qpms/lepaper/examples.lyx

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#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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\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
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placement document
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wide false
sideways false
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\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figs/beyn_lMax_cutoff_Au_sphere.pdf
width 100text%
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
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
\end_layout
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status collapsed
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\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