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Relabel wave number with kappa + other fixes 

Former-commit-id: e005aa094418e3a1cc80ec88768dfd4e9300acca
This commit is contained in:
Marek Nečada 2019-08-07 15:20:45 +03:00
parent 56180ec86b
commit 095525337d
5 changed files with 117 additions and 124 deletions
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14
lepaper/arrayscat.lyx
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@ -485,9 +485,19 @@ These are compatibility macros for the (...)-old files:
\end_inset
-matrix simulations in finite and infinite systems of electromagnetic scatterers
\begin_inset Marginal
status open
\begin_layout Plain Layout
(TODO better title)
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
@ -776,10 +786,6 @@ Truncation notation.
Example results and benchmarks with BEM; figures!
\end_layout
\begin_layout Itemize
Fix and unify notation (mainly indices) in infinite lattices section.
\end_layout
\begin_layout Itemize
Carefully check the transformation directions in sec.

138
lepaper/finite.lyx
View File

@ -104,10 +104,6 @@ name "sec:Finite"
\end_layout
\begin_layout Subsection
Motivation/intro
\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
@ -127,8 +123,8 @@ The expressions appearing in the re-expansions are fairly complicated, and
the implementation of MSTMM is extremely error-prone also due to the various
conventions used in the literature.
Therefore although we do not re-derive from scratch the expressions that
can be found elsewhere in literature, we always state them explicitly in
our convention.
can be found elsewhere in literature, for reader's reference we always
state them explicitly in our convention.
\end_layout
\begin_layout Subsection
@ -137,7 +133,7 @@ Single-particle scattering
\begin_layout Standard
In order to define the basic concepts, let us first consider the case of
EM radiation scattered by a single particle.
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
@ -156,17 +152,7 @@ In order to define the basic concepts, let us first consider the case of
\begin_inset Formula $\openball{R^{>}}{\vect 0}$
\end_inset
, such that
\begin_inset Marginal
status open
\begin_layout Plain Layout
Is there a word for this?
\end_layout
\end_inset
the (non-empty) volume
, 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
@ -187,14 +173,14 @@ ty
\end_inset
in
\begin_inset Formula $\medium$
\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}+k^{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}
\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
@ -233,19 +219,27 @@ todo define
\end_inset
with
\begin_inset Formula $k=k\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
\begin_inset Formula $\kappa=\kappa\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
\end_inset
, as can be derived from the Maxwell's equations
, as can be derived from Maxwell's equations
\begin_inset CommandInset citation
LatexCommand cite
after "???"
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
@ -284,8 +278,8 @@ outgoing
, respectively, defined as follows:
\begin_inset Formula
\begin{align}
\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular}
\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular}
\end{align}
\end_inset
@ -294,7 +288,7 @@ outgoing
\begin_inset Formula
\begin{align}
\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\
\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\
\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(kr\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}
@ -494,7 +488,7 @@ noprefix "false"
around the origin (typically due to presence of a scatterer), one has to
add the outgoing VSWFs
\begin_inset Formula $\vswfrtlm{\tau}lm\left(k\vect r\right)$
\begin_inset Formula $\vswfrtlm{\tau}lm\left(\kappa\vect r\right)$
\end_inset
to have a complete basis of the solutions in the volume
@ -528,7 +522,7 @@ The single-particle scattering problem at frequency
\end_inset
be filled with a homogeneous isotropic medium with wave number
\begin_inset Formula $k\left(\omega\right)$
\begin_inset Formula $\kappa\left(\omega\right)$
\end_inset
.
@ -549,16 +543,16 @@ doplnit frekvence a polohy
\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(k\vect r\right)+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\left(k\vect r\right)\right).\label{eq:E field expansion}
\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 was no scatterer and
If there were no scatterer and
\begin_inset Formula $\closedball{R^{<}}{\vect 0}$
\end_inset
was filled with the same homogeneous medium, the part with the outgoing
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
@ -585,8 +579,8 @@ driving field
\end_inset
.
We also assume that the scatterer is made of optically linear materials,
and hence reacts on the incident field in a linear manner.
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}
@ -614,8 +608,8 @@ transition matrix,
\begin_inset Formula $T$
\end_inset
-matrix, we can solve the single-patricle scatering prroblem simply by substitut
ing appropriate expansion coefficients
-matrix we can solve the single-patricle scatering prroblem simply by substituti
ng appropriate expansion coefficients
\begin_inset Formula $\rcoefftlm{\tau'}{l'}{m'}$
\end_inset
@ -643,8 +637,8 @@ noprefix "false"
\end_inset
) multipole polarisation amplitudes of the scatterer, and this is why we
sometimes refer to the corresponding VSWFs as the electric and magnetic
VSWFs.
sometimes refer to the corresponding VSWFs as to the electric and magnetic
