1763 lines
35 KiB
Plaintext
1763 lines
35 KiB
Plaintext
#LyX 2.0 created this file. For more info see http://www.lyx.org/
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\lyxformat 413
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\begin_document
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\begin_header
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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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\end_inset
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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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\begin_inset FormulaMacro
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\newcommand{\uvec}[1]{\mathbf{\boldsymbol{\hat{#1}}}}
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{\boldsymbol{\hat{\mathbf{#1}}}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ud}{\mathrm{d}}
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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 Standard
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The formulae below might differ depending on the conventions used by various
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authors.
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For instance, Taylor
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\begin_inset CommandInset citation
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LatexCommand cite
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key "taylor_optical_2011"
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\end_inset
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uses normalized spherical wavefunctions
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\begin_inset Formula $\widetilde{\vect M}_{mn}^{(j)},\widetilde{\vect N}_{mn}^{(j)}$
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\end_inset
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which are designed in a way that avoids float number overflow of some of
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the variables during the numerical calculation.
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\end_layout
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\begin_layout Standard
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Beware of various conventions in definitions of Legendre functions etc.
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(the implementation in py-gmm differs, for example, by a factor of
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\begin_inset Formula $(-1)^{m}$
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\end_inset
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from scipy.special.lpmn.
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I think this is also the reason that lead to the
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\begin_inset Quotes eld
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\end_inset
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wrong
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\begin_inset Quotes erd
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\end_inset
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signs in the addition coefficients in my code compared to
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\begin_inset CommandInset citation
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LatexCommand cite
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key "xu_calculation_1996"
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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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T-Matrix method
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\end_layout
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\begin_layout Subsubsection
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VSWF decomposition
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\end_layout
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||
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\begin_layout Standard
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Expressions for VSWF in Xu
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||
\begin_inset CommandInset citation
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||
LatexCommand cite
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after "(2)"
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key "xu_electromagnetic_1995"
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\end_inset
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:
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\end_layout
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||
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\begin_layout Standard
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||
\begin_inset Formula
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\begin{eqnarray}
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\vect M_{mn}^{(J)} & = & \left(i\uvec{\theta}\pi_{mn}(\cos\theta)-\uvec{\phi}\tau_{mn}(\cos\theta)\right)z_{n}^{(J)}(kr)e^{im\phi},\nonumber \\
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\vect N_{mn}^{(J)} & = & \uvec rn(n+1)P_{n}^{m}(\cos\theta)\frac{z_{n}^{(J)}(kr)}{kr}e^{im\phi}\label{eq:vswf}\\
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& & +\left(\uvec{\theta}\tau_{mn}(\cos\theta)+i\uvec{\phi}\pi_{mn}(\cos\theta)\right)\nonumber \\
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& & \phantom{+}\times\frac{1}{kr}\frac{\ud\left(rz_{n}^{(J)}(kr)\right)}{\ud r}e^{im\phi},\nonumber \\
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& = & ...\nonumber
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\end{eqnarray}
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\end_inset
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||
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||
where
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||
\begin_inset Formula $z_{n}^{(J)}$
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||
\end_inset
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||
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denotes
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||
\begin_inset Formula $j_{n},y_{n},h_{n}^{+},h_{n}^{-}$
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||
\end_inset
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||
for
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||
\begin_inset Formula $J=1,2,3,4$
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\end_inset
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||
, respectively, and
|
||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\pi_{mn}(\cos\theta) & = & \frac{m}{\sin\theta}P_{n}^{m}(\cos\theta),\\
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\tau_{mn}(\cos\theta) & = & \frac{\ud P_{n}^{m}(\cos\theta)}{\ud\theta}=-\sin\theta\frac{\ud P_{n}^{m}(\cos\theta)}{\ud\cos\theta}.
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\end{eqnarray*}
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||
\end_inset
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||
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The expressions for
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||
\begin_inset Formula $\vect M_{mn}^{(J)},\vect N_{mn}^{(J)}$
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\end_inset
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are dimensionless.
