1564 lines
35 KiB
Plaintext
1564 lines
35 KiB
Plaintext
#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\shortcut idx
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\end_header
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\begin_body
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\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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Electromagnetic multiple scattering, spherical waves and conventions
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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 Chapter
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Zillion conventions for spherical vector waves
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\end_layout
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\begin_layout Section
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Legendre polynomials and spherical harmonics: messy from the very beginning
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\end_layout
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||
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\begin_layout Standard
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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FIXME check the Condon-Shortley phases.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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Associated Legendre polynomial of degree
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\begin_inset Formula $l\ge0$
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\end_inset
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and order
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\begin_inset Formula $m,$
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\end_inset
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\begin_inset Formula $l\ge m\ge-l$
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\end_inset
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||
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, is given by the recursive relation
|
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\begin_inset Formula
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\[
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P_{l}^{-m}=\underbrace{\left(-1\right)^{m}}_{\mbox{Condon-Shortley phase}}\frac{1}{2^{l}l!}\left(1-x^{2}\right)^{m/2}\frac{\ud^{l+m}}{\ud x^{l+m}}\left(x^{2}-1\right)^{l}.
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\]
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\end_inset
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||
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There is a relation between the positive and negative orders,
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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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\[
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P_{l}^{-m}=\underbrace{\left(-1\right)^{m}}_{\mbox{C.-S. p.}}\frac{\left(l-m\right)!}{\left(l+m\right)!}P_{l}^{m}\left(\cos\theta\right),\quad m\ge0.
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\]
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\end_inset
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||
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The index
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\begin_inset Formula $l$
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\end_inset
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(in certain notations, it is often
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\begin_inset Formula $n$
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||
\end_inset
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||
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) is called
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||
\emph on
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degree
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||
\emph default
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||
, index
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||
\begin_inset Formula $m$
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||
\end_inset
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||
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is the
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\emph on
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order
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\emph default
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.
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These two terms are then transitively used for all the object which build
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on the associated Legendre polynomials, i.e.
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spherical harmonics, vector spherical harmonics, spherical waves etc.
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||
\end_layout
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||
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\begin_layout Subsection
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||
Kristensson
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||
\end_layout
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||
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\begin_layout Standard
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||
Kristensson uses the Condon-Shortley phase, so (sect.
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||
[K]D.2)
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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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||
\[
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Y_{lm}\left(\hat{\vect r}\right)=\left(-1\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}P_{l}^{m}\left(\cos\theta\right)e^{im\phi}
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\]
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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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||
\[
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Y_{lm}^{\dagger}\left(\hat{\vect r}\right)=Y_{lm}^{*}\left(\hat{\vect r}\right)
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||
\]
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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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\[
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Y_{l,-m}\left(\hat{\vect r}\right)=\left(-1\right)^{m}Y_{lm}^{\dagger}\left(\hat{\vect r}\right)
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\]
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\end_inset
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||
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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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||
Orthonormality:
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||
\begin_inset Formula
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||
\[
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||
\int Y_{lm}\left(\hat{\vect r}\right)Y_{l'm'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{ll'}\delta_{mm'}
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||
\]
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||
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||
\end_inset
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||
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||
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||
\end_layout
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||
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\begin_layout Section
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||
Pi and tau
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||
\end_layout
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||
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||
\begin_layout Subsection
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||
Xu
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||
\begin_inset CommandInset label
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||
LatexCommand label
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||
name "subsec:Xu pitau"
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||
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||
\end_inset
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||
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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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||
As in (37)
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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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||
\pi_{mn}\left(\cos\theta\right) & = & \frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\
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\tau_{mn}\left(\cos\theta\right) & = & \frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\right)=-\left(\sin\theta\right)\frac{\ud P_{n}^{m}\left(\cos\theta\right)}{\ud\left(\cos\theta\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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||
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||
\end_layout
|
||
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||
\begin_layout Standard
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||
The expressions
|
||
\begin_inset Formula $\left(\sin\theta\right)^{-1}$
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||
\end_inset
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||
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||
and
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||
\begin_inset Formula $\frac{\ud P_{n}^{m}\left(\cos\theta\right)}{\ud\left(\cos\theta\right)}$
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||
\end_inset
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||
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are singular for
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||
\begin_inset Formula $\cos\theta=\pm1$
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||
\end_inset
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||
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||
, the limits
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||
\begin_inset Formula $\tau_{mn}\left(\pm1\right),\pi_{mn}\left(\pm1\right)$
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||
\end_inset
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||
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||
however exist.
