560 lines
14 KiB
Plaintext
560 lines
14 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\shortcut idx
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\end_header
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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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\begin_layout Standard
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Let
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\begin_inset Formula $\vect r_{1},\vect r_{2}$
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\end_inset
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be two different origins; a regular VSWF with origin
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\begin_inset Formula $\vect r_{1}$
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\end_inset
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can be always expanded in terms of regular VSWFs with origin
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\begin_inset Formula $\vect r_{2}$
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\end_inset
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as follows:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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where an explicit formula for the (regular)
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\emph on
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translation operator
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\emph default
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\begin_inset Formula $\tropr$
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\end_inset
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reads in eq.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:translation operator"
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\end_inset
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below.
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For singular (outgoing) waves, the form of the expansion differs inside
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and outside the ball
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\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$
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\end_inset
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\begin_inset Formula
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\begin{eqnarray}
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\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
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\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}}\\
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\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|}
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\end{cases},\label{eq:singular vswf translation}
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\end{eqnarray}
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\end_inset
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where the singular translation operator
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\begin_inset Formula $\trops$
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\end_inset
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has the same form as
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\begin_inset Formula $\tropr$
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\end_inset
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in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:translation operator"
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\end_inset
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except the regular spherical Bessel functions
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\begin_inset Formula $j_{l}$
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\end_inset
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are replaced with spherical Hankel functions
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\begin_inset Formula $h_{l}^{(1)}$
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\end_inset
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.
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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 note about expansion exactly on the sphere.
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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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As MSTMM deals most of the time with the
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\emph on
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expansion coefficients
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\emph default
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of fields
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\begin_inset Formula $\rcoeffptlm p{\tau}lm,\outcoeffptlm p{\tau}lm$
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\end_inset
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in different origins
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\begin_inset Formula $\vect r_{p}$
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\end_inset
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rather than with the VSWFs directly, let us write down how
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\emph on
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they
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\emph default
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transform under translation.
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Let us assume the field can be in terms of regular waves everywhere, and
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expand it in two different origins
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\begin_inset Formula $\vect r_{p},\vect r_{q}$
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\end_inset
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,
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\begin_inset Formula
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\[
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\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right).
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\]
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\end_inset
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Re-expanding the waves around
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\begin_inset Formula $\vect r_{p}$
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\end_inset
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in terms of waves around
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\begin_inset Formula $\vect r_{q}$
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\end_inset
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using
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:regular vswf translation"
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\end_inset
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,
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\begin_inset Formula
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\[
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\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right)
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\]
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\end_inset
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and comparing to the original expansion around
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\begin_inset Formula $\vect r_{q}$
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\end_inset
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, we obtain
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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For the sake of readability, we introduce a shorthand matrix form for
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:regular vswf coefficient translation"
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\end_inset
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\begin_inset Formula
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\begin{equation}
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\rcoeffp q=\troprp qp\rcoeffp p\label{eq:reqular vswf coefficient vector translation}
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\end{equation}
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\end_inset
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(note the reversed indices; TODO redefine them in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:regular vswf translation"
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\end_inset
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,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:singular vswf translation"
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\end_inset
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? Similarly, if we had only outgoing waves in the original expansion around
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\begin_inset Formula $\vect r_{p}$
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\end_inset
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, we would get
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\begin_inset Formula
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\begin{equation}
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\rcoeffp q=\tropsp qp\outcoeffp p\label{eq:singular to regular vswf coefficient vector translation}
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\end{equation}
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\end_inset
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for the expansion inside the ball
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\begin_inset Formula $\openball{\left\Vert \vect r_{q}-\vect r_{p}\right\Vert }{\vect r_{p}}$
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\end_inset
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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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CHECKME
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\end_layout
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\end_inset
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and
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\begin_inset Formula
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\begin{equation}
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\outcoeffp q=\troprp qp\outcoeffp p\label{eq:singular to singular vswf coefficient vector translation-1}
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\end{equation}
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\end_inset
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outside.
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\end_layout
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\begin_layout Standard
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In our convention, the regular translation operator can be expressed explicitly
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as
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\begin_inset Formula
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\begin{equation}
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\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator}
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\end{equation}
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\end_inset
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The singular operator
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\begin_inset Formula $\trops$
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\end_inset
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for re-expanding outgoing waves into regular ones has the same form except
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the regular spherical Bessel functions
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\begin_inset Formula $j_{l}$
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\end_inset
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in are replaced with spherical Hankel functions
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\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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In our convention, the regular translation operator is unitary,
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\begin_inset Formula $\left(\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)\right)^{-1}=\tropr_{\tau lm;\tau'l'm'}\left(-\vect d\right)=\tropr_{\tau'l'm';\tau lm}^{*}\left(\vect d\right)$
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\end_inset
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,
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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 different notation for the complex conjugation without transposition???
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\end_layout
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\end_inset
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or in the per-particle matrix notation,
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\begin_inset Formula $\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}$
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\end_inset
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.
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Note that truncation at finite multipole degree breaks the unitarity,
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\begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$
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\end_inset
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, which has to be taken into consideration when evaluating quantities such
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as absorption or scattering cross sections.
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\end_layout
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\begin_layout Subsection
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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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Cross-sections (single-particle)
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\end_layout
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\begin_layout Standard
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Extinction, scattering and absorption cross sections of a single particle
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irradiated by a plane wave propagating in direction
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\begin_inset Formula $\uvec k$
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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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||
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\begin_layout Standard
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For a system of many scatterers, Kristensson derives only the scattering
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cross section formula
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||
\begin_inset Formula
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\[
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\sigma_{\mathrm{scat}}\left(\uvec k\right)=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\left\Vert \outcoeffp p\right\Vert ^{2}.
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\]
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\end_inset
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Let us derive the many-particle scattering and absorption cross sections.
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First, let us take a ball circumscribing all the scatterers at once,
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\begin_inset Formula $\openball R{\vect r_{\square}}\supset\particle$
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||
\end_inset
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||
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.
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Outside
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||
\begin_inset Formula $\openball R{\vect r_{\square}}$
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||
\end_inset
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||
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||
, we can describe the EM fields as if there was only a single scatterer,
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\begin_inset Formula
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\[
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\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right),
|
||
\]
|
||
|
||
\end_inset
|
||
|
||
where
|
||
\begin_inset Formula $\rcoeffp{\square},\outcoeffp{\square}$
|
||
\end_inset
|
||
|
||
are the vectors of VSWF expansion coefficients of the incident and total
|
||
scattered fields, respectively, at origin
|
||
\begin_inset Formula $\vect r_{\square}$
|
||
\end_inset
|
||
|
||
.
|
||
In principle, one could evaluate
|
||
\begin_inset Formula $\outcoeffp{\square}$
|
||
\end_inset
|
||
|
||
using the translation operators (REF!!!) and use the single-scatterer formulae
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:extincion CS single"
|
||
|
||
\end_inset
|
||
|
||
–
|
||
\begin_inset CommandInset ref
|
||
LatexCommand eqref
|
||
reference "eq:absorption CS single"
|
||
|
||
\end_inset
|
||
|
||
to obtain the cross sections.
|
||
However, this is not suitable for numerical evaluation with truncation
|
||
in multipole degree; hence we need to express them in terms of particle-wise
|
||
expansions
|
||
\begin_inset Formula $\rcoeffp p,\outcoeffp p$
|
||
\end_inset
|
||
|
||
.
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|