qpms/lepaper/finite.lyx

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\pdf_author "Marek Nečada"
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\begin_body

\begin_layout Section
Finite systems
\begin_inset CommandInset label
LatexCommand label
name "sec:Finite"

\end_inset


\end_layout

\begin_layout Standard
The basic idea of MSTMM is quite simple: the driving electromagnetic field
 incident onto a scatterer is expanded into a vector spherical wavefunction
 (VSWF) basis in which the single scattering problem is solved, and the
 scattered field is then re-expanded into VSWFs centered at the other scatterers.
 Repeating the same procedure with all (pairs of) scatterers yields a set
 of linear equations, solution of which gives the coefficients of the scattered
 field in the VSWF bases.
 Once these coefficients have been found, one can evaluate various quantities
 related to the scattering (such as cross sections or the scattered fields)
 quite easily.
 
\end_layout

\begin_layout Standard
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, for reader's reference we always
 state them explicitly in our convention.
\end_layout

\begin_layout Subsection
Single-particle scattering
\end_layout

\begin_layout Standard
In order to define the basic concepts, let us first consider the case of
 electromagnetic (EM) radiation scattered by a single particle.
 We assume that the scatterer lies inside a closed ball 
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
\end_inset

 of radius 
\begin_inset Formula $R^{<}$
\end_inset

 and center in the origin of the coordinate system (which can be chosen
 that way; the natural choice of 
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
\end_inset

 is the circumscribed ball of the scatterer) and that there exists a larger
 open cocentric ball 
\begin_inset Formula $\openball{R^{>}}{\vect 0}$
\end_inset

, such that the (non-empty) spherical shell 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$
\end_inset

 is filled with a homogeneous isotropic medium with relative electric permittivi
ty 
\begin_inset Formula $\epsilon(\vect r,\omega)=\epsbg(\omega)$
\end_inset

 and magnetic permeability 
\begin_inset Formula $\mu(\vect r,\omega)=\mubg(\omega)$
\end_inset

, and that the whole system is linear, i.e.
 the material properties of neither the medium nor the scatterer depend
 on field intensities.
 Under these assumptions, the EM fields 
\begin_inset Formula $\vect{\Psi}=\vect E,\vect H$
\end_inset

 in 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
\end_inset

 must satisfy the homogeneous vector Helmholtz equation together with the
 transversality condition 
\begin_inset Formula 
\begin{equation}
\left(\nabla^{2}+\kappa^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0,\quad\nabla\cdot\vect{\Psi}\left(\vect r,\vect{\omega}\right)=0\label{eq:Helmholtz eq}
\end{equation}

\end_inset

 
\begin_inset Note Note
status open

\begin_layout Plain Layout
frequency-space Maxwell's equations
\begin_inset Formula 
\begin{align*}
\nabla\times\vect E\left(\vect r,\omega\right)-i\kappa\eta_{0}\eta\vect H\left(\vect r,\omega\right) & =0,\\
\eta_{0}\eta\nabla\times\vect H\left(\vect r,\omega\right)+ik\vect E\left(\vect r,\omega\right) & =0.
\end{align*}

\end_inset


\end_layout

\end_inset

 
\begin_inset Note Note
status open

\begin_layout Plain Layout
todo define 
\begin_inset Formula $\Psi$
\end_inset

, mention transversality
\end_layout

\end_inset

 with wave number
\begin_inset Foot
status open

\begin_layout Plain Layout
Throughout this text, we use the letter 
\begin_inset Formula $\kappa$
\end_inset

 for wave number in order to avoid confusion with Bloch vector 
\begin_inset Formula $\vect k$
\end_inset

 and its magnitude, introduced in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Infinite"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\end_inset

 
\begin_inset Formula $\kappa=\kappa\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
\end_inset

, as can be derived from Maxwell's equations 
\begin_inset CommandInset citation
LatexCommand cite
key "jackson_classical_1998"
literal "false"

\end_inset

.
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO ref to the chapter.
\end_layout

\end_inset

 
\end_layout

\begin_layout Subsubsection
Spherical waves
\end_layout

\begin_layout Standard
Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Helmholtz eq"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be solved by separation of variables in spherical coordinates to give
 the solutions – the 
\emph on
regular
\emph default
 and 
\emph on
outgoing
\emph default
 vector spherical wavefunctions (VSWFs) 
\begin_inset Formula $\vswfrtlm{\tau}lm\left(\kappa\vect r\right)$
\end_inset

 and 
\begin_inset Formula $\vswfouttlm{\tau}lm\left(\kappa\vect r\right)$
\end_inset

, respectively, defined as follows:
\begin_inset Formula 
\begin{align}
\vswfrtlm 1lm\left(\kappa\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
\vswfrtlm 2lm\left(\kappa\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular}
\end{align}

\end_inset


\begin_inset Formula 
\begin{align}
\vswfouttlm 1lm\left(\kappa\vect r\right) & =h_{l}^{\left(1\right)}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
\vswfouttlm 2lm\left(\kappa\vect r\right) & =\frac{1}{kr}\frac{\ud\left(\kappa rh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(\kappa r\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
 & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber 
\end{align}