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||
Labeling
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||
\begin_inset Formula $x\equiv\cos\theta$
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||
\end_inset
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||
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||
,
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||
\begin_inset Formula $\sqrt{\left(1+x\right)\left(1-x\right)}=\sqrt{1-x^{2}}\equiv\sin\theta$
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||
\end_inset
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||
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||
and using the asymptotic expression (DLMF 14.8.2) we obtain that the limits
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are nonzero only for
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||
\begin_inset Formula $m=\pm1$
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||
\end_inset
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||
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||
and
|
||
\begin_inset Formula
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||
\begin{eqnarray*}
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\pi_{1\nu}(+1-) & = & CS\frac{\nu\left(\nu+1\right)}{2}\\
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||
\tau_{1\nu}(+1-) & = & CS\frac{\nu\left(\nu+1\right)}{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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||
and using the parity property
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||
\begin_inset Formula $P_{n}^{m}\left(-x\right)=\left(-1\right)^{m+n}P_{n}^{m}\left(x\right)$
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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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||
\pi_{1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\frac{\nu\left(\nu+1\right)}{2}\\
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\tau_{1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\frac{\nu\left(\nu+1\right)}{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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For
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||
\begin_inset Formula $m=1$
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||
\end_inset
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||
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||
, we simply use the relation
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||
\begin_inset Formula $P_{n}^{-m}=\left(CS\right)^{m}P_{n}^{m}\frac{\left(n-m\right)!}{\left(n+m\right)!}$
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||
\end_inset
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||
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||
to get
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||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\pi_{-1\nu}(+1-) & = & \frac{CS}{2}\\
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||
\tau_{-1\nu}(+1-) & = & -\frac{CS}{2}\\
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||
\pi_{-1\nu}(-1+) & = & -\left(-1\right)^{\nu}\frac{CS}{2}\\
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||
\tau_{-1\nu}(-1+) & = & -\left(-1\right)^{\nu}\frac{CS}{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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||
where
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||
\begin_inset Formula $CS$
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||
\end_inset
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||
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||
is
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||
\begin_inset Formula $-1$
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||
\end_inset
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||
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||
if the Condon-Shortley phase is employed on the level of Legendre polynomials,
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||
1 otherwise.
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||
\end_layout
|
||
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||
\begin_layout Subsection
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||
Taylor
|
||
\end_layout
|
||
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||
\begin_layout Standard
|
||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\tilde{\pi}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\
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\tilde{\tau}_{mn}\left(\cos\theta\right) & = & \sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}\frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\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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||
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||
\end_layout
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||
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||
\begin_layout Standard
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||
The limiting expressions are obtained simply by multiplying the expressions
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||
from sec.
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||
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||
\begin_inset CommandInset ref
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||
LatexCommand ref
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||
reference "subsec:Xu pitau"
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||
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||
\end_inset
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||
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||
by the normalisation factor,
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||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\tilde{\pi}_{1\nu}(+1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\
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||
\tilde{\tau}_{1\nu}(+1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\
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||
\tilde{\pi}_{1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\
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||
\tilde{\tau}_{1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{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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||
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||
\begin_inset Formula
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||
\begin{eqnarray*}
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||
\tilde{\pi}_{-1\nu}(+1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\
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||
\tilde{\tau}_{-1\nu}(+1-) & = & -CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}}{2}\\
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||
\tilde{\pi}_{-1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
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||
\tilde{\tau}_{-1\nu}(-1+) & = & -CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}
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||
\end{eqnarray*}
|
||
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||
\end_inset
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||
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||
i.e.
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||
the expressions for
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||
\begin_inset Formula $m=-1$
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||
\end_inset
|
||
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||
are the same as for
|
||
\begin_inset Formula $m=1$
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||
\end_inset
|
||
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||
except for the sign if Condon-Shortley phase is used on the Legendre polynomial
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||
level.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Vector spherical harmonics (?)
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Kristensson
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Original formulation, sect.
