Merge finite-cs.lyx into finite.lyx

Former-commit-id: c42c118767e8fde87df5655946c3489a70627033
This commit is contained in:
Marek Nečada 2019-07-29 12:41:02 +03:00
parent 3aa4de7e77
commit 64e122b937
3 changed files with 848 additions and 793 deletions

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@ -660,6 +660,27 @@ My implementation.
Maybe put the numerical results separately in the end. Maybe put the numerical results separately in the end.
\end_layout \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_layout Standard
\begin_inset CommandInset include \begin_inset CommandInset include
LatexCommand include LatexCommand include
@ -680,6 +701,10 @@ literal "true"
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset CommandInset include \begin_inset CommandInset include
LatexCommand include LatexCommand include
filename "finite-old.lyx" filename "finite-old.lyx"
@ -690,12 +715,6 @@ literal "true"
\end_layout \end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand include
filename "finite-cs.lyx"
literal "true"
\end_inset \end_inset
@ -713,6 +732,10 @@ literal "true"
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset CommandInset include \begin_inset CommandInset include
LatexCommand include LatexCommand include
filename "infinite-old.lyx" filename "infinite-old.lyx"
@ -721,6 +744,11 @@ literal "true"
\end_inset \end_inset
\end_layout
\end_inset
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard

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@ -1,770 +0,0 @@
#LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 583
\begin_document
\begin_header
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\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\use_package amsmath 1
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\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

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@ -144,7 +144,7 @@ employing point group symmetries and decomposing the problem to decrease
\end_deeper \end_deeper
\end_deeper \end_deeper
\begin_layout Subsection \begin_layout Subsection
Motivation Motivation/intro
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
@ -162,9 +162,9 @@ The basic idea of MSTMM is quite simple: the driving electromagnetic field
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
However, the expressions appearing in the re-expansions are fairly complicated, The expressions appearing in the re-expansions are fairly complicated, and
and the implementation of MSTMM is extremely error-prone also due to the the implementation of MSTMM is extremely error-prone also due to the various
various conventions used in the literature. conventions used in the literature.
Therefore although we do not re-derive from scratch the expressions that Therefore although we do not re-derive from scratch the expressions that
can be found elsewhere in literature, we always state them explicitly in can be found elsewhere in literature, we always state them explicitly in
our convention. our convention.
@ -326,8 +326,8 @@ vector spherical harmonics
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\vsh 1lm & =\\ \vsh 1lm & =TODO\\
\vsh 2lm & =\\ \vsh 2lm & =fixme\\
\vsh 3lm & = \vsh 3lm & =
\end{align*} \end{align*}
@ -452,16 +452,16 @@ noprefix "false"
\end_inset \end_inset
inside a ball inside a ball
\begin_inset Formula $\openball 0R$ \begin_inset Formula $\openball 0{R^{>}}$
\end_inset \end_inset
with radius with radius
\begin_inset Formula $R$ \begin_inset Formula $R^{>}$
\end_inset \end_inset
and center in the origin; however, if the equation is not guaranteed to and center in the origin; however, if the equation is not guaranteed to
hold inside a smaller ball hold inside a smaller ball
\begin_inset Formula $B_{0}\left(R_{<}\right)$ \begin_inset Formula $B_{0}\left(R\right)$
\end_inset \end_inset
around the origin (typically due to presence of a scatterer), one has to around the origin (typically due to presence of a scatterer), one has to
@ -470,7 +470,7 @@ noprefix "false"
\end_inset \end_inset
to have a complete basis of the solutions in the volume 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 \end_inset
. .
@ -492,11 +492,11 @@ The single-particle scattering problem at frequency
\end_inset \end_inset
can be posed as follows: Let a scatterer be enclosed inside the ball 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 \end_inset
and let the whole volume 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 \end_inset
be filled with a homogeneous isotropic medium with wave number be filled with a homogeneous isotropic medium with wave number
@ -598,6 +598,19 @@ noprefix "false"
\end_inset \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 \end_layout
\begin_layout Standard \begin_layout Standard
@ -780,7 +793,7 @@ literal "true"
. .
Let the field in 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 \end_inset
have expansion as in have expansion as in
@ -795,11 +808,11 @@ noprefix "false"
. .
Then the net power transported from Then the net power transported from
\begin_inset Formula $B_{0}\left(R_{<}\right)$ \begin_inset Formula $B_{0}\left(R\right)$
\end_inset \end_inset
to 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 \end_inset
via by electromagnetic radiation is 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 \end_inset
In realistic scattering setups, power is transferred by radiation into 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 \end_inset
and absorbed by the enclosed scatterer, so and absorbed by the enclosed scatterer, so
@ -897,7 +910,7 @@ usual
\end_layout \end_layout
\begin_layout Subsection \begin_layout Subsubsection
Cross-sections (single-particle) Cross-sections (single-particle)
\end_layout \end_layout
@ -957,8 +970,792 @@ reference "eq:plane wave expansion"
Multiple scattering Multiple scattering
\end_layout \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 \begin_layout Subsubsection
Translation operator 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 \end_layout
\begin_layout Subsubsection \begin_layout Subsubsection