diff --git a/worknotes.lyx b/worknotes.lyx index b26ad81..f657533 100644 --- a/worknotes.lyx +++ b/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