|
||
[K]D.3.3
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray}
|
||
\vect A_{1lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\nonumber \\
|
||
& = & \frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect rY_{lm}\left(\hat{\vect r}\right)\right)\nonumber \\
|
||
\vect A_{2lm}\left(\hat{\vect r}\right) & = & \frac{1}{\sqrt{l\left(l+1\right)}}\left(\hat{\vect{\theta}}\frac{\partial}{\partial\phi}Y_{lm}\left(\hat{\vect r}\right)-\hat{\vect{\phi}}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}Y_{lm}\left(\hat{\vect r}\right)\right)\label{eq:vector spherical harmonics Kristensson}\\
|
||
& = & \frac{1}{\sqrt{l\left(l+1\right)}}r\nabla Y_{lm}\left(\hat{\vect r}\right)\nonumber \\
|
||
\vect A_{3lm}\left(\hat{\vect r}\right) & = & \hat{\vect r}Y_{lm}\left(\hat{\vect r}\right)\nonumber
|
||
\end{eqnarray}
|
||
|
||
\end_inset
|
||
|
||
Normalisation:
|
||
\begin_inset Formula
|
||
\[
|
||
\int\vect A_{n}\left(\hat{\vect r}\right)\cdot\vect A_{n'}^{\dagger}\left(\hat{\vect r}\right)\,\ud\Omega=\delta_{nn'}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
Here
|
||
\begin_inset Formula $\mbox{ }^{\dagger}$
|
||
\end_inset
|
||
|
||
means just complex conjugate, apparently (see footnote on p.
|
||
89).
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Jackson
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(9.101)"
|
||
key "jackson_classical_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\[
|
||
\vect X_{lm}(\theta,\phi)=\frac{1}{\sqrt{l(l+1)}}\vect LY_{lm}(\theta,\phi)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(9.119)"
|
||
key "jackson_classical_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\vect L=\frac{1}{i}\left(\vect r\times\vect{\nabla}\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
for its expression in spherical coordinates and other properties check Jackson's
|
||
book around the definitions.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Normalisation
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(9.120)"
|
||
key "jackson_classical_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\[
|
||
\int\vect X_{l'm'}^{*}\cdot\vect X_{lm}\,\ud\Omega=\delta_{ll'}\delta_{mm'}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Local sum rule
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(9.153)"
|
||
key "jackson_classical_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\[
|
||
\sum_{m=-l}^{l}\left|\vect X_{lm}(\theta,\phi)^{2}\right|=\frac{2l+1}{4\pi}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Spherical Bessel functions
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "sec:Spherical-Bessel-functions"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Cf.
|
||
[DLMF] §10.47–60.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The radial dependence of spherical vector waves is given by the spherical
|
||
Bessel functions and their first derivatives.
|
||
Commonly, the following notation is adopted
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
z_{n}^{(1)}(x) & = & j_{n}(x),\\
|
||
z_{n}^{(2)}(x) & = & y_{n}(x),\\
|
||
z_{n}^{(3)}(x) & = & h_{n}^{(1)}(x)=j_{n}(x)+iy_{n}(x),\\
|
||
z_{n}^{(4)}(x) & = & h_{n}^{(2)}(x)=j_{n}(x)-iy_{n}(x).
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
Here,
|
||
\begin_inset Formula $j_{n}$
|
||
\end_inset
|
||
|
||
is the spherical Bessel function of first kind (regular),
|
||
\begin_inset Formula $y_{j}$
|
||
\end_inset
|
||
|
||
is the spherical Bessel function of second kind (singular), and
|
||
\begin_inset Formula $h_{n}^{(1)},h_{n}^{(2)}$
|
||
\end_inset
|
||
|
||
are the Hankel functions a.k.a.
|
||
spherical Bessel functions of third kind.
|
||
In spherical vector waves,
|
||
\begin_inset Formula $j_{n}$
|
||
\end_inset
|
||
|
||
corresponds to regular waves,
|
||
\begin_inset Formula $h^{(1)}$
|
||
\end_inset
|
||
|
||
corresponds (by the usual convention) to outgoing waves, and
|
||
\begin_inset Formula $h^{(2)}$
|
||
\end_inset
|
||
|
||
corresponds to incoming waves.
|
||
To describe scattering, we need two sets of waves with two different types
|
||
of spherical Bessel functions
|
||
\begin_inset Formula $z_{n}^{(J)}$
|
||
\end_inset
|
||
|
||
.
|
||
Most common choice is
|
||
\begin_inset Formula $J=1,3$
|
||
\end_inset
|
||
|
||
, because if we decompose the field into spherical waves centered at
|
||
\begin_inset Formula $\vect r_{0}$
|
||
\end_inset
|
||
|
||
, the field produced by other sources (e.g.
|
||
spherical waves from other scatterers or a plane wave) is always regular
|
||
at
|
||
\begin_inset Formula $\vect r_{0}$
|
||
\end_inset
|
||
|
||
.
|
||
Second choice which makes a bit of sense is
|
||
\begin_inset Formula $J=3,4$
|
||
\end_inset
|
||
|
||
as it leads to a nice expression for the energy transport.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Limiting Forms
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
[DLMF] §10.52:
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
\begin_inset Formula $z\to0$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
j_{n}(z) & \sim & z^{n}/(2n+1)!!\\
|
||
h_{n}^{(1)}(z)\sim iy(z) & \sim & -i\left(2n+1\right)!!/z^{n+1}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Spherical vector waves
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
TODO
|
||
\begin_inset Formula $M,N,\psi,\chi,\widetilde{M},\widetilde{N},u,v,w,\dots$
|
||
\end_inset
|
||
|
||
, sine/cosine convention (B&H), ...
