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
\begin_inset Note Note
status open
\begin_layout Plain Layout
as well as benchmarks with BEM
\end_layout
\end_inset
.
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
Optical response of a square array; finite size effects
\end_layout
\begin_layout Standard
Our first example deals with a plasmonic array made of silver nanoparticles
placed in a square planar configuration.
The nanoparticles have shape of right circular cylinder with 30 nm radius
and 30 nm in height.
The particles are placed with periodicity
\begin_inset Formula $p_{x}=p_{y}=375\,\mathrm{nm}$
\end_inset
into an isotropic medium with a constant refraction index
\begin_inset Formula $n=1.52$
\end_inset
.
For silver, we use Drude-Lorentz model with parameters from
\begin_inset CommandInset citation
LatexCommand cite
key "rakic_optical_1998"
literal "false"
\end_inset
, and the
\begin_inset Formula $T$
\end_inset
-matrix of a single particle we compute using the null-field method (with
cutoff
\begin_inset Formula $l_{\mathrm{max}}=6$
\end_inset
for solving the null-field equations).
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
the optical properties listed in
\begin_inset CommandInset citation
LatexCommand cite
key "johnson_optical_1972"
literal "false"
\end_inset
interpolated with cubical splines.
\end_layout
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
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
\end_inset
\end_layout
\begin_layout Standard
We consider finite arrays with
\begin_inset Formula $N_{x}\times N_{y}=40\times40,70\times70,100\times100$
\end_inset
particles and also the corresponding infinite array, and simulate their
absorption when irradiated by
\begin_inset Note Note
status open
\begin_layout Plain Layout
circularly
\end_layout
\end_inset
plane waves with incidence direction lying in the
\begin_inset Formula $xz$
\end_inset
plane.
We concentrate on the behaviour around the first diffracted order crossing
at the
\begin_inset Formula $\Gamma$
\end_inset
point, which happens around frequency
\begin_inset Formula $2.18\,\mathrm{eV}/\hbar$
\end_inset
.
Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Example rectangular absorption infinite"
plural "false"
caps "false"
noprefix "false"
\end_inset
shows the response for the infinite array for a range of frequencies; here
in particular we used the multipole cutoff
\begin_inset Formula $l_{\mathrm{max}}=3$
\end_inset
for the interparticle interactions, although there is no visible difference
if we use
\begin_inset Formula $l_{\mathrm{max}}=2$
\end_inset
instead due to the small size of the particles.
In Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Example rectangular absorption size comparison"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we compare the response of differently sized array slightly below the
diffracted order crossing.
We see that far from the diffracted orders, all the cross sections are
almost directly proportional to the total number of particles.
However, near the resonances, the size effects become apparent: the lattice
resonances tend to fade away as the size of the array decreases.
Moreover, the proportion between the absorbed and scattered parts changes
as while the small arrays tend to more just scatter the incident light
into different directions, in larger arrays, it is more
\begin_inset Quotes eld
\end_inset
likely
\begin_inset Quotes erd
\end_inset
that the light will scatter many times, each time sacrifying a part of
its energy to the ohmic losses.
\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/inf.pdf
width 45text%
\end_inset
\begin_inset Graphics
filename figs/inf_big_px.pdf
width 45text%
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
Response of an infinite square array of silver nanoparticles with periodicities
\begin_inset Formula $p_{x}=p_{y}=375\,\mathrm{nm}$
\end_inset
to plane waves incident in the
\begin_inset Formula $xz$
\end_inset
-plane.
Left:
\begin_inset Formula $y$
\end_inset
-polarised waves, right:
\begin_inset Formula $x$
\end_inset
-polarised waves.
The images show extinction, scattering and absorption cross section per
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
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%
\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 $p_{x}=p_{y}=375\,\mathrm{nm}$
\end_inset
.
In all cases, the array is illuminated by plane waves linearly polarised
in the
\begin_inset Formula $y$
\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"
noprefix "false"
\end_inset
were computed with quadrupole truncation
\begin_inset Formula $l\le2$
\end_inset
and using the decomposition into the eight irreducible representations
of group
\begin_inset Formula $D_{2h}$
\end_inset
.
The
\begin_inset Formula $100\times100$
\end_inset
array took about 4 h to compute on Dell PowerEdge C4130 with 12 core Xeon
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.
In any case, there is still room for optimisation in the QPMS suite.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
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
\end_inset
\end_layout
\begin_layout Subsection
Lattice mode structure
\end_layout
\begin_layout Subsection
Square lattice
\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 $\vect k=0$
\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_f2.112.23eV_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