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\end_layout
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||
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\begin_layout Standard
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||
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||
\emph on
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||
Note about the case
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||
\begin_inset Formula $\theta\to0,\pi$
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||
\end_inset
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||
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:
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\emph default
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||
There is a divergent
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||
\begin_inset Formula $1/\sin\theta$
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||
\end_inset
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||
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factor in the
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\begin_inset Formula $\pi_{mn}(\cos\theta)$
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||
\end_inset
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||
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||
function.
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||
For
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||
\begin_inset Formula $m=0$
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||
\end_inset
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||
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||
, it is irrelevant because of the
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||
\begin_inset Formula $m$
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||
\end_inset
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||
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||
factor (it would be bad otherwise, because
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||
\begin_inset Formula $P_{n}^{0}(\cos\theta)$
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||
\end_inset
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||
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||
does not go to zero at
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||
\begin_inset Formula $\theta\to0,\pi$
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||
\end_inset
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||
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).
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||
For
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||
\begin_inset Formula $\left|m\right|\ge2$
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\end_inset
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||
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,
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\begin_inset Formula $P_{n}^{m}(x)$
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||
\end_inset
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behaves as
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||
\begin_inset Formula $o(x+1),o(x-1)$
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||
\end_inset
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||
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at
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||
\begin_inset Formula $-1,1$
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||
\end_inset
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||
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||
, so
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||
\begin_inset Formula $P_{n}^{m}(\cos\theta)$
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||
\end_inset
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||
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||
goes like
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||
\begin_inset Formula $o(\theta^{2}),o\left((\theta-\pi)^{2}\right)$
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||
\end_inset
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||
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at
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||
\begin_inset Formula $0,\pi$
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\end_inset
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||
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||
, which safely eliminates the divergent factor.
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||
However, for
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||
\begin_inset Formula $\left|m\right|=1$
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||
\end_inset
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||
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||
, the whole expression
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||
\begin_inset Formula $P_{n}^{m}(\cos\theta)/\sin\theta$
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||
\end_inset
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||
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||
has a finite nonzero limit for
|
||
\begin_inset Formula $\theta\to0,\pi$
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||
\end_inset
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||
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||
.
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||
According to Mathematica (for
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||
\begin_inset Formula $\theta\to\pi,$
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||
\end_inset
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||
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||
Mathematica does not work well, but it can be derived from the
|
||
\begin_inset Formula $\theta\to0$
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||
\end_inset
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||
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||
case and oddness/evenness).
|
||
|
||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\lim_{\theta\to0}\frac{P_{n}^{1}(\cos\theta)}{\sin\theta} & = & -\frac{1}{2}n(1+n),\qquad\lim_{\theta\to0}\frac{P_{n}^{-1}(\cos\theta)}{\sin\theta}=\frac{1}{2},\\
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\lim_{\theta\to\pi}\frac{P_{n}^{1}(\cos\theta)}{\sin\theta} & = & \frac{\left(-1\right)^{n}}{2}n(1+n),\qquad\lim_{\theta\to\pi}\frac{P_{n}^{-1}(\cos\theta)}{\sin\theta}=\frac{\left(-1\right)^{n+1}}{2}.
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||
\end{eqnarray*}
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||
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||
\end_inset
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||
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||
NOT COMPLETELY SURE ABOUT THE SIGN/NORMALIZATION CONVENTION HERE.
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||
IT HAS TO BE CHECKED.
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||
\end_layout
|
||
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||
\begin_layout Standard
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||
Expansions for the scattered fields are
|
||
\begin_inset CommandInset citation
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||
LatexCommand cite
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||
after "(4)"
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||
key "xu_electromagnetic_1995"
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||
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||
\end_inset
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||
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||
:
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||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\vect E_{s}(j) & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[a_{mn}^{j}\vect N_{mn}^{(3)}+b_{mn}^{j}\vect M_{mn}^{(3)}\right],\\
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\vect H_{s}(j) & = & \frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[b_{mn}^{j}\vect N_{mn}^{(3)}+a_{mn}^{j}\vect M_{mn}^{(3)}\right].
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\end{eqnarray*}
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||
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||
\end_inset
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||
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||
These expansions should be OK in SI units (take the Fourier transform of
|
||
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||
\begin_inset Formula $\nabla\times\vect E=-\partial\vect B/\partial t$
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||
\end_inset
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||
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||
and
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||
\begin_inset Formula $\vect B=\mu\vect H$
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||
\end_inset
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||
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||
).