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
There are two mutually orthogonal types of divergence-free (everywhere except
|
||
in the origin for singular waves) spherical vector waves, which I call
|
||
electric and magnetic, given by the type of multipole source to which they
|
||
correspond.
|
||
This is another distinction than the regular/singular/ingoing/outgoing
|
||
waves given by the type of the radial dependence (cf.
|
||
section
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "sec:Spherical-Bessel-functions"
|
||
|
||
\end_inset
|
||
|
||
).
|
||
Oscillating electric current in a tiny rod parallel to its axis will generate
|
||
electric dipole waves (net dipole moment of magnetic current is zero) moment
|
||
, whereas oscillating electric current in a tiny circular loop will generate
|
||
magnetic dipole waves (net dipole moment of electric current is zero).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the usual cases we encounter, the part described by the magnetic waves
|
||
is pretty small.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The expression with Bessel function derivatives appearing below in the electric
|
||
waves can be rewritten using (DLMF 10.51.2)
|
||
\begin_inset Formula
|
||
\[
|
||
\frac{1}{kr}\frac{\ud\left(kr\,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}=\frac{\ud z_{n}^{j}\left(kr\right)}{\ud(kr)}+\frac{z_{n}^{j}\left(kr\right)}{kr}=z_{n-1}^{j}\left(kr\right)-n\frac{z_{n}^{j}\left(kr\right)}{kr}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Taylor
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Definition [T](2.40);
|
||
\begin_inset Formula $\widetilde{\vect N}_{mn}^{(j)},\widetilde{\vect M}_{mn}^{(j)}$
|
||
\end_inset
|
||
|
||
are the electric and magnetic waves, respectively:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\widetilde{\vect N}_{mn}^{(j)} & = & \frac{n(n+1)}{kr}\sqrt{\frac{2n+1}{4\pi}\frac{\left(n-m\right)!}{\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\
|
||
& & +\left[\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr\,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}\\
|
||
\widetilde{\vect M}_{mn}^{(j)} & = & \left[i\tilde{\pi}_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tilde{\tau}_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Xu
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
are the electric and magnetic waves, respectively:
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\vect N_{mn}^{(j)} & = & \frac{n(n+1)}{kr}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\
|
||
& & +\left[\tau_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\pi_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr\,z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}\\
|
||
\vect M_{mn}^{(j)} & = & \left[i\pi_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tau_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Kristensson
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Definition [K](2.4.6);
|
||
\begin_inset Formula $\vect u_{\tau lm},\vect v_{\tau lm},\vect w_{\tau lm}$
|
||
\end_inset
|
||
|
||
are the waves with
|
||
\begin_inset Formula $j=3,1,4$
|
||
\end_inset
|
||
|
||
respectively, i.e.
|
||
outgoing, regular and incoming waves.
|
||
The first index distinguishes between the electric (
|
||
\begin_inset Formula $\tau=2$
|
||
\end_inset
|
||
|
||
) and magnetic (
|
||
\begin_inset Formula $\tau=1$
|
||
\end_inset
|
||
|
||
).
|
||
Kristensson uses a multiindex
|
||
\begin_inset Formula $n\equiv(\tau,l,m)$
|
||
\end_inset
|
||
|
||
to simlify the notation.
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\left(\vect{u/v/w}\right)_{2lm} & = & \frac{1}{kr}\frac{\ud\left(kr\,z_{l}^{(j)}\left(kr\right)\right)}{\ud\,kr}\vect A_{2lm}\left(\hat{\vect r}\right)+\sqrt{l\left(l+1\right)}\frac{z_{l}^{(j)}(kr)}{kr}\vect A_{3lm}\left(\hat{\vect r}\right)\\
|
||
\left(\vect{u/v/w}\right)_{1lm} & = & z_{l}^{(j)}\left(kr\right)\vect A_{1lm}\left(\hat{\vect r}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Kristensson vs.
|
||
Xu
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
As in
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "eq. (36)"
|
||
key "xu_calculation_1996"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
with unnormalised Legendre polynomials:
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\left(\vect{u/v/w}\right)_{1lm} & = & \left(\mbox{CS}\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}\frac{\vect N_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}\\
|
||
\left(\vect{u/v/w}\right)_{1lm} & = & \left(\mbox{CS}\right)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{\left(l-m\right)!}{\left(l+m\right)!}}\frac{\vect M_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
where CS is
|
||
\begin_inset Formula $-1$
|
||
\end_inset
|
||
|
||
in Kristensson's text.
