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Merge branch 'article_wrk' into article 

Former-commit-id: 91ffb5d42066adae1d3193b9c5125f2d546d3314
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Marek Nečada 2020-06-16 22:34:12 +03:00
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57
lepaper/arrayscat.lyx
View File

@ -491,21 +491,12 @@ These are compatibility macros for the (...)-old files:
\end_layout
\begin_layout Title
Many-particle
\begin_inset Formula $T$
\end_inset
-matrix simulations in finite and infinite systems of electromagnetic scatterers
\begin_inset Marginal
status open
\begin_layout Plain Layout
(TODO better title)
\end_layout
\end_inset
-matrix simulations for nanophotonics: symmetries, scattering and lattice
modes
\end_layout
\begin_layout Standard
@ -521,7 +512,7 @@ Multiple-scattering
\end_layout
\begin_layout Itemize
Multiple-scattering
Many-particle
\begin_inset Formula $T$
\end_inset
@ -529,6 +520,13 @@ Multiple-scattering
modes.
\end_layout
\begin_layout Itemize
\begin_inset Formula $T$
\end_inset
-matrix simulations in finite and infinite systems of electromagnetic scatterers
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
@ -617,8 +615,13 @@ The T-matrix multiple scattering method (TMMSM) can be used to solve the
\begin_layout Abstract
Here we extend the method to infinite periodic structures using Ewald-type
lattice summation, and we exploit the possible symmetries of the structure
to further improve its efficiency.
to further improve its efficiency, so that systems containing tens of thousands
of particles can be studied with relative ease.
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Marginal
status open
@ -629,6 +632,11 @@ Should I mention also the cross sections formulae in abstract / intro?
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Abstract
@ -794,6 +802,10 @@ Maybe put the numerical results separately in the end.
\end_layout
\begin_layout Section*
\begin_inset Note Note
status open
\begin_layout Section*
TODO
\end_layout
@ -821,6 +833,12 @@ Truncation notation.
Example results and benchmarks with BEM; figures!
\end_layout
\begin_deeper
\begin_layout Itemize
Given up for BEM, SCUFF-EM too unreliable.
\end_layout
\end_deeper
\begin_layout Itemize
Carefully check the transformation directions in sec.
@ -841,7 +859,18 @@ Check whether everything written is correct also for non-symmorphic space
groups.
\end_layout
\begin_deeper
\begin_layout Itemize
Given up
\end_layout
\end_deeper
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open

577
lepaper/examples.lyx
View File

@ -105,8 +105,17 @@ name "sec:Applications"
\end_layout
\begin_layout Standard
Finally, we present some results obtained with the QPMS suite as well as
benchmarks with BEM.
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
@ -115,20 +124,16 @@ examples
\end_layout
\begin_layout Subsection
Response of a rectangular nanoplasmonic array
Optical response of a square array; finite size effects
\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 $p_{y}=\SI{571}{nm}$
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
@ -136,7 +141,30 @@ Our first example deals with a plasmonic array made of golden nanoparticles
\end_inset
.
For gold, we use the optical properties listed in
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"
@ -145,6 +173,15 @@ 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
@ -157,41 +194,93 @@ Show the mesh as well?
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
We consider finite arrays with
\begin_inset Formula $N_{x}\times N_{y}=\ldots\times\ldots,\ldots\times\ldots,\ldots\times\ldots$
\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 circularly polarised plane waves with energies
from xx to yy and incidence direction lying in the
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.
The results are shown in Figure
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"
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
\begin_inset Marginal
status open
for the interparticle interactions, although there is no visible difference
if we use
\begin_inset Formula $l_{\mathrm{max}}=2$
\end_inset
\begin_layout Plain Layout
Mention lMax.
\end_layout
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
@ -201,37 +290,49 @@ 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
Absorption of rectangular arrays of golden nanoparticles with periodicities
Response of an infinite square array of silver nanoparticles with periodicities
\begin_inset Formula $p_{x}=\SI{621}{nm}$
\begin_inset Formula $p_{x}=p_{y}=375\,\mathrm{nm}$
\end_inset
,
\begin_inset Formula $p_{y}=\SI{571}{nm}$
to plane waves incident in the
\begin_inset Formula $xz$
\end_inset
with a)
\begin_inset Formula $\ldots\times\ldots$
-plane.
Left:
\begin_inset Formula $y$
\end_inset
, b)
\begin_inset Formula $\ldots\times\ldots$
-polarised waves, right:
\begin_inset Formula $x$
\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.
-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"
name "fig:Example rectangular absorption infinite"
\end_inset
@ -245,19 +346,114 @@ name "fig:Example rectangular absorption"
\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
\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 $\left(\vect k,\omega\right)$
@ -283,17 +479,283 @@ TODO also details about the machines used.
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Modes of a rectangular nanoplasmonic array
Lattice mode structure
\end_layout
\begin_layout Subsection
Square lattice
\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.
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_f2.11–2.23eV_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
@ -306,6 +768,11 @@ literal "false"
.
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Effects of multipole cutoff
\end_layout

47
lepaper/infinite.lyx
View File

@ -379,7 +379,13 @@ noprefix "false"
describes the
\emph on
lattice modes.
lattice modes
\emph default
, i.e.
electromagnetic excitations that can sustain themselves for prolonged time
even without external driving
\emph on
.
\emph default
Non-trivial solutions to
@ -1724,7 +1730,32 @@ One pecularity of the two-dimensional case is the two-branchedness of the
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
\end_inset
appearing in the long-range part.
appearing in the long-range part (in the cases
\begin_inset Formula $d=1,3$
\end_inset
the function
\begin_inset Formula $\gamma\left(z\right)$
\end_inset
appears with even powers, and
\begin_inset Formula $\Gamma\left(-j,z\right)$
\end_inset
is meromorphic for integer
\begin_inset Formula $j$
\end_inset
\begin_inset CommandInset citation
LatexCommand cite
after "8.2.9"
key "NIST:DLMF"
literal "false"
\end_inset
).
As a consequence, if we now explicitly label the dependence on the wavenumber,
\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\kappa,\vect k,\vect s\right)$
@ -2108,6 +2139,18 @@ If we assume that
is chosen to represent the (rough) maximum tolerated magnitude of the summand
with regard to target accuracy.
This adjustment means that, in worst-case scenario, with growing wavenumber
one has to include an increasing number of terms in the long-range sum
in order to achieve a given accuracy, the number of terms being proportional
to
\begin_inset Formula $\left|\kappa\right|^{d}$
\end_inset
where
\begin_inset Formula $d$
\end_inset
is the dimension of the lattice.
\begin_inset Note Note
status open

14
lepaper/intro.lyx
View File

@ -540,8 +540,18 @@ reference "sec:Applications"
\end_inset
shows some practical results that can be obtained using QPMS and benchmarks
with BEM.
shows some practical results that can be obtained using QPMS.
\begin_inset Note Note
status open
\begin_layout Plain Layout
and benchmarks with BEM.
\end_layout
\end_inset
\begin_inset Note Note
status open