles, prevent emergence
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of propagating modes).
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However, the spurious frequencies can be made disappear by tuning the parameter
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s of Beyn's algorithm (namely, stricter residual threshold), but that might
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lead to losing legitimate results as well, especially if they are close
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to the integration contour.
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In such cases, it is often helpful to run Beyn's algorithm several times
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with different contours enclosing smaller frequency areas.
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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placement document
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alignment document
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename figs/beyn_lMax_cutoff_Au_sphere.pdf
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width 100text%
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\end_inset
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Consequences of multipole degree cutoff: Eigenfrequencies found with Beyn's
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algorithm for an infinite square lattice of golden spherical nanoparticles
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with varying particle size.
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\begin_inset CommandInset label
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LatexCommand label
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name "square lattice var lMax, r at gamma point Au"
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\end_inset
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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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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Float figure
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placement document
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alignment document
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wide false
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sideways false
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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\begin_inset Graphics
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filename figs/beyn_lMax_cutoff_const_eps_sphere.pdf
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width 100text%
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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Consequences of multipole degree cutoff: Eigenfrequencies found with Beyn's
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algorithm for an infinite square lattice of spherical nanoparticles with
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constant relative permittivity
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\begin_inset Formula $\epsilon=4.0+0.7i$
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\end_inset
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and varying particle size.
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\begin_inset CommandInset label
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LatexCommand label
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name "square lattice var lMax, r at gamma point constant epsilon"
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\end_inset
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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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\end_layout
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\end_body
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\end_document
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