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||
For internal field of a sphere, the (regular-wave) expansion reads
|
||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\vect E_{I}(j) & = & -\sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[d_{mn}^{j}\vect N_{mn}^{(1)}+c_{mn}^{j}\vect M_{mn}^{(1)}\right],\\
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\vect H_{I}(j) & = & -\frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[c_{mn}^{j}\vect N_{mn}^{(1)}+d_{mn}^{j}\vect M_{mn}^{(1)}\right]
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||
\end{eqnarray*}
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||
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||
\end_inset
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||
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||
(note the minus sign; I am not sure about its role) and the incident field
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||
(incl.
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||
field from the other scatterers) is assumed to have the same regular-wave
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||
form
|
||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\vect E_{i}(j) & = & -\sum_{n=1}^{\infty}\sum_{m=-n}^{n}iE_{mn}\left[p_{mn}^{j}\vect N_{mn}^{(1)}+q_{mn}^{j}\vect M_{mn}^{(1)}\right],\\
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||
\vect H_{i}(j) & = & -\frac{k}{\omega\mu}\sum_{n=1}^{\infty}\sum_{m=-n}^{n}E_{mn}\left[q_{mn}^{j}\vect N_{mn}^{(1)}+p_{mn}^{j}\vect M_{mn}^{(1)}\right].
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||
\end{eqnarray*}
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||
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||
\end_inset
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||
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||
Note that
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||
\begin_inset Formula $k/\omega\mu=\sqrt{\varepsilon_{r}\varepsilon_{0}/\mu_{r}\mu_{0}}=1/\eta_{r}\eta_{0}.$
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||
\end_inset
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||
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||
The
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||
\begin_inset Quotes eld
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||
\end_inset
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||
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||
factor
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||
\begin_inset Quotes erd
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||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $H/E$
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||
\end_inset
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||
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||
is thus
|
||
\begin_inset Formula $-ik/\omega\mu=-i\sqrt{\varepsilon_{r}\varepsilon_{0}/\mu_{r}\mu_{0}}$
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||
\end_inset
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||
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||
, which is important in determining the Mie coefficients.
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||
\end_layout
|
||
|
||
\begin_layout Standard
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||
The common multipole-dependent factor is given by
|
||
\begin_inset Formula
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||
\[
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||
E_{mn}=\left|E_{0}\right|i^{n}(2n+1)\frac{\left(n-m\right)!}{\left(n+m\right)!}
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||
\]
|
||
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||
\end_inset
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||
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||
which
|
||
\begin_inset Quotes eld
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||
\end_inset
|
||
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||
is desired for keeping the formulation of the multisphere scattering theory
|
||
consistent with that of the Mie theory.
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||
It ensures that all the expressions in the multisphere theory turn out
|
||
to be identical to those in the Mie theory when one is dealing with a cluster
|
||
containing only one sphere and illuminated by a single plane wave
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
.
|
||
(According to Bohren&Huffman
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(4.37)"
|
||
key "bohren_absorption_1983"
|
||
|
||
\end_inset
|
||
|
||
, the decomposition of a plane wave reads
|
||
\begin_inset Formula
|
||
\[
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||
\vect E=E_{0}\sum_{n=1}^{\infty}i^{n}\frac{2n+1}{n(n+1)}\left(\vect M_{o1n}^{(1)}-i\vect N_{e1n}^{(1)}\right),
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where the even/odd VSWF and
|
||
\begin_inset Formula $m\ge0$
|
||
\end_inset
|
||
|
||
convention is used.)
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
|
||
\emph on
|
||
It should be possible to just take it away and the abovementioned expansions
|
||
are still consistent, are they not?
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "sec. 4A"
|
||
key "xu_electromagnetic_1995"
|
||
|
||
\end_inset
|
||
|
||
, there are formulae for translation of the plane wave between VSWF with
|
||
different origins.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
o
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Mie scattering
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For the exact form of the coefficients following from the boundary conditions
|
||
on the spherical surface, cf.