VSWFs, respectively.
\begin_inset Note Note
status open
@ -705,8 +699,12 @@ literal "false"
\end_inset
, see below.
We typically use the scuff-tmatrix tool from the free software SCUFF-EM
suite
For the numerical evaluation of
\begin_inset Formula $T$
\end_inset
-matrices we typically use the scuff-tmatrix tool from the free software
SCUFF-EM suite
\begin_inset CommandInset citation
LatexCommand cite
key "reid_efficient_2015,SCUFF2"
@ -979,7 +977,7 @@ noprefix "false"
via by electromagnetic radiation is
\begin_inset Formula
\begin{equation}
P=\frac{1}{2k^{2}\eta_{0}\eta}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)=\frac{1}{2k^{2}\eta_{0}\eta}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{}.\label{eq:Power transport}
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
@ -1001,9 +999,9 @@ Plane wave expansion
\begin_layout Standard
In many scattering problems considered in practice, the driving field is
a plane wave.
at least approximately a plane wave.
A transversal (
\begin_inset Formula $\vect k\cdot\vect E_{0}=0$
\begin_inset Formula $\uvec k\cdot\vect E_{0}=0$
\end_inset
) plane wave propagating in direction
@ -1017,7 +1015,7 @@ In many scattering problems considered in practice, the driving field is
can be expanded into regular VSWFs
\begin_inset CommandInset citation
LatexCommand cite
after "7.???"
after "7.7.1"
key "kristensson_scattering_2016"
literal "false"
@ -1026,7 +1024,7 @@ literal "false"
as
\begin_inset Formula
\[
\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right),
\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{i\kappa\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\uvec k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(\kappa\vect r\right),
\]
\end_inset
@ -1034,8 +1032,8 @@ literal "false"
with expansion coefficients
\begin_inset Formula
\begin{eqnarray}
\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\
\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion}
\rcoeffptlm{}1lm\left(\uvec k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\
\rcoeffptlm{}2lm\left(\uvec k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion}
\end{eqnarray}
\end_inset
@ -1113,10 +1111,10 @@ literal "true"
\begin_inset Formula
\begin{eqnarray}
\sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\
\sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\
\sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\
& & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single}
\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
@ -1163,7 +1161,7 @@ If the system consists of multiple scatterers, the EM fields around each
\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 volume
, so there is a non-empty spherical shell
\begin_inset Note Note
status open
@ -1178,11 +1176,11 @@ jaksetometuje?
\end_inset
around each one that contains only the background medium without any scatterers
(we assume that all the volume outside
; 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).
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
@ -1200,7 +1198,7 @@ noprefix "false"
, 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 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}
@ -1362,7 +1360,7 @@ In practice, the multiple-scattering problem is solved in its truncated
-multiindices left.
The truncation degree can vary for different scatterers (e.g.
due to different physical sizes), so the truncated block
\begin_inset Formula $\tropsp pq$
\begin_inset Formula $\left[\tropsp pq\right]_{l_{q}\le L_{q};l_{p}\le L_{q}}$
\end_inset
has shape
@ -1493,7 +1491,7 @@ Let
\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}
\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
@ -1519,9 +1517,9 @@ reference "eq:translation operator"
:
\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|}
\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}
@ -1590,7 +1588,7 @@ they
,
\begin_inset Formula
\[
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right).
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right).
\]
\end_inset
@ -1613,7 +1611,7 @@ reference "eq:regular vswf translation"
,
\begin_inset Formula
\[
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right)
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(\kappa\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)
\]
\end_inset
@ -1625,7 +1623,7 @@ and comparing to the original expansion around
, 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}
\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
@ -2066,7 +2064,7 @@ literal "false"
, we can describe the EM fields as if there was only a single scatterer,
\begin_inset Formula
\[
\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right),
\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{\square}\right)\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\kappa\left(\vect r-\vect r_{\square}\right)\right)\right),
\]
\end_inset
@ -2252,10 +2250,10 @@ noprefix "false"
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 \\
\sigma_{\mathrm{ext}}\left(\uvec k\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\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\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}
@ -2321,7 +2319,7 @@ noprefix "false"
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}
\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
@ -2349,8 +2347,8 @@ noprefix "false"
, 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).
\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