|
||
N.B.
|
||
be careful about the translation coefficients and
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "eq. (81)"
|
||
key "xu_calculation_1996"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
, Xu's text is a bit confusing.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Relation between Kristensson and Taylor
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Kristensson-v-Taylor"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Kristensson's and Taylor's VSWFs seem to differ only by an
|
||
\begin_inset Formula $l$
|
||
\end_inset
|
||
|
||
-dependent normalization factor, and notation of course (n.b.
|
||
the inverse index order)
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\left(\vect{u/v/w}\right)_{2lm} & = & \frac{\widetilde{\vect N}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}\\
|
||
\left(\vect{u/v/w}\right)_{1lm} & = & \frac{\widetilde{\vect M}_{ml}^{(3/1/4)}}{\sqrt{l\left(l+1\right)}}
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Plane wave expansion
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Taylor
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
-polarised,
|
||
\begin_inset Formula $z$
|
||
\end_inset
|
||
|
||
-propagating plane wave,
|
||
\begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$
|
||
\end_inset
|
||
|
||
(CHECK):
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\vect E & = & -i\left(p_{mn}\widetilde{\vect N}_{mn}^{(1)}+q_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)\\
|
||
p_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\tau}_{mn}(1)\\
|
||
q_{mn} & = & E_{0}\frac{4\pi i^{n}}{n(n+1)}\tilde{\pi}_{mn}(1)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
while it can be shown that
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
\tilde{\pi}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}+\delta_{m,-1}\right)\\
|
||
\tilde{\tau}_{mn}(1) & = & -\frac{1}{2}\sqrt{\frac{\left(2n+1\right)\left(n\left(n+1\right)\right)}{4\pi}}\left(\delta_{m,1}-\delta_{m,-1}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Kristensson
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula $x$
|
||
\end_inset
|
||
|
||
-polarised,
|
||
\begin_inset Formula $z$
|
||
\end_inset
|
||
|
||
-propagating plane wave,
|
||
\begin_inset Formula $\vect E=E_{0}\hat{\vect x}e^{i\vect k\cdot\hat{\vect z}}$
|
||
\end_inset
|
||
|
||
(CHECK, ):
|
||
\begin_inset Formula
|
||
\[
|
||
\vect E=\sum_{n}a_{n}\vect v_{n}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
a_{1lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)\\
|
||
a_{2lm} & = & E_{0}i^{l+1}\sqrt{\left(2l+1\right)\pi}\left(\delta_{m,1}+\delta_{m,-1}\right)
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Radiated energy
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In this section I summarize the formulae for power
|
||
\begin_inset Formula $P$
|
||
\end_inset
|
||
|
||
radiated from the system.
|
||
For an absorbing scatterer, this should be negative (n.b.
|
||
sign conventions can be sometimes confusing).
|
||
If the system is excited by a plane wave with intensity
|
||
\begin_inset Formula $E_{0}$
|
||
\end_inset
|
||
|
||
, this can be used to calculate the absorption cross section (TODO check
|
||
if it should be multiplied by the 2),
|
||
\begin_inset Formula
|
||
\[
|
||
\sigma_{\mathrm{abs}}=-\frac{2P}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Kristensson
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "subsec:Radiated enenergy-Kristensson"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Sect.
|
||
[K]2.6.2; here this form of expansion is assumed:
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\vect E\left(\vect r,\omega\right)=k\sqrt{\eta_{0}\eta}\sum_{n}\left(a_{n}\vect v_{n}\left(k\vect r\right)+f_{n}\vect u_{n}\left(k\vect r\right)\right).\label{eq:power-Kristensson-E}
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Here
|
||
\begin_inset Formula $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$
|
||
\end_inset
|
||
|
||
is the wave impedance of free space and
|
||
\begin_inset Formula $\eta=\sqrt{\mu/\varepsilon}$
|
||
\end_inset
|
||
|
||
is the relative wave impedance of the medium.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The radiated power is then (2.28):
|
||
\begin_inset Formula
|
||
\[
|
||
P=\frac{1}{2}\sum_{n}\left(\left|f_{n}\right|^{2}+\Re\left(f_{n}a_{n}^{*}\right)\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
The first term is obviously the power radiated away by the outgoing waves.
|
||
The second term must then be minus the power sucked by the scatterer from
|
||
the exciting wave.
|
||
If the exciting wave is plane, it gives us the extinction cross section
|
||
\begin_inset Formula
|
||
\[
|
||
\sigma_{\mathrm{tot}}=-\frac{\sum_{n}\Re\left(f_{n}a_{n}^{*}\right)}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Taylor
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Here I derive the radiated power in Taylor's convention by applying the
|
||
relations from subsection
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "subsec:Kristensson-v-Taylor"
|
||
|
||
\end_inset
|
||
|
||
to the Kristensson's formulae (sect.