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(12–13)"
|
||
key "xu_electromagnetic_1995"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
For the particular case of spherical nanoparticle, it is important that
|
||
they can be written as
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(14–15)"
|
||
key "xu_electromagnetic_1995"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{alignat*}{1}
|
||
a_{mn}^{j} & =R_{n}^{V}p_{mn}^{j},\quad b_{mn}^{j}=R_{n}^{H}q_{mn}^{j},\\
|
||
c_{mn}^{j} & =T_{n}^{H}q_{mn}^{j},\quad d_{mn}^{j}=T_{n}^{V}p_{mn}^{j},
|
||
\end{alignat*}
|
||
|
||
\end_inset
|
||
|
||
in other words, the Mie coefficients do not depend on
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
but solely on
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
(which is not surprising and probably follows from the Wigner-Eckart theorem).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Respecting the conventions for decomposition in the previous section (i.e.
|
||
there is opposite sign in the scattered part), the reflection and
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
transmission
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
coefficients become (adopted from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(4.52--53)"
|
||
key "bohren_absorption_1983"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
R_{n}^{V} & =\frac{a_{n}}{p_{n}}= & \frac{\mu_{e}m^{2}z^{i}ž^{e}-\mu_{i}z^{e}ž^{i}}{\mu_{e}m^{2}z^{i}ž^{s}-\mu_{i}z^{s}ž^{i}}\\
|
||
R_{n}^{H} & =\frac{b_{n}}{q_{n}}= & \frac{\mu_{i}z^{i}ž^{e}-\mu_{e}z^{e}ž^{i}}{\mu_{i}z^{i}ž^{s}-\mu_{e}z^{s}ž^{i}}\\
|
||
T_{n}^{V} & =\frac{d_{n}}{p_{n}}= & \frac{\mu_{i}mz^{e}ž^{s}-\mu_{i}mz^{s}ž^{e}}{\mu_{e}m^{2}z^{i}ž^{s}-\mu_{i}z^{s}ž^{i}}\\
|
||
T_{n}^{H} & =\frac{c_{n}}{q_{n}}= & \frac{\mu_{i}z^{e}ž^{s}-\mu_{i}z^{s}ž^{e}}{\mu_{i}z^{i}ž^{s}-\mu_{e}z^{s}ž^{i}}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\mu_{i}|\mu_{e}$
|
||
\end_inset
|
||
|
||
is (absolute) permeability of the sphere|envinronment,
|
||
\begin_inset Formula $m=k_{i}/k_{e}=\sqrt{\mu_{i}\varepsilon_{i}/\mu_{e}\varepsilon_{e}}$
|
||
\end_inset
|
||
|
||
, and
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
z^{i} & = & z_{n}^{(J_{i}=1)}(k_{i}R)=j_{n}(k_{i}R),\\
|
||
z^{e} & = & z_{n}^{(J_{e})}(k_{e}R),\\
|
||
z^{s} & = & z_{n}^{(J_{s})}(k_{e}R),\\
|
||
ž^{i/e/s} & = & \frac{\ud(k_{i/e/e}R\cdot z_{n}^{(J_{i/e/e})}(k_{i/e/e}R)}{\ud(k_{i/e/e}R)}.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Translation coefficients
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
A quite detailed study can be found in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "xu_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
, I have not read the recenter one
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "xu_efficient_1998"
|
||
|
||
\end_inset
|
||
|
||
which deals with efficient evaluation of Wigner 3jm symbols and Gaunt coefficie
|
||
nts.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
With the VSWF as in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:vswf"
|
||
|
||
\end_inset
|
||
|
||
and translation relations in the form
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(38,39)"
|
||
