Mie scattering coefficients, might be wrong...

Former-commit-id: 5b52c0e8a218dc1bc9c7e84725d23aa08e2eebf6
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Marek Nečada 2016-02-09 11:37:43 +02:00
parent 5d22179cc9
commit 83a6fe24a8
1 changed files with 79 additions and 4 deletions

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@ -428,6 +428,30 @@ These expansions should be OK in SI units (take the Fourier transform of
\end_inset \end_inset
Note that
\begin_inset Formula $k/\omega\mu=\sqrt{\varepsilon_{r}\varepsilon_{0}/\mu_{r}\mu_{0}}=1/\eta_{r}\eta_{0}.$
\end_inset
The
\begin_inset Quotes eld
\end_inset
factor
\begin_inset Quotes erd
\end_inset
\begin_inset Formula $H/E$
\end_inset
is thus
\begin_inset Formula $-ik/\omega\mu=-i\sqrt{\varepsilon_{r}\varepsilon_{0}/\mu_{r}\mu_{0}}$
\end_inset
, which is important in determining the Mie coefficients.
\end_layout
\begin_layout Standard
The common multipole-dependent factor is given by The common multipole-dependent factor is given by
\begin_inset Formula \begin_inset Formula
\[ \[
@ -524,8 +548,8 @@ key "xu_electromagnetic_1995"
\begin_inset Formula \begin_inset Formula
\begin{alignat*}{1} \begin{alignat*}{1}
a_{mn}^{j} & =a_{n}^{j}p_{mn}^{j},\quad b_{mn}^{j}=b_{n}^{j}q_{mn}^{j},\\ a_{mn}^{j} & =R_{n}^{V}p_{mn}^{j},\quad b_{mn}^{j}=R_{n}^{H}q_{mn}^{j},\\
c_{mn}^{j} & =c_{n}^{j}q_{mn}^{j},\quad d_{mn}^{j}=d_{n}^{j}p_{mn}^{j}, c_{mn}^{j} & =T_{n}^{H}q_{mn}^{j},\quad d_{mn}^{j}=T_{n}^{V}p_{mn}^{j},
\end{alignat*} \end{alignat*}
\end_inset \end_inset
@ -541,6 +565,57 @@ in other words, the Mie coefficients do not depend on
(which is not surprising and probably follows from the Wigner-Eckart theorem). (which is not surprising and probably follows from the Wigner-Eckart theorem).
\end_layout \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 \begin_layout Subsubsection
Translation coefficients Translation coefficients
\end_layout \end_layout
@ -1296,8 +1371,8 @@ Scattering-Taylor.ipynb
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
In the conventions used (no Condon-Shortley phase AFAIK), the following In the conventions used in the code and the corresponding libraries, the
symmetries hold for following symmetries hold for
\begin_inset Formula $J=1$ \begin_inset Formula $J=1$
\end_inset \end_inset