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "subsec:Radiated enenergy-Kristensson"
|
||
|
||
\end_inset
|
||
|
||
).
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Assume the external field decomposed as (here I use tildes even for the
|
||
expansion coefficients in order to avoid confusion with the
|
||
\begin_inset Formula $a_{n}$
|
||
\end_inset
|
||
|
||
in
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:power-Kristensson-E"
|
||
|
||
\end_inset
|
||
|
||
)
|
||
\begin_inset Formula
|
||
\[
|
||
\vect E\left(\vect r,\omega\right)=\sum_{mn}\left[-i\left(\tilde{p}_{mn}\vect{\widetilde{N}}_{mn}^{(1)}+\tilde{q}_{mn}\widetilde{\vect M}_{mn}^{(1)}\right)+i\left(\tilde{a}_{mn}\widetilde{\vect N}_{mn}^{(3)}+\tilde{b}_{mn}\widetilde{\vect M}_{mn}^{(3)}\right)\right]
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
(there is minus between the regular and outgoing part!).
|
||
The coefficients are related to those from
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:power-Kristensson-E"
|
||
|
||
\end_inset
|
||
|
||
as
|
||
\begin_inset Formula
|
||
\[
|
||
\tilde{p}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{2nm},\quad\tilde{q}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{-i\sqrt{n(n+1)}}a_{1nm},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\[
|
||
\tilde{a}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{2nm},\quad\tilde{b}_{mn}=\frac{k\sqrt{\eta_{0}\eta}}{i\sqrt{n(n+1)}}f_{1nm}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
The radiated power is then
|
||
\begin_inset Formula
|
||
\[
|
||
P=\frac{1}{2}\sum_{m,n}\frac{n\left(n+1\right)}{k^{2}\eta_{0}\eta}\left(\left|a_{mn}\right|^{2}+\left|b_{mn}\right|^{2}-\Re\left(a_{mn}p_{mn}^{*}\right)-\Re\left(b_{mn}q_{mn}^{*}\right)\right).
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
If the exciting wave is a plane wave, the extinction cross section is
|
||
\begin_inset Formula
|
||
\[
|
||
\sigma_{\mathrm{tot}}=\frac{1}{\varepsilon\varepsilon_{0}\left|E_{0}\right|^{2}k^{2}\eta_{0}\eta}\sum_{m,n}n(n+1)\left(\Re\left(a_{mn}p_{mn}^{*}\right)+\Re\left(b_{mn}q_{mn}^{*}\right)\right)
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Jackson
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(9.155)"
|
||
key "jackson_classical_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\[
|
||
P=\frac{Z_{0}}{2k^{2}}\sum_{l,m}\left[\left|a_{E}(l,m)\right|^{2}+\left|a_{M}(l,m)\right|^{2}\right]
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Limit solutions
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Far-field asymptotic solution
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
TODO start from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(A7)"
|
||
key "pustovit_plasmon-mediated_2010"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
and Jackson (9.169) and around.
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Near field limit
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Single particle scattering and Mie theory
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The basic idea is simple.
|
||
For an exciting spherical wave (usually the regular wave in whatever convention
|
||
) of a given frequency
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
, type
|
||
\begin_inset Formula $\zeta'$
|
||
\end_inset
|
||
|
||
(electric or magnetic), degree
|
||
\begin_inset Formula $l'$
|
||
\end_inset
|
||
|
||
and order
|
||
\begin_inset Formula $m'$
|
||
\end_inset
|
||
|
||
, the particle responds with waves from the complementary set (e.g.
|
||
outgoing waves), with the same frequency
|
||
\begin_inset Formula $\omega$
|
||
\end_inset
|
||
|
||
, but any type
|
||
\begin_inset Formula $\zeta$
|
||
\end_inset
|
||
|
||
, degree
|
||
\begin_inset Formula $l$
|
||
\end_inset
|
||
|
||
and order
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
, in a way that the Maxwell's equations are satisfied, with the coefficients
|
||
|
||
\begin_inset Formula $T_{l,m;l',m'}^{\zeta,\zeta'}(\omega)$
|
||
\end_inset
|
||
|
||
.