key "xu_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\vect M_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[A_{mn}^{\mu\nu}\vect M_{mn}^{(1)j}+B_{mn}^{\mu\nu}\vect N_{mn}^{(1)j}\right],\quad r\le d_{lj},\\
|
||
\vect N_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[B_{mn}^{\mu\nu}\vect M_{mn}^{(1)j}+A_{mn}^{\mu\nu}\vect N_{mn}^{(1)j}\right],\quad r\le d_{lj},\\
|
||
\vect M_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[A_{mn}^{\mu\nu}\vect M_{mn}^{(J)j}+B_{mn}^{\mu\nu}\vect N_{mn}^{(J)j}\right],\quad r\ge d_{lj},\\
|
||
\vect N_{\mu\nu}^{(J)l} & = & \sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left[B_{mn}^{\mu\nu}\vect M_{mn}^{(J)j}+A_{mn}^{\mu\nu}\vect N_{mn}^{(J)j}\right],\quad r\ge d_{lj},
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
the translation coefficients (which should in fact be also labeled with
|
||
their origin indices
|
||
\begin_inset Formula $l,j$
|
||
\end_inset
|
||
|
||
) are
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(82,83)"
|
||
key "xu_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{multline*}
|
||
A_{mn}^{\mu\nu}=\\
|
||
\frac{(-1)^{m}i^{\nu+n}(n+2)_{n-1}\left(\nu+2\right)_{\nu+1}(n+\nu+m-\mu)!}{4n(n+\nu+1)_{n+\nu}(n-m)!(\nu+m)!}\\
|
||
\times e^{i(\mu-m)\phi_{lj}}\sum_{q=0}^{q_{\mathrm{max}}}(-1)^{q}\left[n(n+1)+\nu(\nu+1)-p(p+1)\right]\\
|
||
\times\tilde{a}_{1q}\begin{pmatrix}z_{p}^{(J)}(kd_{lj})\\
|
||
j_{p}(kd_{lj})
|
||
\end{pmatrix}P_{p}^{\mu-m}(\cos\theta_{lj}),\qquad\begin{pmatrix}r\le d_{lj}\\
|
||
r\ge d_{lj}
|
||
\end{pmatrix};
|
||
\end{multline*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{multline*}
|
||
B_{mn}^{\mu\nu}=\\
|
||
\frac{(-1)^{m}i^{\nu+n+1}(n+2)_{n+1}\left(\nu+2\right)_{\nu+1}(n+\nu+m-\mu+1)!}{4n(n+1)(n+m+1)(n+\nu+2)_{n+\nu+1}(n-m)!(\nu+m)!}\\
|
||
\times e^{i(\mu-m)\phi_{lj}}\sum_{q=0}^{Q_{\mathrm{max}}}(-1)^{q}\Big\{2(n+1)(\nu-\mu)\tilde{a}_{2q}-\\
|
||
-\left[p(p+3)-\nu(\nu+1)-n(n+3)-2\mu(n+1)\right]\tilde{a}_{3q}\Big\}\\
|
||
\times\begin{pmatrix}z_{p+1}^{(J)}(kd_{lj})\\
|
||
j_{p+1}(kd_{lj})
|
||
\end{pmatrix}P_{p+1}^{\mu-m}(\cos\theta_{lj}),\qquad\begin{pmatrix}r\le d_{lj}\\
|
||
r\ge d_{lj}
|
||
\end{pmatrix};
|
||
\end{multline*}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(79,80)"
|
||
key "xu_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\tilde{a}_{1q} & = & a(-m,n,\mu,\nu,n+\nu-2q)/a(-m,n,\mu,\nu,n+\nu),\\
|
||
\tilde{a}_{2q} & = & a(-m-1,n+1,\mu+1,\nu,n+\nu+1-2q)/\\
|
||
& & /a(-m-1,n+1,\mu+1,\nu,n+\nu+1),\\
|
||
\tilde{a}_{3q} & = & a(-m,n+1,\mu,\nu,n+\nu+1-2q)/\\
|
||
& & /a(-m,n+1,\mu,\nu,\mu+\nu+1),
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
p & = & n+\nu-2q\\
|
||
q_{\max} & = & \min\left(n,\nu,\frac{n+\nu-\left|m-\mu\right|}{2}\right),\\
|
||
Q_{\max} & = & \min\left(n+1,\nu,\frac{n+\nu+1-\left|m-\mu\right|}{2}\right),
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
where the parentheses with lower index mean most likely the Pochhammer symbol
|
||
/
|
||
\emph on
|
||
rising
|
||
\emph default
|
||
factorial
|
||
\begin_inset Formula
|
||
\[
|
||
\left(x\right)_{n}=x(x+1)(x+2)\dots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}=\frac{\Gamma(x+n)}{\Gamma(x)},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
which is damn confusing (because this can also mean the falling factorial,
|
||
cf.
