From b14e776c34648c1874d5f33383efdf00f3817605 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 29 Jul 2019 10:51:43 +0300 Subject: [PATCH] Single particle scattering done Former-commit-id: c10e7db98d7f8c2eacbb8c7ef818371c61d2a7f4 --- lepaper/finite-cs.lyx | 136 +---------------------- lepaper/finite.lyx | 247 +++++++++++++++++++++++++++++++++++++++++- 2 files changed, 244 insertions(+), 139 deletions(-) diff --git a/lepaper/finite-cs.lyx b/lepaper/finite-cs.lyx index a6bfc45..6d52637 100644 --- a/lepaper/finite-cs.lyx +++ b/lepaper/finite-cs.lyx @@ -94,36 +94,6 @@ \begin_body -\begin_layout Subsection -Dual vector spherical harmonics -\end_layout - -\begin_layout Standard -For evaluation of expansion coefficients of incident fields, it is useful - to introduce „dual“ vector spherical harmonics -\begin_inset Formula $\vshD{\tau}lm$ -\end_inset - - defined by duality relation -\begin_inset Formula -\begin{equation} -\iint\vshD{\tau'}{l'}{m'}\left(\uvec r\right)\cdot\vsh{\tau}lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual vsh} -\end{equation} - -\end_inset - -(complex conjugation not implied in the dot product here). - In our convention, we have -\begin_inset Formula -\[ -\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right). -\] - -\end_inset - - -\end_layout - \begin_layout Subsection Translation operators \end_layout @@ -440,39 +410,6 @@ better wording Plane wave expansion coefficients \end_layout -\begin_layout Standard -A transversal ( -\begin_inset Formula $\vect k\cdot\vect E_{0}=0$ -\end_inset - -) plane wave propagating in direction -\begin_inset Formula $\uvec k$ -\end_inset - - with (complex) amplitude -\begin_inset Formula $\vect E_{0}$ -\end_inset - - can be expanded into regular VSWFs [REF KRIS] -\begin_inset Formula -\[ -\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right), -\] - -\end_inset - -with expansion coefficients -\begin_inset Formula -\begin{eqnarray} -\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\ -\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion} -\end{eqnarray} - -\end_inset - - -\end_layout - \begin_layout Subsection Multiple-scattering problem \end_layout @@ -489,78 +426,7 @@ Multiple-scattering problem \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 -Cross-sections (single-particle) -\end_layout - -\begin_layout Standard -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 -\begin_inset CommandInset citation -LatexCommand cite -after "sect. 7.8.2" -key "kristensson_scattering_2016" -literal "true" - -\end_inset - - -\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\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\ -\sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\ -\sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\ - & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single} -\end{eqnarray} - -\end_inset - -where -\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$ -\end_inset - - is the vector of plane wave expansion coefficients as in -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:plane wave expansion" - -\end_inset - -. - +Cross-sections (many scatterers) \end_layout \begin_layout Standard diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 1f8c54c..f8f359f 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -516,9 +516,9 @@ doplnit frekvence a polohy \begin_inset Formula -\[ -\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right). -\] +\begin{equation} +\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).\label{eq:E field expansion} +\end{equation} \end_inset @@ -600,6 +600,19 @@ noprefix "false" . \end_layout +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TOOD H-field expansion here? +\end_layout + +\end_inset + + +\end_layout + \begin_layout Standard \begin_inset Formula $T$ \end_inset @@ -690,6 +703,16 @@ The magnitude of the \end_inset will also be negligible. +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO when it will not be negligible +\end_layout + +\end_inset + + \end_layout \begin_layout Standard @@ -711,7 +734,223 @@ A rule of thumb to choose the \end_layout \begin_layout Subsubsection -Absorbed power +Power transport +\end_layout + +\begin_layout Standard +For convenience, let us introduce a short-hand matrix notation for the expansion + coefficients and related quantities, so that we do not need to write the + indices explicitly; so for example, eq. + +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:T-matrix definition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + would be written as +\begin_inset Formula $\outcoeffp{}=T\rcoeffp{}$ +\end_inset + +, where +\begin_inset Formula $\rcoeffp{},\outcoeffp{}$ +\end_inset + + are column vectors with the expansion coefficients. + Transposed and complex-conjugated matrices are labeled with the +\begin_inset Formula $\dagger$ +\end_inset + + superscript. +\end_layout + +\begin_layout Standard +With this notation, we state an important result about power transport, + derivation of which can be found in +\begin_inset CommandInset citation +LatexCommand cite +after "sect. 7.3" +key "kristensson_scattering_2016" +literal "true" + +\end_inset + +. + Let the field in +\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\end_inset + + have expansion as in +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:E field expansion" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + Then the net power transported from +\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\end_inset + + to +\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\end_inset + + via by electromagnetic radiation is +\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 + +In realistic scattering setups, power is transferred by radiation into +\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\end_inset + + and absorbed by the enclosed scatterer, so +\begin_inset Formula $P$ +\end_inset + + is negative and its magnitude equals to power absorbed by the scatterer. +\end_layout + +\begin_layout Subsubsection +Plane wave expansion +\end_layout + +\begin_layout Standard +In many scattering problems considered in practice, the driving field is + a plane wave. + A transversal ( +\begin_inset Formula $\vect k\cdot\vect E_{0}=0$ +\end_inset + +) plane wave propagating in direction +\begin_inset Formula $\uvec k$ +\end_inset + + with (complex) amplitude +\begin_inset Formula $\vect E_{0}$ +\end_inset + + can be expanded into regular VSWFs [REF KRIS] as +\begin_inset Formula +\[ +\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right), +\] + +\end_inset + +with expansion coefficients +\begin_inset Formula +\begin{eqnarray} +\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\ +\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion} +\end{eqnarray} + +\end_inset + +where +\begin_inset Formula $\vshD{\tau}lm$ +\end_inset + + are the +\begin_inset Quotes eld +\end_inset + +dual +\begin_inset Quotes erd +\end_inset + + vector spherical harmonics defined by duality relation with the +\begin_inset Quotes eld +\end_inset + +usual +\begin_inset Quotes erd +\end_inset + + vector spherical harmonics +\begin_inset Formula +\begin{equation} +\iint\vshD{\tau'}{l'}{m'}\left(\uvec r\right)\cdot\vsh{\tau}lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual vsh} +\end{equation} + +\end_inset + +(complex conjugation not implied in the dot product here). + In our convention, we have +\begin_inset Formula +\[ +\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right). +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +Cross-sections (single-particle) +\end_layout + +\begin_layout Standard +With the +\begin_inset Formula $T$ +\end_inset + +-matrix and expansion coefficients of plane waves in hand, we can state + the expressions for cross-sections of a single scatterer. + 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 +\begin_inset CommandInset citation +LatexCommand cite +after "sect. 7.8.2" +key "kristensson_scattering_2016" +literal "true" + +\end_inset + + +\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\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\ +\sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\ +\sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\ + & & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single} +\end{eqnarray} + +\end_inset + +where +\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$ +\end_inset + + is the vector of plane wave expansion coefficients as in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:plane wave expansion" + +\end_inset + +. + \end_layout \begin_layout Subsection