From 64e122b93757c28d9b19ab3dc2aefe728d663ed8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 29 Jul 2019 12:41:02 +0300 Subject: [PATCH] Merge finite-cs.lyx into finite.lyx Former-commit-id: c42c118767e8fde87df5655946c3489a70627033 --- lepaper/arrayscat.lyx | 40 +- lepaper/finite-cs.lyx | 770 -------------------------------------- lepaper/finite.lyx | 831 +++++++++++++++++++++++++++++++++++++++++- 3 files changed, 848 insertions(+), 793 deletions(-) delete mode 100644 lepaper/finite-cs.lyx diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index b4cb7c7..f00e958 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -660,6 +660,27 @@ My implementation. Maybe put the numerical results separately in the end. \end_layout +\begin_layout Section +TODOs +\end_layout + +\begin_layout Itemize +Consistent notation of balls. + How is the difference between two cocentric balls called? +\end_layout + +\begin_layout Itemize +Abstract. +\end_layout + +\begin_layout Itemize +Example results. +\end_layout + +\begin_layout Itemize +Concrete comparison with other methods. +\end_layout + \begin_layout Standard \begin_inset CommandInset include LatexCommand include @@ -680,6 +701,10 @@ literal "true" \end_layout \begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset CommandInset include LatexCommand include filename "finite-old.lyx" @@ -690,12 +715,6 @@ literal "true" \end_layout -\begin_layout Standard -\begin_inset CommandInset include -LatexCommand include -filename "finite-cs.lyx" -literal "true" - \end_inset @@ -713,6 +732,10 @@ literal "true" \end_layout \begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset CommandInset include LatexCommand include filename "infinite-old.lyx" @@ -721,6 +744,11 @@ literal "true" \end_inset +\end_layout + +\end_inset + + \end_layout \begin_layout Standard diff --git a/lepaper/finite-cs.lyx b/lepaper/finite-cs.lyx deleted file mode 100644 index 6d52637..0000000 --- a/lepaper/finite-cs.lyx +++ /dev/null @@ -1,770 +0,0 @@ -#LyX 2.4 created this file. For more info see https://www.lyx.org/ -\lyxformat 583 -\begin_document -\begin_header -\save_transient_properties true -\origin unavailable -\textclass article -\use_default_options true -\maintain_unincluded_children false -\language english -\language_package default -\inputencoding utf8 -\fontencoding auto -\font_roman "default" "TeX Gyre Pagella" -\font_sans "default" "default" -\font_typewriter "default" "default" -\font_math "auto" "auto" -\font_default_family default -\use_non_tex_fonts false -\font_sc false -\font_roman_osf true -\font_sans_osf false -\font_typewriter_osf false -\font_sf_scale 100 100 -\font_tt_scale 100 100 -\use_microtype false -\use_dash_ligatures false -\graphics default -\default_output_format default -\output_sync 0 -\bibtex_command default -\index_command default -\float_placement class -\float_alignment class -\paperfontsize default -\spacing single -\use_hyperref true -\pdf_title "Sähköpajan päiväkirja" -\pdf_author "Marek Nečada" -\pdf_bookmarks true -\pdf_bookmarksnumbered false -\pdf_bookmarksopen false -\pdf_bookmarksopenlevel 1 -\pdf_breaklinks false -\pdf_pdfborder false -\pdf_colorlinks false -\pdf_backref false -\pdf_pdfusetitle true -\papersize default -\use_geometry false -\use_package amsmath 1 -\use_package amssymb 1 -\use_package cancel 1 -\use_package esint 1 -\use_package mathdots 1 -\use_package mathtools 1 -\use_package mhchem 1 -\use_package stackrel 1 -\use_package stmaryrd 1 -\use_package undertilde 1 -\cite_engine basic -\cite_engine_type default -\biblio_style plain -\use_bibtopic false -\use_indices false -\paperorientation portrait -\suppress_date false -\justification true -\use_refstyle 1 -\use_minted 0 -\use_lineno 0 -\index Index -\shortcut idx -\color #008000 -\end_index -\secnumdepth 3 -\tocdepth 3 -\paragraph_separation indent -\paragraph_indentation default -\is_math_indent 0 -\math_numbering_side default -\quotes_style english -\dynamic_quotes 0 -\papercolumns 1 -\papersides 1 -\paperpagestyle default -\tablestyle default -\tracking_changes false -\output_changes false -\html_math_output 0 -\html_css_as_file 0 -\html_be_strict false -\end_header - -\begin_body - -\begin_layout Subsection -Translation operators -\end_layout - -\begin_layout Standard -Let -\begin_inset Formula $\vect r_{1},\vect r_{2}$ -\end_inset - - be two different origins; a regular VSWF with origin -\begin_inset Formula $\vect r_{1}$ -\end_inset - - can be always expanded in terms of regular VSWFs with origin -\begin_inset Formula $\vect r_{2}$ -\end_inset - - as follows: -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right)=\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right),\label{eq:regular vswf translation} -\end{equation} - -\end_inset - -where an explicit formula for the (regular) -\emph on -translation operator -\emph default - -\begin_inset Formula $\tropr$ -\end_inset - - reads in eq. - -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:translation operator" - -\end_inset - - below. - For singular (outgoing) waves, the form of the expansion differs inside - and outside the ball -\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$ -\end_inset - - -\begin_inset Formula -\begin{eqnarray} -\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases} -\sum_{\tau'l'm'}\trops_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfouttlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right), & \vect r\in\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}\\ -\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right), & \vect r\notin\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\left|\vect r_{1}\right|} -\end{cases},\label{eq:singular vswf translation} -\end{eqnarray} - -\end_inset - -where the singular translation operator -\begin_inset Formula $\trops$ -\end_inset - - has the same form as -\begin_inset Formula $\tropr$ -\end_inset - - in -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:translation operator" - -\end_inset - - except the regular spherical Bessel functions -\begin_inset Formula $j_{l}$ -\end_inset - - are replaced with spherical Hankel functions -\begin_inset Formula $h_{l}^{(1)}$ -\end_inset - -. - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -TODO note about expansion exactly on the sphere. -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard -As MSTMM deals most of the time with the -\emph on -expansion coefficients -\emph default - of fields -\begin_inset Formula $\rcoeffptlm p{\tau}lm,\outcoeffptlm p{\tau}lm$ -\end_inset - - in different origins -\begin_inset Formula $\vect r_{p}$ -\end_inset - - rather than with the VSWFs directly, let us write down how -\emph on -they -\emph default - transform under translation. - Let us assume the field can be in terms of regular waves everywhere, and - expand it in two different origins -\begin_inset Formula $\vect r_{p},\vect r_{q}$ -\end_inset - -, -\begin_inset Formula -\[ -\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right). -\] - -\end_inset - -Re-expanding the waves around -\begin_inset Formula $\vect r_{p}$ -\end_inset - - in terms of waves around -\begin_inset Formula $\vect r_{q}$ -\end_inset - - using -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:regular vswf translation" - -\end_inset - -, -\begin_inset Formula -\[ -\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right) -\] - -\end_inset - -and comparing to the original expansion around -\begin_inset Formula $\vect r_{q}$ -\end_inset - -, we obtain -\begin_inset Formula -\begin{equation} -\rcoeffptlm q{\tau'}{l'}{m'}=\sum_{\tau,l,m}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\rcoeffptlm p{\tau}lm.\label{eq:regular vswf coefficient translation} -\end{equation} - -\end_inset - -For the sake of readability, we introduce a shorthand matrix form for -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:regular vswf coefficient translation" - -\end_inset - - -\begin_inset Formula -\begin{equation} -\rcoeffp q=\troprp qp\rcoeffp p\label{eq:reqular vswf coefficient vector translation} -\end{equation} - -\end_inset - -(note the reversed indices; TODO redefine them in -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:regular vswf translation" - -\end_inset - -, -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:singular vswf translation" - -\end_inset - -? Similarly, if we had only outgoing waves in the original expansion around - -\begin_inset Formula $\vect r_{p}$ -\end_inset - -, we would get -\begin_inset Formula -\begin{equation} -\rcoeffp q=\tropsp qp\outcoeffp p\label{eq:singular to regular vswf coefficient vector translation} -\end{equation} - -\end_inset - -for the expansion inside the ball -\begin_inset Formula $\openball{\left\Vert \vect r_{q}-\vect r_{p}\right\Vert }{\vect r_{p}}$ -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -CHECKME -\end_layout - -\end_inset - - and -\begin_inset Formula -\begin{equation} -\outcoeffp q=\troprp qp\outcoeffp p\label{eq:singular to singular vswf coefficient vector translation-1} -\end{equation} - -\end_inset - -outside. -\end_layout - -\begin_layout Standard -In our convention, the regular translation operator can be expressed explicitly - as -\begin_inset Formula -\begin{equation} -\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator} -\end{equation} - -\end_inset - -The singular operator -\begin_inset Formula $\trops$ -\end_inset - - for re-expanding outgoing waves into regular ones has the same form except - the regular spherical Bessel functions -\begin_inset Formula $j_{l}$ -\end_inset - - in are replaced with spherical Hankel functions -\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -In our convention, the regular translation operator is unitary, -\begin_inset Formula $\left(\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)\right)^{-1}=\tropr_{\tau lm;\tau'l'm'}\left(-\vect d\right)=\tropr_{\tau'l'm';\tau lm}^{*}\left(\vect d\right)$ -\end_inset - -, -\begin_inset Note Note -status open - -\begin_layout Plain Layout -todo different notation for the complex conjugation without transposition??? -\end_layout - -\end_inset - - or in the per-particle matrix notation, -\begin_inset Formula -\begin{equation} -\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity} -\end{equation} - -\end_inset - -. - Note that truncation at finite multipole degree breaks the unitarity, -\begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$ -\end_inset - -, which has to be taken into consideration when evaluating quantities such - as absorption or scattering cross sections. - Similarly, the full regular operators can be composed -\begin_inset Note Note -status open - -\begin_layout Plain Layout -better wording -\end_layout - -\end_inset - -, -\begin_inset Formula -\begin{equation} -\troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition} -\end{equation} - -\end_inset - - but truncation breaks this, -\begin_inset Formula $\truncated{\troprp ac}L\ne\truncated{\troprp ab}L\truncated{\troprp bc}L.$ -\end_inset - - -\end_layout - -\begin_layout Subsection -Plane wave expansion coefficients -\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 -Cross-sections (many scatterers) -\end_layout - -\begin_layout Standard -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 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 - -. - Outside -\begin_inset Formula $\openball R{\vect r_{\square}}$ -\end_inset - -, we can describe the EM fields as if there was only a single scatterer, -\begin_inset Formula -\[ -\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right), -\] - -\end_inset - -where -\begin_inset Formula $\rcoeffp{\square},\outcoeffp{\square}$ -\end_inset - - are the vectors of VSWF expansion coefficients of the incident and total - scattered fields, respectively, at origin -\begin_inset Formula $\vect r_{\square}$ -\end_inset - -. - In principle, one could evaluate -\begin_inset Formula $\outcoeffp{\square}$ -\end_inset - - using the translation operators (REF!!!) and use the single-scatterer formulae - -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:extincion CS single" - -\end_inset - -– -\begin_inset CommandInset ref -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. - However, this is not suitable for numerical evaluation with truncation - in multipole degree; hence we need to express them in terms of particle-wise - expansions -\begin_inset Formula $\rcoeffp p,\outcoeffp p$ -\end_inset - -. - The original incident field re-expanded around -\begin_inset Formula $p$ -\end_inset - --th particle reads according to -\begin_inset CommandInset ref -LatexCommand ref -reference "eq:regular vswf translation" -plural "false" -caps "false" -noprefix "false" - -\end_inset - - -\begin_inset Formula -\begin{equation} -\rcoeffincp p=\troprp p{\square}\rcoeffp{\square}\label{eq:a_inc local from global} -\end{equation} - -\end_inset - -whereas the contributions of fields scattered from each particle expanded - around the global origin -\begin_inset Formula $\vect r_{\square}$ -\end_inset - - is, according to -\begin_inset CommandInset ref -LatexCommand ref -reference "eq:singular vswf translation" -plural "false" -caps "false" -noprefix "false" - -\end_inset - -, -\begin_inset Formula -\begin{equation} -\outcoeffp{\square}=\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q.\label{eq:f global from local} -\end{equation} - -\end_inset - -Using the unitarity -\begin_inset CommandInset ref -LatexCommand ref -reference "eq:regular translation unitarity" -plural "false" -caps "false" -noprefix "false" - -\end_inset - - and composition -\begin_inset CommandInset ref -LatexCommand ref -reference "eq:regular translation composition" -plural "false" -caps "false" -noprefix "false" - -\end_inset - - properties, one has -\begin_inset Formula -\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}}\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 -\end_document diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index f8f359f..9e8f2f6 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -144,7 +144,7 @@ employing point group symmetries and decomposing the problem to decrease \end_deeper \end_deeper \begin_layout Subsection -Motivation +Motivation/intro \end_layout \begin_layout Standard @@ -162,9 +162,9 @@ The basic idea of MSTMM is quite simple: the driving electromagnetic field \end_layout \begin_layout Standard -However, the expressions appearing in the re-expansions are fairly complicated, - and the implementation of MSTMM is extremely error-prone also due to the - various conventions used in the literature. +The expressions appearing in the re-expansions are fairly complicated, and + the implementation of MSTMM is extremely error-prone also due to the various + conventions used in the literature. Therefore although we do not re-derive from scratch the expressions that can be found elsewhere in literature, we always state them explicitly in our convention. @@ -326,8 +326,8 @@ vector spherical harmonics \begin_inset Formula \begin{align*} -\vsh 1lm & =\\ -\vsh 2lm & =\\ +\vsh 1lm & =TODO\\ +\vsh 2lm & =fixme\\ \vsh 3lm & = \end{align*} @@ -452,16 +452,16 @@ noprefix "false" \end_inset inside a ball -\begin_inset Formula $\openball 0R$ +\begin_inset Formula $\openball 0{R^{>}}$ \end_inset with radius -\begin_inset Formula $R$ +\begin_inset Formula $R^{>}$ \end_inset and center in the origin; however, if the equation is not guaranteed to hold inside a smaller ball -\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\begin_inset Formula $B_{0}\left(R\right)$ \end_inset around the origin (typically due to presence of a scatterer), one has to @@ -470,7 +470,7 @@ noprefix "false" \end_inset to have a complete basis of the solutions in the volume -\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset . @@ -492,11 +492,11 @@ The single-particle scattering problem at frequency \end_inset can be posed as follows: Let a scatterer be enclosed inside the ball -\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\begin_inset Formula $B_{0}\left(R\right)$ \end_inset and let the whole volume -\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset be filled with a homogeneous isotropic medium with wave number @@ -598,6 +598,19 @@ noprefix "false" \end_inset . + The outgoing VSWF expansion coefficients +\begin_inset Formula $\outcoefftlm{\tau}lm$ +\end_inset + + are related to the induced electric ( +\begin_inset Formula $\tau=1$ +\end_inset + +) and magnetic ( +\begin_inset Formula $\tau=2$ +\end_inset + +) multipole polarisation amplitudes of the scatterer. \end_layout \begin_layout Standard @@ -780,7 +793,7 @@ literal "true" . Let the field in -\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset have expansion as in @@ -795,11 +808,11 @@ noprefix "false" . Then the net power transported from -\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\begin_inset Formula $B_{0}\left(R\right)$ \end_inset to -\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset via by electromagnetic radiation is @@ -811,7 +824,7 @@ P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\ri \end_inset In realistic scattering setups, power is transferred by radiation into -\begin_inset Formula $B_{0}\left(R_{<}\right)$ +\begin_inset Formula $B_{0}\left(R\right)$ \end_inset and absorbed by the enclosed scatterer, so @@ -897,7 +910,7 @@ usual \end_layout -\begin_layout Subsection +\begin_layout Subsubsection Cross-sections (single-particle) \end_layout @@ -957,8 +970,792 @@ reference "eq:plane wave expansion" Multiple scattering \end_layout +\begin_layout Standard +If the system consists of multiple scatterers, the EM fields around each + one can be expanded in analogous way. + Let +\begin_inset Formula $\mathcal{P}$ +\end_inset + + be an index set labeling the scatterers. + We enclose each scatterer in a ball +\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)$ +\end_inset + + such that the balls do not touch, +\begin_inset Formula $B_{\vect r_{p}}\left(R_{p}\right)\cap B_{\vect r_{q}}\left(R_{q}\right)=\emptyset;p,q\in\mathcal{P}$ +\end_inset + +, +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO bacha, musejí být uzavřené! +\end_layout + +\end_inset + +so there is a non-empty volume +\begin_inset Note Note +status open + +\begin_layout Plain Layout +jaksetometuje? +\end_layout + +\end_inset + + +\begin_inset Formula $\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right)$ +\end_inset + + around each one that contains only the background medium without any scatterers. + Then the EM field inside each such volume can be expanded in a way similar + to +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:E field expansion" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, using VSWFs with origins shifted to the centre of the volume: +\begin_inset Formula +\begin{align} +\vect E\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lm\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)\right),\label{eq:E field expansion multiparticle}\\ + & \vect r\in\openball{\vect r_{p}}{R_{p}^{>}}\backslash B_{\vect r_{p}}\left(R_{p}\right).\nonumber +\end{align} + +\end_inset + +Unlike the single scatterer case, the incident field coefficients +\begin_inset Formula $\rcoeffptlm p{\tau}lm$ +\end_inset + + here are not only due to some external driving field that the particle + does not influence but they also contain the contributions of fields scattered + from +\emph on +all other scatterers +\emph default +: +\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 + +where +\begin_inset Formula $\rcoeffincp p$ +\end_inset + + represents the part due to the external driving that the scatterers can + not influence, and +\begin_inset Formula $\tropsp pq$ +\end_inset + + is a +\emph on +translation operator +\emph default + defined below in Sec. + +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:Translation-operator" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, that contains the re-expansion coefficients of the outgoing waves in