\end_inset

where 
\begin_inset Formula $\vect r=r\uvec r=r\left(\sin\theta\left(\uvec x\cos\phi+\uvec y\sin\phi\right)+\uvec z\cos\theta\right)$
\end_inset

; 
\begin_inset Formula $j_{l}\left(x\right),h_{l}^{\left(1\right)}\left(x\right)=j_{l}\left(x\right)+iy_{l}\left(x\right)$
\end_inset

 are the regular spherical Bessel function and spherical Hankel function
 of the first kind
\begin_inset Foot
status open

\begin_layout Plain Layout
The interpretation of 
\begin_inset Formula $\vswfouttlm{\tau}lm\left(\kappa\vect r\right)$
\end_inset

 containing spherical Hankel functions of the first kind as 
\emph on
outgoing
\emph default
 waves at positive frequencies is associated with a specific choice of sign
 in the exponent of time-frequency transformation, 
\begin_inset Formula $\psi\left(t\right)=\left(2\pi\right)^{-\pi/2}\int\psi\left(\omega\right)e^{-i\omega t}\,\ud\omega$
\end_inset

.
 This matters especially when considering materials with gain or loss: in
 this convention, lossy materials will have refractive index (and wavenumber
 
\begin_inset Formula $\kappa$
\end_inset

, at a given positive frequency) with 
\emph on
positive
\emph default
 imaginary part, and gain materials will have it negative and, for example,
 Drude-Lorenz model of a lossy medium will have poles in the lower complex
 half-plane.
\end_layout

\end_inset

, respectively, as in 
\begin_inset CommandInset citation
LatexCommand cite
after "§10.47"
key "NIST:DLMF"
literal "false"

\end_inset

, and 
\begin_inset Formula $\vsh{\tau}lm$
\end_inset

 are the 
\emph on
vector spherical harmonics
\emph default

\begin_inset Formula 
\begin{align}
\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\
\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\
\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition}
\end{align}

\end_inset

Note that the regular waves 
\begin_inset Formula $\vswfrtlm{\tau}lm$
\end_inset

 (with fields expressed in cartesian coordinates) have all well-defined
 limits in the origin, and except for the 
\begin_inset Quotes eld
\end_inset

electric dipolar
\begin_inset Quotes erd
\end_inset

 waves 
\begin_inset Formula $\vswfrtlm 21m$
\end_inset

, they vanish.
 In our convention, the (scalar) spherical harmonics 
\begin_inset Formula $\ush lm$
\end_inset

 are identical to those in 
\begin_inset CommandInset citation
LatexCommand cite
after "14.30.1"
key "NIST:DLMF"
literal "false"

\end_inset

, i.e.
\begin_inset Formula 
\[
\ush lm\left(\uvec r\right)=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
\]

\end_inset

where 
\begin_inset Formula $ $
\end_inset

 importantly, the Ferrers functions 
\begin_inset Formula $\dlmfFer lm$
\end_inset

 defined as in 
\begin_inset CommandInset citation
LatexCommand cite
after "§14.3(i)"
key "NIST:DLMF"
literal "false"

\end_inset

 do already contain the Condon-Shortley phase 
\begin_inset Formula $\left(-1\right)^{m}$
\end_inset

.
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO názornější definice.
\end_layout

\end_inset

 For later use, we also introduce 
\begin_inset Quotes eld
\end_inset

dual
\begin_inset Quotes erd
\end_inset

 spherical harmonics 
\begin_inset Formula $\ushD lm$
\end_inset

 defined by duality relation with the 
\begin_inset Quotes eld
\end_inset

usual
\begin_inset Quotes erd
\end_inset

 spherical harmonics 
\begin_inset Formula 
\begin{equation}
\iint\ushD{l'}{m'}\left(\uvec r\right)\ush lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual spherical harmonics}
\end{equation}

\end_inset

 and corresponding dual vector spherical harmonics 
\begin_inset Formula 
\begin{equation}
\iint\vshD{\tau'}{l'}{m'}\left(\uvec r\right)\cdot\vsh{\tau}lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual vsh}
\end{equation}

\end_inset

(complex conjugation not implied in the dot product here).
 In our convention, we have
\begin_inset Formula 
\begin{align*}
\ush lm\left(\uvec r\right) & =\left(\ush lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\ush l{-m}\left(\uvec r\right).\\
\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{align*}

\end_inset


\end_layout

\begin_layout Standard
The convention for VSWFs used here is the same as in 
\begin_inset CommandInset citation
LatexCommand cite
key "kristensson_spherical_2014"
literal "false"

\end_inset

; over other conventions used elsewhere in literature, it has several fundamenta
l advantages – most importantly, the translation operators introduced later
 in eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:reqular vswf coefficient vector translation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 are unitary, and it gives the simplest possible expressions for power transport
 and cross sections without additional 
\begin_inset Formula $l,m$
\end_inset

