From a6e9cbc351485a55e231f4da71d92d6b4d8f2529 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Sun, 28 Jul 2019 11:26:12 +0300 Subject: [PATCH] Finite system cross sections done Former-commit-id: bf5afd2dfbcadf4aede6550ffd5e0fc5d030e3e1 --- lepaper/finite-cs.lyx | 150 +++++++++++++++++++++++++++++++++++++++++- 1 file changed, 147 insertions(+), 3 deletions(-) diff --git a/lepaper/finite-cs.lyx b/lepaper/finite-cs.lyx index 7cf17b1..28072a4 100644 --- a/lepaper/finite-cs.lyx +++ b/lepaper/finite-cs.lyx @@ -469,6 +469,21 @@ with expansion coefficients \end_inset +\end_layout + +\begin_layout Subsection +Multiple-scattering problem +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\rcoeffp p=\rcoeffincp p+\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q\label{eq:particle total incident field coefficient a} +\end{equation} + +\end_inset + + \end_layout \begin_layout Subsection @@ -501,9 +516,14 @@ Cross-sections (single-particle) \end_layout \begin_layout Standard -Extinction, scattering and absorption cross sections of a single particle - irradiated by a plane wave propagating in direction +Assuming a non-lossy background medium, extinction, scattering and absorption + cross sections of a single scatterer irradiated by a plane wave propagating + in direction \begin_inset Formula $\uvec k$ +\end_inset + + and (complex) amplitude +\begin_inset Formula $\vect E_{0}$ \end_inset are @@ -538,6 +558,7 @@ reference "eq:plane wave expansion" \end_inset . + \end_layout \begin_layout Standard @@ -746,12 +767,135 @@ noprefix "false" \sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\outcoeffp p^{\dagger}\troprp pq\outcoeffp q\nonumber \\ & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\rcoeffp p^{\dagger}\Tp p^{\dagger}\troprp pq\Tp q\rcoeffp q,\label{eq:scattering CS multi}\\ \sigma_{\mathrm{abs}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\Re\left(\outcoeffp p^{\dagger}\left(\rcoeffincp p+\sum_{q\in\mathcal{P}}\troprp pq\outcoeffp q\right)\right)\nonumber \\ - & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}TODO,\label{eq:absorption CS multi} + & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\mbox{TODO}.\label{eq:absorption CS multi} \end{eqnarray} \end_inset +An alternative approach to derive the absorption cross section is via a + power transport argument. + Note the direct proportionality between absorption cross section +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:absorption CS single" +plural "false" +caps "false" +noprefix "false" +\end_inset + + and net radiated power for single scatterer +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Power transport" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset Formula $\sigma_{\mathrm{abs}}=-\eta_{0}\eta P/2\left\Vert \vect E_{0}\right\Vert ^{2}$ +\end_inset + +. + In the many-particle setup (with non-lossy background medium, so that only + the particles absorb), the total absorbed power is equal to the sum of + absorbed powers on each particle, +\begin_inset Formula $-P=\sum_{p\in\mathcal{P}}-P_{p}$ +\end_inset + +. + Using the power transport formula +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:Power transport" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + particle-wise gives +\begin_inset Formula +\begin{equation} +\sigma_{\mathrm{abs}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\left(\Re\left(\rcoeffp p^{\dagger}\outcoeffp p\right)+\left\Vert \outcoeffp p\right\Vert ^{2}\right)\label{eq:absorption CS multi alternative} +\end{equation} + +\end_inset + +which seems different from +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:absorption CS multi" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, but using +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:particle total incident field coefficient a" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, we can rewrite it as +\begin_inset Formula +\begin{align*} +\sigma_{\mathrm{abs}}\left(\uvec k\right) & =-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\Re\left(\outcoeffp p^{\dagger}\left(\rcoeffp p+\outcoeffp p\right)\right)\\ + & =-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\Re\left(\outcoeffp p^{\dagger}\left(\rcoeffincp p+\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q+\outcoeffp p\right)\right). +\end{align*} + +\end_inset + +It is easy to show that all the terms of +\begin_inset Formula $\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\tropsp pq\outcoeffp q$ +\end_inset + + containing the singular spherical Bessel functions +\begin_inset Formula $y_{l}$ +\end_inset + + are imaginary, +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO better formulation +\end_layout + +\end_inset + + so that actually +\begin_inset Formula $\sum_{p\in\mathcal{P}}\Re\left(\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\tropsp pq\outcoeffp q+\outcoeffp p^{\dagger}\outcoeffp p\right)=\sum_{p\in\mathcal{P}}\Re\left(\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\troprp pq\outcoeffp q+\outcoeffp p^{\dagger}\outcoeffp p\right)=\sum_{p\in\mathcal{P}}\Re\left(\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\outcoeffp p^{\dagger}\troprp pq\outcoeffp q+\outcoeffp p^{\dagger}\troprp pp\outcoeffp p\right)=\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\outcoeffp p^{\dagger}\troprp pq\outcoeffp q,$ +\end_inset + + proving that the expressions in +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:absorption CS multi" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:absorption CS multi alternative" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + are equal. \end_layout \end_body