From 81fdc50dc0632e1c46f3943c4e24312d8fb84790 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Sun, 28 Jul 2019 00:48:30 +0300 Subject: [PATCH] =?UTF-8?q?Jdu=20sp=C3=83ti?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: 0801757927c86093e3621a3cf1e803df21318c4f --- lepaper/finite-cs.lyx | 117 +++++++++++++++++++++++++++++++++++++----- 1 file changed, 104 insertions(+), 13 deletions(-) diff --git a/lepaper/finite-cs.lyx b/lepaper/finite-cs.lyx index 834bbd3..7cf17b1 100644 --- a/lepaper/finite-cs.lyx +++ b/lepaper/finite-cs.lyx @@ -469,6 +469,31 @@ with expansion coefficients \end_inset +\end_layout + +\begin_layout Subsection +Power transport +\end_layout + +\begin_layout Standard +Radiated power +\begin_inset CommandInset citation +LatexCommand cite +after "sect. 7.3" +key "kristensson_scattering_2016" +literal "true" + +\end_inset + + +\begin_inset Formula +\begin{equation} +P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)=\frac{1}{2k^{2}\eta_{0}\eta}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{}.\label{eq:Power transport} +\end{equation} + +\end_inset + + \end_layout \begin_layout Subsection @@ -519,20 +544,15 @@ reference "eq:plane wave expansion" For a system of many scatterers, Kristensson \begin_inset CommandInset citation LatexCommand cite +after "sect. 9.2.2" key "kristensson_scattering_2016" literal "false" \end_inset - derives only the scattering cross section formula -\begin_inset Formula -\[ -\sigma_{\mathrm{scat}}\left(\uvec k\right)=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\left\Vert \outcoeffp p\right\Vert ^{2}. -\] - -\end_inset - -Let us derive the many-particle scattering and absorption cross sections. + derives only the extinction cross section formula. + Let us re-derive it together with the many-particle scattering and absorption + cross sections. First, let us take a ball circumscribing all the scatterers at once, \begin_inset Formula $\openball R{\vect r_{\square}}\supset\particle$ \end_inset @@ -577,6 +597,10 @@ reference "eq:extincion CS single" LatexCommand eqref reference "eq:absorption CS single" +\end_inset + + with +\begin_inset Formula $\rcoeffp{}=\rcoeffp{\square},\outcoeffp{}=\outcoeffp{\square}$ \end_inset to obtain the cross sections. @@ -654,13 +678,80 @@ noprefix "false" properties, one has \begin_inset Formula -\[ -\rcoeffp{\square}^{\dagger}\outcoeffp{\square}=\rcoeffincp p^{\dagger}\sum_{q\in\mathcal{P}}\troprp pqf_{q}. -\] +\begin{align} +\rcoeffp{\square}^{\dagger}\outcoeffp{\square} & =\rcoeffincp p^{\dagger}\troprp p{\square}\troprp{\square}q\outcoeffp q=\rcoeffincp p^{\dagger}\sum_{q\in\mathcal{P}}\troprp pqf_{q}\nonumber \\ + & =\sum_{q\in\mathcal{P}}\left(\troprp qp\rcoeffincp p\right)^{\dagger}f_{q}=\sum_{q\in\mathcal{P}}\rcoeffincp q^{\dagger}f_{q},\label{eq:atf form multiparticle} +\end{align} \end_inset - +where only the last expression is suitable for numerical evaluation with + truncated matrices, because the previous ones contain a translation operator + right next to an incident field coefficient vector (see Sec. + TODO). + Similarly, +\begin_inset Formula +\begin{align} +\left\Vert \outcoeffp{\square}\right\Vert ^{2} & =\outcoeffp{\square}^{\dagger}\outcoeffp{\square}=\sum_{p\in\mathcal{P}}\left(\troprp{\square}p\outcoeffp p\right)^{\dagger}\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q\nonumber \\ + & =\sum_{p\in\mathcal{P}}\sum_{q\in\mathcal{P}}\outcoeffp p^{\dagger}\troprp pq\outcoeffp q.\label{eq:f squared form multiparticle} +\end{align} + +\end_inset + +Substituting +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:atf form multiparticle" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:f squared form multiparticle" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + into +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:scattering CS single" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:absorption CS single" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, we get the many-particle expressions for extinction, scattering and absorption + cross sections suitable for numerical evaluation: +\begin_inset Formula +\begin{eqnarray} +\sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\sum_{p\in\mathcal{P}}\rcoeffincp p^{\dagger}\outcoeffp p=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\sum_{p\in\mathcal{P}}\rcoeffincp p^{\dagger}\left(\Tp p+\Tp p^{\dagger}\right)\rcoeffp p,\label{eq:extincion CS multi}\\ +\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} +\end{eqnarray} + +\end_inset + + \end_layout \end_body