-dependent factors (for that reason, we also call our VSWFs as 
\emph on
power-normalised
\emph default
).
 Power-normalisation and unitary translation operators are possible to achieve
 also with real spherical harmonics – such a convention is used in 
\begin_inset CommandInset citation
LatexCommand cite
key "kristensson_scattering_2016"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
Its solutions (TODO under which conditions? What vector space do the SVWFs
 actually span? Check Comment 9.2 and Appendix f.9.1 in Kristensson)
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO small note about cartesian multipoles, anapoles etc.
 (There should be some comparing paper that the Russians at META 2018 mentioned.)
\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
T-matrix definition
\end_layout

\begin_layout Standard
The regular VSWFs 
\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
\end_inset

 would constitute a basis for solutions of the Helmholtz equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Helmholtz eq"
plural "false"
caps "false"
noprefix "false"

\end_inset

 inside a ball 
\begin_inset Formula $\openball{R^{>}}{\vect0}$
\end_inset

 with radius 
\begin_inset Formula $R^{>}$
\end_inset

 and center in the origin, were it filled with homogeneous isotropic medium;
 however, if the equation is not guaranteed to hold inside a smaller ball
 
\begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset

 around the origin (typically due to presence of a scatterer), one has to
 add the outgoing VSWFs 
\begin_inset Formula $\vswfouttlm{\tau}lm\left(\kappa\vect r\right)$
\end_inset

 to have a complete basis of the solutions in the volume 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$
\end_inset

.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Vnitřní koule uzavřená? Jak se řekne mezikulí anglicky?
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The single-particle scattering problem at frequency 
\begin_inset Formula $\omega$
\end_inset

 can be posed as follows: Let a scatterer be enclosed inside the ball 
\begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset

 and let the whole volume 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
\end_inset

 be filled with a homogeneous isotropic medium with wave number 
\begin_inset Formula $\kappa\left(\omega\right)$
\end_inset

.
 Inside 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$
\end_inset

, the electric field can be expanded as
\begin_inset Note Note
status open

\begin_layout Plain Layout
doplnit frekvence a polohy
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm\left(\kappa\vect r\right)+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(\kappa\vect r\right)\right).\label{eq:E field expansion}
\end{equation}

\end_inset

If there were no scatterer and 
\begin_inset Formula $\closedball{R^{<}}{\vect0}$
\end_inset

 were filled with the same homogeneous medium, the part with the outgoing
 VSWFs would vanish and only the part 
\begin_inset Formula $\vect E_{\mathrm{inc}}=\sum_{\tau lm}\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm$
\end_inset

 due to sources outside 
\begin_inset Formula $\openball{R^{>}}{\vect0}$
\end_inset

 would remain.
 Let us assume that the 
\begin_inset Quotes eld
\end_inset

driving field
\begin_inset Quotes erd
\end_inset

 is given, so that presence of the scatterer does not affect 
\begin_inset Formula $\vect E_{\mathrm{inc}}$
\end_inset

 and is fully manifested in the latter part, 
\begin_inset Formula $\vect E_{\mathrm{scat}}=\sum_{\tau lm}\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm$
\end_inset

.
 We also assume that the scatterer is made of optically linear materials
 and hence reacts to the incident field in a linear manner.
 This gives a linearity constraint between the expansion coefficients
\begin_inset Formula 
\begin{equation}
\outcoefftlm{\tau}lm=\sum_{\tau'l'm'}T_{\tau lm}^{\tau'l'm'}\rcoefftlm{\tau'}{l'}{m'}\label{eq:T-matrix definition}
\end{equation}

\end_inset

where the 
\begin_inset Formula $T_{\tau lm}^{\tau'l'm'}=T_{\tau lm}^{\tau'l'm'}\left(\omega\right)$
\end_inset

 are the elements of the 
\emph on
transition matrix,
\emph default
 a.k.a.
 
\begin_inset Formula $T$
\end_inset

-matrix.
 It completely describes the scattering properties of a linear scatterer,
 so with the knowledge of the 
\begin_inset Formula $T$
\end_inset

-matrix we can solve the single-particle scattering problem simply by substituti
ng appropriate expansion coefficients 
\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
\end_inset

 of the driving field into 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix definition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The outgoing VSWF expansion coefficients 
\begin_inset Formula $\outcoefftlm{\tau}lm$
\end_inset

 are the effective induced electric (
\begin_inset Formula $\tau=2$
\end_inset

) and magnetic (
\begin_inset Formula $\tau=1$
\end_inset

) multipole polarisation amplitudes of the scatterer, and this is why we
 sometimes refer to the corresponding VSWFs as to the electric and magnetic
 VSWFs, respectively.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO mention the pseudovector character of magnetic VSWFs.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
TOOD H-field expansion here?
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $T$
\end_inset

-matrices of particles with certain simple geometries (most famously spherical)
 can be obtained analytically 
\begin_inset CommandInset citation
LatexCommand cite
key "kristensson_scattering_2016,mie_beitrage_1908"
literal "false"

