#LyX 2.4 created this file. For more info see https://www.lyx.org/ \lyxformat 584 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass article \use_default_options true \maintain_unincluded_children false \language english \language_package default \inputencoding utf8 \fontencoding auto \font_roman "default" "TeX Gyre Pagella" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_roman_osf true \font_sans_osf false \font_typewriter_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \float_placement class \float_alignment class \paperfontsize default \spacing single \use_hyperref true \pdf_author "Marek Nečada" \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder false \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \use_minted 0 \use_lineno 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style english \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tablestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Section Applications \begin_inset CommandInset label LatexCommand label name "sec:Applications" \end_inset \end_layout \begin_layout Standard Finally, we present some results obtained with the QPMS suite as well as benchmarks with BEM. Scripts to reproduce these results are available under the \family typewriter examples \family default directory of the QPMS source repository. \end_layout \begin_layout Subsection Response of a rectangular nanoplasmonic array \end_layout \begin_layout Standard Our first example deals with a plasmonic array made of golden nanoparticles placed in a rectangular planar configuration. The nanoparticles have shape of right circular cylinder with radius 50 nm and height 50 nm. The particles are placed with periodicities \begin_inset Formula $p_{x}=\SI{621}{nm}$ \end_inset , \begin_inset Formula $p_{y}=\SI{571}{nm}$ \end_inset into an isotropic medium with a constant refraction index \begin_inset Formula $n=1.52$ \end_inset . For gold, we use the optical properties listed in \begin_inset CommandInset citation LatexCommand cite key "johnson_optical_1972" literal "false" \end_inset interpolated with cubical splines. The particles' cylindrical shape is approximated with a triangular mesh with XXX boundary elements. \begin_inset Marginal status open \begin_layout Plain Layout Show the mesh as well? \end_layout \end_inset \end_layout \begin_layout Standard We consider finite arrays with \begin_inset Formula $N_{x}\times N_{y}=\ldots\times\ldots,\ldots\times\ldots,\ldots\times\ldots$ \end_inset particles and also the corresponding infinite array, and simulate their absorption when irradiated by circularly polarised plane waves with energies from xx to yy and incidence direction lying in the \begin_inset Formula $xz$ \end_inset plane. The results are shown in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Example rectangular absorption" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Marginal status open \begin_layout Plain Layout Mention lMax. \end_layout \end_inset \begin_inset Float figure placement document alignment document wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout Absorption of rectangular arrays of golden nanoparticles with periodicities \begin_inset Formula $p_{x}=\SI{621}{nm}$ \end_inset , \begin_inset Formula $p_{y}=\SI{571}{nm}$ \end_inset with a) \begin_inset Formula $\ldots\times\ldots$ \end_inset , b) \begin_inset Formula $\ldots\times\ldots$ \end_inset , c) \begin_inset Formula $\ldots\times\ldots$ \end_inset and d) infinitely many particles, irradiated by circularly polarised plane waves. e) Absoption profile of a single nanoparticle. \begin_inset CommandInset label LatexCommand label name "fig:Example rectangular absorption" \end_inset \end_layout \end_inset \end_layout \end_inset We compared the \begin_inset Formula $\ldots\times\ldots$ \end_inset case with a purely BEM-based solution obtained using the \family typewriter scuff-scatter \family default utility. TODO WHAT DO WE GET? \end_layout \begin_layout Standard In the infinite case, we benchmarked against a pseudorandom selection of \begin_inset Formula $\left(\vect k,\omega\right)$ \end_inset pairs and the difference was TODO WHAT? We note that evaluating one \begin_inset Formula $\left(\vect k,\omega\right)$ \end_inset pair took xxx miliseconds with MSTMM and truncation degree \begin_inset Formula $L=?$ \end_inset , the same took xxx hours with BEM. \begin_inset Marginal status open \begin_layout Plain Layout TODO also details about the machines used. More info about time also at least for the largest case. \end_layout \end_inset \end_layout \begin_layout Subsection Modes of a rectangular nanoplasmonic array \end_layout \begin_layout Standard Next, we study the eigenmode problem of the same rectangular arrays. The system is lossy, therefore the eigenfrequencies are complex and we need to have a model of the material optical properties also for complex frequencies. So in this case we use the Drude-Lorentz model for gold with parameters as in \begin_inset CommandInset citation LatexCommand cite key "rakic_optical_1998" literal "false" \end_inset . \end_layout \begin_layout Subsubsection Effects of multipole cutoff \end_layout \begin_layout Standard In order to