32
lepaper/infinite.lyx
View File

@ -605,7 +605,7 @@ reference "eq:W definition"
\end_inset
is the asymptotic behaviour of the translation operator,
\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect R_{\vect b}\right|}$
\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect m}\right|^{-1}e^{i\kappa\left|\vect R_{\vect m}\right|}$
\end_inset
that does not in the strict sense converge for any
@ -831,8 +831,8 @@ Check sign of s
\begin_layout Standard
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{k^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2l}\ud\xi\\
+\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right),\label{eq:Ewald in 3D short-range part}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa/4\xi^{2}}\xi^{2l}\ud\xi\\
+\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right),\label{eq:Ewald in 3D short-range part}
\end{multline}
\end_inset
@ -861,26 +861,6 @@ Poznámka ohledně zahození radiální části u kulových fcí?
\end_inset
\begin_inset Marginal
status open
\begin_layout Plain Layout
N.B.
here
\begin_inset Formula $\vect k$
\end_inset
is the Bloch vector while
\begin_inset Formula $k$
\end_inset
is the wavenumber.
Relabel to make this distinction clear.
\end_layout
\end_inset
The long-range part for cases
\begin_inset Formula $d=1,2$
\end_inset
@ -902,8 +882,8 @@ check sign of
\begin_inset Formula
\begin{multline}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{k^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
\times\sum_{\vect K\in\Lambda^{*}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/k\right)^{l-2j}\Gamma\left(-j,\frac{k^{2}\gamma\left(\left|\vect k+\vect K\right|/k\right)}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\left(\gamma\left(\left|\vect k+\vect K\right|/k\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
\times\sum_{\vect K\in\Lambda^{*}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(-j,\frac{k^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D}
\end{multline}
\end_inset
@ -915,7 +895,7 @@ and for
\begin_inset Formula
\begin{equation}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{k\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\frac{\left(\left|\vect k+\vect K\right|/k\right)^{l}}{k^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(k^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right).\label{eq:Ewald in 3D long-range part 3D}
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\frac{\left(\left|\vect k+\vect K\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right).\label{eq:Ewald in 3D long-range part 3D}
\end{equation}
\end_inset