\end_inset

; for particles with smooth surfaces one can find them numerically using
 the 
\emph on
null-field method
\emph default
 
\begin_inset CommandInset citation
LatexCommand cite
key "waterman_new_1969,waterman_symmetry_1971,kristensson_scattering_2016"
literal "false"

\end_inset

 which works well in the most typical cases, but for less common parameter
 ranges (such as concave shapes, extreme values of aspect ratios or relative
 refractive index) they might suffer from serious numerical instabilities
 
\begin_inset CommandInset citation
LatexCommand cite
after "Sect. 5.8.4"
key "mishchenko_scattering_2002"
literal "false"

\end_inset

.
 In general, elements of the 
\begin_inset Formula $T$
\end_inset

-matrix can be obtained by simulating scattering of a regular spherical
 wave 
\begin_inset Formula $\vswfrtlm{\tau}lm$
\end_inset

 and projecting the scattered fields (or induced currents, depending on
 the method) onto the outgoing VSWFs 
\begin_inset Formula $\vswfouttlm{\tau'}{l'}{m'}$
\end_inset

.
 In practice, one can compute only a finite number of elements with a cut-off
 value 
\begin_inset Formula $L$
\end_inset

 on the multipole degree, 
\begin_inset Formula $l,l'\le L$
\end_inset

, see below.
 
\end_layout

\begin_layout Standard
For the numerical evaluation of 
\begin_inset Formula $T$
\end_inset

-matrices for simple axially symmetric scatterers in QPMS, we typically
 use the null-field equations, and for more complicated scatterers we use
 the 
\family typewriter
scuff-tmatrix
\family default
 tool from the free software SCUFF-EM suite 
\begin_inset CommandInset citation
LatexCommand cite
key "reid_efficient_2015,SCUFF2"
literal "false"

\end_inset

.
\begin_inset Foot
status open

\begin_layout Plain Layout
Note that the upstream versions of SCUFF-EM contain a bug that renders almost
 all 
\begin_inset Formula $T$
\end_inset

-matrix results wrong; we found and fixed the bug in our fork available
 at 
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/texnokrates/scuff-em"
literal "false"

\end_inset

 in revision 
\begin_inset CommandInset href
LatexCommand href
name "g78689f5"
target "https://github.com/texnokrates/scuff-em/commit/78689f5514072853aa5cad455ce15b3e024d163d"
literal "false"

\end_inset

.
 However, the 
\begin_inset CommandInset href
LatexCommand href
name "bugfix"
target "https://github.com/HomerReid/scuff-em/pull/197"
literal "false"

\end_inset

 has not been merged into upstream by the time of writing this article.
 
\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
T-matrix compactness, cutoff validity
\end_layout

\begin_layout Standard
The magnitude of the 
\begin_inset Formula $T$
\end_inset

-matrix elements depends heavily on the scatterer's size compared to the
 wavelength.
 Typically,
\begin_inset Foot
status open

\begin_layout Plain Layout
It has been proven that the 
\begin_inset Formula $T$
\end_inset

-matrix of a bounded scatterer is a compact operator for 
\emph on
acoustic
\emph default
 scattering problems 
\begin_inset CommandInset citation
LatexCommand cite
key "ganesh_convergence_2012"
literal "false"

\end_inset

.
 While we conjecture that this holds also for bounded electromagnetic scatterers
, we are not aware of a definitive proof.
\end_layout

\end_inset

 from certain multipole degree onwards, 
\begin_inset Formula $l,l'>L$
\end_inset

, the elements of the 
\begin_inset Formula $T$
\end_inset

-matrix are negligible, so truncating the 
\begin_inset Formula $T$
\end_inset

-matrix at finite multipole degree 
\begin_inset Formula $L$
\end_inset

 gives a good approximation of the actual infinite-dimensional operator.
 If the incident field is well-behaved, i.e.
 the expansion coefficients 
\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
\end_inset

 do not take excessive values for 
\begin_inset Formula $l'>L$
\end_inset

, the scattered field expansion coefficients 
\begin_inset Formula $\outcoefftlm{\tau}lm$
\end_inset

 with 
\begin_inset Formula $l>L$
\end_inset

 will also be negligible.
\end_layout

\begin_layout Standard
A rule of thumb to choose the 
\begin_inset Formula $L$
\end_inset

 with desired 
\begin_inset Formula $T$
\end_inset

-matrix element accuracy 
\begin_inset Formula $\delta$
\end_inset

 can be obtained from the spherical Bessel function expansion around zero
 
\begin_inset CommandInset citation
LatexCommand cite
after "10.52.1"
key "NIST:DLMF"
literal "false"

\end_inset

 by requiring that 
\begin_inset Formula $\delta\gg\left(nR\right)^{L}/\left(2L+1\right)!!$
\end_inset

