qpms/lepaper/finite.lyx

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\pdf_author "Marek Nečada"
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\begin_body

\begin_layout Section
Finite systems
\end_layout

\begin_layout Itemize
motivation (classes of problems that this can solve: response to external
 radiation, resonances, ...)
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
theory
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
T-matrix definition, basics
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
How to get it?
\end_layout

\end_deeper
\begin_layout Itemize
translation operators (TODO think about how explicit this should be, but
 I guess it might be useful to write them to write them explicitly (but
 in the shortest possible form) in the normalisation used in my program)
\end_layout

\begin_layout Itemize
employing point group symmetries and decomposing the problem to decrease
 the computational complexity (maybe separately)
\end_layout

\end_deeper
\end_deeper
\begin_layout Subsection
Motivation
\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
However, 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.
\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
 EM radiation scattered by a single particle.
 We assume that the scatterer lies inside a closed sphere 
\begin_inset Formula $\particle$
\end_inset

, the space outside this volume 
\begin_inset Formula $\medium$
\end_inset

 is filled with an homogeneous isotropic medium with relative electric permittiv
ity 
\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 $\medium$
\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}
\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)-ik\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 
\begin_inset Formula $k=k\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
\end_inset

, as can be derived from the Maxwell's equations [REF Jackson?].
 
\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 $\vswfrtlm{\tau}lm\left(k\vect r\right)$
\end_inset

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

, 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),\\
\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),
\end{align*}

\end_inset


\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),\\
\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),\\
 & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
\end{align*}

\end_inset

where 
\begin_inset Formula $\vect r=r\uvec r$
\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, respectively, as in [DLMF §10.47], 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 & =\\
\vsh 2lm & =\\
\vsh 3lm & =
\end{align*}

\end_inset

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

 are identical to those in [DLMF 14.30.1], i.e.
\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

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

 defined as in [DLMF §14.3(i)] 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

 
\end_layout

\begin_layout Standard
The convention for VSWFs used here is the same as in [Kristensson 2014];
 over other conventions used elsewhere in literature, it has several fundamental
 advantages – most importantly, the translation operators introduced later
 in eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:translation op def"
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
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

\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 Subsubsection
Absorbed power
\end_layout

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

\begin_layout Subsection
Multiple scattering
\end_layout

\begin_layout Subsubsection
Translation operator
\end_layout

\begin_layout Subsubsection
Numerical complexity, comparison to other methods
\end_layout

\end_body
\end_document