demonstrate some of the consequences of multipole cutoff, we consider a square lattice with periodicity \begin_inset Formula $p_{x}=p_{y}=580\,\mathrm{nm}$ \end_inset filled with spherical golden nanoparticles (with Drude-Lorentz model for permittivity; one sphere per unit cell) embedded in a medium with a constant refractive index \begin_inset Formula $n=1.52$ \end_inset . We vary the multipole cutoff \begin_inset Formula $l_{\max}=1,\dots,5$ \end_inset and the particle radius \begin_inset Formula $r=50\,\mathrm{nm},\dots,300\,\mathrm{nm}$ \end_inset (note that right end of this interval is unphysical, as the spheres touch at \begin_inset Formula $r=290\,\mathrm{nm}$ \end_inset ) We look at the lattice modes at the \begin_inset Formula $\Gamma$ \end_inset point right below the diffracted order crossing at 1.406 eV using Beyn's algorithm; the integration contour for Beyn's algorithm being a circle with centre at \begin_inset Formula $\omega=\left(1.335+0i\right)\mathrm{eV}/\hbar$ \end_inset and radius \begin_inset Formula $70.3\,\mathrm{meV}/\hbar$ \end_inset , and 410 sample points. We classify each of the found modes as one of the ten irreducible representatio ns of the corresponding little group at the \begin_inset Formula $\Gamma$ \end_inset point, \begin_inset Formula $D_{4h}$ \end_inset . \end_layout \begin_layout Standard The real and imaginary parts of the obtained mode frequencies are shown in Fig. \begin_inset CommandInset ref LatexCommand ref reference "square lattice var lMax, r at gamma point Au" plural "false" caps "false" noprefix "false" \end_inset . The most obvious (and expected) effect of the cutoff is the reduction of the number of modes found: the case \begin_inset Formula $l_{\max}=1$ \end_inset (dipole-dipole approximation) contains only the modes with nontrivial dipole excitations ( \begin_inset Formula $x,y$ \end_inset dipoles in \begin_inset Formula $\mathrm{E}'$ \end_inset and \begin_inset Formula $z$ \end_inset dipole in \begin_inset Formula $\mathrm{A_{2}''})$ \end_inset . For relatively small particle sizes, the main effect of increasing \begin_inset Formula $l_{\max}$ \end_inset is making the higher multipolar modes accessible at all. As the particle radius increases, there start to appear more non-negligible elements in the \begin_inset Formula $T$ \end_inset -matrix, and the cutoff then affects the mode frequencies as well. \end_layout \begin_layout Standard Another effect related to mode finding is, that increasing \begin_inset Formula $l_{\max}$ \end_inset leads to overall decrease of the lowest singular values of the mode problem matrix \begin_inset Formula $M\left(\omega,\vect k\right)$ \end_inset , so that they are very close to zero for a large frequency area, making it harder to determine the exact roots of the mode equation \begin_inset CommandInset ref LatexCommand eqref reference "eq:lattice mode equation" plural "false" caps "false" noprefix "false" \end_inset , which might lead to some spurious results: Fig. \begin_inset CommandInset ref LatexCommand ref reference "square lattice var lMax, r at gamma point Au" plural "false" caps "false" noprefix "false" \end_inset shows modes with positive imaginary frequencies for \begin_inset Formula $l_{\max}\ge3$ \end_inset , which is unphysical (positive imaginary frequency means effective losses of the medium, which, together with the lossy particles, prevent emergence of propagating modes). However, the spurious frequencies can be made disappear by tuning the parameter s of Beyn's algorithm (namely, stricter residual threshold), but that might lead to losing legitimate results as well, especially if they are close to the integration contour. In such cases, it is often helpful to run Beyn's algorithm several times with different contours enclosing smaller frequency areas. \end_layout \begin_layout Standard \begin_inset Float figure placement document alignment document wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset Graphics filename figs/beyn_lMax_cutoff_Au_sphere.pdf width 100text% \end_inset \end_layout \end_inset Consequences of multipole degree cutoff: Eigenfrequencies found with Beyn's algorithm for an infinite square lattice of golden spherical nanoparticles with varying particle size. \begin_inset CommandInset label LatexCommand label name "square lattice var lMax, r at gamma point Au" \end_inset \end_layout \end_inset \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Float figure placement document alignment document wide false sideways false status collapsed \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \begin_inset Graphics filename figs/beyn_lMax_cutoff_const_eps_sphere.pdf width 100text% \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout Consequences of multipole degree cutoff: Eigenfrequencies found with Beyn's algorithm for an infinite square lattice of spherical nanoparticles with constant relative permittivity \begin_inset Formula $\epsilon=4.0+0.7i$ \end_inset and varying particle size. \begin_inset CommandInset label LatexCommand label name "square lattice var lMax, r at gamma point constant epsilon" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \end_body \end_document