, where 
\begin_inset Formula $R$
\end_inset

 is the scatterer radius and 
\begin_inset Formula $n$
\end_inset

 its (maximum) refractive index.
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
\begin_inset Formula 
\begin{align*}
\left(2n+1\right)!! & =\frac{\left(2n+1\right)!}{2^{n}n!},\\
\delta\gtrsim & \frac{R^{L}}{\left(2L+1\right)!!}=\frac{\left(2R\right)^{L}L!}{\left(2L+1\right)!}
\end{align*}

\end_inset

Stirling 
\begin_inset Formula $n!\approx\sqrt{2\pi n}\left(n/e\right)^{n}$
\end_inset

 so 
\begin_inset Newline newline
\end_inset


\begin_inset Formula 
\begin{align*}
\delta & \gtrsim\left(2R\right)^{L}\frac{\sqrt{2\pi L}\left(\frac{L}{e}\right)^{L}}{\sqrt{2\pi\left(2L+1\right)}\left(\frac{2L+1}{e}\right)^{2L+1}}\\
\delta & \gtrsim\left(2R\right)^{L}\frac{\sqrt{L}\left(\frac{L}{e}\right)^{L}}{\sqrt{2L+1}\left(\frac{2L+1}{e}\right)^{2L+1}}\\
\log\delta & \gtrsim L\log2+L\log R+\frac{1}{2}\log L-\frac{1}{2}\log\left(2L+1\right)+L\log L-L\log e-\left(2L+1\right)\log\left(2L+1\right)+\left(2L+1\right)\log e\\
\log\delta & \gtrsim L\log2+L\log R+\left(L+\frac{1}{2}\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2L+1\right)+\left(L+1\right)\\
 & >L\log2+L\log R+\left(L+\frac{1}{2}\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2L\right)+\left(L+1\right)\\
 & =L\log2+L\log R-\left(L+1\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2\right)+\left(L+1\right)\\
 & =L\log2+L\log R-\left(L+1\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2\right)+\left(L+1\right)
\end{align*}

\end_inset

too complicated, watabout
\begin_inset Formula 
\[
\delta\gtrsim\left(2R\right)^{L}\frac{L^{L+1/2}e^{L}}{\left(2L\right)^{2L}}=\frac{R^{L}e^{L}}{2^{L}}L^{L+1/2}
\]

\end_inset


\begin_inset Formula 
\[
\log\delta\gtrsim L\log\frac{ReL}{2}
\]

\end_inset


\begin_inset Formula 
\[
\log\delta\gtrsim L\log\frac{ReL}{2}
\]

\end_inset

yäk
\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Power transport
\end_layout

\begin_layout Standard
For convenience, let us introduce a short-hand matrix notation for the expansion
 coefficients and related quantities, so that we do not need to write the
 indices explicitly; so for example, eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix definition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 would be written as 
\begin_inset Formula $\outcoeffp{}=T\rcoeffp{}$
\end_inset

, where 
\begin_inset Formula $\rcoeffp{},\outcoeffp{}$
\end_inset

 are column vectors with the expansion coefficients.
 Transposed and complex-conjugated matrices are labeled with the 
\begin_inset Formula $\dagger$
\end_inset

 superscript.
\end_layout

\begin_layout Standard
With this notation, we state an important result about power transport,
 derivation of which can be found in 
\begin_inset CommandInset citation
LatexCommand cite
after "sect. 7.3"
key "kristensson_scattering_2016"
literal "true"

\end_inset

.
 Let the field in 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
\end_inset

 have expansion as in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:E field expansion"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Then the net power transported from 
\begin_inset Formula $\openball{R^{<}}{\vect 0}$
\end_inset

 to 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
\end_inset

 by electromagnetic radiation is
\begin_inset Formula 
\begin{equation}
P=\frac{1}{2\kappa^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)=\frac{1}{2\kappa^{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

where 
\begin_inset Formula $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$
\end_inset

 and 
\begin_inset Formula $\eta=\sqrt{\mu/\varepsilon}$
\end_inset

 are wave impedance of vacuum and relative wave impedance of the medium
 in 
\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$
\end_inset

, respectively.
 Here 
\family roman
\series medium
\shape up
\size normal
\emph off
\nospellcheck off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none

\begin_inset Formula $P$
\end_inset

 is well-defined only when 
\begin_inset Formula $\kappa^{2}\eta$
\end_inset

 is real.

