Single particle scattering done
Former-commit-id: c10e7db98d7f8c2eacbb8c7ef818371c61d2a7f4
This commit is contained in:
Marek Nečada
parent
80ea82a33f
commit
b14e776c34
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@ -94,36 +94,6 @@
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\begin_body
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\begin_layout Subsection
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Dual vector spherical harmonics
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\end_layout
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\begin_layout Standard
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For evaluation of expansion coefficients of incident fields, it is useful
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to introduce „dual“ vector spherical harmonics
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\begin_inset Formula $\vshD{\tau}lm$
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\end_inset
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defined by duality relation
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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(complex conjugation not implied in the dot product here).
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In our convention, we have
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\begin_inset Formula
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\[
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\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right).
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Translation operators
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\end_layout
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@ -440,39 +410,6 @@ better wording
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Plane wave expansion coefficients
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\end_layout
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\begin_layout Standard
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A transversal (
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\begin_inset Formula $\vect k\cdot\vect E_{0}=0$
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\end_inset
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) plane wave propagating in direction
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\begin_inset Formula $\uvec k$
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\end_inset
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with (complex) amplitude
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\begin_inset Formula $\vect E_{0}$
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\end_inset
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can be expanded into regular VSWFs [REF KRIS]
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\begin_inset Formula
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\[
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\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right),
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\]
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\end_inset
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with expansion coefficients
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\begin_inset Formula
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\begin{eqnarray}
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\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\
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\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}
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\end{eqnarray}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Multiple-scattering problem
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\end_layout
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@ -489,78 +426,7 @@ Multiple-scattering problem
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\end_layout
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\begin_layout Subsection
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Power transport
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\end_layout
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\begin_layout Standard
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Radiated power
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\begin_inset CommandInset citation
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LatexCommand cite
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after "sect. 7.3"
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key "kristensson_scattering_2016"
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literal "true"
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\end_inset
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Cross-sections (single-particle)
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\end_layout
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\begin_layout Standard
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Assuming a non-lossy background medium, extinction, scattering and absorption
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cross sections of a single scatterer irradiated by a plane wave propagating
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in direction
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\begin_inset Formula $\uvec k$
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\end_inset
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and (complex) amplitude
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\begin_inset Formula $\vect E_{0}$
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\end_inset
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are
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\begin_inset CommandInset citation
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LatexCommand cite
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after "sect. 7.8.2"
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key "kristensson_scattering_2016"
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literal "true"
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\end_inset
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\begin_inset Formula
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\begin{eqnarray}
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\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}\\
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\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}\\
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\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 \\
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& & =\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}
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\end{eqnarray}
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\end_inset
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where
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\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$
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\end_inset
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is the vector of plane wave expansion coefficients as in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:plane wave expansion"
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\end_inset
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.
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Cross-sections (many scatterers)
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\end_layout
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\begin_layout Standard
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@ -516,9 +516,9 @@ doplnit frekvence a polohy
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\begin_inset Formula
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\[
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\vect E\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\left(\rcoefftlm{\tau}lm\vswfrtlm{\tau}lm+\outcoefftlm{\tau}lm\vswfouttlm{\tau}lm\right).
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\]
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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@ -600,6 +600,19 @@ noprefix "false"
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.
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TOOD H-field expansion here?
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula $T$
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\end_inset
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@ -690,6 +703,16 @@ The magnitude of the
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\end_inset
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will also be negligible.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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TODO when it will not be negligible
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -711,7 +734,223 @@ A rule of thumb to choose the
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\end_layout
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\begin_layout Subsubsection
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Absorbed power
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Power transport
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\end_layout
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\begin_layout Standard
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For convenience, let us introduce a short-hand matrix notation for the expansion
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coefficients and related quantities, so that we do not need to write the
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indices explicitly; so for example, eq.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:T-matrix definition"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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would be written as
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\begin_inset Formula $\outcoeffp{}=T\rcoeffp{}$
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\end_inset
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, where
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\begin_inset Formula $\rcoeffp{},\outcoeffp{}$
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\end_inset
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are column vectors with the expansion coefficients.
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Transposed and complex-conjugated matrices are labeled with the
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\begin_inset Formula $\dagger$
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\end_inset
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superscript.
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\end_layout
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\begin_layout Standard
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With this notation, we state an important result about power transport,
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derivation of which can be found in
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\begin_inset CommandInset citation
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LatexCommand cite
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after "sect. 7.3"
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key "kristensson_scattering_2016"
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literal "true"
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\end_inset
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.
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Let the field in
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\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
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\end_inset
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have expansion as in
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:E field expansion"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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Then the net power transported from
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\begin_inset Formula $B_{0}\left(R_{<}\right)$
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\end_inset
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to
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\begin_inset Formula $\openball 0R\backslash B_{0}\left(R_{<}\right)$
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\end_inset
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via by electromagnetic radiation is
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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In realistic scattering setups, power is transferred by radiation into
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\begin_inset Formula $B_{0}\left(R_{<}\right)$
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\end_inset
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and absorbed by the enclosed scatterer, so
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\begin_inset Formula $P$
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\end_inset
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is negative and its magnitude equals to power absorbed by the scatterer.
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\end_layout
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\begin_layout Subsubsection
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Plane wave expansion
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\end_layout
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\begin_layout Standard
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In many scattering problems considered in practice, the driving field is
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a plane wave.
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A transversal (
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\begin_inset Formula $\vect k\cdot\vect E_{0}=0$
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\end_inset
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) plane wave propagating in direction
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\begin_inset Formula $\uvec k$
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\end_inset
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with (complex) amplitude
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\begin_inset Formula $\vect E_{0}$
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\end_inset
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can be expanded into regular VSWFs [REF KRIS] as
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\begin_inset Formula
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\[
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\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right),
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\]
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\end_inset
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with expansion coefficients
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\begin_inset Formula
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\begin{eqnarray}
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\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\
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\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}
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\end{eqnarray}
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\end_inset
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where
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\begin_inset Formula $\vshD{\tau}lm$
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\end_inset
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are the
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\begin_inset Quotes eld
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\end_inset
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dual
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\begin_inset Quotes erd
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\end_inset
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vector spherical harmonics defined by duality relation with the
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\begin_inset Quotes eld
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\end_inset
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usual
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\begin_inset Quotes erd
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\end_inset
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vector spherical harmonics
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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(complex conjugation not implied in the dot product here).
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In our convention, we have
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\begin_inset Formula
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\[
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\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right).
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\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Cross-sections (single-particle)
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\end_layout
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\begin_layout Standard
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With the
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\begin_inset Formula $T$
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\end_inset
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-matrix and expansion coefficients of plane waves in hand, we can state
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the expressions for cross-sections of a single scatterer.
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Assuming a non-lossy background medium, extinction, scattering and absorption
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cross sections of a single scatterer irradiated by a plane wave propagating
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in direction
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\begin_inset Formula $\uvec k$
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\end_inset
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and (complex) amplitude
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\begin_inset Formula $\vect E_{0}$
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\end_inset
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are
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\begin_inset CommandInset citation
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LatexCommand cite
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after "sect. 7.8.2"
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key "kristensson_scattering_2016"
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literal "true"
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\end_inset
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\begin_inset Formula
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\begin{eqnarray}
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\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}\\
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\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}\\
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\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 \\
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& & =\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}
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\end{eqnarray}
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\end_inset
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where
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\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$
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\end_inset
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is the vector of plane wave expansion coefficients as in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:plane wave expansion"
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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