origin + +\begin_inset Formula $\vect r_{q}$ +\end_inset + + into regular waves in origin +\begin_inset Formula $\vect r_{p}$ +\end_inset + +. +\end_layout + \begin_layout Subsubsection Translation operator +\begin_inset CommandInset label +LatexCommand label +name "subsec:Translation-operator" + +\end_inset + + +\end_layout + +\begin_layout Standard +Let +\begin_inset Formula $\vect r_{1},\vect r_{2}$ +\end_inset + + be two different origins; a regular VSWF with origin +\begin_inset Formula $\vect r_{1}$ +\end_inset + + can be always expanded in terms of regular VSWFs with origin +\begin_inset Formula $\vect r_{2}$ +\end_inset + + as follows: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right)=\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right),\label{eq:regular vswf translation} +\end{equation} + +\end_inset + +where an explicit formula for the (regular) +\emph on +translation operator +\emph default + +\begin_inset Formula $\tropr$ +\end_inset + + reads in eq. + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:translation operator" + +\end_inset + + below. + For singular (outgoing) waves, the form of the expansion differs inside + and outside the ball +\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$ +\end_inset + + +\begin_inset Formula +\begin{eqnarray} +\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases} +\sum_{\tau'l'm'}\trops_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfouttlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right), & \vect r\in\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}\\ +\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right), & \vect r\notin\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\left|\vect r_{1}\right|} +\end{cases},\label{eq:singular vswf translation} +\end{eqnarray} + +\end_inset + +where the singular translation operator +\begin_inset Formula $\trops$ +\end_inset + + has the same form as +\begin_inset Formula $\tropr$ +\end_inset + + in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:translation operator" + +\end_inset + + except the regular spherical Bessel functions +\begin_inset Formula $j_{l}$ +\end_inset + + are replaced with spherical Hankel functions +\begin_inset Formula $h_{l}^{(1)}$ +\end_inset + +. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO note about expansion exactly on the sphere. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +As MSTMM deals most of the time with the +\emph on +expansion coefficients +\emph default + of fields +\begin_inset Formula $\rcoeffptlm p{\tau}lm,\outcoeffptlm p{\tau}lm$ +\end_inset + + in different origins +\begin_inset Formula $\vect r_{p}$ +\end_inset + + rather than with the VSWFs directly, let us write down how +\emph on +they +\emph default + transform under translation. + Let us assume the field can be in terms of regular waves everywhere, and + expand it in two different origins +\begin_inset Formula $\vect r_{p},\vect r_{q}$ +\end_inset + +, +\begin_inset Formula +\[ +\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right). +\] + +\end_inset + +Re-expanding the waves around +\begin_inset Formula $\vect r_{p}$ +\end_inset + + in terms of waves around +\begin_inset Formula $\vect r_{q}$ +\end_inset + + using +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:regular vswf translation" + +\end_inset + +, +\begin_inset Formula +\[ +\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right) +\] + +\end_inset + +and comparing to the original expansion around +\begin_inset Formula $\vect r_{q}$ +\end_inset + +, we obtain +\begin_inset Formula +\begin{equation} +\rcoeffptlm q{\tau'}{l'}{m'}=\sum_{\tau,l,m}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\rcoeffptlm p{\tau}lm.\label{eq:regular vswf coefficient translation} +\end{equation} + +\end_inset + +For the sake of readability, we introduce a shorthand matrix form for +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:regular vswf coefficient translation" + +\end_inset + + +\begin_inset Formula +\begin{equation} +\rcoeffp q=\troprp qp\rcoeffp p\label{eq:reqular vswf coefficient vector translation} +\end{equation} + +\end_inset + +(note the reversed indices; TODO redefine them in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:regular vswf translation" + +\end_inset + +, +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:singular vswf translation" + +\end_inset + +? Similarly, if we had only outgoing waves in the original expansion around + +\begin_inset Formula $\vect r_{p}$ +\end_inset + +, we would get +\begin_inset Formula +\begin{equation} +\rcoeffp q=\tropsp qp\outcoeffp p\label{eq:singular to regular vswf coefficient vector translation} +\end{equation} + +\end_inset + +for the expansion inside the ball +\begin_inset Formula $\openball{\left\Vert \vect r_{q}-\vect r_{p}\right\Vert }{\vect r_{p}}$ +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +CHECKME +\end_layout + +\end_inset + + and +\begin_inset Formula +\begin{equation} +\outcoeffp q=\troprp