\family default
\series default
\shape default
\size default
\emph default
\nospellcheck default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
 In realistic scattering setups, power is transferred by radiation into
 
\begin_inset Formula $\openball{R^{<}}{\vect 0}$
\end_inset

 and absorbed by the enclosed scatterer, so 
\begin_inset Formula $P$
\end_inset

 is negative and its magnitude equals to power absorbed by the scatterer.
 In other words, the hermitian operator 
\begin_inset Formula $\Pi=\Tp{}^{\dagger}\Tp{}+\left(\Tp{}^{\dagger}+\Tp{}\right)/2$
\end_inset

 must be negative (semi-)definite for a particle without gain.
 This provides a simple but very useful sanity check on the numerically
 obtained 
\begin_inset Formula $T$
\end_inset

-matrices: non-negligible positive eigenvalues of 
\begin_inset Formula $\Pi$
\end_inset

 indicate either too drastic multipole truncation or another problem with
 the 
\begin_inset Formula $T$
\end_inset

-matrix.
\end_layout

\begin_layout Subsubsection
Plane wave expansion
\end_layout

\begin_layout Standard
In many scattering problems considered in practice, the driving field is
 at least approximately a plane wave.
 A transversal (
\begin_inset Formula $\uvec k\cdot\vect E_{0}=0$
\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 
\begin_inset CommandInset citation
LatexCommand cite
after "7.7.1"
key "kristensson_scattering_2016"
literal "false"

\end_inset

 as
\begin_inset Formula 
\[
\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{i\kappa\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\uvec k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(\kappa\vect r\right),
\]

\end_inset

where the expansion coefficients are obtained from the scalar products of
 the amplitude and corresponding dual vector spherical harmonics
\begin_inset Formula 
\begin{eqnarray}
\rcoeffptlm{}1lm\left(\uvec k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right)\cdot\vect E_{0},\nonumber \\
\rcoeffptlm{}2lm\left(\uvec k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right)\cdot\vect E_{0}.\label{eq:plane wave expansion}
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Subsubsection
Cross-sections (single-particle)
\end_layout

\begin_layout Standard
With the 
\begin_inset Formula $T$
\end_inset

-matrix and expansion coefficients of plane waves in hand, we can state
 the expressions for cross-sections of a single scatterer.
 Assuming a non-lossy background medium, extinction, scattering and absorption
 cross sections of a single scatterer irradiated by a plane wave propagating
 in direction 
\begin_inset Formula $\uvec k$
\end_inset

 and (complex) amplitude 
\begin_inset Formula $\vect E_{0}$
\end_inset

 are 
\begin_inset CommandInset citation
LatexCommand cite
after "sect. 7.8.2"
key "kristensson_scattering_2016"
literal "true"

\end_inset


\begin_inset Formula 
\begin{eqnarray}
\sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2\kappa^{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}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{\kappa^{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}{\kappa^{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}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single}
\end{eqnarray}

\end_inset

where 
\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$
\end_inset

 is the vector of plane wave expansion coefficients as in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:plane wave expansion"

\end_inset

.
 
\end_layout

\begin_layout Subsection
Multiple scattering
\begin_inset CommandInset label
LatexCommand label
name "subsec:Multiple-scattering"

\end_inset


\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 closed ball 
\begin_inset Formula $\closedball{R_{p}}{\vect r_{p}}$
\end_inset

 such that the balls do not touch, 
\begin_inset Formula $\closedball{R_{p}}{\vect r_{p}}\cap\closedball{R_{q}}{\vect r_{q}}=\emptyset;p,q\in\mathcal{P}$
\end_inset

, so there is a non-empty spherical shell
\begin_inset Note Note
status open

\begin_layout Plain Layout
jaksetometuje?
\end_layout

\end_inset

 
\begin_inset Formula $\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}$
\end_inset

 around each one that contains only the background medium without any scatterers
; we assume that all the relevant volume outside 
\begin_inset Formula $\bigcap_{p\in\mathcal{P}}\closedball{R_{p}}{\vect r_{p}}$
\end_inset

 is filled with the same background medium.
 Then the EM field inside each 
\begin_inset Formula $\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}$
\end_inset

 can be expanded in a way similar to 
\begin_inset CommandInset ref
LatexCommand eqref
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(\kappa\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lm\vswfouttlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{p}\right)\right)\right),\label{eq:E field expansion multiparticle}\\
 & \vect r\in\mezikuli{R_{p}}{R_{p}^{>}}{\vect r_{p}}.\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

.
 For each scatterer, we also have its 
\begin_inset Formula $T$
\end_inset

-matrix relation as in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix definition"
plural "false"
caps "false"
noprefix "false"

\end_inset

,
\begin_inset Formula 
\[
\outcoeffp q=T_{q}\rcoeffp q.
\]

\end_inset

Together with 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:particle total incident field coefficient a"
plural "false"
caps "false"
noprefix "false"

\end_inset

, this gives rise to a set of linear equations
\begin_inset Formula 
\begin{equation}
\outcoeffp p-T_{p}\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q=T_{p}\rcoeffincp p,\quad p\in\mathcal{P}\label{eq:Multiple-scattering problem}
\end{equation}

\end_inset

which defines the multiple-scattering problem.
 If all the 
\begin_inset Formula $p,q$
\end_inset

-indexed vectors and matrices (note that without truncation, they are infinite-d
imensional) are arranged into blocks of even larger vectors and matrices,
 this can be written in a short-hand form
\begin_inset Formula 
\begin{equation}
\left(I-T\trops\right)\outcoeff=T\rcoeffinc\label{eq:Multiple-scattering problem block form}
\end{equation}

\end_inset

where 
\begin_inset Formula $I$
\end_inset

 is the identity matrix, 
\begin_inset Formula $T$
\end_inset

 is a block-diagonal matrix containing all the individual 
\begin_inset Formula $T$
\end_inset

-matrices, and 
\begin_inset Formula $\trops$
\end_inset

 contains the individual 
\begin_inset Formula $\tropsp pq$
\end_inset

matrices as the off-diagonal blocks, whereas the diagonal blocks are set
 to zeros.
 
