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Single particle scattering done 

Former-commit-id: c10e7db98d7f8c2eacbb8c7ef818371c61d2a7f4
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Marek Nečada 2019-07-29 10:51:43 +03:00
parent 80ea82a33f
commit b14e776c34
2 changed files with 244 additions and 139 deletions
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136
lepaper/finite-cs.lyx
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@ -94,36 +94,6 @@
\begin_body \begin_body
\begin_layout Subsection
Dual vector spherical harmonics
\end_layout
\begin_layout Standard
For evaluation of expansion coefficients of incident fields, it is useful
to introduce „dual“ vector spherical harmonics
\begin_inset Formula $\vshD{\tau}lm$
\end_inset
defined by duality relation
\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
\[
\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right).
\]
\end_inset
\end_layout
\begin_layout Subsection \begin_layout Subsection
Translation operators Translation operators
\end_layout \end_layout
@ -440,39 +410,6 @@ better wording
Plane wave expansion coefficients Plane wave expansion coefficients
\end_layout \end_layout
\begin_layout Standard
A transversal (
\begin_inset Formula $\vect k\cdot\vect E_{0}=0$
\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 [REF KRIS]
\begin_inset Formula
\[
\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right),
\]
\end_inset
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}
\end{eqnarray}
\end_inset
\end_layout
\begin_layout Subsection \begin_layout Subsection
Multiple-scattering problem Multiple-scattering problem
\end_layout \end_layout
@ -489,78 +426,7 @@ Multiple-scattering problem
\end_layout \end_layout
\begin_layout Subsection \begin_layout Subsection
Power transport Cross-sections (many scatterers)
\end_layout
\begin_layout Standard
Radiated power
\begin_inset CommandInset citation
LatexCommand cite
after "sect. 7.3"
key "kristensson_scattering_2016"
literal "true"
\end_inset
\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}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsection
Cross-sections (single-particle)
\end_layout
\begin_layout Standard
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}{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}
\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 \end_layout
\begin_layout Standard \begin_layout Standard

247
lepaper/finite.lyx
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@ -516,9 +516,9 @@ doplnit frekvence a polohy
\begin_inset Formula \begin_inset Formula
\[ \begin{equation}
\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right). \vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).\label{eq:E field expansion}
\] \end{equation}
\end_inset \end_inset
@ -600,6 +600,19 @@ noprefix "false"
. .
\end_layout \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_layout Standard
\begin_inset Formula $T$ \begin_inset Formula $T$
\end_inset \end_inset
@ -690,6 +703,16 @@ The magnitude of the
\end_inset \end_inset
will also be negligible. will also be negligible.
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO when it will not be negligible
\end_layout
\end_inset
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard
@ -711,7 +734,223 @@ A rule of thumb to choose the
\end_layout \end_layout
\begin_layout Subsubsection \begin_layout Subsubsection
Absorbed power 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 ref
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 $\openball 0R\backslash B_{0}\left(R_{<}\right)$
\end_inset
have expansion as in
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:E field expansion"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Then the net power transported from
\begin_inset Formula $B_{0}\left(R_{<}\right)$
\end_inset
to
\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
\end_inset
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}
\end{equation}
\end_inset
In realistic scattering setups, power is transferred by radiation into
\begin_inset Formula $B_{0}\left(R_{<}\right)$
\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.
\end_layout
\begin_layout Subsubsection
Plane wave expansion
\end_layout
\begin_layout Standard
In many scattering problems considered in practice, the driving field is
a plane wave.
A transversal (
\begin_inset Formula $\vect k\cdot\vect E_{0}=0$
\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 [REF KRIS] as
\begin_inset Formula
\[
\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right),
\]
\end_inset
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}
\end{eqnarray}
\end_inset
where
\begin_inset Formula $\vshD{\tau}lm$
\end_inset
are the
\begin_inset Quotes eld
\end_inset
dual
\begin_inset Quotes erd
\end_inset
vector spherical harmonics defined by duality relation with the
\begin_inset Quotes eld
\end_inset
usual
\begin_inset Quotes erd
\end_inset
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
\[
\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right).
\]
\end_inset
\end_layout
\begin_layout Subsection
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}{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}
\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 \end_layout
\begin_layout Subsection \begin_layout Subsection