From d53214e3bfd41c44875f123cc27c551146c2413e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 8 Jun 2020 06:02:46 +0300 Subject: [PATCH] WIP examples Former-commit-id: b7b33fa78bea10655c9a447f8b9c1f20ce9a3edd --- lepaper/arrayscat.lyx | 21 ++ lepaper/examples.lyx | 581 +++++++++++++++++++++++++++++++++++++----- lepaper/infinite.lyx | 12 + lepaper/intro.lyx | 14 +- 4 files changed, 569 insertions(+), 59 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 3e19445..46b563c 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -794,6 +794,10 @@ 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 +825,12 @@ 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. @@ -841,6 +851,17 @@ 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 Itemize \begin_inset Note Note status open diff --git a/lepaper/examples.lyx b/lepaper/examples.lyx index 056c0d1..33b18ec 100644 --- a/lepaper/examples.lyx +++ b/lepaper/examples.lyx @@ -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" @@ -145,7 +173,16 @@ literal "false" \end_inset interpolated with cubical splines. - The particles' cylindrical shape is approximated with a triangular mesh +\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 @@ -157,42 +194,94 @@ 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 -. - -\begin_inset Marginal -status open + 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_layout Plain Layout -Mention lMax. -\end_layout + 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 @@ -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 diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 4f5683b..8d3ac86 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -2081,6 +2081,18 @@ 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 diff --git a/lepaper/intro.lyx b/lepaper/intro.lyx index 6da2b52..34b4c97 100644 --- a/lepaper/intro.lyx +++ b/lepaper/intro.lyx @@ -476,8 +476,18 @@ 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