Examples draft
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Marek Nečada
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@ -10,6 +10,22 @@
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/HES6WJTP/(Wiley science paperback series) Craig F. Bohren, Donald R. Huffman-Absorption and scattering of light by small particles-Wiley-VCH (1998).djvu}
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}
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@article{johnson_optical_1972,
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title = {Optical {{Constants}} of the {{Noble Metals}}},
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volume = {6},
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abstract = {The optical constants n and k were obtained for the noble metals (copper, silver, and gold) from reflection and transmission measurements on vacuum-evaporated thin films at room temperature, in the spectral range 0.5-6.5 eV. The film-thickness range was 185-500 {\AA}. Three optical measurements were inverted to obtain the film thickness d as well as n and k. The estimated error in d was {$\pm$} 2 {\AA}, and that in n, k was less than 0.02 over most of the spectral range. The results in the film-thickness range 250-500 {\AA} were independent of thickness, and were unchanged after vacuum annealing or aging in air. The free-electron optical effective masses and relaxation times derived from the results in the near infrared agree satisfactorily with previous values. The interband contribution to the imaginary part of the dielectric constant was obtained by subtracting the free-electron contribution. Some recent theoretical calculations are compared with the results for copper and gold. In addition, some other recent experiments are critically compared with our results.},
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number = {12},
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urldate = {2015-10-06},
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journal = {Phys. Rev. B},
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doi = {10.1103/PhysRevB.6.4370},
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url = {http://link.aps.org/doi/10.1103/PhysRevB.6.4370},
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author = {Johnson, P. B. and Christy, R. W.},
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month = dec,
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year = {1972},
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pages = {4370-4379},
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file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/ANQIIJA5/PhysRevB.6.html}
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}
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@misc{SCUFF2,
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title = {{{SCUFF}}-{{EM}}},
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url = {http://homerreid.dyndns.org/scuff-EM/},
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@ -99,6 +99,12 @@
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\begin_body
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\SI}[2]{}
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{#1\,\mathrm{#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\uoft}[1]{\mathfrak{F}#1}
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\end_inset
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@ -747,8 +753,8 @@ 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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TODOs
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\begin_layout Section*
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TODO
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\end_layout
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\begin_layout Itemize
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@ -104,5 +104,214 @@ 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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Scripts to reproduce these results are available under the
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\family typewriter
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examples
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\family default
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directory of the QPMS source repository.
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The benchmarks require SCUFF-EM of version xxx
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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Add the version when possible.
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\end_layout
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\end_inset
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or newer.
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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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\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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\end_inset
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into an isotropic medium with a constant refraction index
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\begin_inset Formula $n=1.52$
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\end_inset
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.
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For gold, we use the optical properties listed in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "johnson_optical_1972"
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literal "false"
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\end_inset
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interpolated with cubical splines.
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The particles' cylindrical shape is approximated with a triangular mesh
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with XXX boundary elements.
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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Show the mesh as well?
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\end_layout
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\end_inset
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\end_layout
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\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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\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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\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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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Example rectangular absorption"
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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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\begin_layout Plain Layout
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Mention lMax.
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\end_layout
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\end_inset
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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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\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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\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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\end_inset
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with a)
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\begin_inset Formula $\ldots\times\ldots$
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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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\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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\begin_inset CommandInset label
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LatexCommand label
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name "fig:Example rectangular absorption"
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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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\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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In the infinite case, we benchmarked against a pseudorandom selection of
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\begin_inset Formula $\left(\vect k,\omega\right)$
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\end_inset
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pairs and the difference was TODO WHAT? We note that evaluating one
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\begin_inset Formula $\left(\vect k,\omega\right)$
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\end_inset
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pair took xxx miliseconds with MSTMM and truncation degree
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\begin_inset Formula $L=?$
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\end_inset
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, the same took xxx hours with BEM.
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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TODO also details about the machines used.
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More info about time also at least for the largest case.
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\end_layout
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\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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\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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So in this case we use the Drude-Lorentz model for gold with parameters
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as in [TODO REF].
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\end_layout
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\end_body
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\end_document
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@ -318,14 +318,14 @@ noprefix "false"
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and the properties of the gradient operator under coordinate transforms,
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vector spherical harmonics
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\begin_inset Formula $\vsh2lm,\vsh3lm$
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\begin_inset Formula $\vsh 2lm,\vsh 3lm$
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\end_inset
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transform in the same way,
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\begin_inset Formula
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\begin{align*}
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\left(\groupop g\vsh2lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh2l{m'}\left(\uvec r\right),\\
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\left(\groupop g\vsh3lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh3l{m'}\left(\uvec r\right),
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\left(\groupop g\vsh 2lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
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\left(\groupop g\vsh 3lm\right)\left(\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
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\end{align*}
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\end_inset
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@ -337,8 +337,8 @@ status open
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align*}
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\left(\groupop g\vsh2lm\right)\left(\uvec r\right) & =R_{g}\vsh2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh2l{m'}\left(\uvec r\right),\\
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\left(\groupop g\vsh3lm\right)\left(\uvec r\right) & =R_{g}\vsh2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh3l{m'}\left(\uvec r\right),
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\left(\groupop g\vsh 2lm\right)\left(\uvec r\right) & =R_{g}\vsh 2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
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\left(\groupop g\vsh 3lm\right)\left(\uvec r\right) & =R_{g}\vsh 2lm\left(R_{g}^{-1}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
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\end{align*}
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\end_inset
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@ -349,14 +349,14 @@ status open
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\end_inset
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but the remaining set
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\begin_inset Formula $\vsh1lm$
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\begin_inset Formula $\vsh 1lm$
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\end_inset
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transforms differently due to their pseudovector nature stemming from the
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cross product in their definition:
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\begin_inset Formula
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\[
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\left(\groupop g\vsh1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh1l{m'}\left(\uvec r\right),
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\left(\groupop g\vsh 1lm\right)\left(\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 1l{m'}\left(\uvec r\right),
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\]
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\end_inset
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@ -411,8 +411,8 @@ noprefix "false"
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:
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\begin_inset Formula
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\begin{align*}
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\left(\groupop g\vswfouttlm1lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm1l{m'}\left(\vect r\right),\\
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\left(\groupop g\vswfouttlm2lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm2l{m'}\left(\vect r\right),
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\left(\groupop g\vswfouttlm 1lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
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\left(\groupop g\vswfouttlm 2lm\right)\left(\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right),
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\end{align*}
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\end_inset
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@ -581,7 +581,18 @@ noprefix "false"
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\end_inset
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, we have (CHECK THIS CAREFULLY AND EXPLAIN)
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, we have
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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Check this carefully.
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Maybe explain in more detail?
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{multline}
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\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
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