1028 lines
21 KiB
Plaintext
1028 lines
21 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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\shortcut idx
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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
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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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as well as benchmarks with BEM
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\end_layout
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\end_inset
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.
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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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Optical response of a square array; finite size effects
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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 silver nanoparticles
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placed in a square planar configuration.
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The nanoparticles have shape of right circular cylinder with 30 nm radius
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and 30 nm in height.
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The particles are placed with periodicity
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\begin_inset Formula $p_{x}=p_{y}=375\,\mathrm{nm}$
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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 silver, we use Drude-Lorentz model with parameters from
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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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, and the
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\begin_inset Formula $T$
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\end_inset
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-matrix of a single particle we compute using the null-field method (with
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cutoff
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\begin_inset Formula $l_{\mathrm{max}}=6$
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\end_inset
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for solving the null-field equations).
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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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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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\end_layout
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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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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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\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 $N_{x}\times N_{y}=40\times40,70\times70,100\times100$
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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
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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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circularly
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\end_layout
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\end_inset
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||
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plane waves with incidence direction lying in the
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\begin_inset Formula $xz$
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\end_inset
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||
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plane.
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||
We concentrate on the behaviour around the first diffracted order crossing
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||
at the
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\begin_inset Formula $\Gamma$
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||
\end_inset
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||
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||
point, which happens around frequency
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||
\begin_inset Formula $2.18\,\mathrm{eV}/\hbar$
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||
\end_inset
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||
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.
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||
Figure
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "fig:Example rectangular absorption infinite"
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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 the response for the infinite array for a range of frequencies; here
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in particular we used the multipole cutoff
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\begin_inset Formula $l_{\mathrm{max}}=3$
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||
\end_inset
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||
|
||
for the interparticle interactions, although there is no visible difference
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||
if we use
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||
\begin_inset Formula $l_{\mathrm{max}}=2$
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||
\end_inset
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||
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||
instead due to the small size of the particles.
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||
In Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:Example rectangular absorption size comparison"
|
||
plural "false"
|
||
caps "false"
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||
noprefix "false"
|
||
|
||
\end_inset
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||
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||
, we compare the response of differently sized array slightly below the
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||
diffracted order crossing.
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||
We see that far from the diffracted orders, all the cross sections are
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||
almost directly proportional to the total number of particles.
|
||
However, near the resonances, the size effects become apparent: the lattice
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||
resonances tend to fade away as the size of the array decreases.
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||
Moreover, the proportion between the absorbed and scattered parts changes
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||
as while the small arrays tend to more just scatter the incident light
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||
into different directions, in larger arrays, it is more
|
||
\begin_inset Quotes eld
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||
\end_inset
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||
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likely
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||
\begin_inset Quotes erd
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||
\end_inset
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||
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||
that the light will scatter many times, each time sacrifying a part of
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its energy to the ohmic losses.
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||
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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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||
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||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename figs/inf.pdf
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||
width 45text%
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\end_inset
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||
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\begin_inset Graphics
|
||
filename figs/inf_big_px.pdf
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width 45text%
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\end_inset
|
||
|
||
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Response of an infinite square array of silver nanoparticles with periodicities
|
||
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||
\begin_inset Formula $p_{x}=p_{y}=375\,\mathrm{nm}$
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||
\end_inset
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||
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||
to plane waves incident in the
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\begin_inset Formula $xz$
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||
\end_inset
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||
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||
-plane.
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||
Left:
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||
\begin_inset Formula $y$
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||
\end_inset
|
||
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||
-polarised waves, right:
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||
\begin_inset Formula $x$
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||
\end_inset
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||
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||
-polarised waves.
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||
The images show extinction, scattering and absorption cross section per
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||
unit cell.
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:Example rectangular absorption infinite"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
placement document
|
||
alignment document
|
||
wide false
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||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename figs/sqlat_scattering_cuts.pdf
|
||
width 90col%
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||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Comparison of optical responses of differently sized square arrays of silver
|
||
nanoparticles with the same periodicity
|
||
\begin_inset Formula $p_{x}=p_{y}=375\,\mathrm{nm}$
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||
\end_inset
|
||
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||
.
|
||
In all cases, the array is illuminated by plane waves linearly polarised
|
||
in the
|
||
\begin_inset Formula $y$
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||
\end_inset
|
||
|
||
-direction, with constant frequency
|
||
\begin_inset Formula $2.15\,\mathrm{eV}/\hbar$
|
||
\end_inset
|
||
|
||
.
|
||
The cross sections are normalised by the total number of particles in the
|
||
array.
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:Example rectangular absorption size comparison"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The finite-size cases in Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:Example rectangular absorption size comparison"
|
||
plural "false"
|
||
caps "false"
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||
noprefix "false"
|
||
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||
\end_inset
|
||
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||
were computed with quadrupole truncation
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||
\begin_inset Formula $l\le2$
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||
\end_inset
|
||
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||
and using the decomposition into the eight irreducible representations
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||
of group
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||
\begin_inset Formula $D_{2h}$
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||
\end_inset
|
||
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||
.
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||
The
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||
\begin_inset Formula $100\times100$
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||
\end_inset
|
||
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||
array took about 4 h to compute on Dell PowerEdge C4130 with 12 core Xeon
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||
E5 2680 v3 2.50GHz, requiring about 20 GB of RAM.
|
||
For smaller systems, the computation time decreases quickly, as the main
|
||
bottleneck is the LU factorisation.