|
||
Wikipedia); and Xu does not bother explaining the notation
|
||
\emph on
|
||
anywhere
|
||
\emph default
|
||
.
|
||
The fact that it is the rising factorial has been checked by comparing
|
||
|
||
\begin_inset Formula $a_{0}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(78)"
|
||
key "xu_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
to some implementation from the internets
|
||
\begin_inset Foot
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
\begin_inset CommandInset href
|
||
LatexCommand href
|
||
name "https://raw.githubusercontent.com/michael-hartmann/gaunt/master/gaunt.py"
|
||
target "https://raw.githubusercontent.com/michael-hartmann/gaunt/master/gaunt.py"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The implementation should be checked with
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "Table II"
|
||
key "xu_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Equations for the scattering problem
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The linear system for the scattering problem reads
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(30)"
|
||
key "xu_electromagnetic_1995"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
a_{mn}^{j} & = & a_{n}^{j}\left\{ p_{mn}^{j,j}-\sum_{l\neq j}^{(1,L)}\sum_{\nu=1}^{\infty}\sum_{\mu=-\nu}^{\nu}\left[a_{\mu\nu}^{l}A_{mn}^{\mu\nu;lj}+b_{\mu\nu}^{l}B_{mn}^{\mu\nu;lj}\right]\right\} \\
|
||
b_{mn}^{j} & = & b_{n}^{j}\left\{ q_{mn}^{j,j}-\sum_{l\neq j}^{(1,L)}\sum_{\nu=1}^{\infty}\sum_{\mu=-\nu}^{\nu}\left[a_{\mu\nu}^{l}B_{mn}^{\mu\nu;lj}+b_{\mu\nu}^{l}A_{mn}^{\mu\nu;lj}\right]\right\}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $p_{mn}^{j,j},q_{mn}^{j,j}$
|
||
\end_inset
|
||
|
||
are the expansion coefficients of the initial incident waves in the
|
||
\begin_inset Formula $j$
|
||
\end_inset
|
||
|
||
-th particle's coordinate system
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "sec. 4A"
|
||
key "xu_electromagnetic_1995"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\emph on
|
||
TODO expressions for
|
||
\begin_inset Formula $p_{mn}^{j,j},q_{mn}^{j,j}$
|
||
\end_inset
|
||
|
||
in the case of dipole initial wave.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Solving the linear system
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "sec. 5"
|
||
key "xu_electromagnetic_1995"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
T-Matrix resummation (multiple scatterers)
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Boundary element method
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
BEM→TM
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Cf.
|
||
SCUFF-TMATRIX (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sub:SCUFF-TMATRIX"
|
||
|
||
\end_inset
|
||
|
||
)
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Available software
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
TODO which of them can calculate the VSWF translation coefficients?
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
SCUFF-EM
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "reid_scuff-em_2015"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
|
||
\family typewriter
|
||
SCUFF-TMATRIX
|
||
\family default
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sub:SCUFF-TMATRIX"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
|
||
\family typewriter
|
||
SCUFF-SCATTER
|
||
\family default
|
||
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sub:SCUFF-SCATTER"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Caveats
|
||
\end_layout
|
||
|
||
\begin_layout Description
|
||
Units.
|
||
|
||
\family typewriter
|
||
SCUFF-SCATTER
|
||
\family default
|
||
's Angular frequencies specified using the
|
||
\family typewriter
|
||
--Omega
|
||
\family default
|
||
or
|
||
\family typewriter
|
||
--OmegaFile
|
||
\family default
|
||
arguments are interpreted in units of
|
||
\begin_inset Formula $c/1\,\mathrm{μm}=3\cdot10^{14}\,\mathrm{rad/s}$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Foot
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\family typewriter
|
||
\begin_inset CommandInset href
|
||
LatexCommand href
|
||
name "http://homerreid.dyndns.org/scuff-EM/scuff-scatter/scuffScatterExamples.shtml"
|
||
target "http://homerreid.dyndns.org/scuff-EM/scuff-scatter/scuffScatterExamples.shtml"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\emph on
|
||
TODO what are the output units?
