WIP examples
Former-commit-id: b7b33fa78bea10655c9a447f8b9c1f20ce9a3edd
This commit is contained in:
Marek Nečada
parent
07fe3a4645
commit
d53214e3bf
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@ -794,6 +794,10 @@ Maybe put the numerical results separately in the end.
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\end_layout
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\begin_layout Section*
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\begin_inset Note Note
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status open
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\begin_layout Section*
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TODO
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\end_layout
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@ -821,6 +825,12 @@ Truncation notation.
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Example results and benchmarks with BEM; figures!
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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Given up for BEM, SCUFF-EM too unreliable.
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\end_layout
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\end_deeper
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\begin_layout Itemize
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Carefully check the transformation directions in sec.
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@ -841,6 +851,17 @@ Check whether everything written is correct also for non-symmorphic space
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groups.
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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Given up
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\end_layout
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\end_deeper
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\end_inset
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\end_layout
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\begin_layout Itemize
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\begin_inset Note Note
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status open
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@ -105,8 +105,17 @@ name "sec:Applications"
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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 as well as
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benchmarks with BEM.
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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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@ -115,20 +124,16 @@ examples
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\end_layout
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\begin_layout Subsection
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Response of a rectangular nanoplasmonic array
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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 golden nanoparticles
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placed in a rectangular planar configuration.
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The nanoparticles have shape of right circular cylinder with radius 50
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nm and height 50 nm.
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The particles are placed with periodicities
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\begin_inset Formula $p_{x}=\SI{621}{nm}$
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\end_inset
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,
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\begin_inset Formula $p_{y}=\SI{571}{nm}$
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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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@ -136,7 +141,30 @@ Our first example deals with a plasmonic array made of golden nanoparticles
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\end_inset
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.
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For gold, we use the optical properties listed in
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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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@ -145,7 +173,16 @@ literal "false"
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\end_inset
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interpolated with cubical splines.
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The particles' cylindrical shape is approximated with a triangular mesh
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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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@ -157,42 +194,94 @@ Show the mesh as well?
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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 $N_{x}\times N_{y}=\ldots\times\ldots,\ldots\times\ldots,\ldots\times\ldots$
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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 circularly polarised plane waves with energies
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from xx to yy and incidence direction lying in the
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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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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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plane.
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The results are shown in Figure
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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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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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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Example rectangular absorption"
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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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.
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\begin_inset Marginal
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status open
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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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\begin_layout Plain Layout
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Mention lMax.
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\end_layout
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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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instead due to the small size of the particles.
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In Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Example rectangular absorption size comparison"
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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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, 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.
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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
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\begin_inset Quotes eld
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\end_inset
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likely
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\begin_inset Quotes erd
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\end_inset
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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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\begin_inset Float figure
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placement document
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alignment document
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@ -201,37 +290,49 @@ sideways false
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename figs/inf.pdf
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width 45text%
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\end_inset
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\begin_inset Graphics
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filename figs/inf_big_px.pdf
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width 45text%
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\end_inset
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Absorption of rectangular arrays of golden nanoparticles with periodicities
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Response of an infinite square array of silver nanoparticles with periodicities
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\begin_inset Formula $p_{x}=\SI{621}{nm}$
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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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\begin_inset Formula $p_{y}=\SI{571}{nm}$
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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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with a)
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\begin_inset Formula $\ldots\times\ldots$
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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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, b)
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\begin_inset Formula $\ldots\times\ldots$
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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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, c)
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\begin_inset Formula $\ldots\times\ldots$
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\end_inset
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and d) infinitely many particles, irradiated by circularly polarised plane
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waves.
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e) Absoption profile of a single nanoparticle.
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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.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:Example rectangular absorption"
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name "fig:Example rectangular absorption infinite"
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\end_inset
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@ -245,19 +346,114 @@ name "fig:Example rectangular absorption"
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\end_inset
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We compared the
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\begin_inset Formula $\ldots\times\ldots$
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\end_inset
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case with a purely BEM-based solution obtained using the
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\family typewriter
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scuff-scatter
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\family default
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utility.
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TODO WHAT DO WE GET?
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\end_layout
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\begin_layout Standard
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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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\begin_layout Plain Layout
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\end_layout
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename figs/sqlat_scattering_cuts.pdf
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width 90col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Comparison of optical responses of differently sized square arrays of silver
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nanoparticles with the same 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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.
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In all cases, the array is illuminated by plane waves linearly polarised
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in the
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\begin_inset Formula $y$
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\end_inset
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-direction, with constant frequency
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\begin_inset Formula $2.15\,\mathrm{eV}/\hbar$
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\end_inset
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.
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The cross sections are normalised by the total number of particles in the
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array.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:Example rectangular absorption size comparison"
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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 Plain Layout
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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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The finite-size cases in Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Example rectangular absorption size comparison"
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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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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.
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For smaller systems, the computation time decreases quickly, as the main
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bottleneck is the LU factorisation.
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In any case, there is still room for optimisation in the QPMS suite.
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\end_layout
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\begin_layout Standard
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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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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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@ -283,17 +479,283 @@ TODO also details about the machines used.
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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 Subsection
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Modes of a rectangular nanoplasmonic array
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Lattice mode structure
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\end_layout
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\begin_layout Subsection
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Square lattice
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\end_layout
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\begin_layout Standard
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Next, we study the eigenmode problem of the same rectangular arrays.
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The system is lossy, therefore the eigenfrequencies are complex and we
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need to have a model of the material optical properties also for complex
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frequencies.
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Next, we study the lattice mode problem of the same square arrays.
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First we consider the mode problem exactly at the
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\begin_inset Formula $\Gamma$
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\end_inset
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point,
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\begin_inset Formula $\vect k=0$
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\end_inset
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.
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Before proceeding with more sophisticated methods, it is often helpful
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to look at the singular values of mode problem matrix
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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\end_inset
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from the lattice mode equation
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:lattice mode equation"
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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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, as shown in Fig.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:square lattice real interval SVD"
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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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.
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This can be always done, even with tabulated/interpolated material properties
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and/or
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\begin_inset Formula $T$
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\end_inset
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-matrices.
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An additional insight, especially in the high-symmetry points of the Brillouin
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zone, is provided by decomposition of the matrix into irreps – in this
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case of group
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\begin_inset Formula $D_{4h}$
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\end_inset
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, which corresponds to the point group symmetry of the array at the
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\begin_inset Formula $\Gamma$
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\end_inset
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point.
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Although on the picture none of the SVDs hits manifestly zero, we see two
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prominent dips in the
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\begin_inset Formula $E'$
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\end_inset
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and
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\begin_inset Formula $A_{2}''$
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\end_inset
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irrep subspaces, which is a sign of an actual solution nearby in the complex
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plane.
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Moreover, there might be some less obvious minima in the very vicinity
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of the diffracted order crossing which do not appear in the picture due
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to rough frequency sampling.
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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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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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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
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width 80col%
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Singular values of the mode problem matrix
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\begin_inset Formula $\truncated{M\left(\omega,\vect k=0\right)}3$
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\end_inset
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for a real frequency interval.
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The irreducible representations of
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\begin_inset Formula $D_{4h}$
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\end_inset
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are labeled with different colors.
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The density of the data points on the horizontal axis is
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\begin_inset Formula $1/\mathrm{meV}$
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\end_inset
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.
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:square lattice real interval SVD"
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\end_inset
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\end_layout
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|
||||
\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
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
|
|
|||
|
|
@ -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
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue