998 lines
16 KiB
Plaintext
998 lines
16 KiB
Plaintext
#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\end_header
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\begin_body
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\begin_layout Standard
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\lang finnish
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\begin_inset FormulaMacro
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\newcommand{\ket}[1]{\left|#1\right\rangle }
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\bra}[1]{\left\langle #1\right|}
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{\boldsymbol{#1}}}
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{\boldsymbol{\mathbf{#1}}}
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\end_inset
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\end_layout
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\begin_layout Title
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Technical notes on quantum electromagnetic multiple scattering
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\end_layout
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\begin_layout Author
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Marek Nečada
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\end_layout
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\begin_layout Affiliation
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COMP Centre of Excellence, Department of Applied Physics, Aalto University,
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P.O.
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Box 15100, Fi-00076 Aalto, Finland
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\end_layout
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\begin_layout Date
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\begin_inset ERT
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status open
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\begin_layout Plain Layout
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\backslash
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today
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Abstract
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...
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\end_layout
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\begin_layout Section
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Theory of quantum electromagnetic multiple scattering
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\end_layout
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\begin_layout Subsection
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Incoherent pumping
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\end_layout
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\begin_layout Standard
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Cf.
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Wubs
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\begin_inset CommandInset citation
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LatexCommand cite
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key "wubs_multiple-scattering_2004"
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\end_inset
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, Delga
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\begin_inset CommandInset citation
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LatexCommand cite
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key "delga_quantum_2014,delga_theory_2014"
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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General initial states
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\end_layout
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\begin_layout Standard
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Look at
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\begin_inset CommandInset citation
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LatexCommand cite
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key "landau_computational_2015"
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\end_inset
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for an inspiration for solving the LS equation with an arbitrary initial
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state.
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\end_layout
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\begin_layout Section
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Computing classical Green's functions
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\end_layout
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\begin_layout Subsection
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Boundary element method
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\end_layout
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\begin_layout Subsection
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T-Matrix method
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\end_layout
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\begin_layout Subsection
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T-Matrix resummation (multiple scatterers)
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\end_layout
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\begin_layout Subsection
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BEM→TM
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\end_layout
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\begin_layout Standard
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Cf.
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SCUFF-TMATRIX (
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sub:SCUFF-TMATRIX"
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\end_inset
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)
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\end_layout
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\begin_layout Section
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Available software
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\end_layout
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\begin_layout Itemize
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TODO which of them can calculate the VSWF translation coefficients?
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\end_layout
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\begin_layout Subsection
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SCUFF-EM
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\begin_inset CommandInset citation
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LatexCommand cite
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key "reid_scuff-em_2015"
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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\family typewriter
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SCUFF-TMATRIX
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\family default
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\begin_inset CommandInset label
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LatexCommand label
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name "sub:SCUFF-TMATRIX"
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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\family typewriter
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SCUFF-SCATTER
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\family default
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\begin_inset CommandInset label
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LatexCommand label
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name "sub:SCUFF-SCATTER"
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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Caveats
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\end_layout
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\begin_layout Description
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Units.
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\family typewriter
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SCUFF-SCATTER
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\family default
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's Angular frequencies specified using the
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\family typewriter
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--Omega
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\family default
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or
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\family typewriter
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--OmegaFile
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\family default
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arguments are interpreted in units of
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\begin_inset Formula $c/1\,\mathrm{μm}=3\cdot10^{14}\,\mathrm{rad/s}$
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\end_inset
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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\family typewriter
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\begin_inset CommandInset href
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LatexCommand href
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name "http://homerreid.dyndns.org/scuff-EM/scuff-scatter/scuffScatterExamples.shtml"
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target "http://homerreid.dyndns.org/scuff-EM/scuff-scatter/scuffScatterExamples.shtml"
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\end_inset
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\end_layout
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\end_inset
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.
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\emph on
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TODO what are the output units?
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\end_layout
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\begin_layout Subsection
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MSTM
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\begin_inset CommandInset citation
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LatexCommand cite
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key "mackowski_mstm_2013"
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\end_inset
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\end_layout
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\begin_layout Itemize
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The incident field is a gaussian beam or a plane wave in the vanilla code
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(no multipole radiation as input!).
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\end_layout
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\begin_layout Itemize
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The bulk of the useful code is in the
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\family typewriter
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mstm-modules-v3.0.f90
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\family default
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file.
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\end_layout
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\begin_layout Itemize
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For solving the interaction equations
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(14)"
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key "mackowski_mstm_2013"
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\end_inset
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, the BCGM (biconjugate gradient method) is used.
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(According to Wikipedia, this method is numerically unstable but has a
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stabilized version (stabilized BCGM).)
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\end_layout
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\begin_layout Itemize
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According to the manual
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\begin_inset CommandInset citation
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LatexCommand cite
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after "2.3"
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key "mackowski_mstm_2013"
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\end_inset
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, they use some method (rotational-axial translation decomposition of the
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translation operation), which
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\begin_inset Quotes eld
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\end_inset
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reduces the operation from an
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\begin_inset Formula $L_{S}^{4}$
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\end_inset
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process to
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\begin_inset Formula $L_{S}^{3}$
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\end_inset
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process where
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\begin_inset Formula $L_{S}$
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\end_inset
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is the truncation order of the expansion
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\begin_inset Quotes erd
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\end_inset
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(more details can probably be found at
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\begin_inset CommandInset citation
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LatexCommand cite
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after "around (68)"
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key "mackowski_calculation_1996"
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\end_inset
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.
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\emph on
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Not sure if this holds also for nonspherical particles, I should either
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read carefully
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\emph default
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\begin_inset CommandInset citation
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LatexCommand cite
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key "mackowski_calculation_1996"
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\end_inset
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\emph on
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or look into
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\begin_inset CommandInset citation
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LatexCommand cite
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key "mishchenko_electromagnetic_2003"
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\end_inset
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which is also cited in the manual.
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\end_layout
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\end_deeper
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\begin_layout Itemize
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By default spheres, it is possible to add own T-Matrix coefficients instead.
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\emph on
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Is it then possible to insert a T-Matrix of an arbitrary shape, or is it
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somehow limited to
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\begin_inset Quotes eld
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\end_inset
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spherical-like
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\begin_inset Quotes erd
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\end_inset
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particles?
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\end_layout
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\end_deeper
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\begin_layout Itemize
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Why the heck are the T-matrix options listed in the
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\begin_inset Quotes eld
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\end_inset
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Options for random orientation calculations
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\begin_inset Quotes erd
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\end_inset
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? Well, it seems that for fixed orientation, it is not possible to specify
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the T-matrix, cf.
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the description of
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\family typewriter
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fixed_or_random_orientation
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\family default
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option in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "3.2.3"
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key "mackowski_mstm_2013"
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection
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Interesting subroutines
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\end_layout
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\begin_layout Itemize
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\family typewriter
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rottranfarfield
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\family default
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: it states
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\begin_inset Quotes eld
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\end_inset
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far field formula for outgoing vswf translation
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\begin_inset Quotes erd
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\end_inset
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.
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What is that and how does it differ from whatever else (near field?) formula?
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\end_layout
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\begin_layout Subsection
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py_gmm
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\begin_inset CommandInset citation
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LatexCommand cite
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key "pellegrini_py_gmm_2015"
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\end_inset
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\end_layout
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\begin_layout Itemize
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Fortran code, already (partially) pythonized using
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\family typewriter
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f2py
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\family default
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by the authors(?); under GNU GPLv3.
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This could save my day.
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\end_layout
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\begin_layout Itemize
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Lots of unnecessary code duplication (see e.g.
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\family typewriter
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coeff_sp2
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\family default
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and
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\family typewriter
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coeff_sp2_dip
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\family default
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subroutines).
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\end_layout
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\begin_layout Itemize
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Has comments!!! (Sometimes they are slightly inaccurate due to the copy-pasting,
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but it is still one of the most readable FORTRAN codes I have seen.)
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\end_layout
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\begin_layout Itemize
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The subroutines seem not to be bloated with dependencies on static/global
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variables, so they should be quite easily reusable.
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\end_layout
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\begin_layout Itemize
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The FORTRAN code was apparently used in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "pellegrini_interacting_2007"
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\end_inset
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.
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Uses the multiple-scattering formalism described in
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\begin_inset CommandInset citation
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LatexCommand cite
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key "xu_efficient_1998"
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection
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Interesting subroutines
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\end_layout
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\begin_layout Standard
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Mie scattering:
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\end_layout
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\begin_layout Itemize
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\family typewriter
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coeff_sp2
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\family default
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: calculation of the Mie scattering coefficients (
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\begin_inset Formula $\overline{a}_{n}^{l},\overline{b}_{n}^{l}$
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\end_inset
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as in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(1), (2), \\ldots"
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key "pellegrini_py_gmm_2015"
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\end_inset
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), for a set of spheres (therefore all the parameters have +1 dimension).
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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What does the input parameter
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\family typewriter
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v_req
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\family default
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(
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\emph on
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vettore raggi equivalenti
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\emph default
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) mean?
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\end_layout
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\begin_layout Itemize
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How do I put in the environment permittivity?
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\end_layout
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\begin_layout Itemize
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\family typewriter
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m_epseq
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\family default
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are real and imaginary parts of the permittivity (which are then transformed
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into complex
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\family typewriter
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v_epsc
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\family default
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)
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\end_layout
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\begin_layout Itemize
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\family typewriter
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ref_index
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\family default
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is the environment refractive index (called
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\family typewriter
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n_matrix
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\family default
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in the example ipython notebook)
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\end_layout
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\begin_layout Itemize
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\family typewriter
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v_req
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\family default
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are the sphere radii?
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\end_layout
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\begin_layout Itemize
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\family typewriter
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nstop
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\family default
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is the maximum order of the
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\begin_inset Formula $n$
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\end_inset
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-expansion
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\end_layout
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\begin_layout Itemize
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\family typewriter
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neq
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\family default
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is ns, number of spheres for which the calculation is performed apparently,
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it is connected to some
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\begin_inset Quotes eld
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\end_inset
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dirty hack to interface fortran and python properly
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\begin_inset Quotes erd
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\end_inset
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(cf.
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\family typewriter
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gmm_f2py_module.f90
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\family default
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)
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\end_layout
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\end_deeper
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\begin_layout Section
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Code integration
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\end_layout
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\begin_layout Section
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Testing and reproduction of foreign results
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\end_layout
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\begin_layout Subsection
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Delga PRL
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\begin_inset CommandInset citation
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LatexCommand cite
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key "delga_quantum_2014"
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\end_inset
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\end_layout
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\begin_layout Subsubsection
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Parameters
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\end_layout
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\begin_layout Itemize
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Surrounding lossless dielectric
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\series bold
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medium
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\series default
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with permittivity
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\begin_inset Formula $\epsilon_{d}=2.13$
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\end_inset
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.
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\end_layout
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\begin_layout Itemize
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\series bold
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QEs:
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\series default
|
|
dipole moment
|
|
\begin_inset Formula $\mu=0.19\, e\cdot\mathrm{nm}=9.12\,\mathrm{D}$
|
|
\end_inset
|
|
|
|
, count
|
|
\begin_inset Formula $N\in\left\{ 1,50,100,200\right\} $
|
|
\end_inset
|
|
|
|
, radial orientation,
|
|
\begin_inset Formula $h=1\,\mathrm{nm}$
|
|
\end_inset
|
|
|
|
above the sphere (except for Fig.
|
|
5 where variable), natural frequency
|
|
\begin_inset Formula $\Omega_{n}=\omega_{0}-i\gamma_{\mathrm{QE}}/2,$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $\omega_{0}=$
|
|
\end_inset
|
|
|
|
varies,
|
|
\begin_inset Formula $\gamma_{\mathrm{QE}}=15\,\mathrm{meV}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
|
|
\series bold
|
|
Sphere:
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
radius
|
|
\begin_inset Formula $a=7\,\mathrm{nm}$
|
|
\end_inset
|
|
|
|
,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Drude model
|
|
\begin_inset Formula $\epsilon_{m}(\omega)=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega\left(\omega+i\gamma_{p}\right)}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
Drude parameters
|
|
\begin_inset Formula $\omega_{p}=9\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $\epsilon_{\infty}=4.6$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $\gamma_{p}=0.1\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Itemize
|
|
background permittivity
|
|
\begin_inset Formula $\epsilon_{d}(\omega)=2.13$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
(approximate?; not really a parameter) LSP resonances
|
|
\begin_inset Formula $\omega_{l}=\omega_{p}/\sqrt{\epsilon_{\infty}+\left(1+1/l\right)\epsilon_{d}}$
|
|
\end_inset
|
|
|
|
; particularly,
|
|
\begin_inset Formula $\omega_{1}\approx3.0236\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $\omega_{2}\approx3.2236\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $\omega_{3}\approx3.30\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $\omega_{4}\approx3.34\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $\omega_{5}\approx3.364\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $\omega_{\infty}\approx3.4692\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Itemize
|
|
|
|
\series bold
|
|
Detector:
|
|
\series default
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
Far field:
|
|
\begin_inset Formula $1\,\mathrm{\mu m}$
|
|
\end_inset
|
|
|
|
away from the center of the nanoparticle along the
|
|
\begin_inset Formula $y$
|
|
\end_inset
|
|
|
|
axis (Fig.
|
|
3).
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Near field: position not specified in the paper; but in Fig.
|
|
4(b) there are
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
polarization spectra
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
instead of
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
light spectra
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
(eq.
|
|
4) in Fig.
|
|
4(a).
|
|
Does this mean that they are evaluated somewhere in/on the sphere? Or in
|
|
the particle? The latter is likely, as it is given by
|
|
\begin_inset Formula $P_{n}\left(\omega\right)=\left\langle \sigma_{n}^{+}\left(-\omega\right)\sigma_{n}^{-}(\omega)\right\rangle $
|
|
\end_inset
|
|
|
|
(cf.
|
|
the column below Fig.
|
|
3).
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Subsubsection
|
|
Testing
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In my
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
old
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
code, there no splitting observable around
|
|
\begin_inset Formula $\omega\approx\omega_{0}\approx\omega_{\infty}\approx3.46\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
.
|
|
This is perhaps because the couplings to the higher multipoles is miscalculated
|
|
(too small).
|
|
No splitting around the NP dipole (
|
|
\begin_inset Formula $\approx3,02\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
) should be OK for single QE in far field (cf.
|
|
Fig.
|
|
3).
|
|
And there are yet the
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
switched axes
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
...
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
If I set the dipole reflection coefficients RH[1], RV[1] to zero, and multiply
|
|
the the quadrupole reflection coefficients RH[2], RV[2] by
|
|
\begin_inset Formula $10^{6}$
|
|
\end_inset
|
|
|
|
, the peak at
|
|
\begin_inset Formula $3.0\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
dissapears and a tiny(!) peak appears around the (expected) position of
|
|
|
|
\begin_inset Formula $3.0\,\mathrm{eV}$
|
|
\end_inset
|
|
|
|
.
|
|
Have I fucked up the Mie reflection coefficients? Sounds like if I forgot
|
|
a factor of
|
|
\begin_inset Formula $c$
|
|
\end_inset
|
|
|
|
somewhere.
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Delga JoO
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "delga_theory_2014"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Parameters
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
|
|
\series bold
|
|
QEs:
|
|
\series default
|
|
dipole moment
|
|
\begin_inset Formula $\mu=0.38\, e\cdot\mathrm{nm}=18.24\,\mathrm{D}$
|
|
\end_inset
|
|
|
|
(double), otherwise the same parameters as in
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "delga_quantum_2014"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
|
|
\series bold
|
|
Sphere:
|
|
\series default
|
|
as in
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "delga_quantum_2014"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
|
|
\series bold
|
|
Detector:
|
|
\series default
|
|
not stated in the paper
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
|
|
\series bold
|
|
Numerics:
|
|
\series default
|
|
looking at the leftmost ball in Fig.
|
|
3, it seems that their SVW cutoff is around 12.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
TODO
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Päivi's suggestion: suppress the dipole and let it interact only with the
|
|
higher multipoles.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset bibtex
|
|
LatexCommand bibtex
|
|
bibfiles "dipdip"
|
|
options "apsrev"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|