Mie scattering coefficients, might be wrong...

Former-commit-id: 5b52c0e8a218dc1bc9c7e84725d23aa08e2eebf6

worknotes.lyx

@@ -428,6 +428,30 @@ These expansions should be OK in SI units (take the Fourier transform of
 
 \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
 \begin_inset Formula 
 \[
@@ -524,8 +548,8 @@ key "xu_electromagnetic_1995"
 
 \begin_inset Formula 
 \begin{alignat*}{1}
-a_{mn}^{j} & =a_{n}^{j}p_{mn}^{j},\quad b_{mn}^{j}=b_{n}^{j}q_{mn}^{j},\\
-c_{mn}^{j} & =c_{n}^{j}q_{mn}^{j},\quad d_{mn}^{j}=d_{n}^{j}p_{mn}^{j},
+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
@@ -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).
 \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
@@ -1296,8 +1371,8 @@ Scattering-Taylor.ipynb
 \end_layout
 
 \begin_layout Standard
-In the conventions used (no Condon-Shortley phase AFAIK), the following
- symmetries hold for 
+In the conventions used in the code and the corresponding libraries, the
+ following symmetries hold for 
 \begin_inset Formula $J=1$
 \end_inset