#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 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 \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 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 \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 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 . \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