\end_layout

\begin_layout Standard
We note that eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"

\end_inset

 with zero right-hand side describes the normal modes of the system; the
 methods mentioned later in Section 
\begin_inset CommandInset ref
LatexCommand eqref
reference "sec:Infinite"
plural "false"
caps "false"
noprefix "false"

\end_inset

 for solving the band structure of a periodic system can be used as well
 for finding the resonant frequencies of a finite system.
\end_layout

\begin_layout Standard
In practice, the multiple-scattering problem is solved in its truncated
 form, in which all the 
\begin_inset Formula $l$
\end_inset

-indices related to a given scatterer 
\begin_inset Formula $p$
\end_inset

 are truncated as 
\begin_inset Formula $l\le L_{p}$
\end_inset

, leaving only 
\begin_inset Formula $N_{p}=2L_{p}\left(L_{p}+2\right)$
\end_inset

 different 
\begin_inset Formula $\tau lm$
\end_inset

-multi-indices left.
 The truncation degree can vary for different scatterers (e.g.
\begin_inset space \space{}
\end_inset

due to different physical sizes), so the truncated block 
\begin_inset Formula $\left[\tropsp pq\right]_{l_{q}\le L_{q};l_{p}\le L_{q}}$
\end_inset

 has shape 
\begin_inset Formula $N_{p}\times N_{q}$
\end_inset

, not necessarily square.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
Such truncation of the translation operator 
\begin_inset Formula $\tropsp pq$
\end_inset

 is justified by the fact on the left, TODO
\end_layout

\end_inset


\end_layout

\begin_layout Standard
If no other type of truncation is done, there remain 
\begin_inset Formula $2L_{p}\left(L_{p}+2\right)$
\end_inset

 different 
\begin_inset Formula $\tau lm$
\end_inset

-multi-indices for the 
\begin_inset Formula $p$
\end_inset

-th scatterer, so that the truncated version of the matrix 
\begin_inset Formula $\left(I-T\trops\right)$
\end_inset

 is a square matrix with 
\begin_inset Formula $\left(\sum_{p\in\mathcal{P}}N_{p}\right)^{2}$
\end_inset

 elements in total.
 The truncated problem 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can then be solved using standard numerical linear algebra methods (typically,
 by LU factorisation of the 
\begin_inset Formula $\left(I-T\trops\right)$
\end_inset

 matrix at a given frequency, and then solving with Gauss elimination for
 as many different incident 
\begin_inset Formula $\rcoeffinc$
\end_inset

 vectors as needed).
\end_layout

\begin_layout Standard
Alternatively, the multiple scattering problem can be formulated in terms
 of the regular field expansion coefficients,
\begin_inset Formula 
\begin{align*}
\rcoeffp p-\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pqT_{q}\rcoeffp q & =\rcoeffincp p,\quad p\in\mathcal{P},\\
\left(I-\trops T\right)\rcoeff & =\rcoeffinc,
\end{align*}

\end_inset

but this form is less suitable for numerical calculations due to the fact
 that the regular VSWF expansion coefficients on both sides of the equation
 are typically non-negligible even for large multipole degree 
\begin_inset Formula $l$
\end_inset

, hence the truncation is not justified in this case.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO less bulshit.
\end_layout

\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(\kappa\left(\vect r-\vect r_{1}\right)\right)=\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(\kappa\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 translation operator 
\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 $\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}$
\end_inset

:
\begin_inset Formula 
\begin{eqnarray}
\vswfouttlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
\sum_{\tau'l'm'}\trops_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfouttlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{2}\right)\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(\kappa\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{2}\right)\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.
 We assume the field can be expressed 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 
\[
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right).
\]

\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 
\[
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)
\]

\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(\kappa\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
\begin_inset Note Note
status open

\begin_layout Plain Layout
; 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

?
\end_layout

\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 elements can be expressed
 explicitly as 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
\begin_inset Formula 
\begin{align}
\tropr_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\nonumber \\
\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator-1}
\end{align}

\end_inset


\end_layout

\end_inset


\begin_inset Formula 
\begin{align}
\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\label{eq:translation operator}
\end{align}

\end_inset

and analogously the elements of the singular operator 
\begin_inset Formula $\trops$
\end_inset

, having spherical Hankel functions (
\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
\end_inset

) in the radial part instead of the regular Bessel functions,
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
\begin_inset Formula 
\begin{align}
\trops_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\nonumber \\
\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular-1}
\end{align}

\end_inset


\end_layout

\end_inset


\begin_inset Formula 
\begin{align}
\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\label{eq:translation operator singular}
\end{align}