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
MSTM
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "mackowski_mstm_2013"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The incident field is a gaussian beam or a plane wave in the vanilla code
|
||
(no multipole radiation as input!).
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The bulk of the useful code is in the
|
||
\family typewriter
|
||
mstm-modules-v3.0.f90
|
||
\family default
|
||
file.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
For solving the interaction equations
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(14)"
|
||
key "mackowski_mstm_2013"
|
||
|
||
\end_inset
|
||
|
||
, the BCGM (biconjugate gradient method) is used.
|
||
(According to Wikipedia, this method is numerically unstable but has a
|
||
stabilized version (stabilized BCGM).)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
According to the manual
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "2.3"
|
||
key "mackowski_mstm_2013"
|
||
|
||
\end_inset
|
||
|
||
, they use some method (rotational-axial translation decomposition of the
|
||
translation operation), which
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
reduces the operation from an
|
||
\begin_inset Formula $L_{S}^{4}$
|
||
\end_inset
|
||
|
||
process to
|
||
\begin_inset Formula $L_{S}^{3}$
|
||
\end_inset
|
||
|
||
process where
|
||
\begin_inset Formula $L_{S}$
|
||
\end_inset
|
||
|
||
is the truncation order of the expansion
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
(more details can probably be found at
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "around (68)"
|
||
key "mackowski_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\end_layout
|
||
|
||
\begin_deeper
|
||
\begin_layout Itemize
|
||
|
||
\emph on
|
||
Not sure if this holds also for nonspherical particles, I should either
|
||
read carefully
|
||
\emph default
|
||
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "mackowski_calculation_1996"
|
||
|
||
\end_inset
|
||
|
||
|
||
\emph on
|
||
or look into
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "mishchenko_electromagnetic_2003"
|
||
|
||
\end_inset
|
||
|
||
which is also cited in the manual.
|
||
\end_layout
|
||
|
||
\end_deeper
|
||
\begin_layout Itemize
|
||
By default spheres, it is possible to add own T-Matrix coefficients instead.
|
||
|
||
\end_layout
|
||
|
||
\begin_deeper
|
||
\begin_layout Itemize
|
||
|
||
\emph on
|
||
Is it then possible to insert a T-Matrix of an arbitrary shape, or is it
|
||
somehow limited to
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
spherical-like
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
particles?
|
||
\end_layout
|
||
|
||
\end_deeper
|
||
\begin_layout Itemize
|
||
Why the heck are the T-matrix options listed in the
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
Options for random orientation calculations
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
? Well, it seems that for fixed orientation, it is not possible to specify
|
||
the T-matrix, cf.
|
||
the description of
|
||
\family typewriter
|
||
fixed_or_random_orientation
|
||
\family default
|
||
option in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "3.2.3"
|
||
key "mackowski_mstm_2013"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Interesting subroutines
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
rottranfarfield
|
||
\family default
|
||
: it states
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
far field formula for outgoing vswf translation
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
.
|
||
What is that and how does it differ from whatever else (near field?) formula?
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
py_gmm
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "pellegrini_py_gmm_2015"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Fortran code, already (partially) pythonized using
|
||
\family typewriter
|
||
f2py
|
||
\family default
|
||
by the authors(?); under GNU GPLv3.
|
||
This could save my day.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Lots of unnecessary code duplication (see e.g.
|
||
|
||
\family typewriter
|
||
coeff_sp2
|
||
\family default
|
||
and
|
||
\family typewriter
|
||
coeff_sp2_dip
|
||
\family default
|
||
subroutines).
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Has comments!!! (Sometimes they are slightly inaccurate due to the copy-pasting,
|
||
but it is still one of the most readable FORTRAN codes I have seen.)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The subroutines seem not to be bloated with dependencies on static/global
|
||
variables, so they should be quite easily reusable.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The FORTRAN code was apparently used in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "pellegrini_interacting_2007"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
Uses the multiple-scattering formalism described in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "xu_efficient_1998"
|
||
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Interesting subroutines
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Mie scattering:
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
coeff_sp2
|
||
\family default
|
||
: calculation of the Mie scattering coefficients (
|
||
\begin_inset Formula $\overline{a}_{n}^{l},\overline{b}_{n}^{l}$
|
||
\end_inset
|
||
|
||
as in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(1), (2), \\ldots"
|
||
key "pellegrini_py_gmm_2015"
|
||
|
||
\end_inset
|
||
|
||
), for a set of spheres (therefore all the parameters have +1 dimension).
|
||
\end_layout
|
||
|
||
\begin_deeper
|
||
\begin_layout Itemize
|
||
What does the input parameter
|
||
\family typewriter
|
||
v_req
|
||
\family default
|
||
(
|
||
\emph on
|
||
vettore raggi equivalenti
|
||
\emph default
|
||
) mean?