|
||
This yields one row in the scattering matrix (often called the
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix)
|
||
\begin_inset Formula $T(\omega)$
|
||
\end_inset
|
||
|
||
, which fully characterizes the scattering properties of the particle (in
|
||
the linear regime, of course).
|
||
Analytical expression for the matrix is known for spherical scatterer,
|
||
otherwise it is computed numerically (using DDA, BEM or whatever).
|
||
So if we have the two sets of spherical wave functions
|
||
\begin_inset Formula $\vect f_{lm}^{J_{1},\zeta}$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\vect f_{lm}^{J_{2},\zeta}$
|
||
\end_inset
|
||
|
||
and the full
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
exciting
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
wave has electric field given as
|
||
\begin_inset Formula
|
||
\[
|
||
\vect E_{\mathrm{inc}}=\sum_{\zeta'=\mathrm{E,M}}\sum_{l',m'}c_{l'm'}^{\zeta'}\vect f_{l'm'}^{\zeta'},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
the
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
scattered
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
field will be
|
||
\begin_inset Formula
|
||
\[
|
||
\vect E_{\mathrm{scat}}=\sum_{\zeta',l',m'}\sum_{\zeta,l,m}T_{l,m;l',m'}^{\zeta,\zeta'}c_{l'm'}^{\zeta'}\vect f_{lm}^{\zeta},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
and the total field around the scaterer is
|
||
\begin_inset Formula $\vect E=\vect E_{\mathrm{ext}}+\vect E_{\mathrm{scat}}$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Mie theory – full version
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
-matrix for a spherical particle is type-, degree- and order- diagonal,
|
||
that is,
|
||
\begin_inset Formula $T_{l',m';l,m}^{\zeta',\zeta}(\omega)=0$
|
||
\end_inset
|
||
|
||
if
|
||
\begin_inset Formula $l\ne l'$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $m\ne m'$
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Formula $\zeta\ne\zeta'$
|
||
\end_inset
|
||
|
||
.
|
||
Moreover, it does not depend on
|
||
\begin_inset Formula $m$
|
||
\end_inset
|
||
|
||
, so
|
||
\begin_inset Formula
|
||
\[
|
||
T_{l,m;l',m'}^{\zeta,\zeta'}(\omega)=T_{l}^{\zeta}\left(\omega\right)\delta_{\zeta'\zeta}\delta_{l'l}\delta_{m'm}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where for the usual choice
|
||
\begin_inset Formula $J_{1}=1,J_{2}=3$
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula
|
||
\begin{eqnarray*}
|
||
T_{l}^{E}\left(\omega\right) & = & TODO,\\
|
||
T_{l}^{M}(\omega) & = & TODO.
|
||
\end{eqnarray*}
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Long wave approximation for spherical nanoparticle
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
TODO start from
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(A11)"
|
||
key "pustovit_plasmon-mediated_2010"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
and around.
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Note on transforming T-matrix conventions
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
T-matrix depends on the used conventions as well.
|
||
This is not apparent for the Mie case as the T-matrix for a sphere is
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
diagonal
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
.
|
||
But for other shapes, dipole incoming field can induce also higher-order
|
||
multipoles in the nanoparticle, etc.
|
||
The easiest way to determine the transformation properties is to write
|
||
down the total scattered electric field for both conventions in the form
|
||
\begin_inset Formula
|
||
\[
|
||
\vect E_{\mathrm{scat}}=\sum_{n'}\sum_{n}T_{n'}^{n}c^{n'}\vect f_{n}=\sum_{n'}\sum_{n}\widetilde{T}_{n'}^{n}\widetilde{c}^{n'}\widetilde{\vect f}_{n}
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where we merged all the indices into single multiindex
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
or
|
||
\begin_inset Formula $n'$
|
||
\end_inset
|
||
|
||
.
|
||
This way of writing immediately suggest how to transform the T-matrix into
|
||
the new convention if we know the transformation properties of the base
|
||
waves and expansion coefficients, as it reminds the notation used in geometry
|
||
–
|
||
\begin_inset Formula $c^{\alpha}$
|
||
\end_inset
|
||
|
||
are
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
vector coordinates
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $\vect f_{\alpha}$
|
||
\end_inset
|
||
|
||
are
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
base vectors
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
.
|
||
Obviously, T-matrix is then
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
tensor of type (1,1)
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
, and it transforms as vector coordinates (i.e.
|
||
wave expansion coefficients) in the
|
||
\begin_inset Formula $n$
|
||
\end_inset
|
||
|
||
(outgoing wave) indices and as form coordinates in the
|
||
\begin_inset Formula $n'$
|
||
\end_inset
|
||
|
||
(regular/illuminating wave) indices.
