VSWF definitions

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Marek Nečada 2019-07-28 23:39:56 +03:00
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@ -155,7 +155,19 @@ The basic idea of MSTMM is quite simple: the driving electromagnetic field
Repeating the same procedure with all (pairs of) scatterers yields a set Repeating the same procedure with all (pairs of) scatterers yields a set
of linear equations, solution of which gives the coefficients of the scattered of linear equations, solution of which gives the coefficients of the scattered
field in the VSWF bases. field in the VSWF bases.
However, 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 \end_layout
\begin_layout Subsection \begin_layout Subsection
@ -185,12 +197,40 @@ ity
, and that the whole system is linear, i.e. , and that the whole system is linear, i.e.
the material properties of neither the medium nor the scatterer depend the material properties of neither the medium nor the scatterer depend
on field intensities. on field intensities.
Under these assumptions, the EM fields in Under these assumptions, the EM fields
\begin_inset Formula $\vect{\Psi}=\vect E,\vect H$
\end_inset
in
\begin_inset Formula $\medium$ \begin_inset Formula $\medium$
\end_inset \end_inset
must satisfy the homogeneous vector Helmholtz equation must satisfy the homogeneous vector Helmholtz equation together with the
\begin_inset Formula $\left(\nabla^{2}+k^{2}\right)\Psi\left(\vect r,\vect{\omega}\right)=0$ 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 \end_inset
@ -208,29 +248,165 @@ todo define
\end_inset \end_inset
with with
\begin_inset Formula $k=TODO$ \begin_inset Formula $k=k\left(\omega\right)=\omega\sqrt{\mubg(\omega)\epsbg(\omega)}/c_{0}$
\end_inset \end_inset
[TODO REF Jackson?]. , as can be derived from the Maxwell's equations [REF Jackson?].
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 Subsubsection
Spherical waves
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
Throughout this text, we will use the same normalisation conventions as Equation
in \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 \begin_inset CommandInset citation
LatexCommand cite LatexCommand cite
key "kristensson_scattering_2016" key "kristensson_scattering_2016"
literal "true" literal "false"
\end_inset \end_inset
. .
\end_layout \end_layout
\begin_layout Subsubsection \begin_layout Standard
Spherical waves 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_layout
\begin_layout Standard \begin_layout Standard