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
How do I put in the environment permittivity?
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
m_epseq
|
||
\family default
|
||
are real and imaginary parts of the permittivity (which are then transformed
|
||
into complex
|
||
\family typewriter
|
||
v_epsc
|
||
\family default
|
||
)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
ref_index
|
||
\family default
|
||
is the environment refractive index (called
|
||
\family typewriter
|
||
n_matrix
|
||
\family default
|
||
in the example ipython notebook)
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
v_req
|
||
\family default
|
||
are the sphere radii?
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
nstop
|
||
\family default
|
||
is the maximum order of the
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
-expansion
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\family typewriter
|
||
neq
|
||
\family default
|
||
is ns, number of spheres for which the calculation is performed apparently,
|
||
it is connected to some
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
dirty hack to interface fortran and python properly
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
(cf.
|
||
|
||
\family typewriter
|
||
gmm_f2py_module.f90
|
||
\family default
|
||
)
|
||
\end_layout
|
||
|
||
\end_deeper
|
||
\begin_layout Section
|
||
Implementation / code integration
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
There are several scipy functions to compute the Legendre polynomials.
|
||
lpmv is ufunc, whereas lpmn is not; lpmn can, however, compute also the
|
||
derivatives.
|
||
This is a bit annoying, because I have to iterate the positions with a
|
||
for loop.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The default gsl legendre function (gsl_sf_legendre_array) without additional
|
||
parameters has opposite sign than the scipy.special.lpmn, and it should contain
|
||
the Condon-Shortley phase; thus scipy.special.lpmn probably does NOT include
|
||
the CS phase.
|
||
But again, this should hopefully play no role.
|
||
The overall normalisation, on the other hand, plays huge role.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Scattering-Taylor.ipynb
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the conventions used in the code and the corresponding libraries, the
|
||
following symmetries hold for
|
||
\begin_inset Formula $J=1$
|
||
\end_inset
|
||
|
||
(regular wavefunctions):
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\widetilde{\vect M}_{m,n}^{(1)} & = & (-1)^{m}\widetilde{\vect M}_{-m,n}^{(1)},\\
|
||
\widetilde{\vect N}_{m,n}^{(1)} & = & (-1)^{m}\widetilde{\vect N}_{-m,n}^{(1)}.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Testing and reproduction of foreign results
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Delga PRL
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "delga_quantum_2014"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
Parameters
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
Surrounding lossless dielectric
|
||
\series bold
|
||
medium
|
||
\series default
|
||
with permittivity
|
||
\begin_inset Formula $\epsilon_{d}=2.13$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
|
||
\series bold
|
||
QEs:
|
||
\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
|
||
Misc
|
||
\end_layout
|
||
|
||
\begin_layout Itemize
|
||
The
|
||
\begin_inset Quotes eld
|
||
\end_inset
|
||
|
||
zero limits
|
||
\begin_inset Quotes erd
|
||
\end_inset
|
||
|
||
of
|
||
\begin_inset Formula $\tilde{\pi},\tilde{\tau}$
|
||
\end_inset
|
||
|
||
functions in Taylor's normalisation can be expressed as
|
||
\lang finnish
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\lim_{\theta\to0}\tilde{\pi}_{mn}\left(\cos\theta\right) & = & -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}+\delta_{m,-1})\\
|
||
\lim_{\theta\to0}\tilde{\tau}_{mn}\left(\cos\theta\right) & = & -\frac{1}{2}\sqrt{\frac{2n+1}{4\pi}n\left(n+1\right)}(\delta_{m,1}-\delta_{m,-1})
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
bibfiles "dipdip"
|
||
options "apsrev"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|