\end_inset

where the constant factors in our convention read 
\begin_inset Marginal
status open

\begin_layout Plain Layout
TODO check once again carefully for possible phase factors.
\end_layout

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Original Kristensson's 
\begin_inset Formula $C,D's$
\end_inset

from F.7:
\begin_inset Formula 
\begin{multline*}
C_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sqrt{\frac{\varepsilon_{m}\varepsilon_{m'}}{4}}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
D_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sqrt{\frac{\varepsilon_{m}\varepsilon_{m'}}{4}}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'.
\end{multline*}

\end_inset

where I have found a 
\begin_inset Formula $-i$
\end_inset

 factor in the 
\begin_inset Formula $\tau\ne\tau'$
\end_inset

 coefficients, so I force it here: 
\begin_inset Formula 
\begin{multline*}
C_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
D_{ml,m'l'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'.
\end{multline*}

\end_inset

 TODO check influence of the 
\begin_inset Formula $\varepsilon_{m}$
\end_inset

s, whether they can be just removed as above.
 If we take our definition of spherical harmonics, 
\begin_inset Formula 
\[
\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
\]

\end_inset

so 
\begin_inset Formula 
\[
\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}=\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right)
\]

\end_inset

and taking into account that we use the CS phase 
\begin_inset Formula $\dlmfFer lm\left(\cos\theta\right)=\left(-1\right)^{m}P_{l}^{m}$
\end_inset

, and that 
\begin_inset Formula $\left(-1\right)^{m+m'}=\left(-1\right)^{m-m'}$
\end_inset

 we have
\end_layout

\begin_layout Plain Layout
\begin_inset Formula 
\begin{multline*}
C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'.
\end{multline*}

\end_inset


\begin_inset Formula 
\begin{multline*}
C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right),\\
D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'.
\end{multline*}

\end_inset


\begin_inset Formula 
\begin{multline*}
C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\\
D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'.
\end{multline*}

\end_inset

and finally 
\begin_inset Formula 
\begin{multline*}
C_{ml,m'l'}\left(\vect d\right)=\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\\
D_{ml,m'l'}\left(\vect d\right)=-i\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'.
\end{multline*}

\end_inset


\end_layout

\end_inset


\begin_inset Formula 
\[
\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}=\begin{cases}
A_{lm;l'm'}^{\lambda} & \tau=\tau',\\
B_{lm;l'm'}^{\lambda} & \tau\ne\tau',
\end{cases}
\]

\end_inset


\begin_inset Formula 
\begin{multline}
A_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right),\\
B_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}.\label{eq:translation operator constant factors}
\end{multline}

\end_inset

Here 
\begin_inset Formula $\begin{pmatrix}l_{1} & l_{2} & l_{3}\\
m_{1} & m_{2} & m_{3}
\end{pmatrix}$
\end_inset

 is the 
\begin_inset Formula $3j$
\end_inset

 symbol defined as in 
\begin_inset CommandInset citation
LatexCommand cite
after "§34.2"
key "NIST:DLMF"
literal "false"

\end_inset

.
 Importantly for practical calculations, these rather complicated coefficients
 need to be evaluated only once up to the highest truncation order, 
\begin_inset Formula $l,l'\le L$
\end_inset

.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
TODO write more here.
\end_layout

\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 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 containing all the scatterers at once, 
\begin_inset Formula $\openball R{\vect r_{\square}}\supset\bigcup_{p\in\mathcal{P}}\closedball{R_{p}}{\vect r_{p}}$
\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 
\[
\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{\square}\right)\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{\square}\right)\right)\right),
\]

\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 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 eqref
reference "eq:reqular vswf coefficient vector 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 eqref
reference "eq:singular to regular vswf coefficient vector 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 eqref
reference "eq:regular translation unitarity"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and composition 
\begin_inset CommandInset ref
LatexCommand eqref
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
\begin_inset Note Note
status open

\begin_layout Plain Layout
(see Sec.
 TODO)
\end_layout

\end_inset

.
 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 eqref
reference "eq:atf form multiparticle"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:f squared form multiparticle"
plural "false"
caps "false"
noprefix "false"

\end_inset

 into 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:scattering CS single"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
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,\hat{\vect E}_{0}\right) & = & -\frac{1}{\kappa^{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,\hat{\vect E}_{0}\right) & = & \frac{1}{\kappa^{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}{\kappa^{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,\hat{\vect E}_{0}\right) & = & -\frac{1}{\kappa^{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 eqref
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 eqref
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 eqref
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}{\kappa^{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 eqref
reference "eq:absorption CS multi"
plural "false"
caps "false"
noprefix "false"

\end_inset

, but using 
\begin_inset CommandInset ref
LatexCommand eqref
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}{\kappa^{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}{\kappa^{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 eqref
reference "eq:absorption CS multi"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:absorption CS multi alternative"
plural "false"
caps "false"
noprefix "false"

\end_inset

 are equal.
\end_layout

\end_body
\end_document