37
lepaper/intro.lyx
View File

@ -107,7 +107,16 @@ name "sec:Introduction"
\begin_layout Standard
The problem of electromagnetic response of a system consisting of many relativel
y small, compact scatterers in various geometries, and its numerical solution,
is relevant to many branches of nanophotonics (TODO refs).
is relevant to many branches of nanophotonics.
\begin_inset Marginal
status open
\begin_layout Plain Layout
Some refs here?
\end_layout
\end_inset
The most commonly used general approaches used in computational electrodynamics
are often unsuitable for simulating systems with larger number of scatterers
due to their computational complexity: differential methods such as the
@ -118,7 +127,7 @@ y small, compact scatterers in various geometries, and its numerical solution,
with dense matrices containing couplings between each pair of DoF.
Therefore, a common (frequency-domain) approach to get an approximate solution
of the scattering problem for many small particles has been the coupled
dipole approximation (CDA) where drastic reduction of the number of DoF
dipole approximation (CDA) where a drastic reduction of the number of DoF
is achieved by approximating individual scatterers to electric dipoles
(characterised by a polarisability tensor) coupled to each other through
Green's functions.
@ -133,8 +142,8 @@ CDA is easy to implement and demands relatively little computational resources
to quantitative errors.
The other one, more subtle, manifests itself in photonic crystal-like structure
s used in nanophotonics: there are modes in which the particles' electric
dipole moments completely vanish due to symmetry, regardless of how small
the particles are, and the excitations have quadrupolar or higher-degree
dipole moments completely vanish due to symmetry, and regardless of how
small the particles are, the excitations have quadrupolar or higher-degree
multipolar character.
These modes typically appear at the band edges where interesting phenomena
such as lasing or Bose-Einstein condensation have been observed
@ -152,13 +161,13 @@ literal "false"
The natural way to overcome both limitations of CDA mentioned above is to
include higher multipoles into account.
Instead of polarisability tensor, the scattering properties of an individual
particle are then described a more general
particle are then described with more general
\begin_inset Formula $T$
\end_inset
-matrix, and different particles' multipole excitations are coupled together
via translation operators, a generalisation of the Green's functions in
CDA.
via translation operators, a generalisation of the Green's functions used
in CDA.
This is the idea behind the
\emph on
multiple-scattering
@ -227,8 +236,8 @@ literal "false"
During the process, it became apparent that although the size of the arrays
we were able to simulate with MSTMM was far larger than with other methods,
sometimes we were unable to match the full size of our physical arrays
(typically consisting of tens of thousands of metallic nanoparticles) due
to memory constraints.
(typically consisting of tens of thousands of metallic nanoparticles) mainly
due to memory constraints.
Moreover, to distinguish the effects attributable to the finite size of
the arrays, it became desirable to simulate also
\emph on
@ -247,7 +256,7 @@ Here we address both challenges: we extend the method on infinite periodic
of the system to decompose the problem into several substantially smaller
ones, which 1) reduces the demands on computational resources, hence speeds
up the computations and allows for simulations of larger systems, and 2)
provides better understanding of modes in periodic systems.
provides better understanding of modes, mainly in periodic systems.
\end_layout
\begin_layout Standard
@ -273,8 +282,8 @@ TODO refs to the code repositories once it is published.
s.
Moreover, it includes the improvements covered in this paper, enabling
to simulate even larger systems and also infinite structures with periodicity
in one, two or three dimensions, which can be e.g.
used for quickly evaluating dispersions of such structures
in one, two or three dimensions, which can be used e.g.
for quickly evaluating dispersions of such structures
\begin_inset Marginal
status open
@ -327,7 +336,7 @@ reference "sec:Infinite"
\end_inset
we develop the theory for infinite periodic structures.
In section
In Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Symmetries"
@ -339,7 +348,7 @@ noprefix "false"
we apply group theory on MSTMM to utilise the symmetries of the simulated
system.
Finally, section
Finally, Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Applications"

18
lepaper/symmetries.lyx
View File

@ -473,7 +473,7 @@ noprefix "false"
of the electric field around origin in a rotated/reflected system,
\begin_inset Formula
\[
\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\vect r\right)\right),
\]
\end_inset
@ -595,8 +595,8 @@ Check this carefully.
\begin_inset Formula
\begin{multline}
\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
+\left.\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
\left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
+\left.\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
\end{multline}
\end_inset
@ -750,12 +750,12 @@ With these transformation properties in hand, we can proceed to the effects
\begin_inset Formula
\begin{align*}
\left(\groupop g\vect E\right)\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
& \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right)\\
& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right.\\
& \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right)\\
& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{q}\right)\right)\right.\\
& \quad+\left.\outcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{q}\right)\right)\right)
\left(\groupop g\vect E\right)\left(\omega,\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
& \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right)\\
& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right.\\
& \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right)\\
& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right.\\
& \quad+\left.\outcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right)
\end{align*}
\end_inset