qpms/lepaper/finite-cs.lyx

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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\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