qp\outcoeffp p\label{eq:singular to singular vswf coefficient vector translation-1} +\end{equation} + +\end_inset + +outside. +\end_layout + +\begin_layout Standard +In our convention, the regular translation operator can be expressed explicitly + as +\begin_inset Formula +\begin{equation} +\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator} +\end{equation} + +\end_inset + +The singular operator +\begin_inset Formula $\trops$ +\end_inset + + for re-expanding outgoing waves into regular ones has the same form except + the regular spherical Bessel functions +\begin_inset Formula $j_{l}$ +\end_inset + + in are replaced with spherical Hankel functions +\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +In our convention, the regular translation operator is unitary, +\begin_inset Formula $\left(\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)\right)^{-1}=\tropr_{\tau lm;\tau'l'm'}\left(-\vect d\right)=\tropr_{\tau'l'm';\tau lm}^{*}\left(\vect d\right)$ +\end_inset + +, +\begin_inset Note Note +status open + +\begin_layout Plain Layout +todo different notation for the complex conjugation without transposition??? +\end_layout + +\end_inset + + or in the per-particle matrix notation, +\begin_inset Formula +\begin{equation} +\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity} +\end{equation} + +\end_inset + +. + Note that truncation at finite multipole degree breaks the unitarity, +\begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$ +\end_inset + +, which has to be taken into consideration when evaluating quantities such + as absorption or scattering cross sections. + Similarly, the full regular operators can be composed +\begin_inset Note Note +status open + +\begin_layout Plain Layout +better wording +\end_layout + +\end_inset + +, +\begin_inset Formula +\begin{equation} +\troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition} +\end{equation} + +\end_inset + + but truncation breaks this, +\begin_inset Formula $\truncated{\troprp ac}L\ne\truncated{\troprp ab}L\truncated{\troprp bc}L.$ +\end_inset + + +\end_layout + +\begin_layout Subsubsection +Cross-sections (many scatterers) +\end_layout + +\begin_layout Standard +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 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 + +. + Outside +\begin_inset Formula $\openball R{\vect r_{\square}}$ +\end_inset + +, we can describe the EM fields as if there was only a single scatterer, +\begin_inset Formula +\[ +\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right), +\] + +\end_inset + +where +\begin_inset Formula $\rcoeffp{\square},\outcoeffp{\square}$ +\end_inset + + are the vectors of VSWF expansion coefficients of the incident and total + scattered fields, respectively, at origin +\begin_inset Formula $\vect r_{\square}$ +\end_inset + +. + In principle, one could evaluate +\begin_inset Formula $\outcoeffp{\square}$ +\end_inset + + using the translation operators (REF!!!) and use the single-scatterer formulae + +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:extincion CS single" + +\end_inset + +– +\begin_inset CommandInset ref +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. + However, this is not suitable for numerical evaluation with truncation + in multipole degree; hence we need to express them in terms of particle-wise + expansions +\begin_inset Formula $\rcoeffp p,\outcoeffp p$ +\end_inset + +. + The original incident field re-expanded around +\begin_inset Formula $p$ +\end_inset + +-th particle reads according to +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:regular vswf translation" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + +\begin_inset Formula +\begin{equation} +\rcoeffincp p=\troprp p{\square}\rcoeffp{\square}\label{eq:a_inc local from global} +\end{equation} + +\end_inset + +whereas the contributions of fields scattered from each particle expanded + around the global origin +\begin_inset Formula $\vect r_{\square}$ +\end_inset + + is, according to +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:singular vswf translation" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset Formula +\begin{equation} +\outcoeffp{\square}=\sum_{q\in\mathcal{P}}\troprp{\square}q\outcoeffp q.\label{eq:f global from local} +\end{equation} + +\end_inset + +Using the unitarity +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:regular translation unitarity" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and composition +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:regular translation composition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + properties, one has +\begin_inset Formula +\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 \\ +\label{eq:absorption CS multi} +\end{eqnarray} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula $=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\mbox{TODO}.$ +\end_inset + + +\end_layout + +\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 \begin_layout Subsubsection