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||
In any case, there is still room for optimisation in the QPMS suite.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
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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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||
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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||
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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
|
||
\begin_inset Formula $L=?$
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||
\end_inset
|
||
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||
, the same took xxx hours with BEM.
|
||
\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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||
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||
\end_inset
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||
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||
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||
\end_layout
|
||
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||
\end_inset
|
||
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||
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||
\end_layout
|
||
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||
\begin_layout Subsection
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||
Lattice mode structure
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||
\end_layout
|
||
|
||
\begin_layout Subsection
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||
Square lattice
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||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Next, we study the lattice mode problem of the same square arrays.
|
||
First we consider the mode problem exactly at the
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
point,
|
||
\begin_inset Formula $\vect k=0$
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||
\end_inset
|
||
|
||
.
|
||
Before proceeding with more sophisticated methods, it is often helpful
|
||
to look at the singular values of mode problem matrix
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)$
|
||
\end_inset
|
||
|
||
from the lattice mode equation
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:lattice mode equation"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
, as shown in Fig.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:square lattice real interval SVD"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
This can be always done, even with tabulated/interpolated material properties
|
||
and/or
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrices.
|
||
An additional insight, especially in the high-symmetry points of the Brillouin
|
||
zone, is provided by decomposition of the matrix into irreps – in this
|
||
case of group
|
||
\begin_inset Formula $D_{4h}$
|
||
\end_inset
|
||
|
||
, which corresponds to the point group symmetry of the array at the
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
point.
|
||
Although on the picture none of the SVDs hits manifestly zero, we see two
|
||
prominent dips in the
|
||
\begin_inset Formula $E'$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $A_{2}''$
|
||
\end_inset
|
||
|
||
irrep subspaces, which is a sign of an actual solution nearby in the complex
|
||
plane.
|
||
Moreover, there might be some less obvious minima in the very vicinity
|
||
of the diffracted order crossing which do not appear in the picture due
|
||
to rough frequency sampling.
|
||
\begin_inset Float figure
|
||
placement document
|
||
alignment document
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename figs/cyl_r30nm_h30nm_p375nmx375nm_mAg_bg1.52_φ0_θ(-0.0075_0.0075)π_ψ0.5π_χ0π_f2.11–2.23eV_L3.pdf
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Singular values of the mode problem matrix
|
||
\begin_inset Formula $\truncated{M\left(\omega,\vect k=0\right)}3$
|
||
\end_inset
|
||
|
||
for a real frequency interval.
|
||
The irreducible representations of
|
||
\begin_inset Formula $D_{4h}$
|
||
\end_inset
|
||
|
||
are labeled with different colors.
|
||
The density of the data points on the horizontal axis is
|
||
\begin_inset Formula $1/\mathrm{meV}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:square lattice real interval SVD"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As we have used only analytical ingredients in
|
||
\begin_inset Formula $M\left(\omega,\vect k\right)$
|
||
\end_inset
|
||
|
||
, the matrix is itself analytical, hence Beyn's algorithm can be used to
|
||
search for complex mode frequencies, which is shown in Figure
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:square lattice beyn dispersion"
|
||
plural "false"
|
||
caps "false"
|
||
noprefix "false"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
The number of the frequency point found is largely dependent on the parameters
|
||
used in Beyn's algorithm, mostly the integration contour in the frequency
|
||
space.
|
||
Here we used ellipses discretised by 250 points each, with edges nearly
|
||
touching the empty lattice diffracted orders (from either above or below
|
||
in the real part), and with major axis covering 1/5 of the interval between
|
||
two diffracted orders.
|
||
At the
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
point, the algorithm finds the actual complex positions of the suspected
|
||
|
||
\begin_inset Formula $E'$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $A_{2}''$
|
||
\end_inset
|
||
|
||
modes without a problem, as well as their continuations to the other nearby
|
||
values of
|
||
\begin_inset Formula $\vect k$
|
||
\end_inset
|
||
|
||
.
|
||
However, for further
|
||
\begin_inset Formula $\vect k$
|
||
\end_inset
|
||
|
||
it might
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
lose track
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
, especially as the modes cross the diffracted orders.
|
||
As a result, the parameters of Beyn's algorithm often require manual tuning
|
||
based on the observed behaviour.
|
||
|
||
\begin_inset Float figure
|
||
placement document
|
||
alignment document
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Graphics
|
||
filename figs/sqlat_beyn_dispersion.pdf
|
||
width 80col%
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
Solutions of the lattice mode problem
|
||
\begin_inset Formula $\truncated{M\left(\omega,\vect k\right)}3$
|
||
\end_inset
|
||
|
||
found using Beyn's method nearby the first diffracted order crossing at
|
||
the
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
point for
|
||
\begin_inset Formula $k_{y}=0$
|
||
\end_inset
|
||
|
||
.
|
||
At the
|
||
\begin_inset Formula $\Gamma$
|
||
\end_inset
|
||
|
||
point, they are classified according to the irreducible representations
|
||
of
|
||
\begin_inset Formula $D_{4h}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "fig:square lattice beyn dispersion"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
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
|
||
|
||
\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
|
||
\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
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status collapsed
|
||
|
||
\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
|