|
||
Form coordinates change in the same waves as base vectors
|
||
\end_layout
|
||
|
||
\begin_layout Subsection
|
||
Kristensson to Taylor
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
For instance, let us transform between from the Kristensson's to Taylor's
|
||
convention.
|
||
We know that the Taylor's base vectors are
|
||
\begin_inset Quotes sld
|
||
\end_inset
|
||
|
||
larger
|
||
\begin_inset Quotes srd
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula $\widetilde{\vect N}_{ml}^{(3/1/4)}=\sqrt{l(l+1)}\left(\vect{u/v/w}\right)_{2lm}$
|
||
\end_inset
|
||
|
||
etc, so the coefficients must be smaller by the reciprocal factor, e.g.
|
||
|
||
\begin_inset Formula $\tilde{a}_{ml}=f_{2lm}/\sqrt{l(l+1)}$
|
||
\end_inset
|
||
|
||
(now we assume that there are no other prefactors in the expansion of the
|
||
field, they are already included in the coefficients).
|
||
Then the T-matrix in the Taylor's convention (tilded) can be calculated
|
||
from the T-matrix in the Kristensson's convention as
|
||
\begin_inset Formula
|
||
\[
|
||
\underbrace{\widetilde{T}_{\zeta'l'm'}^{\zeta lm}}_{\mbox{Taylor}}=\frac{\sqrt{l'(l'+1)}}{\sqrt{l(l+1)}}\underbrace{T_{\zeta'l'm'}^{\zeta lm}}_{\mbox{Krist.}}\,_{\leftarrow\mbox{illuminating}}^{\leftarrow\mbox{outgoing}}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Subsubsection
|
||
scuff-tmatrix output
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Indices of the outgoing wave (without primes) come first, illuminating regular
|
||
wave (with primes) second in the output files of scuff-tmatrix.
|
||
It seems that it at least in the electric part, the output of scuff-tmatrix
|
||
is equivalent to the Kristensson's convention.
|
||
Not sure whether it is also true for the E-M cross terms.
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Green's functions
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
xyz pure free-space dipole waves in terms of SVWF
|
||
\end_layout
|
||
|
||
\begin_layout Section
|
||
Mie decomposition of Green's function for single nanoparticle
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Translation of spherical waves: getting insane
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
Cruzan's formulation, Xu's normalisation
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(59)"
|
||
key "xu_efficient_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
:
|
||
\begin_inset Formula
|
||
\[
|
||
B_{m,n,\mu,\nu}=\underbrace{\left(-1\right)^{-m}\frac{\left(2\nu+1\right)\left(n+m\right)!\left(\nu-\mu\right)!}{2n\left(n+1\right)\left(n-m\right)!\left(\nu+\mu\right)!}\sum_{q=1}^{Q_{max}^{-m,n,\mu,\nu}}i^{p+1}\sqrt{\left(\left(p+1\right)^{2}-\left(n-\nu\right)^{2}\right)\left(\left(n+\nu+1\right)^{2}-\left(p+1\right)^{2}\right)}b_{-m,n,\mu,\nu}^{p,p+1}}_{\mbox{(without the \ensuremath{\sum})}\equiv B_{m,n,\mu,\nu}^{q}}z_{p+1}P_{p+1}e^{i\left(\mu-m\right)\phi},
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
after "(28,5,60,61)"
|
||
key "xu_efficient_1998"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Formula $p\equiv n+\nu-2q$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula $Q_{max}^{-m,n,\mu,\nu}\equiv\min\left(n,\nu,\frac{n+\nu+1-\left|\mu-m\right|}{2}\right)$
|
||
\end_inset
|
||
|
||
,
|
||
\begin_inset Formula
|
||
\[
|
||
b_{-m,n,\mu,\nu}^{p,p+1}\equiv\left(-1\right)^{\mu-m}\left(2p+3\right)\sqrt{\frac{\left(n-m\right)!\left(\nu+\mu\right)!\left(p+m-\mu+1\right)!}{\left(n+m\right)!\left(\nu-\mu\right)!\left(p-m+\mu+1\right)!}}\begin{pmatrix}n & \nu & p+1\\
|
||
-m & \mu & m-\mu
|
||
\end{pmatrix}\begin{pmatrix}n & \nu & p\\
|
||
0 & 0 & 0
|
||
\end{pmatrix}.
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Multiple scattering: nice linear algebra born from all the mess
|
||
\end_layout
|
||
|
||
\begin_layout Chapter
|
||
Quantisation of quasistatic modes of a sphere
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
bibfiles "Electrodynamics"
|
||
options "plain"
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|